diff --git "a/202511/qaEval_202511_all.json" "b/202511/qaEval_202511_all.json" new file mode 100644--- /dev/null +++ "b/202511/qaEval_202511_all.json" @@ -0,0 +1,7930 @@ +[ + { + "id": "2511.20502v2", + "paper_link": "http://arxiv.org/abs/2511.20502v2", + "theorems_cnt": 1, + "theorem": { + "env_name": "teo", + "content": "\\label{A}\n Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$. Let $\\alpha = f'(p) \\in (0,1)$. Then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.", + "start_pos": 9131, + "end_pos": 9553, + "label": "A" + }, + "ref_dict": { + "A": "\\begin{teo}\\label{A}\n Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$. Let $\\alpha = f'(p) \\in (0,1)$. Then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.\n\\end{teo}" + }, + "pre_theorem_intro_text_len": 2630, + "pre_theorem_intro_text": "Let $\\mathbb{D}$ be the unit disk, let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be \na holomorphic self map of $\\mathbb{D}$, and consider the dynamical system given by its iterates $\\{ f^n \\}_{n\\in \\mathbb{N}}$.\nThe Denjoy-Wolff Theorem states that if $f$ is not conjugate to a rotation, then there exists a point $p \\in \\overline{\\mathbb{D}}$ such that the orbit of every point in $\\mathbb{D}$ converges to $p$, that is, $f^n(z) \\to p$ as $n \\to \\infty$ for every $z \\in \\mathbb{D}$. We say that $p$ is the {\\em Denjoy-Wolff point} of $f$.\n\nIn the case where $f$ is an {\\em inner function}, that is, for $\\lambda$-almost every $\\zeta\\in\\partial \\mathbb{D}$ the radial limit \\[ f^*(\\zeta)=\\lim_{r\\to1} f(r\\zeta)\\] belongs to $\\partial\\mathbb{D}$ (where $\\lambda$ denotes the normalized Lebesgue measure on $\\partial\\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension\n$$\nf^{*} \\colon \\partial\\mathbb{D} \\longrightarrow \\partial\\mathbb{D}.\n$$\nAssuming that the Denjoy-Wolff point $p$ lies in $\\partial\\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.\nIn the seminal work of Aaronson, Doreing and Mañé \\cite{Aaronson78, mane_dynamics_1991}, this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\\lambda$-almost every point on $\\partial\\mathbb{D}$ converges to the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ if and only if \\[\\sum_{n\\geq 0} 1-|f^n(0|<\\infty.\\]\n\nIt is well-known that {\\em hyperbolic} inner functions (i.e. for which the angular derivative $f'(p)\\in (0,1)$, see Section \\ref{sect-hyperbolic-inner}) always satisfy the condition above. Going one step further, one may ask at which\nrate do these orbits approach the Denjoy–Wolff point.\n\nIf the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that $f$ behaves like the map $z \\mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\\partial \\mathbb{D}$ in a neighborhood of $p$. \n\nThe case when the Denjoy–Wolff point is a singularity is much more complicated, due to the lack of normal forms. In this paper, we establish an explicit rate of convergence for hyperbolic inner functions, covering the case when the Denjoy–Wolff point is a singularity. Namely, if we denote $A(p;a,b) \\coloneqq \\{ z \\in \\mathbb{C}\\colon a <|z-p| < b\\}$, we prove the following. \\\\", + "context": "Let $\\mathbb{D}$ be the unit disk, let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be \na holomorphic self map of $\\mathbb{D}$, and consider the dynamical system given by its iterates $\\{ f^n \\}_{n\\in \\mathbb{N}}$.\nThe Denjoy-Wolff Theorem states that if $f$ is not conjugate to a rotation, then there exists a point $p \\in \\overline{\\mathbb{D}}$ such that the orbit of every point in $\\mathbb{D}$ converges to $p$, that is, $f^n(z) \\to p$ as $n \\to \\infty$ for every $z \\in \\mathbb{D}$. We say that $p$ is the {\\em Denjoy-Wolff point} of $f$.\n\nIn the case where $f$ is an {\\em inner function}, that is, for $\\lambda$-almost every $\\zeta\\in\\partial \\mathbb{D}$ the radial limit \\[ f^*(\\zeta)=\\lim_{r\\to1} f(r\\zeta)\\] belongs to $\\partial\\mathbb{D}$ (where $\\lambda$ denotes the normalized Lebesgue measure on $\\partial\\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension\n$$\nf^{*} \\colon \\partial\\mathbb{D} \\longrightarrow \\partial\\mathbb{D}.\n$$\nAssuming that the Denjoy-Wolff point $p$ lies in $\\partial\\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.\nIn the seminal work of Aaronson, Doreing and Mañé \\cite{Aaronson78, mane_dynamics_1991}, this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\\lambda$-almost every point on $\\partial\\mathbb{D}$ converges to the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ if and only if \\[\\sum_{n\\geq 0} 1-|f^n(0|<\\infty.\\]\n\nIt is well-known that {\\em hyperbolic} inner functions (i.e. for which the angular derivative $f'(p)\\in (0,1)$, see Section \\ref{sect-hyperbolic-inner}) always satisfy the condition above. Going one step further, one may ask at which\nrate do these orbits approach the Denjoy–Wolff point.\n\nIf the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that $f$ behaves like the map $z \\mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\\partial \\mathbb{D}$ in a neighborhood of $p$.\n\nThe case when the Denjoy–Wolff point is a singularity is much more complicated, due to the lack of normal forms. In this paper, we establish an explicit rate of convergence for hyperbolic inner functions, covering the case when the Denjoy–Wolff point is a singularity. Namely, if we denote $A(p;a,b) \\coloneqq \\{ z \\in \\mathbb{C}\\colon a <|z-p| < b\\}$, we prove the following. \\\\", + "full_context": "Let $\\mathbb{D}$ be the unit disk, let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be \na holomorphic self map of $\\mathbb{D}$, and consider the dynamical system given by its iterates $\\{ f^n \\}_{n\\in \\mathbb{N}}$.\nThe Denjoy-Wolff Theorem states that if $f$ is not conjugate to a rotation, then there exists a point $p \\in \\overline{\\mathbb{D}}$ such that the orbit of every point in $\\mathbb{D}$ converges to $p$, that is, $f^n(z) \\to p$ as $n \\to \\infty$ for every $z \\in \\mathbb{D}$. We say that $p$ is the {\\em Denjoy-Wolff point} of $f$.\n\nIn the case where $f$ is an {\\em inner function}, that is, for $\\lambda$-almost every $\\zeta\\in\\partial \\mathbb{D}$ the radial limit \\[ f^*(\\zeta)=\\lim_{r\\to1} f(r\\zeta)\\] belongs to $\\partial\\mathbb{D}$ (where $\\lambda$ denotes the normalized Lebesgue measure on $\\partial\\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension\n$$\nf^{*} \\colon \\partial\\mathbb{D} \\longrightarrow \\partial\\mathbb{D}.\n$$\nAssuming that the Denjoy-Wolff point $p$ lies in $\\partial\\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.\nIn the seminal work of Aaronson, Doreing and Mañé \\cite{Aaronson78, mane_dynamics_1991}, this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\\lambda$-almost every point on $\\partial\\mathbb{D}$ converges to the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ if and only if \\[\\sum_{n\\geq 0} 1-|f^n(0|<\\infty.\\]\n\nIt is well-known that {\\em hyperbolic} inner functions (i.e. for which the angular derivative $f'(p)\\in (0,1)$, see Section \\ref{sect-hyperbolic-inner}) always satisfy the condition above. Going one step further, one may ask at which\nrate do these orbits approach the Denjoy–Wolff point.\n\nIf the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that $f$ behaves like the map $z \\mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\\partial \\mathbb{D}$ in a neighborhood of $p$.\n\nThe case when the Denjoy–Wolff point is a singularity is much more complicated, due to the lack of normal forms. In this paper, we establish an explicit rate of convergence for hyperbolic inner functions, covering the case when the Denjoy–Wolff point is a singularity. Namely, if we denote $A(p;a,b) \\coloneqq \\{ z \\in \\mathbb{C}\\colon a <|z-p| < b\\}$, we prove the following. \\\\\n\nIn the case where $f$ is an {\\em inner function}, that is, for $\\lambda$-almost every $\\zeta\\in\\partial \\mathbb{D}$ the radial limit \\[ f^*(\\zeta)=\\lim_{r\\to1} f(r\\zeta)\\] belongs to $\\partial\\mathbb{D}$ (where $\\lambda$ denotes the normalized Lebesgue measure on $\\partial\\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension\n$$\nf^{*} \\colon \\partial\\mathbb{D} \\longrightarrow \\partial\\mathbb{D}.\n$$\nAssuming that the Denjoy-Wolff point $p$ lies in $\\partial\\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.\nIn the seminal work of Aaronson, Doreing and Mañé \\cite{Aaronson78, mane_dynamics_1991}, this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\\lambda$-almost every point on $\\partial\\mathbb{D}$ converges to the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ if and only if \\[\\sum_{n\\geq 0} 1-|f^n(0|<\\infty.\\]\n\nIf the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that $f$ behaves like the map $z \\mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\\partial \\mathbb{D}$ in a neighborhood of $p$.\n\nWithout being precise, we say that a shrinking target is a collection of arcs of $\\partial\\mathbb{D}$ shrinking in length, and the problem is to determine whether orbits hit these arcs almost surely. Indeed, the arcs $A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n})$ in Theorem \\ref{A} form a shrinking target, and we want to conclude that it is hit almost surely. Via Möbius transformations, we convert our autonomous system into a non-autonomous one fixing $0$, to be able to apply the criteria in \\cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely. We make use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.\n\\\n\nIn this section, we prove Theorem \\ref{A}, which states that if $f \\colon \\mathbb{D} \\to \\mathbb{D}$ is a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$, $\\alpha = f'(p) \\in (0,1)$, then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.\n\n\\begin{lemma}\\label{cotas+}\n Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic self map of $\\mathbb{D}$ and let $p \\in \\partial\\mathbb{D}$ be its Denjoy-Wolff point. Let $\\alpha = f'(p)$. Then, for every $\\delta > 0$, there exists a real constant $C \\geq 1$ such that $$\\frac{1}{C} (\\alpha - \\delta)^n \\leq |f^n(0) - p| \\leq C \\alpha^n,$$ for $n$ large enough.\n\nWe now prove that given $0 < \\varepsilon < 1$, $$(f^*)^n(\\zeta) \\in D(p, \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$, where $\\alpha = f'(p) \\in (0,1)$.\n\nConsider the arc $J_n(\\varepsilon) = D(p,\\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$, $n \\in \\mathbb{N}$, and define \n \\begin{align*}\n E(\\varepsilon) &\\coloneqq \\{ \\zeta \\in \\partial\\mathbb{D}\\colon f^n(\\zeta) \\in J_n(\\varepsilon) \\text{\\ for all $n$ large enough}\\}\\\\\n &= \\{ \\zeta \\in \\partial\\mathbb{D} \\colon (f^n(\\zeta)) \\text{\\ fails to hit\\ } (J_n(\\varepsilon)^c)\\},\n \\end{align*}\n where $J_n(\\varepsilon)^c = \\partial\\mathbb{D}\\setminus J_n(\\varepsilon)$. We will prove that $E(\\varepsilon)$ has full measure.\n\nWe finally prove that given $0 < \\varepsilon < 1$, $$(f^*)^n(\\zeta) \\not \\in D(p, \\alpha^{(1+\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.\n\n\\begin{teo}\\label{A}\n Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$. Let $\\alpha = f'(p) \\in (0,1)$. Then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.\n\\end{teo}", + "post_theorem_intro_text_len": 1298, + "post_theorem_intro_text": "\\\nThe proof of the theorem is based on the concept of \\textit{shrinking targets} of $\\partial\\mathbb{D}$ and their hitting properties (see Section \\ref{sec:shrinking}), developed in \\cite{benini_shrinking_2024} to analyze the recurrent behavior of compositions of inner functions fixing 0. \n\nWithout being precise, we say that a shrinking target is a collection of arcs of $\\partial\\mathbb{D}$ shrinking in length, and the problem is to determine whether orbits hit these arcs almost surely. Indeed, the arcs $A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n})$ in Theorem \\ref{A} form a shrinking target, and we want to conclude that it is hit almost surely. Via Möbius transformations, we convert our autonomous system into a non-autonomous one fixing $0$, to be able to apply the criteria in \\cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely. We make use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.\n\\\n\n{\\bf Acknowledgements. } The authors gratefully acknowledge the Barcelona Introduction to Mathematical Research (BIMR) program at the Centre de Recerca Matemàtica (CRM) for providing an excellent research environment and support during the development of this work.", + "sketch": "The proof of Theorem~\\ref{A} is \"based on the concept of \\textit{shrinking targets} of $\\partial\\mathbb{D}$ and their hitting properties\" (Section~\\ref{sec:shrinking}), as developed in \\cite{benini_shrinking_2024}. The arcs $A(p;\\alpha^{(1+\\varepsilon)n},\\alpha^{(1-\\varepsilon)n})$ appearing in Theorem~\\ref{A} \"form a shrinking target\", and the goal is to \"conclude that it is hit almost surely\" by boundary orbits. \"Via Möbius transformations\", the authors \"convert\" the autonomous system into \"a non-autonomous one fixing $0$\" in order to \"apply the criteria in \\cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely\". The argument also \"make[s] use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.\"", + "expanded_sketch": "The proof of Theorem~\\ref{A} is \"based on the concept of \\textit{shrinking targets} of $\\partial\\mathbb{D}$ and their hitting properties\" (Section~\\ref{sec:shrinking}), as developed in \\cite{benini_shrinking_2024}. The arcs $A(p;\\alpha^{(1+\\varepsilon)n},\\alpha^{(1-\\varepsilon)n})$ appearing in the main theorem \"form a shrinking target\", and the goal is to \"conclude that it is hit almost surely\" by boundary orbits. \"Via Möbius transformations\", the authors \"convert\" the autonomous system into \"a non-autonomous one fixing $0$\" in order to \"apply the criteria in \\cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely\". The argument also \"make[s] use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.\"", + "expanded_theorem": "\\label{A}\n Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$. Let $\\alpha = f'(p) \\in (0,1)$. Then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.,", + "theorem_type": [ + "Universal", + "Asymptotic or Limit" + ], + "mcq": { + "question": "Let \\(\\mathbb D\\) be the unit disk, and let \\(f:\\mathbb D\\to\\mathbb D\\) be a hyperbolic inner function whose Denjoy--Wolff point is \\(p\\in \\partial\\mathbb D\\). Let \\(\\alpha=f'(p)\\in(0,1)\\), and let \\(f^*\\) denote the radial boundary map, defined for \\(\\lambda\\)-almost every \\(\\zeta\\in\\partial\\mathbb D\\) by \\(f^*(\\zeta)=\\lim_{r\\to1^-}f(r\\zeta)\\), where \\(\\lambda\\) is normalized Lebesgue measure on \\(\\partial\\mathbb D\\). For \\(a,b>0\\), define the annulus centered at \\(p\\) by \\(A(p;a,b)=\\{z\\in\\mathbb C: a<|z-p| 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{cor:eq_Grassmannian}\n\\mathcal{N}_{c, \\beta}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}", + "start_pos": 183309, + "end_pos": 183830, + "label": "cor:Grasssmannian" + }, + "ref_dict": { + "eq:WeilIntegrationFormula*": "\\begin{equation} \\label{eq:WeilIntegrationFormula*}\n\\forall \\, f \\in L^1(\\widetilde{X}), \\quad \\int_{\\Omega} S_\\chi f \\, \\dd \\mu_{\\Omega} = \\int_{\\widetilde{X}} f \\, \\dd \\lambda_{\\widetilde{X}}.\n\\end{equation}", + "thm:Critical": "\\begin{thmx}[Effective counting at the Diophantine exponent] \\label{thm:Critical}\nLet $d = \\dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{eq:Thm_Critical}\n\\mathcal{N}_{c, \\beta_{\\chi}}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}\n\\end{thmx}", + "cor:eq_Grassmannian": "\\begin{equation} \\label{cor:eq_Grassmannian}\n\\mathcal{N}_{c, \\beta}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}", + "thm:Effective": "\\begin{thmx} [Effective equidistribution of maximal compact subgroup orbits] \\label{thm:Effective}\nSuppose the action of $G$ on $\\Omega = G/\\G$ has a spectral gap. Let $A = \\{a(y) : y \\in \\R_+^{\\times} \\}$ be an $\\Ad$-diagonalizable one-parameter subgroup of $G$ such that $a(e)$ projects non-trivially to each simple factor of $G$. Then there exist a constant $c > 0$ and an integer $r \\geq 1$ such that for every compact subset $Q \\subset \\Omega$, for all $f \\in C^{\\infty}(K)$, $\\phi \\in C_c^\\infty(\\Omega)$, $x \\in Q$, and $y \\geq 1$, we have\n\\begin{equation} \\label{eq:Single-Eq}\n\\int_K f(k) \\phi \\bigl(a(y) k x \\bigr) \\dd \\mu_K(k) = \\int_K f \\dd \\mu_K \\int_{\\Omega} \\phi \\dd \\mu_{\\Omega} + O \\big ( y^{-c} \\cS_r(f) \\cS_r(\\phi) \\big ),\n\\end{equation}\nand, for all $f \\in C^{\\infty}(K)$, $\\phi_1, \\phi_2 \\in C_c^\\infty(\\Omega)$, $x_1, x_2 \\in Q$, \nand $y_2 \\geq y_1 \\geq 1$, we have\n\\begin{multline} \\label{eq:Double-Eq}\n\\int_K f(k) \\phi_{1}\\bigl(a(y_1) k x_1 \\bigr) \\phi_{2} \\bigl(a(y_2) k x_2 \\bigr) d \\mu_K(k) = \\int_K f \\dd \\mu_K \\int_{\\Omega} \\phi_1 \\dd \\mu_{\\Omega} \\int_{\\Omega} \\phi_2 \\dd \\mu_{\\Omega} \\\\ + O \\big ( \\min \\{y_1, y_2/y_1 \\}^{-c} \\cS_r(f) \\cS_r(\\phi_1) \\cS_r(\\phi_2) \\big ),\n\\end{multline} \nwhere the implicit constant in $O(\\cdot)$ depends only on $Q$.\n\\end{thmx}", + "thm:L1": "\\begin{thmx} [$L^1$-integrability] \\label{thm:L1} \nThe following assertions are equivalent.\n\\begin{enumerate}[label=(\\arabic*)]\n\\item The Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^1(\\Omega)$.\n\\item There exists a unique (up to scaling) $G$-invariant Radon measure $\\lambda_{\\widetilde{X}}$ on $\\widetilde{X}$, the Siegel transform $S_\\chi$ extends to a bounded operator $S_{\\chi} : L^1(\\widetilde{X}) \\rightarrow L^1(\\Omega)$, and $\\lambda_{\\widetilde{X}}$ can be normalized so that we have a convergent mean value formula:\n\\begin{equation} \\label{eq:WeilIntegrationFormula*}\n\\forall \\, f \\in L^1(\\widetilde{X}), \\quad \\int_{\\Omega} S_\\chi f \\, \\dd \\mu_{\\Omega} = \\int_{\\widetilde{X}} f \\, \\dd \\lambda_{\\widetilde{X}}.\n\\end{equation}\n\\item The Lie group $L = \\bL(\\R)$ is unimodular and $\\G_L = \\G \\cap L$ is a lattice in $L$.\n\\item The parabolic $\\Q$-subgroup $\\bP$ of $\\bG$ is maximal.\n\\item There exists $\\varepsilon > 0$ such that $S_\\chi$ maps $B_{c}^{\\infty}(\\widetilde{X})$ into $L^{1+\\varepsilon}(\\Omega)$.\n\\end{enumerate}\n\\end{thmx}", + "eq:Proof_Sketch": "\\begin{equation} \\label{eq:Proof_Sketch}\n\\int_K \\left | S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\G) - \\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(e^N)^+) \\right |^{1+\\varepsilon} \\, \\dd \\mu_{K}(k) \\, \\ll \\, N,\n\\end{equation}", + "eq:SchmidtCountingFunction": "\\begin{equation} \\label{eq:SchmidtCountingFunction}\n\\cN_\\psi(x,T) = \\# \\left \\{ (p,q) \\in \\Z \\times \\N : 0 \\leq q x - p < \\psi(q), \\, 1 \\leq q < T \\right \\}\n\\end{equation}", + "eq:primitive_Siegel_Transform": "\\begin{equation} \\label{eq:primitive_Siegel_Transform}\n\\forall \\, g \\in \\mathrm{SL}_n(\\mathbb{R}), \\qquad S f(g\\Z^n) = \\sum_{\\bm{v} \\in \\cP(\\Z^n)} f (g \\bm{v}).\n\\end{equation}", + "thm:L2": "\\begin{thmx} [$L^2$-integrability] \\label{thm:L2} \nSuppose that the Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^{2}(\\Omega)$. Then \n\\[\n\\forall \\, w \\in W, \\quad X^*(\\bL_w^\\circ)_{\\Q} = \\{1\\}.\n\\]\nIn particular, the parabolic $\\Q$-subgroup $\\bP$ is maximal and the simple root $\\alpha \\in \\Delta$, satisfying that $\\bP = \\bP_{\\Delta \\smallsetminus \\{\\alpha \\}}$, has at most one neighbor in the associated Dynkin diagram.\n\\end{thmx}", + "eq:Norm": "\\begin{equation} \\label{eq:Norm}\n\\cS_r(\\phi) = \\sum_{\\deg(\\cD) \\leq r} \\left \\| \\cD \\phi \\right \\|_{\\infty},\n\\end{equation}", + "thm:Linfty": "\\begin{thmx} [$L^{\\infty}$-integrability] \\label{thm:Linfty} \nThe following assertions are equivalent.\n\\begin{enumerate} [label=(\\arabic*)]\n\\item The Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^{\\infty}(\\Omega)$.\n\\item The $\\mathbb{Q}$-rank of $\\mathbf{G}$ is $1$.\n\\item The discrete group $\\G_L$ is a cocompact lattice in $L$.\n\\end{enumerate}\n\\end{thmx}", + "eq:Proof_Sketch_Decomposition": "\\begin{equation} \\label{eq:Proof_Sketch_Decomposition}\n\\cE_{\\beta_{\\chi}}(e^N)^+ = \\bigsqcup_{i=0}^{N-1} a(e^{\\beta_{\\chi}})^{-i} \\cF. \n\\end{equation}", + "eq:primitive_Siegel_Transform_variance": "\\begin{equation} \\label{eq:primitive_Siegel_Transform_variance}\n\\int_{\\Omega} \\left| S f - \\frac{1}{\\zeta(n)} \\int_{\\R^n} f \\, \\dd \\lambda_{\\R^n} \\right|^2 \\dd \\mu_{\\Omega} \\, \\ll \\, \\int_{\\R^n} |f|^2 \\, \\dd \\lambda_{\\R^n}.\n\\end{equation}" + }, + "pre_theorem_intro_text_len": 22775, + "pre_theorem_intro_text": "The Siegel transform, introduced in 1945 by Siegel \\cite{Siegel45}, maps a function of sufficient decay on the Euclidean space $\\mathbb{R}^n$ to a function on the moduli space of unimodular lattices $\\Omega = \\mathrm{SL}_n(\\mathbb{R}) / \\mathrm{SL}_n(\\mathbb{Z})$. Let ${\\mathcal P}(\\mathbb{Z}^n)$ denote the set of primitive elements of $\\mathbb{Z}^n$ and let $B_c^{\\infty}(\\mathbb{R}^n)$ be the space of Borel measurable bounded compactly supported functions $f : \\mathbb{R}^n \\rightarrow \\mathbb{C}$. Then, for every $f \\in B_c^{\\infty}(\\mathbb{R}^n)$, the primitive Siegel transform $S f : \\Omega \\rightarrow \\mathbb{C}$ of $f$ is defined by\n\\begin{equation} \\label{eq:primitive_Siegel_Transform}\n\\forall \\, g \\in \\mathrm{SL}_n(\\mathbb{R}), \\qquad S f(g\\mathbb{Z}^n) = \\sum_{\\bm{v} \\in {\\mathcal P}(\\mathbb{Z}^n)} f (g \\bm{v}).\n\\end{equation}\nLet $\\mu_{\\Omega}$ be the unique $\\SL_{n}(\\mathbb{R})$-invariant probability measure on $\\Omega$, let $\\zeta$ be the Riemann zeta function and let $\\lambda_{\\mathbb{R}^n}$ be the usual Lebesgue measure on $\\mathbb{R}^n$. Siegel's mean value formula \\cite{Siegel45} expresses the average of $Sf$ in terms of the average of $f$: \n\\begin{equation} \\label{eq:primitive_Siegel_Transform_formula}\n\\int_{\\Omega} S f \\, \\,\\mathrm{d} \\mu_{\\Omega} = \\frac{1}{\\zeta(n)} \\int_{\\mathbb{R}^n} f \\, \\,\\mathrm{d} \\lambda_{\\mathbb{R}^n}.\n\\end{equation}\nLater, extending Siegel’s result, Rogers \\cite{Rogers55} proved a $k$-th moment formula for the Siegel transform for $k$ up to $n-1$. A remarkable application of the second moment formula to the geometry of numbers was given by Schmidt \\cite{Schmidt60b}, who derived an asymptotic formula for counting lattice points in an expanding family of sets in $\\mathbb{R}^n$ from the variance bound\n\\begin{equation} \\label{eq:primitive_Siegel_Transform_variance}\n\\int_{\\Omega} \\left| S f - \\frac{1}{\\zeta(n)} \\int_{\\mathbb{R}^n} f \\, \\,\\mathrm{d} \\lambda_{\\mathbb{R}^n} \\right|^2 \\,\\mathrm{d} \\mu_{\\Omega} \\, \\ll \\, \\int_{\\mathbb{R}^n} |f|^2 \\, \\,\\mathrm{d} \\lambda_{\\mathbb{R}^n}.\n\\end{equation}\nIn other words, the \\emph{centered} Siegel transform $\\overline{S} f = Sf - 1/\\zeta(n) \\int_{\\mathbb{R}^n} f \\, \\,\\mathrm{d} \\lambda_{\\mathbb{R}^n}$ extends to a bounded linear operator \n\\[\n\\overline{S} : L^2(\\mathbb{R}^n) \\rightarrow L^2(\\Omega). \n\\]\nThe variance bound \\eqref{eq:primitive_Siegel_Transform_variance} also yields an alternative proof of Schmidt’s strengthening \\cite{Schmidt60a} of Khintchine’s theorem \\cite{Khintchine26} in metric Diophantine approximation on $\\mathbb{R}^n$ and its projective counterpart $\\mathbb{P}(\\mathbb{R}^n)$. There has been an active line of research extending classical results in Diophantine approximation from Euclidean space to other varieties, such as spheres \\cite{AG22, KM15, KY23, Ouaggag23}, projective quadrics \\cite{SK18, deSaxce22b, FKMS22}, Grassmannians \\cite{deSaxce22a}, and more general flag varieties \\cite{deSaxce20}. \n\nThe purpose of this paper is to study fundamental integrability properties of a natural extension of the Siegel transform \\eqref{eq:primitive_Siegel_Transform} from the Euclidean space to the setting of generalized flag varieties. Our results have an application to metric Diophantine approximation on rank-one flag varieties. The proof of this application relies in addition on the effective single and double equidistribution property for expanding orbits of maximal compact subgroups, a result of independent interest. \n\n\\subsection{Main results}\nLet ${\\bf G}$ be a connected simply-connected almost $\\mathbb{Q}$-simple $\\mathbb{Q}$-group and let ${\\bf P}$ be a proper parabolic $\\mathbb{Q}$-subgroup of ${\\bf G}$. We denote algebraic varieties defined over $\\mathbb{Q}$ by bold letters and their sets of real points by ordinary letters. For instance, we write $G = {\\bf G}(\\mathbb{R})$ to denote the group of real points of ${\\bf G}$. Let $\\Gamma \\subset {\\bf G}(\\mathbb{Q})$ be an arithmetic subgroup of $G$. Let $\\pi_{\\chi} : {\\bf G} \\rightarrow \\GL(\\bV_{\\chi})$ be an irreducible representation defined over $\\mathbb{Q}$ which is generated by a line $\\bD_{\\chi}$ defined over $\\mathbb{Q}$ of highest weight $\\chi$ such that ${\\bf P} = \\mathrm{Stab}_{{\\bf G}} (\\bD_{\\chi})$ (Section~\\ref{sec:Reps}). In particular, the space of real points $X = {\\bf X}(\\mathbb{R})$ of the generalized flag variety ${\\bf X} = {\\bf G} / {\\bf P}$ embeds into the projective space $\\mathbb{P}(V_{\\chi})$. We fix a highest weight vector $\\bm{e}_{\\chi} \\in \\bD_{\\chi}(\\mathbb{Q})$ and define $\\widetilde{X}$ to be the orbit $\\widetilde{X} = G \\, \\bm{e}_{\\chi} \\subset V_{\\chi}$. We refer to $\\widetilde{X}$ as the \\emph{cone over $X$ relative to $\\chi$}. Fix a $\\Gamma$-stable lattice $\\bV_{\\chi}(\\mathbb{Z}) \\subset \\bV_{\\chi}(\\mathbb{Q})$ of $V_{\\chi}$ and denote by $\\cP_{\\chi}$ the set of primitive elements of $\\bV_{\\chi}(\\mathbb{Z}) \\cap \\widetilde{X}$. Let $B_c^{\\infty}(\\widetilde{X})$ be the space of Borel measurable bounded compactly supported complex-valued functions $f : \\widetilde{X} \\rightarrow \\mathbb{C}$.\n\\begin{definition} [Siegel transform]\nFor every $f \\in B_c^{\\infty}(\\widetilde{X})$, we define the \\emph{Siegel transform} $S_{\\chi} f : \\Omega \\rightarrow \\mathbb{C}$ of $f$ by\n\\begin{equation*} \\label{def:Siegel-Transform}\n\\forall \\, g \\in G, \\quad S_{\\chi} f(g \\Gamma) = \\sum_{\\bm{v} \\in \\cP_{\\chi}} f(g \\bm{v}). \n\\end{equation*} \n\\end{definition}\n\nLet $\\mu_{\\Omega}$ be the unique $G$-invariant Borel probability measure on the homogeneous space $\\Omega = G/\\Gamma$. We will answer the question: \\emph{For any $p = 1,2, \\infty$, what are necessary and sufficient conditions for $S_{\\chi}$ to map $B_c^{\\infty}(\\widetilde{X})$ into $L^p(\\Omega)$?} \n\nIn our first result, the equivalences $(1)$ - $(4)$ are likely known to experts; the formula \\eqref{eq:WeilIntegrationFormula*} below is a consequence of a general integration formula due to Weil \\cite[Theorem~2.51]{Folland15}. Let ${\\bf L} = \\mathrm{Stab}_{{\\bf G}} (\\bm{e}_{\\chi}) \\subset {\\bf P}$. \n\n\\begin{thmx} [$L^1$-integrability] \\label{thm:L1} \nThe following assertions are equivalent.\n\\begin{enumerate}[label=(\\arabic*)]\n\\item The Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^1(\\Omega)$.\n\\item There exists a unique (up to scaling) $G$-invariant Radon measure $\\lambda_{\\widetilde{X}}$ on $\\widetilde{X}$, the Siegel transform $S_\\chi$ extends to a bounded operator $S_{\\chi} : L^1(\\widetilde{X}) \\rightarrow L^1(\\Omega)$, and $\\lambda_{\\widetilde{X}}$ can be normalized so that we have a convergent mean value formula:\n\\begin{equation} \\label{eq:WeilIntegrationFormula*}\n\\forall \\, f \\in L^1(\\widetilde{X}), \\quad \\int_{\\Omega} S_\\chi f \\, \\,\\mathrm{d} \\mu_{\\Omega} = \\int_{\\widetilde{X}} f \\, \\,\\mathrm{d} \\lambda_{\\widetilde{X}}.\n\\end{equation}\n\\item The Lie group $L = {\\bf L}(\\mathbb{R})$ is unimodular and $\\G_L = \\Gamma \\cap L$ is a lattice in $L$.\n\\item The parabolic $\\mathbb{Q}$-subgroup ${\\bf P}$ of ${\\bf G}$ is maximal.\n\\item There exists $\\varepsilon > 0$ such that $S_\\chi$ maps $B_{c}^{\\infty}(\\widetilde{X})$ into $L^{1+\\varepsilon}(\\Omega)$.\n\\end{enumerate}\n\\end{thmx}\n\n\\begin{thmx} [$L^{\\infty}$-integrability] \\label{thm:Linfty} \nThe following assertions are equivalent.\n\\begin{enumerate} [label=(\\arabic*)]\n\\item The Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^{\\infty}(\\Omega)$.\n\\item The $\\mathbb{Q}$-rank of $\\mathbf{G}$ is $1$.\n\\item The discrete group $\\G_L$ is a cocompact lattice in $L$.\n\\end{enumerate}\n\\end{thmx}\n\nLet $\\bP_0$ be a minimal parabolic $\\mathbb{Q}$-subgroup of ${\\bf G}$ contained in ${\\bf P}$ and let ${\\bf T}$ be a maximal $\\mathbb{Q}$-split torus of ${\\bf G}$ contained in $\\bP_0$. Let $\\Phi$ be the root system of ${\\bf G}$ relative to ${\\bf T}$ and let $\\Delta \\subset \\Phi$ be the corresponding set of simple roots. For each subset $\\theta$ of $\\Delta$, write $\\bP_{\\theta}$ for the associated standard parabolic $\\mathbb{Q}$-subgroup of ${\\bf G}$. Let $W$ be the Weyl group of ${\\bf G}$ relative to ${\\bf T}$. For every $w \\in W$, we define $\\bL_w = {\\bf L} \\cap x_w {\\bf L} x_w^{-1}$, where $x_w \\in \\cN_{{\\bf G}}({\\bf T})(\\mathbb{Q})$ is a representative of $w$, and denote by $X^*(\\bL_w^{\\circ})_{\\mathbb{Q}}$ the group of $\\mathbb{Q}$-characters of its identity component. \n\n\\begin{thmx} [$L^2$-integrability] \\label{thm:L2} \nSuppose that the Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^{2}(\\Omega)$. Then \n\\[\n\\forall \\, w \\in W, \\quad X^*(\\bL_w^\\circ)_{\\mathbb{Q}} = \\{1\\}.\n\\]\nIn particular, the parabolic $\\mathbb{Q}$-subgroup ${\\bf P}$ is maximal and the simple root $\\alpha \\in \\Delta$, satisfying that ${\\bf P} = \\bP_{\\Delta \\smallsetminus \\{\\alpha \\}}$, has at most one neighbor in the associated Dynkin diagram.\n\\end{thmx}\n\nWe were unable to determine whether the converse statement is true: \\emph{Assuming that for every $w \\in W$, we have $X^*(\\bL_w^\\circ)_{\\mathbb{Q}} = \\{1\\}$, does the Siegel transform $S_\\chi$ map $B_c^{\\infty}(\\widetilde{X})$ into $L^2(\\Omega)$?} \n\nFor every $f \\in B_c^{\\infty}(\\widetilde{X})$, define the \\emph{centered Siegel transform} $\\overline{S}_{\\chi} f : \\Omega \\rightarrow \\mathbb{C}$ of $f$ by\n\\[\n\\forall \\, g \\in G, \\qquad \\overline{S}_{\\chi} f(g\\Gamma) = S_{\\chi}f(g\\Gamma) - \\int_{\\widetilde{X}} f \\, \\,\\mathrm{d} \\lambda_{\\widetilde{X}}.\n\\] \nBeyond the case $p=q=1$ in Theorem~\\ref{thm:L1}, it is natural to ask for which pairs $p,q\\in[1,+\\infty]$ the Siegel transform, or its centered counterpart, extends to a bounded linear operator $L^p(\\widetilde{X}) \\to L^q(\\Omega)$. More specifically, we would like to include the following question, due to Saxc\\'e, suggesting a fractional version of the variance bound \\eqref{eq:primitive_Siegel_Transform_variance} that also takes into account point~(5) of Theorem~\\ref{thm:L1} as well as Theorem~\\ref{thm:L2}:\n\\emph{Assuming that the parabolic $\\mathbb{Q}$-subgroup ${\\bf P}$ is maximal, does there exist $\\varepsilon>0$ such that the centered Siegel transform $\\overline{S}_{\\chi}$ extends to a bounded linear operator}\n\\[\n\\overline{S}_{\\chi} : L^{1+\\varepsilon}(\\widetilde{X}) \\to L^{1+\\varepsilon}(\\Omega)\\,?\n\\]\n\n\\subsection{Effective equidistribution of maximal compact subgroup orbits}\nThe fact that the Siegel transform $S_{\\chi}$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^{1+\\varepsilon}(\\Omega)$ for some small $\\varepsilon > 0$ when the parabolic subgroup ${\\bf P}$ is maximal (Theorem~\\ref{thm:L1}), together with the effective single and double equidistribution property for translated orbits of maximal compact subgroups (Theorem~\\ref{thm:Effective}), are the key analytic inputs for our Schmidt-type counting theorem at the Diophantine exponent on rank-one flag varieties. Before describing this application, let us state here our equidistribution result, which will be derived from an effective multiple equidistribution result for expanding translates of horospherical orbits due to Shi (see \\cite[Theorem~1.5]{Shi21}). Let $K \\subset G$ be a maximal compact subgroup, equipped with the Haar probability measure $\\mu_K$. We write $\\cS_r$ ($r \\in \\mathbb{N}^*)$ for the degree $r$ Sobolev norms on $C_c^{\\infty}(\\Omega)$ and $C^{\\infty}(K)$ as defined in \\eqref{eq:Norm}.\n\n\\begin{thmx} [Effective equidistribution of maximal compact subgroup orbits] \\label{thm:Effective}\nSuppose the action of $G$ on $\\Omega = G/\\Gamma$ has a spectral gap. Let $A = \\{a(y) : y \\in \\R_+^{\\times} \\}$ be an $\\Ad$-diagonalizable one-parameter subgroup of $G$ such that $a(e)$ projects non-trivially to each simple factor of $G$. Then there exist a constant $c > 0$ and an integer $r \\geq 1$ such that for every compact subset $Q \\subset \\Omega$, for all $f \\in C^{\\infty}(K)$, $\\phi \\in C_c^\\infty(\\Omega)$, $x \\in Q$, and $y \\geq 1$, we have\n\\begin{equation} \\label{eq:Single-Eq}\n\\int_K f(k) \\phi \\bigl(a(y) k x \\bigr) \\,\\mathrm{d} \\mu_K(k) = \\int_K f \\,\\mathrm{d} \\mu_K \\int_{\\Omega} \\phi \\,\\mathrm{d} \\mu_{\\Omega} + O \\big ( y^{-c} \\cS_r(f) \\cS_r(\\phi) \\big ),\n\\end{equation}\nand, for all $f \\in C^{\\infty}(K)$, $\\phi_1, \\phi_2 \\in C_c^\\infty(\\Omega)$, $x_1, x_2 \\in Q$, \nand $y_2 \\geq y_1 \\geq 1$, we have\n\\begin{multline} \\label{eq:Double-Eq}\n\\int_K f(k) \\phi_{1}\\bigl(a(y_1) k x_1 \\bigr) \\phi_{2} \\bigl(a(y_2) k x_2 \\bigr) d \\mu_K(k) = \\int_K f \\,\\mathrm{d} \\mu_K \\int_{\\Omega} \\phi_1 \\,\\mathrm{d} \\mu_{\\Omega} \\int_{\\Omega} \\phi_2 \\,\\mathrm{d} \\mu_{\\Omega} \\\\ + O \\big ( \\min \\{y_1, y_2/y_1 \\}^{-c} \\cS_r(f) \\cS_r(\\phi_1) \\cS_r(\\phi_2) \\big ),\n\\end{multline} \nwhere the implicit constant in $O(\\cdot)$ depends only on $Q$.\n\\end{thmx}\n\n\\subsection{Application to Diophantine approximation on flag varieties}\nLet us now state our Schmidt-type counting theorem at the Diophantine exponent on rank-one flag varieties, which uses Theorems~\\ref{thm:L1} and \\ref{thm:Effective} as inputs. Many classical results in Diophantine approximation on the real line $\\mathbb{R}$ or in Euclidean space $\\mathbb{R}^n$ admit a dynamical reinterpretation in terms of properties of certain diagonal orbits in the space of lattices $\\Omega = \\SL_n(\\mathbb{R}) / \\SL_n(\\mathbb{Z})$; this is known as Dani's correspondence \\cite{Dani85}. Via this dynamical reinterpretation and building on influential work of Margulis, Kleinbock and others \\cite{KM96, KM98, KM99, FKMS22}, Saxc\\'e \\cite{deSaxce20} extended analogues of classical results to generalized flag varieties ${\\bf X} = {\\bf G} / {\\bf P}$ defined over $\\mathbb{Q}$. First examples of such varieties include projective $n$-space $\\mathbb{P}^n(\\mathbb{R})$, the Grassmann variety $\\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ of $\\ell$-dimensional subspaces of $\\mathbb{R}^n$, projective quadric hypersurfaces (that is, the solution set in $\\mathbb{P}^n(\\mathbb{R})$ of a non-degenerate rational quadratic form in $n+1$ variables), and more general flag varieties, parametrizing flags of subspaces of a Euclidean space. \n\nLet $\\psi : \\mathbb{N} \\rightarrow (0,+\\infty)$ be a non-increasing function. By Khintchine's theorem~\\cite{Khintchine26}, the inequality\n\\[\n0 \\leq q x - p < \\psi(q)\n\\]\nhas infinitely (resp. at most finitely) many solutions $(p,q) \\in \\mathbb{Z} \\times \\mathbb{N}$ for almost every $x \\in \\mathbb{R}$, if the series $\\sum_{q = 1}^{\\infty} \\psi(q)$ diverges (resp. converges). In the divergence case, Schmidt~\\cite{Schmidt60a} strengthened Khintchine's theorem. More precisely, for every $x \\in \\mathbb{R}$ and $T \\geq 1$, he considered the counting function\n\\begin{equation} \\label{eq:SchmidtCountingFunction}\n\\cN_\\psi(x,T) = \\# \\left \\{ (p,q) \\in \\mathbb{Z} \\times \\mathbb{N} : 0 \\leq q x - p < \\psi(q), \\, 1 \\leq q < T \\right \\}\n\\end{equation}\nand showed that for almost every $x \\in \\mathbb{R}$, $\\cN_\\psi(x,T)$ is asymptotically equal to $\\sum_{1\\leq q < T} \\psi(q)$ as $T$ goes to infinity, with an explicit error term. In fact, Schmidt's result holds not only for the real line, but also for the Euclidean space $\\mathbb{R}^n$ of any dimension $n \\geq 1$. \n\nOur goal is to prove a version of this theorem, where the Euclidean space $\\mathbb{R}^n$ is replaced by the space of real points $X = {\\bf X}(\\mathbb{R})$ of the generalized flag variety ${\\bf X} = {\\bf G} / {\\bf P}$ defined over $\\mathbb{Q}$. We assume that ${\\bf P}$ is a maximal parabolic $\\mathbb{Q}$-subgroup of ${\\bf G}$ with abelian unipotent radical. In particular, ${\\bf X}$ has $\\mathbb{Q}$-rank $1$ and there exists a unique simple root $\\alpha \\in \\Delta$ such that ${\\bf P} = \\bP_{\\Delta \\smallsetminus \\{\\alpha\\}}$. Let $Y$ be the unique element in the Lie algebra of ${\\bf T}(\\mathbb{R})$ such that \n\\[\n\\alpha(Y) = -1 \\quad \\text{and} \\quad \\beta(Y) = 0 \\quad \\text{for all } \\beta \\in \\Delta \\smallsetminus \\{\\alpha\\}.\n\\]\nWe suppose that the element $\\exp(Y)$ projects non-trivially to each simple factor of $G$. Let $K$ be a maximal compact subgroup of $G$. Let $\\sigma_X$ be the unique $K$-invariant probability measure on $X$. We equip $X$ with a $K$-invariant Riemannian distance $d(\\cdot, \\cdot)$ and the set of rational points ${\\bf X}(\\mathbb{Q})$ with a height function $H_\\chi$ associated to an irreducible rational representation $\\pi_{\\chi} : {\\bf G} \\rightarrow \\GL(\\bV_{\\chi})$ which is generated by a unique rational line $\\bD_{\\chi}$ of highest weight $\\chi$ such that $\\mathrm{Stab}_{{\\bf G}} (\\bD_{\\chi}) = {\\bf P}$ (see Section \\ref{sec:Reps}). By \\cite[Th\\'eor\\`emes 2.4.5 et 3.2.1]{deSaxce20}, there exists a rational number $\\beta_\\chi \\in \\Q_{>0}$ such that, for every $c > 0$ and for $\\sigma_X$-almost every $x \\in X$, the inequality\n\\begin{equation} \\label{eq:DiophantineExpo}\nd(x,v) < c \\, H_\\chi (v)^{-\\tau}\n\\end{equation}\nadmits infinitely (resp. at most finitely) many solutions $v \\in {\\bf X}(\\mathbb{Q})$, if $\\tau \\leq \\beta_\\chi$ (resp. $\\tau > \\beta_\\chi$). We refer to $\\beta_\\chi$ as the \\emph{Diophantine exponent} of $X$ relative to $\\chi$ and to $\\tau \\in [0,\\beta_{\\chi}]$ as an \\emph{approximation exponent}.\n\nIn analogy to \\eqref{eq:SchmidtCountingFunction}, for every constant $c > 0$, approximation exponent $\\tau \\in [0, \\beta_{\\chi}]$, element $x \\in X$, and parameter $T \\geq 1$, we define \n\\[\n\\cN_{c,\\tau}(x,T) = \\# \\left \\{ v \\in {\\bf X}(\\mathbb{Q}) : d(x,v) < c \\, H_{\\chi}(v)^{-\\tau}, \\, 1 \\leq H_{\\chi}(v) < T \\right \\}.\n\\]\nIn \\cite{Pfitscher24}, we provided an almost-sure asymptotic formula for $\\cN_{c,\\tau}(x,T)$ as $T \\rightarrow +\\infty$, with an explicit error term in the case where $\\tau \\in [0, \\beta_{\\chi})$. Our method did not yield an effective estimate when counting \\emph{at the Diophantine exponent}, that is, when $\\tau = \\beta_{\\chi}$. In our application, we upgrade our previous result to an effective asymptotic estimate. Our approach is inspired by a recent effective counting result due to Ouaggag \\cite[Theorem~1.2]{Ouaggag23} for spheres, and our result may be viewed as a substantial generalization thereof. \n\n\\begin{thmx}[Effective counting at the Diophantine exponent] \\label{thm:Critical}\nLet $d = \\dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{eq:Thm_Critical}\n\\mathcal{N}_{c, \\beta_{\\chi}}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}\n\\end{thmx}\n\n\\subsection{Proof sketch of Theorem \\ref{thm:Critical}}\nLet us illustrate Theorem~\\ref{thm:Critical} and its proof in the special case of the Grassmann variety $X = \\mathrm{Gr}_{\\ell, n}(\\mathbb{R})$ ($\\ell, n \\in \\mathbb{N}, \\, 1 \\leq \\ell < n$), parametrizing $\\ell$-dimensional subspaces of the Euclidean space $\\mathbb{R}^n$; this theorem also applies to projective quadric hypersurfaces and we refer the reader to \\cite[Sections $1.2$ and $8$]{Pfitscher24} and the references therein. The argument involves introducing the Siegel transform in this specific setting, studying its analytic properties, and establishing equidistribution of expanding translates of orbits of maximal compact subgroups. \n\nLet ${\\bf G} = \\SL_n$, let ${\\bf T} \\leq {\\bf G}$ be the maximal $\\mathbb{Q}$-split $\\mathbb{Q}$-torus given by the subgroup of ${\\bf G}$ consisting of all diagonal matrices, and let $\\bP_0$ be the Borel subgroup of ${\\bf G}$ consisting of all upper-triangular matrices. Let $\\Phi = \\Phi({\\bf G}, {\\bf T})$ be the associated root system with ordering induced by $\\bP_0$, $\\Delta = \\{\\alpha_1, \\dots, \\alpha_{n-1}\\}$ the set of simple roots, and $\\{\\lambda_{1}, \\dots, \\lambda_{n-1} \\}$ the set of fundamental $\\mathbb{Q}$-weights. Fix $\\alpha_{\\ell} \\in \\Delta$ and let $\\chi = \\lambda_{\\ell}$ be the associated fundamental $\\mathbb{Q}$-weight. Recall that for all $a = \\diag(a_1, \\dots, a_n) \\in {\\bf T}$, we have $\\chi(a) = a_1 \\cdots a_{\\ell}$. Let ${\\bf P} = \\bP_{\\Delta \\smallsetminus \\{\\alpha_{\\ell}\\}}$ be the corresponding standard parabolic $\\mathbb{Q}$-subgroup. Then ${\\bf P}$ is the stabilizer in ${\\bf G}$ of the rational line spanned by the pure tensor $\\bm{e}_{\\chi} = \\bm{e}_1 \\wedge \\dots \\wedge \\bm{e}_{\\ell}$ in the $\\ell$-th exterior power of the standard representation of ${\\bf G}$. The Siegel transform in this case is defined as follows. Let $\\widetilde{X} = G \\, \\bm{e}_{\\chi} \\subset \\bigwedge^\\ell \\mathbb{R}^n$: this is the set of all non-zero pure tensors of $\\bigwedge^\\ell \\mathbb{R}^n$. Let ${\\bf L} = \\mathrm{Stab}_{{\\bf G}} ( \\bm{e}_{\\chi})$, $\\Gamma = \\SL_n(\\mathbb{Z})$ and let $\\cP_{\\chi}$ be the set of all primitive elements of $\\bigwedge^\\ell \\mathbb{Z}^n$ that are contained in $\\widetilde{X}$. The group $\\Gamma$ acts transitively on $\\cP_{\\chi}$: $\\cP_{\\chi} = \\Gamma \\, \\bm{e}_{\\chi} \\cong \\Gamma/\\G_L$. Therefore, for every $f \\in B_c^{\\infty}(\\widetilde{X})$, the Siegel transform $S_{\\chi} f : G / \\Gamma \\rightarrow \\mathbb{C}$ is given by\n\\[\n\\forall \\, g \\in G, \\qquad S_{\\chi} f(g \\Gamma) = \\sum_{\\bm{v} \\in \\cP_{\\chi}} f(g \\bm{v}) = \\sum_{\\gamma \\in \\Gamma / \\G_L} f(g \\gamma \\bm{e}_{\\chi}).\n\\]\nThen $X = G/P$, viewed as a subvariety of $\\mathbb{P}(\\bigwedge^\\ell \\mathbb{R}^n)$ via the embedding $g P \\mapsto g [\\bm{e}_{\\chi}]$ (here $[\\bm{e}_{\\chi}]$ denotes the projectivization of $\\bm{e}_{\\chi}$), is the Grassmann variety $\\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ of $\\ell$-dimensional subspaces of $\\mathbb{R}^n$. This is in accordance with Schmidt's paper \\cite{Schmidt67}, where he used the Pl\\\"ucker embedding to define the height $H(v)$ of a rational subspace $v$ of $\\mathbb{R}^n$: for $v \\in \\mathrm{Gr}_{\\ell,n}(\\mathbb{Q})$ pick $\\bm{v} \\in \\cP_{\\chi}$ with $v = [\\bm{v}]$ and set $H(v) = \\|\\bm{v}\\|$, where $\\|\\cdot \\|$ denotes the $\\SO_n(\\mathbb{R})$-invariant norm on $\\bigwedge^{\\ell} \\mathbb{R}^n$ induced from the standard Euclidean norm on $\\mathbb{R}^n$. The distance used on $X$ is the usual Riemannian distance and we equip $X$ with the unique probability measure $\\sigma_X$ invariant under the action of the maximal compact subgroup $K = \\SO_n(\\mathbb{R}) \\leq G$. We study the approximation of a real subspace chosen randomly according to $\\sigma_X$ by rational subspaces. Write $d$ for the dimension $\\dim_\\mathbb{R} X = \\ell(n-\\ell)$. The Diophantine exponent of $X = \\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ with respect to $\\chi = \\lambda_{\\ell}$ is given by $\\beta_{\\chi} = \\frac{n}{\\ell(n-\\ell)}$ (see \\cite[Th\\'eor\\`eme~1]{deSaxce22a}). We wish to determine the asymptotic behavior of the counting function\n\\begin{equation} \\label{eq:CountingFunction}\n\\cN_{c,\\beta_{\\chi}}(x,T) = \\# \\left \\{ v \\in \\mathrm{Gr}_{\\ell,n}(\\mathbb{Q}) : d(x,v) < c \\, H(v)^{-\\beta_{\\chi}}, \\, H(v) < T \\right \\}\n\\end{equation}\nas $T \\rightarrow + \\infty$, for $\\sigma_X$-almost every $x \\in X$. In fact, Theorem \\ref{thm:Critical} takes the following form in this special case.", + "context": "Let $\\psi : \\mathbb{N} \\rightarrow (0,+\\infty)$ be a non-increasing function. By Khintchine's theorem~\\cite{Khintchine26}, the inequality\n\\[\n0 \\leq q x - p < \\psi(q)\n\\]\nhas infinitely (resp. at most finitely) many solutions $(p,q) \\in \\mathbb{Z} \\times \\mathbb{N}$ for almost every $x \\in \\mathbb{R}$, if the series $\\sum_{q = 1}^{\\infty} \\psi(q)$ diverges (resp. converges). In the divergence case, Schmidt~\\cite{Schmidt60a} strengthened Khintchine's theorem. More precisely, for every $x \\in \\mathbb{R}$ and $T \\geq 1$, he considered the counting function\n\\begin{equation} \\label{eq:SchmidtCountingFunction}\n\\cN_\\psi(x,T) = \\# \\left \\{ (p,q) \\in \\mathbb{Z} \\times \\mathbb{N} : 0 \\leq q x - p < \\psi(q), \\, 1 \\leq q < T \\right \\}\n\\end{equation}\nand showed that for almost every $x \\in \\mathbb{R}$, $\\cN_\\psi(x,T)$ is asymptotically equal to $\\sum_{1\\leq q < T} \\psi(q)$ as $T$ goes to infinity, with an explicit error term. In fact, Schmidt's result holds not only for the real line, but also for the Euclidean space $\\mathbb{R}^n$ of any dimension $n \\geq 1$.\n\n\\begin{thmx}[Effective counting at the Diophantine exponent] \\label{thm:Critical}\nLet $d = \\dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{eq:Thm_Critical}\n\\mathcal{N}_{c, \\beta_{\\chi}}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}\n\\end{thmx}\n\n\\subsection{Proof sketch of Theorem \\ref{thm:Critical}}\nLet us illustrate Theorem~\\ref{thm:Critical} and its proof in the special case of the Grassmann variety $X = \\mathrm{Gr}_{\\ell, n}(\\mathbb{R})$ ($\\ell, n \\in \\mathbb{N}, \\, 1 \\leq \\ell < n$), parametrizing $\\ell$-dimensional subspaces of the Euclidean space $\\mathbb{R}^n$; this theorem also applies to projective quadric hypersurfaces and we refer the reader to \\cite[Sections $1.2$ and $8$]{Pfitscher24} and the references therein. The argument involves introducing the Siegel transform in this specific setting, studying its analytic properties, and establishing equidistribution of expanding translates of orbits of maximal compact subgroups.\n\nLet ${\\bf G} = \\SL_n$, let ${\\bf T} \\leq {\\bf G}$ be the maximal $\\mathbb{Q}$-split $\\mathbb{Q}$-torus given by the subgroup of ${\\bf G}$ consisting of all diagonal matrices, and let $\\bP_0$ be the Borel subgroup of ${\\bf G}$ consisting of all upper-triangular matrices. Let $\\Phi = \\Phi({\\bf G}, {\\bf T})$ be the associated root system with ordering induced by $\\bP_0$, $\\Delta = \\{\\alpha_1, \\dots, \\alpha_{n-1}\\}$ the set of simple roots, and $\\{\\lambda_{1}, \\dots, \\lambda_{n-1} \\}$ the set of fundamental $\\mathbb{Q}$-weights. Fix $\\alpha_{\\ell} \\in \\Delta$ and let $\\chi = \\lambda_{\\ell}$ be the associated fundamental $\\mathbb{Q}$-weight. Recall that for all $a = \\diag(a_1, \\dots, a_n) \\in {\\bf T}$, we have $\\chi(a) = a_1 \\cdots a_{\\ell}$. Let ${\\bf P} = \\bP_{\\Delta \\smallsetminus \\{\\alpha_{\\ell}\\}}$ be the corresponding standard parabolic $\\mathbb{Q}$-subgroup. Then ${\\bf P}$ is the stabilizer in ${\\bf G}$ of the rational line spanned by the pure tensor $\\bm{e}_{\\chi} = \\bm{e}_1 \\wedge \\dots \\wedge \\bm{e}_{\\ell}$ in the $\\ell$-th exterior power of the standard representation of ${\\bf G}$. The Siegel transform in this case is defined as follows. Let $\\widetilde{X} = G \\, \\bm{e}_{\\chi} \\subset \\bigwedge^\\ell \\mathbb{R}^n$: this is the set of all non-zero pure tensors of $\\bigwedge^\\ell \\mathbb{R}^n$. Let ${\\bf L} = \\mathrm{Stab}_{{\\bf G}} ( \\bm{e}_{\\chi})$, $\\Gamma = \\SL_n(\\mathbb{Z})$ and let $\\cP_{\\chi}$ be the set of all primitive elements of $\\bigwedge^\\ell \\mathbb{Z}^n$ that are contained in $\\widetilde{X}$. The group $\\Gamma$ acts transitively on $\\cP_{\\chi}$: $\\cP_{\\chi} = \\Gamma \\, \\bm{e}_{\\chi} \\cong \\Gamma/\\G_L$. Therefore, for every $f \\in B_c^{\\infty}(\\widetilde{X})$, the Siegel transform $S_{\\chi} f : G / \\Gamma \\rightarrow \\mathbb{C}$ is given by\n\\[\n\\forall \\, g \\in G, \\qquad S_{\\chi} f(g \\Gamma) = \\sum_{\\bm{v} \\in \\cP_{\\chi}} f(g \\bm{v}) = \\sum_{\\gamma \\in \\Gamma / \\G_L} f(g \\gamma \\bm{e}_{\\chi}).\n\\]\nThen $X = G/P$, viewed as a subvariety of $\\mathbb{P}(\\bigwedge^\\ell \\mathbb{R}^n)$ via the embedding $g P \\mapsto g [\\bm{e}_{\\chi}]$ (here $[\\bm{e}_{\\chi}]$ denotes the projectivization of $\\bm{e}_{\\chi}$), is the Grassmann variety $\\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ of $\\ell$-dimensional subspaces of $\\mathbb{R}^n$. This is in accordance with Schmidt's paper \\cite{Schmidt67}, where he used the Pl\\\"ucker embedding to define the height $H(v)$ of a rational subspace $v$ of $\\mathbb{R}^n$: for $v \\in \\mathrm{Gr}_{\\ell,n}(\\mathbb{Q})$ pick $\\bm{v} \\in \\cP_{\\chi}$ with $v = [\\bm{v}]$ and set $H(v) = \\|\\bm{v}\\|$, where $\\|\\cdot \\|$ denotes the $\\SO_n(\\mathbb{R})$-invariant norm on $\\bigwedge^{\\ell} \\mathbb{R}^n$ induced from the standard Euclidean norm on $\\mathbb{R}^n$. The distance used on $X$ is the usual Riemannian distance and we equip $X$ with the unique probability measure $\\sigma_X$ invariant under the action of the maximal compact subgroup $K = \\SO_n(\\mathbb{R}) \\leq G$. We study the approximation of a real subspace chosen randomly according to $\\sigma_X$ by rational subspaces. Write $d$ for the dimension $\\dim_\\mathbb{R} X = \\ell(n-\\ell)$. The Diophantine exponent of $X = \\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ with respect to $\\chi = \\lambda_{\\ell}$ is given by $\\beta_{\\chi} = \\frac{n}{\\ell(n-\\ell)}$ (see \\cite[Th\\'eor\\`eme~1]{deSaxce22a}). We wish to determine the asymptotic behavior of the counting function\n\\begin{equation} \\label{eq:CountingFunction}\n\\cN_{c,\\beta_{\\chi}}(x,T) = \\# \\left \\{ v \\in \\mathrm{Gr}_{\\ell,n}(\\mathbb{Q}) : d(x,v) < c \\, H(v)^{-\\beta_{\\chi}}, \\, H(v) < T \\right \\}\n\\end{equation}\nas $T \\rightarrow + \\infty$, for $\\sigma_X$-almost every $x \\in X$. In fact, Theorem \\ref{thm:Critical} takes the following form in this special case.", + "full_context": "Let $\\psi : \\mathbb{N} \\rightarrow (0,+\\infty)$ be a non-increasing function. By Khintchine's theorem~\\cite{Khintchine26}, the inequality\n\\[\n0 \\leq q x - p < \\psi(q)\n\\]\nhas infinitely (resp. at most finitely) many solutions $(p,q) \\in \\mathbb{Z} \\times \\mathbb{N}$ for almost every $x \\in \\mathbb{R}$, if the series $\\sum_{q = 1}^{\\infty} \\psi(q)$ diverges (resp. converges). In the divergence case, Schmidt~\\cite{Schmidt60a} strengthened Khintchine's theorem. More precisely, for every $x \\in \\mathbb{R}$ and $T \\geq 1$, he considered the counting function\n\\begin{equation} \\label{eq:SchmidtCountingFunction}\n\\cN_\\psi(x,T) = \\# \\left \\{ (p,q) \\in \\mathbb{Z} \\times \\mathbb{N} : 0 \\leq q x - p < \\psi(q), \\, 1 \\leq q < T \\right \\}\n\\end{equation}\nand showed that for almost every $x \\in \\mathbb{R}$, $\\cN_\\psi(x,T)$ is asymptotically equal to $\\sum_{1\\leq q < T} \\psi(q)$ as $T$ goes to infinity, with an explicit error term. In fact, Schmidt's result holds not only for the real line, but also for the Euclidean space $\\mathbb{R}^n$ of any dimension $n \\geq 1$.\n\n\\begin{thmx}[Effective counting at the Diophantine exponent] \\label{thm:Critical}\nLet $d = \\dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{eq:Thm_Critical}\n\\mathcal{N}_{c, \\beta_{\\chi}}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}\n\\end{thmx}\n\n\\subsection{Proof sketch of Theorem \\ref{thm:Critical}}\nLet us illustrate Theorem~\\ref{thm:Critical} and its proof in the special case of the Grassmann variety $X = \\mathrm{Gr}_{\\ell, n}(\\mathbb{R})$ ($\\ell, n \\in \\mathbb{N}, \\, 1 \\leq \\ell < n$), parametrizing $\\ell$-dimensional subspaces of the Euclidean space $\\mathbb{R}^n$; this theorem also applies to projective quadric hypersurfaces and we refer the reader to \\cite[Sections $1.2$ and $8$]{Pfitscher24} and the references therein. The argument involves introducing the Siegel transform in this specific setting, studying its analytic properties, and establishing equidistribution of expanding translates of orbits of maximal compact subgroups.\n\nLet ${\\bf G} = \\SL_n$, let ${\\bf T} \\leq {\\bf G}$ be the maximal $\\mathbb{Q}$-split $\\mathbb{Q}$-torus given by the subgroup of ${\\bf G}$ consisting of all diagonal matrices, and let $\\bP_0$ be the Borel subgroup of ${\\bf G}$ consisting of all upper-triangular matrices. Let $\\Phi = \\Phi({\\bf G}, {\\bf T})$ be the associated root system with ordering induced by $\\bP_0$, $\\Delta = \\{\\alpha_1, \\dots, \\alpha_{n-1}\\}$ the set of simple roots, and $\\{\\lambda_{1}, \\dots, \\lambda_{n-1} \\}$ the set of fundamental $\\mathbb{Q}$-weights. Fix $\\alpha_{\\ell} \\in \\Delta$ and let $\\chi = \\lambda_{\\ell}$ be the associated fundamental $\\mathbb{Q}$-weight. Recall that for all $a = \\diag(a_1, \\dots, a_n) \\in {\\bf T}$, we have $\\chi(a) = a_1 \\cdots a_{\\ell}$. Let ${\\bf P} = \\bP_{\\Delta \\smallsetminus \\{\\alpha_{\\ell}\\}}$ be the corresponding standard parabolic $\\mathbb{Q}$-subgroup. Then ${\\bf P}$ is the stabilizer in ${\\bf G}$ of the rational line spanned by the pure tensor $\\bm{e}_{\\chi} = \\bm{e}_1 \\wedge \\dots \\wedge \\bm{e}_{\\ell}$ in the $\\ell$-th exterior power of the standard representation of ${\\bf G}$. The Siegel transform in this case is defined as follows. Let $\\widetilde{X} = G \\, \\bm{e}_{\\chi} \\subset \\bigwedge^\\ell \\mathbb{R}^n$: this is the set of all non-zero pure tensors of $\\bigwedge^\\ell \\mathbb{R}^n$. Let ${\\bf L} = \\mathrm{Stab}_{{\\bf G}} ( \\bm{e}_{\\chi})$, $\\Gamma = \\SL_n(\\mathbb{Z})$ and let $\\cP_{\\chi}$ be the set of all primitive elements of $\\bigwedge^\\ell \\mathbb{Z}^n$ that are contained in $\\widetilde{X}$. The group $\\Gamma$ acts transitively on $\\cP_{\\chi}$: $\\cP_{\\chi} = \\Gamma \\, \\bm{e}_{\\chi} \\cong \\Gamma/\\G_L$. Therefore, for every $f \\in B_c^{\\infty}(\\widetilde{X})$, the Siegel transform $S_{\\chi} f : G / \\Gamma \\rightarrow \\mathbb{C}$ is given by\n\\[\n\\forall \\, g \\in G, \\qquad S_{\\chi} f(g \\Gamma) = \\sum_{\\bm{v} \\in \\cP_{\\chi}} f(g \\bm{v}) = \\sum_{\\gamma \\in \\Gamma / \\G_L} f(g \\gamma \\bm{e}_{\\chi}).\n\\]\nThen $X = G/P$, viewed as a subvariety of $\\mathbb{P}(\\bigwedge^\\ell \\mathbb{R}^n)$ via the embedding $g P \\mapsto g [\\bm{e}_{\\chi}]$ (here $[\\bm{e}_{\\chi}]$ denotes the projectivization of $\\bm{e}_{\\chi}$), is the Grassmann variety $\\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ of $\\ell$-dimensional subspaces of $\\mathbb{R}^n$. This is in accordance with Schmidt's paper \\cite{Schmidt67}, where he used the Pl\\\"ucker embedding to define the height $H(v)$ of a rational subspace $v$ of $\\mathbb{R}^n$: for $v \\in \\mathrm{Gr}_{\\ell,n}(\\mathbb{Q})$ pick $\\bm{v} \\in \\cP_{\\chi}$ with $v = [\\bm{v}]$ and set $H(v) = \\|\\bm{v}\\|$, where $\\|\\cdot \\|$ denotes the $\\SO_n(\\mathbb{R})$-invariant norm on $\\bigwedge^{\\ell} \\mathbb{R}^n$ induced from the standard Euclidean norm on $\\mathbb{R}^n$. The distance used on $X$ is the usual Riemannian distance and we equip $X$ with the unique probability measure $\\sigma_X$ invariant under the action of the maximal compact subgroup $K = \\SO_n(\\mathbb{R}) \\leq G$. We study the approximation of a real subspace chosen randomly according to $\\sigma_X$ by rational subspaces. Write $d$ for the dimension $\\dim_\\mathbb{R} X = \\ell(n-\\ell)$. The Diophantine exponent of $X = \\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ with respect to $\\chi = \\lambda_{\\ell}$ is given by $\\beta_{\\chi} = \\frac{n}{\\ell(n-\\ell)}$ (see \\cite[Th\\'eor\\`eme~1]{deSaxce22a}). We wish to determine the asymptotic behavior of the counting function\n\\begin{equation} \\label{eq:CountingFunction}\n\\cN_{c,\\beta_{\\chi}}(x,T) = \\# \\left \\{ v \\in \\mathrm{Gr}_{\\ell,n}(\\mathbb{Q}) : d(x,v) < c \\, H(v)^{-\\beta_{\\chi}}, \\, H(v) < T \\right \\}\n\\end{equation}\nas $T \\rightarrow + \\infty$, for $\\sigma_X$-almost every $x \\in X$. In fact, Theorem \\ref{thm:Critical} takes the following form in this special case.\n\n\\begin{thmx}[Effective counting at the Diophantine exponent] \\label{thm:Critical}\nLet $d = \\dim X$ be the dimension of $X$ and let $c > 0$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{eq:Thm_Critical}\n\\mathcal{N}_{c, \\beta_{\\chi}}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}\n\\end{thmx}\n\n\\begin{proof}\nWe shall need the following consequence of the proof of \\cite[Theorem~C]{Pfitscher24}. For every $T \\geq 1$, consider the function \n\\[\n\\cN(T) = \\# \\left \\{v \\in \\bX(\\Q) : H_{\\chi}(v) < T \\right \\}\n\\]\ncounting rational points in $X$ of height $< T$. Let $\\beta_{\\chi} \\in \\Q_{>0}$ be the Diophantine exponent of $X$ with respect to $\\chi$ (see \\cite[D\\'efinition~2.4.1 et Th\\'eor\\`eme~2.4.5]{deSaxce20}) and let $d = \\dim X$ be the dimension of $X$. Then, as $T \\rightarrow + \\infty$, we have $\\cN(T) \\sim T^{\\beta_{\\chi} d}$. Since there is a one-to-one correspondence between points in $\\bX(\\Q)$ and lines passing through $\\cP_{\\chi}$, by the definition of the height function $H_{\\chi}$, we also have that, as $T \\rightarrow + \\infty$\n\\begin{equation} \\label{eq:Number_Primitive_Bounded_Height}\n\\# \\left \\{\\bm{v} \\in \\cP_{\\chi} : \\|\\bm{v}\\| < T \\right \\} \\asymp T^{\\beta_{\\chi} d}.\n\\end{equation}\nFix $f \\in B_{c}^{\\infty}(\\widetilde{X})$ and pick $r = r(\\supp(f)) \\geq 1$ such that $\\supp(f)$ is contained in $B_{\\widetilde{X}}(r) = \\{\\bm{v} \\in \\widetilde{X} : \\|\\bm{v}\\| < r \\}$. The proof now proceeds using reduction theory as presented, for instance, in \\cite[Section~12, Theorem~13.1]{Borel69}. By a slight abuse of notation, we let $\\mathfrak{a}$ be the Lie algebra of $T^{\\circ}$ and, for every $\\tau \\geq 0$, let $\\mathfrak{a}_{\\tau} = \\{Y \\in \\mathfrak{a} : \\forall \\, \\beta \\in \\Delta, \\, \\beta(Y) \\leq \\tau \\}$. We set $A_{\\tau} = \\exp \\, \\mathfrak{a}_{\\tau}$ and an note that $\\mathfrak{a}^- = \\mathfrak{a}_0$ is the negative Weyl chamber of $\\mathfrak{a}$ with respect to $\\Delta$. Let $\\bM_0$ be the largest $\\Q$-anisotropic $\\Q$-subgroup of the centralizer $\\cZ_{\\bG}(\\bT)^{\\circ}$ in $\\bG$ of $\\bT$ and let $\\bU_0$ be the unipotent radical of the minimal parabolic $\\Q$-subgroup $\\bP_0$. There exist $\\tau > 0$, a compact subset $\\bm{\\omega}$ of $M_0 U_0$, and a finite subset $C \\subset \\bG(\\Q)$ such that the Siegel set $\\mathfrak{S} = K \\, A_{\\tau} \\, \\bm{\\omega}$ satisfies\n\\[\nG = \\mathfrak{S} \\, C \\, \\G.\n\\]\nIn particular, we can express, though not uniquely, each $g \\in G$ as $g = k a n c \\gamma$ with $k \\in K$, $a \\in A_{\\tau}$, $n \\in \\bm{\\omega}$, $c \\in C$, and $\\gamma \\in \\G$. Fix any norm $\\|\\cdot \\|_{\\mathfrak{a}}$ on $\\mathfrak{a}$ and, for $r_0 > 0$, let $B_{\\mathfrak{a}}(r_0)$ denote the corresponding ball centered at the origin with radius $r_0$. Let $r_0 > 0$ be such that $\\mathfrak{a}_{\\tau}$ is contained in $\\mathfrak{a}^- + B_{\\mathfrak{a}}(r_0)$. Let $k \\in K$, $n \\in \\bm{\\omega}$, $a \\in A_{\\tau}$, $c \\in C$, and $\\gamma \\in \\G$. We express $a = a^- \\, \\exp(O(1))$ with $a^- \\in \\exp(\\mathfrak{a}^-)$. Using that $\\lambda_\\chi$ is right $\\G$-invariant, that $K$ is compact, that $\\bigcup_{a \\in A_{\\tau}} a \\bm{\\omega}a^{-1}$ is relatively compact (see \\cite[Lemma~12.2]{Borel69}), and that $C \\subset \\bG(\\Q)$ consists of rational elements and is finite, we have\n\\begin{equation} \\label{eq:Proof-lem:Upper-Bound-Siegel}\n\\lambda_\\chi(k a n c \\gamma \\G) \\asymp \\lambda_\\chi(a^- \\G).\n\\end{equation}\nBy the description of the $\\Q$-weights of the representation $\\pi_{\\chi}$ in \\eqref{eq:Q-weights}, for every $\\Q$-weight $\\mu$ of $\\pi_{\\chi}$, we have\n\\[\n\\chi(a^-) \\leq \\mu(a^-). \n\\]\nHence, since we assumed $\\bV_{\\chi}(\\Z)$ to be spanned over $\\Z$ by an orthonormal basis consisting of weight vectors for the action of $T$, we have $\\lambda_\\chi(a^-) = \\chi(a^-)$. Thus, for every $\\bm{v} \\in V_{\\chi}$, we have $\\lambda_\\chi(a^-) \\| \\bm{v} \\| \\leq \\|a^- \\bm{v} \\|$. Using that the norm $\\| \\cdot \\|$ on $V_{\\chi}$ is $K$-invariant, that $\\bigcup_{a \\in A_{\\tau}} a \\bm{\\omega}a^{-1}$ is relatively compact, and that $C \\subset \\bG(\\Q)$ is finite, there exists a constant $C_0 \\geq 1$, independent of $f$, such that, for every $g \\in G$ with Siegel decomposition $g = k a n c \\gamma$ (and writing $a = a^- \\exp(O(1))$ as above), we have\n\\begin{align*}\n|S_{\\chi} f(g \\G)| &\\leq \\|f\\|_{\\infty} \\, \\# \\big \\{ \\bm{v} \\in \\cP_{\\chi} : \\| g \\bm{v} \\| < r \\big \\} \\\\\n&\\leq \\|f\\|_{\\infty} \\, \\# \\big \\{ \\bm{v} \\in \\cP_{\\chi} : \\| \\bm{v} \\| < C_0 \\, \\lambda_\\chi(a^-)^{-1} \\, r \\big \\}.\n\\end{align*}\nBy the estimate in \\eqref{eq:Number_Primitive_Bounded_Height}, we further have \n\\[\n\\# \\big \\{ \\bm{v} \\in \\cP_{\\chi} : \\| \\bm{v} \\| < C_0 \\, \\lambda_\\chi(a^-)^{-1} \\, r \\big \\} \\, \\ll_{\\supp(f)} \\, \\lambda_\\chi(a^-)^{-\\beta_{\\chi} d}.\n\\]\nThis together with \\eqref{eq:Proof-lem:Upper-Bound-Siegel} now implies that\n\\[\n|S_{\\chi} f(g\\G)| \\ll_{\\supp(f)} \\|f\\|_{\\infty} \\, \\lambda_\\chi(g \\G)^{-\\beta_\\chi d},\n\\]\nfinishing the proof of the lemma.\n\\end{proof}\n\nLet us now define these sets that approximate $\\cE_{\\beta_{\\chi}}(T)$. We recall from Section \\ref{sec:Distance} that the map $\\ku^- \\rightarrow X$ sending $u \\mapsto \\exp(u) x_0$ restricts to a diffeomorphism from a neighborhood of $1 \\in \\ku^-$ to a neighborhood of $x_0 \\in X$. In particular, any $\\bm{v} \\in \\widetilde{X}$, such that $[\\bm{v}]$ is close to $x_0$, defines an element $u_{\\bm{v}}^-$ in the Lie algebra $\\ku^-$ by $[\\bm{v}] = \\exp(u_{\\bm{v}}^-) x_0$. The adjoint action of $a(y) \\in A$ on $\\ku^- = T_{x_0} X$ acts by scalar multiplication: for all $y \\in \\R_+^\\times$, $\\Ad(a(y)) u^- = y \\, u^-$. Observe that \n\\[\n[a(y) \\bm{v}] = a(y) [\\bm{v}] = a(y) \\exp(u_{\\bm{v}}^-) a(y) a(y)^{-1} x_0 = \\exp(\\Ad(a(y)) u_{\\bm{v}}^-) x_0.\n\\]\nBut we also have $[a(y) \\bm{v}] = \\exp(u_{a(y) \\bm{v}}^- ) x_0$. By uniqueness, this gives the relation\n\\begin{equation} \\label{eq:LieAlgebraDiagonal}\nu_{a(y) \\bm{v}}^- = y \\, u_{\\bm{v}}^-.\n\\end{equation}\nMoreover, by the distance estimate \\eqref{eq:Distance_Estimate}, there exists a constant $C_0 > 0$ such that \n\\[\n\\left | d(x_0, [\\bm{v}]) - \\|u_{\\bm{v}}^-\\|_{\\ku^-} \\right | \\leq C_0 \\, \\|u_{\\bm{v}}^-\\|_{\\ku^-}^2.\n\\]\nLet $\\pi^+ : V_\\chi \\rightarrow V_\\chi$ be the orthogonal projection onto $\\R\\bm{e}_{\\chi}$ and we abbreviate $\\pi^+(\\bm{v})$ by $\\bm{v}^+$. For every $T \\geq 1$ and $c > 0$ close to $1$, we will work with the sets\n\\[\n\\cE_{T, c}^+ = \\{\\bm{v} \\in \\widetilde{X} : \\|u_{\\bm{v}}^-\\|_{\\ku^-} < c \\, \\|\\bm{v}^+\\|^{-\\beta_{\\chi}}, 1 \\leq \\|\\bm{v}^+\\| < c \\, T \\}.\n\\] \nBy enlarging $C_0$ if necessary, we can assume that $\\|\\bm{v}^+\\| \\geq C_0^{-1} \\|\\bm{v}\\|$ as soon as $d(x_0, [\\bm{v}]) < 1$. For every integer $\\ell \\geq 1$, we let \n\\[\nQ_{\\ell} = \\{\\bm{v} \\in \\widetilde{X} : \\|\\bm{v}\\| \\leq C_0 \\, \\ell \\}\n\\]\nand we define\n\\[\n\\widehat{c}_{\\ell} = \\left ( 1 + C_0 \\, \\ell^{-\\beta_{\\chi}} \\right )^{-(1+\\beta_{\\chi})} \\in (0,1).\n\\]\nIn particular, we have $\\widehat{c}_{\\ell} \\nearrow 1$ as $\\ell \\rightarrow + \\infty$.", + "post_theorem_intro_text_len": 7966, + "post_theorem_intro_text": "Let us now go through the main steps of the argument. For simplicity, we assume that $c = 1$ and we write $\\cN_{\\beta_{\\chi}}(x,T) = \\cN_{1,\\beta_{\\chi}}(x,T)$. The first observation is that the quantity $\\cN_{\\beta_{\\chi}}(x,T)$ can be understood as the Siegel transform of the indicator function of a certain subset $\\cE_{\\beta_{\\chi}}(T) \\subset \\widetilde{X}$ evaluated at a certain point in $\\Omega = G/\\Gamma$: we can associate to each $x \\in X$ an element $k_{x} \\in K$ such that \n\\[\n\\cN_{\\beta_{\\chi}}(x,T) = \\# (\\cP_{\\chi} \\cap k_{x} \\, \\cE_{\\beta_{\\chi}}(T)) = S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)} (k_{x}^{-1} \\Gamma).\n\\]\nBy Theorem \\ref{thm:L1}, since ${\\bf P} = \\bP_{\\Delta \\smallsetminus \\{\\alpha_{\\ell}\\}}$ is maximal, the group $L = {\\bf L}(\\mathbb{R})$ is unimodular, $\\widetilde{X} = G / L$ admits a unique up to scaling Radon measure $\\lambda_{\\widetilde{X}}$ and the expected value of $S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta}(T)}$, viewed as a random variable on $\\Omega$, is given by\n\\[\n\\int_{\\Omega} S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)} \\, \\,\\mathrm{d} \\mu_{\\Omega} = \\int_{\\widetilde{X}} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)} \\, \\,\\mathrm{d} \\lambda_{\\widetilde{X}} = \\lambda_{\\widetilde{X}} (\\cE_{\\beta_{\\chi}}(T)). \n\\]\nThe hope is that, for $\\sigma_X$-almost every $x \\in X$, the quantity $\\cN_{\\beta_{\\chi}}(x,T)$ is asymptotically equal to the volume $\\lambda_{\\widetilde{X}} (\\cE_{\\beta_{\\chi}}(T))$, as $T \\rightarrow + \\infty$, and this is what we will show. In fact, the main term on the right-hand side in \\eqref{cor:eq_Grassmannian} is just the explicit value of the (main term of the) volume $\\lambda_{\\widetilde{X}} (\\cE_{\\beta_{\\chi}}(T))$. In order to prove the desired asymptotic estimate, we will exploit the special geometry of the set $\\cE_{\\beta_{\\chi}}(T)$. In fact, this set can be approximated by a set $\\cE_{\\beta_{\\chi}}(T)^{+}$ that admits a simple decomposition under the action of the diagonal subgroup \n\\[\n\\forall \\, y \\in \\R_+^\\times, \\quad a(y) = \\diag \\big (\\underbrace{y^{-(n-\\ell)/n}, \\dots, y^{-(n-\\ell)/n}}_{\\text{$\\ell$ times}}, \\underbrace{y^{\\ell/n}, \\dots, y^{\\ell/n}}_{\\text{$n-\\ell$ times}} \\big ).\n\\]\nIndeed, there exists a subset ${\\mathcal F} \\subset \\widetilde{X}$ such that for all integers $N \\geq 1$ \n\\begin{equation} \\label{eq:Proof_Sketch_Decomposition}\n\\cE_{\\beta_{\\chi}}(e^N)^+ = \\bigsqcup_{i=0}^{N-1} a(e^{\\beta_{\\chi}})^{-i} {\\mathcal F}. \n\\end{equation}\n\n\\begin{figure}[htbp]\n\\includegraphics[scale=0.6]{figure_counting.png}\n\\caption{The set $\\cE_{\\beta_{\\chi}}(T)^+$ for the group $G = \\SL_2(\\mathbb{R})$, the flag variety the real projective line $X = \\mathbb{P}^1(\\mathbb{R}) = \\mathrm{Gr}_{1,2}(\\mathbb{R})$, the punctured affine cone $\\widetilde{X} = \\mathbb{R}^2 \\setminus \\{0\\}$ above $X$, and the set $\\cP_{\\chi} = \\mathcal{P}(\\mathbb{Z}^2)$ of primitive elements of $\\mathbb{Z}^2$. Rational approximations to a point $x \\in X$ of height bounded by $T$ correspond to primitive lattice points of $\\mathbb{Z}^2$ in the red region $k_x \\cE_{\\beta_{\\chi}}(T)$, where $k_x \\in \\SO_2(\\mathbb{R})$ is a rotation such that $x = k_x [\\bm{e}_1]$. The action of $a(y) = \\diag ( y^{-1/2}, y^{1/2})$ with $y > 1$ on $\\widetilde{X} = \\mathbb{R}^2 \\setminus \\{0\\}$ contracts the line through $\\bm{e}_\\chi = \\bm{e}_1$ and expands the line through $\\bm{e}_2$. The domain $\\cE_{\\beta_{\\chi}}(T)^+$ can be decomposed into translates of the elementary domain ${\\mathcal F}$ under the action of $a(y)$. The hope is that for $x$ chosen randomly according to the Lebesgue measure on $X$ the number of primitive lattice points in the red region $k_x \\cE_{\\beta_{\\chi}}(T)$, that is, the quantity $\\# (\\cP_{\\chi} \\cap k_x \\, \\cE_{\\beta_{\\chi}}(T))$, which is the classical primitive Siegel transform of the indicator function $\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}$ evaluated at the rotated lattice $k_x^{-1} \\mathbb{Z}^2$, is approximately given (up to a scalar) by the volume of $\\cE_{\\beta_{\\chi}}(T)$.}\n\\label{fig:4}\n\\end{figure}\n\nOn the level of the Siegel transform this yields the sum decomposition\n\\[\nS_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}(k_x^{-1} \\Gamma) = \\sum_{i=0}^{N-1} \\mathbbm{1}_{{\\mathcal F}}(a(e^{\\beta_{\\chi}})^{i} k_x^{-1} \\Gamma).\n\\]\nFrom now on, we simply view $S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}(k \\Gamma)$ as a random variable on the probability space $(K, \\mu_K)$, where $\\mu_K$ is the Haar probability measure of $K$. \nUp to dividing the right-hand side by $N$, it is a Birkhoff sum, but we will not take this viewpoint. Instead, we shall try to bound a quantity related to the variance of $S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}$ and then conclude by a Borel-Cantelli argument. More specifically, we shall bound a $(1+\\varepsilon)$-moment, for some $\\varepsilon > 0$, of the centered Siegel transform $S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+} - \\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(T)^+)$, viewed as a random variable on $(K,\\mu_K)$: if we can show that for some $\\varepsilon > 0$ and all $N \\geq 1$, \n\\begin{equation} \\label{eq:Proof_Sketch}\n\\int_K \\left | S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\Gamma) - \\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(e^N)^+) \\right |^{1+\\varepsilon} \\, \\,\\mathrm{d} \\mu_{K}(k) \\, \\ll \\, N,\n\\end{equation}\nthen there exists $c > 0$ and $\\nu(\\varepsilon) \\in (0,1)$ such that for $\\mu_K$-almost every $k \\in K$, \n\\[\nS_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\Gamma) = c \\, N \\, \\left ( 1 + O_x(N^{-\\nu(\\varepsilon)})\\right ),\n\\]\nas required. Due to integrability issues of the Siegel transform at this level of generality (see Theorems \\ref{thm:L1} and \\ref{thm:L2}), we are forced to work with $1 + \\varepsilon$ for some small $\\varepsilon > 0$ instead of $2$, which would represent the usual variance. \nUsing the decomposition \\eqref{eq:Proof_Sketch_Decomposition}, we express the argument in the integral of \\eqref{eq:Proof_Sketch} as\n\\begin{equation*} \nS_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\Gamma) - \\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(e^N)^+)\n= \\sum_{i=0}^{N-1} \\bigg ( S_{\\chi} \\mathbbm{1}_{{\\mathcal F}} (a(e^{\\beta_{\\chi}}) k_x^{-1} \\Gamma) - \\lambda_{\\widetilde{X}}({\\mathcal F}) \\bigg ).\n\\end{equation*}\nand obtain the bound in \\eqref{eq:Proof_Sketch} using the effective single and double equidistribution property of expanding translates of $K$-orbits. In particular, we will need to work with smooth compactly supported functions that, on translated $K$-orbits, approximate the Siegel transform $S_{\\chi} \\mathbbm{1}_{{\\mathcal F}}$, which typically is neither smooth nor compactly supported. \n\n\\subsection{Notation and conventions}\nWe use the Landau notation $O(\\cdot)$ and the Vinogradov symbol $\\ll$. Given $A, B > 0$, we use the notation $A \\gg B$ for $B \\ll A$, and $A \\asymp B$ for $A \\ll B \\ll A$. We use subscripts to indicate the dependence of the constant on parameters. \nFor simplicity of exposition, we will work with the set of complex points of an algebraic variety defined over $\\mathbb{Q}$, and refer to it simply as the variety itself when no confusion arises. For instance, we write $G = {\\bf G}(\\mathbb{R})$ and ${\\bf G} = {\\bf G}(\\mathbb{C})$ to denote the groups of real and complex points of ${\\bf G}$, respectively. Given a discrete subgroup $\\Gamma \\leq G$ and a closed subgroup $H \\leq G$, we write $\\G_H$ for $\\Gamma \\cap H$. Discrete groups are always equipped with the counting measure. \n\n\\vspace{5mm}\n\\textbf{Acknowledgments}. \nI am very grateful to Nicolas de Saxc\\'e for introducing me to this topic, during my doctoral thesis under his supervision, and for sharing with me crucial insights that contributed to the proofs of Theorems \\ref{thm:L2} and \\ref{thm:Linfty}. I also thank Shucheng Yu for useful discussions, and Fr\\'ed\\'eric Paulin for numerous corrections and suggestions that led, in particular, to the removal of a restrictive hypothesis in Theorem \\ref{thm:Critical} (namely, that $\\beta_{\\chi} \\leq 1$).", + "sketch": "Let us now go through the main steps of the argument (for simplicity, $c=1$). One first rewrites $\\cN_{\\beta_{\\chi}}(x,T)$ as a Siegel transform: for $x\\in X$ one associates $k_x\\in K$ such that\n\\[\n\\cN_{\\beta_{\\chi}}(x,T)=\\#(\\cP_{\\chi}\\cap k_x\\cE_{\\beta_{\\chi}}(T))=S_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}(k_x^{-1}\\Gamma).\n\\]\nBy Theorem \\ref{thm:L1}, the expected value (on $\\Omega$) of this random variable is\n\\[\n\\int_{\\Omega} S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}\\,d\\mu_{\\Omega}=\\int_{\\widetilde{X}}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}\\,d\\lambda_{\\widetilde{X}}=\\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(T)).\n\\]\nThe goal is to show that for $\\sigma_X$-almost every $x$, $\\cN_{\\beta_{\\chi}}(x,T)$ is asymptotically equal to this volume as $T\\to+\\infty$; moreover, “the main term on the right-hand side in \\eqref{cor:eq_Grassmannian} is just the explicit value of the (main term of the) volume $\\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(T))$.”\n\nTo get an effective almost-everywhere estimate, one exploits the geometry of $\\cE_{\\beta_{\\chi}}(T)$ by approximating it with a set $\\cE_{\\beta_{\\chi}}(T)^+$ that decomposes under the diagonal subgroup $a(y)$: there exists ${\\mathcal F}\\subset\\widetilde{X}$ such that for integers $N\\ge1$,\n\\[\n\\cE_{\\beta_{\\chi}}(e^N)^+=\\bigsqcup_{i=0}^{N-1} a(e^{\\beta_{\\chi}})^{-i}{\\mathcal F}. \\tag{\\ref{eq:Proof_Sketch_Decomposition}}\n\\]\nOn the Siegel-transform level this gives\n\\[\nS_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}(k_x^{-1}\\Gamma)=\\sum_{i=0}^{N-1}\\mathbbm{1}_{{\\mathcal F}}(a(e^{\\beta_{\\chi}})^i k_x^{-1}\\Gamma),\n\\]\nand one views $S_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}(k\\Gamma)$ as a random variable on $(K,\\mu_K)$.\n\nRather than using a Birkhoff-sum viewpoint, the argument “bound[s] a quantity related to the variance … and then conclude[s] by a Borel–Cantelli argument,” namely a $(1+\\varepsilon)$-moment of the centered Siegel transform. Concretely, if for some $\\varepsilon>0$ and all $N\\ge1$,\n\\[\n\\int_K\\Big|S_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\Gamma)-\\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(e^N)^+)\\Big|^{1+\\varepsilon}\\,d\\mu_K(k)\\ll N, \\tag{\\ref{eq:Proof_Sketch}}\n\\]\nthen one deduces that for $\\mu_K$-almost every $k\\in K$,\n\\[\nS_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\Gamma)=c\\,N\\,(1+O_x(N^{-\\nu(\\varepsilon)})).\n\\]\nBecause of “integrability issues of the Siegel transform … we are forced to work with $1+\\varepsilon$ … instead of $2$.” Using the decomposition \\eqref{eq:Proof_Sketch_Decomposition}, the centered sum is written as a sum of $N$ terms involving $S_{\\chi}\\mathbbm{1}_{\\mathcal F}$, and the bound \\eqref{eq:Proof_Sketch} is obtained “using the effective single and double equidistribution property of expanding translates of $K$-orbits,” after approximating $S_{\\chi}\\mathbbm{1}_{\\mathcal F}$ by “smooth compactly supported functions” on translated $K$-orbits.", + "expanded_sketch": "Let us now go through the main steps of the argument (for simplicity, $c=1$). One first rewrites $\\cN_{\\beta_{\\chi}}(x,T)$ as a Siegel transform: for $x\\in X$ one associates $k_x\\in K$ such that\n\\[\n\\cN_{\\beta_{\\chi}}(x,T)=\\#(\\cP_{\\chi}\\cap k_x\\cE_{\\beta_{\\chi}}(T))=S_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}(k_x^{-1}\\Gamma).\n\\]\nWe first recall the following theorem.\n\\begin{thmx} [$L^1$-integrability] \\label{thm:L1} \nThe following assertions are equivalent.\n\\begin{enumerate}[label=(\\arabic*)]\n\\item The Siegel transform $S_\\chi$ maps $B_c^{\\infty}(\\widetilde{X})$ into $L^1(\\Omega)$.\n\\item There exists a unique (up to scaling) $G$-invariant Radon measure $\\lambda_{\\widetilde{X}}$ on $\\widetilde{X}$, the Siegel transform $S_\\chi$ extends to a bounded operator $S_{\\chi} : L^1(\\widetilde{X}) \\rightarrow L^1(\\Omega)$, and $\\lambda_{\\widetilde{X}}$ can be normalized so that we have a convergent mean value formula:\n\\begin{equation} \\label{eq:WeilIntegrationFormula*}\n\\forall \\, f \\in L^1(\\widetilde{X}), \\quad \\int_{\\Omega} S_\\chi f \\, \\dd \\mu_{\\Omega} = \\int_{\\widetilde{X}} f \\, \\dd \\lambda_{\\widetilde{X}}.\n\\end{equation}\n\\item The Lie group $L = \\bL(\\R)$ is unimodular and $\\G_L = \\G \\cap L$ is a lattice in $L$.\n\\item The parabolic $\\Q$-subgroup $\\bP$ of $\\bG$ is maximal.\n\\item There exists $\\varepsilon > 0$ such that $S_\\chi$ maps $B_{c}^{\\infty}(\\widetilde{X})$ into $L^{1+\\varepsilon}(\\Omega)$.\n\\end{enumerate}\n\\end{thmx}\nBy this theorem, the expected value (on $\\Omega$) of this random variable is\n\\[\n\\int_{\\Omega} S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}\\,d\\mu_{\\Omega}=\\int_{\\widetilde{X}}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)}\\,d\\lambda_{\\widetilde{X}}=\\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(T)).\n\\]\nThe goal is to show that for $\\sigma_X$-almost every $x$, $\\cN_{\\beta_{\\chi}}(x,T)$ is asymptotically equal to this volume as $T\\to+\\infty$; moreover, the main term in the volume asymptotic is given by the explicit formula\n\\begin{equation} \\label{cor:eq_Grassmannian}\n\\mathcal{N}_{c, \\beta}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}\n\nTo get an effective almost-everywhere estimate, one exploits the geometry of $\\cE_{\\beta_{\\chi}}(T)$ by approximating it with a set $\\cE_{\\beta_{\\chi}}(T)^+$ that decomposes under the diagonal subgroup $a(y)$: there exists ${\\mathcal F}\\subset\\widetilde{X}$ such that for integers $N\\ge1$,\n\\begin{equation} \\label{eq:Proof_Sketch_Decomposition}\n\\cE_{\\beta_{\\chi}}(e^N)^+ = \\bigsqcup_{i=0}^{N-1} a(e^{\\beta_{\\chi}})^{-i} \\cF. \n\\end{equation}\nOn the Siegel-transform level this gives\n\\[\nS_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}(k_x^{-1}\\Gamma)=\\sum_{i=0}^{N-1}\\mathbbm{1}_{{\\mathcal F}}(a(e^{\\beta_{\\chi}})^i k_x^{-1}\\Gamma),\n\\]\nand one views $S_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(T)^+}(k\\Gamma)$ as a random variable on $(K,\\mu_K)$.\n\nRather than using a Birkhoff-sum viewpoint, the argument “bound[s] a quantity related to the variance … and then conclude[s] by a Borel–Cantelli argument,” namely a $(1+\\varepsilon)$-moment of the centered Siegel transform. Concretely, if for some $\\varepsilon>0$ and all $N\\ge1$,\n\\begin{equation} \\label{eq:Proof_Sketch}\n\\int_K \\left | S_{\\chi} \\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\G) - \\lambda_{\\widetilde{X}}(\\cE_{\\beta_{\\chi}}(e^N)^+) \\right |^{1+\\varepsilon} \\, \\dd \\mu_{K}(k) \\, \\ll \\, N,\n\\end{equation}\nthen one deduces that for $\\mu_K$-almost every $k\\in K$,\n\\[\nS_{\\chi}\\mathbbm{1}_{\\cE_{\\beta_{\\chi}}(e^N)^+}(k\\Gamma)=c\\,N\\,(1+O_x(N^{-\\nu(\\varepsilon)})).\n\\]\nBecause of “integrability issues of the Siegel transform … we are forced to work with $1+\\varepsilon$ … instead of $2$.” Using the decomposition above, the centered sum is written as a sum of $N$ terms involving $S_{\\chi}\\mathbbm{1}_{\\mathcal F}$, and the bound above is obtained “using the effective single and double equidistribution property of expanding translates of $K$-orbits,” after approximating $S_{\\chi}\\mathbbm{1}_{\\mathcal F}$ by “smooth compactly supported functions” on translated $K$-orbits.", + "expanded_theorem": "\\label{cor:Grasssmannian}\nFix integers $1 \\leq \\ell < n$ and let $X = \\mathrm{Gr}_{\\ell,n}(\\mathbb{R})$ be the Grassmann variety of $\\ell$-dimensional subspaces in $\\mathbb{R}^n$. Then there exists an explicit constant $\\varkappa > 0$ and $\\varepsilon > 0$ such that for $\\sigma_X$-almost every $x \\in X$, as $T \\rightarrow + \\infty$,\n\\begin{equation} \\label{cor:eq_Grassmannian}\n\\mathcal{N}_{c, \\beta}(x, T) = \\varkappa \\, c^d \\, \\ln (T) \\left ( 1 + O_x(\\ln(T)^{- \\varepsilon}) \\right ).\n\\end{equation}", + "theorem_type": [ + "Asymptotic or Limit", + "Existential–Universal" + ], + "mcq": { + "question": "Fix integers \\(1\\le \\ell0\\), let \\(\\mathcal N_{c,\\beta}(x,T)\\) denote the counting function for rational \\(\\ell\\)-planes in \\(X\\) of height less than \\(T\\) that approximate \\(x\\) at the critical exponent \\(\\beta\\). As \\(T\\to+\\infty\\), which asymptotic statement holds for \\(\\mathcal N_{c,\\beta}(x,T)\\) for \\(\\sigma_X\\)-almost every \\(x\\in X\\)?", + "correct_choice": { + "label": "A", + "text": "There exist explicit constants \\(\\varkappa>0\\) and \\(\\varepsilon>0\\) such that for \\(\\sigma_X\\)-almost every \\(x\\in X\\),\n\\[\n\\mathcal N_{c,\\beta}(x,T)=\\varkappa\\,c^d\\,\\ln(T)\\left(1+O_x\\bigl((\\ln T)^{-\\varepsilon}\\bigr)\\right)\n\\]\nas \\(T\\to+\\infty\\)." + }, + "choices": [ + { + "label": "B", + "text": "There exist explicit constants \\(\\varkappa>0\\) and \\(\\varepsilon>0\\) such that for every \\(x\\in X\\),\n\\[\n\\mathcal N_{c,\\beta}(x,T)=\\varkappa\\,c^d\\,\\ln(T)\\left(1+O_x\\bigl((\\ln T)^{-\\varepsilon}\\bigr)\\right)\n\\]\nas \\(T\\to+\\infty\\)." + }, + { + "label": "C", + "text": "There exists an explicit constant \\(\\varkappa>0\\) such that for \\(\\sigma_X\\)-almost every \\(x\\in X\\),\n\\[\n\\mathcal N_{c,\\beta}(x,T)\\sim \\varkappa\\,c^d\\,\\ln(T)\n\\]\nas \\(T\\to+\\infty\\)." + }, + { + "label": "D", + "text": "There exist explicit constants \\(\\varkappa>0\\) and \\(\\varepsilon>0\\) such that for \\(\\sigma_X\\)-almost every \\(x\\in X\\),\n\\[\n\\mathcal N_{c,\\beta}(x,T)=\\varkappa\\,c^d\\,T\\left(1+O_x\\bigl(T^{-\\varepsilon}\\bigr)\\right)\n\\]\nas \\(T\\to+\\infty\\)." + }, + { + "label": "E", + "text": "There exist explicit constants \\(\\varkappa>0\\) and \\(\\varepsilon>0\\) such that for \\(\\sigma_X\\)-almost every \\(x\\in X\\),\n\\[\n\\mathcal N_{c,\\beta}(x,T)=\\varkappa\\,c^d\\,\\ln(T)\\left(1+O_x\\bigl(T^{-\\varepsilon}\\bigr)\\right)\n\\]\nas \\(T\\to+\\infty\\)." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "almost_everywhere_quantifier", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "counting_estimate", + "tampered_component": "effective_error_term", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "counting_estimate", + "tampered_component": "logarithmic_main_term_scale", + "template_used": "boundary_range" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "error_decay_variable_mismatch", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not state the asymptotic form, growth scale, quantifier, or error term. It only asks for the correct asymptotic at the critical exponent, so the answer is not leaked." + }, + "TAS": { + "score": 1, + "justification": "The item is close to theorem recall: it asks which asymptotic statement holds for the counting function. However, it is not a pure restatement because the options force distinctions between almost-everywhere vs everywhere, effective vs noneffective asymptotics, and logarithmic vs polynomial growth." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is required to choose the strongest valid conclusion among nearby alternatives, especially between the correct effective statement and the weaker true asymptotic. Still, the question mainly tests recognition/recall of a known theorem rather than deeper generative derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically close and target realistic failure modes: overstrengthening the quantifier (B), choosing a weaker true conclusion instead of the strongest one (C), using the wrong main growth scale (D), and mismatching the decay variable in the error term (E). They are distinct and plausibly confusable." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with no answer leakage and strong distractors, but it leans more toward precise theorem recall than genuinely generative mathematical reasoning." + } + }, + { + "id": "2511.12595v2", + "paper_link": "http://arxiv.org/abs/2511.12595v2", + "theorems_cnt": 3, + "theorem": { + "env_name": "theorem", + "content": "\\label{open}\nLet $[a_1,b_1],\\cdots,[a_k,b_k]$ be $k$ disjoint intervals. Then as $g\\to \\infty$, the vector of random variables\n$$(N_{g,[a_1,b_1]},\\cdots,N_{g,[a_k,b_k]}):\\H_g(1^{2g-2})\\to \\mathbb{N}_0^k$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=8\\pi(b_i^2-a_i^2)$$\nfor $i=1,\\cdots,k$. That is,\n$$\\mathbb{P}(N_{g,[a_1,b_1]}=n_1,\\cdots,N_{g,[a_k,b_k]}=n_k)=\\prod_{i=1}^k\\frac{\\lambda_{[a_i,b_i]}^{n_i}e^{-\\lambda_{[a_i,b_i]}}}{n_i!}$$", + "start_pos": 23973, + "end_pos": 24547, + "label": "open" + }, + "ref_dict": { + "claim": "\\begin{theorem}[The method of moment from \\cite{bollobas2001random}, Theorem 1.23]\\label{moment}\nLet $\\left\\{(\\Omega_i,\\mathbb{P}_i)\\right\\}_{i\\in \\mathbb{N}}$ be a sequence of probability spaces. \nFor $m\\in \\mathbb{N}$, let $N_{1,i},\\cdots,N_{m,i}:\\Omega_i\\to \\mathbb{N}_0$ be random variables for all $i\\in \\mathbb{N}$\nand suppose there exist $\\lambda_1,\\cdots,\\lambda_m\\in(0,\\infty)$ such that:\n$$\\lim_{i\\to \\infty}\\mathbb{E}[(N_{1,i})_{k_1}\\cdots(N_{m,i})_{k_m}]=\\lambda_1^{k_1}\\cdots\\lambda_m^{k_m}$$\nfor all $k_1\\cdots k_m\\in \\mathbb{N}$. Then\n$$\\lim_{i\\to \\infty}\\mathbb{P}[(N_{1,i})=n_1,\\cdots,(N_{m,i})=n_m]=\\prod^k_{j=1}\\frac{\\lambda_j^{n_j} e^{-\\lambda_j}}{n_j!}.$$\nThat is, $(N_{1,i},\\cdots,N_{m,i}):\\Omega_i \\to \\mathbb{N}^m_0$ converges jointly in distribution to a vector which is independently Poisson distributed with means \n$\\lambda_1,\\cdots,\\lambda_m$.\n\n\\end{theorem}\n\n\\section{Some claims}\\label{claim}\nBefore the proof of the main theorem, we have to make some claims which will simplify the situation.\nWe consider the stratum whose zero order is $O(1)$, and it inplies $|\\kappa|=O(g)$. \n\n\\begin{clm} \\label{clm1}\nA closed saddle connection is non-separable.\n\\end{clm}\n\\begin{proof}\nFor a closed connection $\\gamma$ of $(X,\\omega)$, if it's separable, then it cuts $X$ into $X_1$ and $X_2$, and $\\partial X_1=\\partial X_2=\\gamma$.\nBy Stokes theorem, $\\int_\\gamma \\omega=0$, but since $\\gamma$ is a saddle connection its length is $|\\int_\\gamma \\omega|$, which is a contradiction.\n\\end{proof}\n\n\\begin{clm} \\label{clm2}\nThe probability measure of the subset of $\\H_g(\\kappa)$ which has a closed saddle connection involving $q$ cylinders of multiplicity $p$ \ngoes to $0$ as $g\\to \\infty$, where $p>1$ or $q>0$.\n\\end{clm}\n\n\\begin{proof}\nFrom Theorem \\ref{Siegel of closed}, \nthe Siegel-Veech constant of configuration with closed saddle connection involving $q$ cylinders of multiplicity $p$ is $O(\\frac{1}{g^{2p+q-2}})$ . \nDenote by $\\mathcal{C}^{p,q}_g(\\kappa)[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ the subset of $\\H_g(\\kappa)$ consisting of surfaces which have a closed saddle connection \nof length in the interval $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$\ninvolving $q$ cylinders of multiplicity $p$.\nThen for $p\\geq 2$ or $q\\geq 1$, we have\n $$\\V(\\mathcal{C}^{p,q}_g(\\kappa)[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}])\\leq |\\kappa|O(\\frac{1}{g^{q}})\\frac{b^2-a^2}{g}\\V(\\H_g(\\kappa))=O(\\frac{1}{g})\\V(\\H_g(\\kappa)$$\nwhere the coefficient $|\\kappa|$ is the selection of the zero that the closed saddle connection goes through. \nSo when $g\\to \\infty$ its probability measure goes to $0$ \nand it suffices to consider the closed saddle connection with multiplicity $1$ and no cylinders around it.\n\\end{proof}\n\n\\begin{clm} \\label{clm3}\nThe angle of a closed saddle connection at the zero which it connects is odd multiples of $\\pi$.\n\\end{clm}\n\n\\begin{proof}\nFor an Abelian differential $(X,\\omega)$, if it has a closed saddle connection $\\gamma$ connecting a zero $p$, if its angle at $p$ is $2k\\pi$,\nthen the holonomy of $\\gamma$ and $-\\gamma$ have same direction. But\n$$\\int_\\gamma \\omega=-\\int_{-\\gamma} \\omega.$$\nIt implies $\\int_\\gamma \\omega=0$, which is a contradiction.\n\\end{proof}\n\n\\begin{clm} \\label{clm4}\nIf a closed saddle connection has angle $\\pi$ at one side, it must have a cylinder at the side.\n\\end{clm}\n\n\\begin{proof}\nLet $\\gamma$ be a colsed saddle connection on $(X,\\omega)$ which has angle $\\pi$ at one side. Since $(X,\\omega)$ is oriented, its normal bundle of $\\gamma$ at the side is oriented.\nConsider the exponential map from the bundle to $(X,\\omega)$, which is well-defined on $\\gamma \\times [0,s]$ for some $s$ small sufficiently and the image is a cylinder.\n\n\\end{proof}\n\nCombine claim $2,3$ and $4$, we have that for a closed saddle connection $\\gamma$ on $(X,\\omega)$ in principal stratum, \nsince its total angle around the zero is $4\\pi$ and has to be divided into odd multiples of $\\pi$, \nit must have angle $\\pi$ at one side of $\\gamma$ and $3\\pi$ at another.\nSo there is a cyliner with $\\gamma$ as a boundary. As for the other boundary, if it consists of some open saddle connections, \nthen these saddle connections have angles $\\pi$. That means there are at least two non-homologous saddle connections have angle $\\pi$,\nwhich occurs on a set of measure zero. \nSo we only need to consider the cylinder bounded by curves homologous to $\\gamma$.\nThen from Claim \\ref{clm2} the probability measure of $\\mathcal{C}_g(1^{2g-2})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ has limit\n $$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g(1^{2g-2})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}])}{\\V(\\H_g(1^{2g-2})}=O(\\frac{1}{g})\\to 0.$$\n\nSo we need to consider the question on stratum with higher order zeros. Next we will consider $\\H_g(2^{g-1})$ firstly.\n\n\\section{Surgery}\\label{surgery}\nThis section we will introduce some surgeries that, when combined, can collapse a closed saddle connection.\n\\subsection{Open up a higher-order zero}\nMasur, Rafi and Randecker introduce a surgery in \\cite{masur2024lengths} which is a variation of a surgery from \\cite{eskin2003moduli}. \nThis surgery can collapse a saddle connection which is not closed and has multiplicity $1$. \n\nLet $(X,\\omega)$ be an Abelian differential, $\\sigma$ be an open saddle connection with endpoints $v_1$ and $v_2$, \nwhose orders are $n_1$ and $n_2$ respectively.\nSince the total angle at $v_1$ is $2(n_1+1)\\pi$, \none can extend $\\sigma$ from $v_1$ along its direction to $v_3$ and denote by $\\sigma'$ the geodesic segment $v_1v_3$ \nsuch that $\\ell_\\omega(\\sigma)=\\ell_\\omega(\\sigma')$\nand the angle between $\\sigma$ and $\\sigma'$ is\n$2k_1\\pi$ and $2k_2\\pi$, where $k_1+k_2=n_1+1$. \nIf $\\sigma'$ does not go through other zeros of $(X,\\omega)$, \nwhich means $\\sigma'$ is a ray from $v_1$, the following surgery can be carried out.\n\nCut along $\\sigma+\\sigma'$ and denote the two copies of $\\sigma+\\sigma'$ and $v_1$ by $\\sigma^{\\pm}+(\\sigma')^{\\pm}$ and $v_1^{\\pm}$. \nThen glue $\\sigma^+$ and $(\\sigma')^+$, $\\sigma^-$ and $(\\sigma')^-$. \nThis surgery reduces the order of $v_1$ and constructs a new Abelian differential $(X',\\omega')$. \nOn $(X',\\omega')$, the total angles at $v_1^+$, $v_1^-$ and $v_2$ are $2k_1\\pi$, $2k_2\\pi$, and $2(n_2+2)\\pi$ respectively,\nthat is, the orders of $v_1^+$, $v_1^-$ and $v_2$ are $k_1-1$, $k_2-1$, and $n_2+1$.\nParticularly, if $v_1$ is a simple zero, the surgery collapses the saddle connection. We call this surgery \\emph{collapsing surgery}.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.6\\linewidth]{collapsing}\n \\caption{The collapsing surgery}\n \\label{collapsing}\n\\end{figure}\n\nIn our situation, we need to use the inverse of collapsing surgery which we call \\emph{opening surgery}.\nFrom Claim \\ref{clm2}, it suffices to consider the subset $\\mathcal{C}^{1,0}_g(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nFor an Abelian differential $(X,\\omega)$ in $\\mathcal{C}^{1,0}_g(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$, \nchoose a closed saddle connection $\\gamma$ with length in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ at a double zero $p$.\nWe can reverse the surgery introduced above as follows.\n\nSince the total angle at $p$ is $6\\pi$, from Claim \\ref{clm3} and Claim \\ref{clm4}, the angles on both side of $\\gamma$ at $p$ must be $3\\pi$.\nFix the orientation of $\\gamma$, choose two rays $\\sigma^+$ and $\\sigma^-$ from $p$ such that \n$\\ell_\\omega(\\sigma^+)=\\ell_\\omega(\\sigma^-)=\\ell_\\omega(\\gamma)$,\nand the angle between $\\gamma$ and $\\sigma^+$\nis $\\pi$, the angle between $\\gamma$ and $\\sigma^-$ is $-\\pi$, where the sign is consistent with the orientation of the surface.\nDenote by $q^+$ and $q^-$ the other endpoints of $\\sigma^+$ and $\\sigma^-$.\nThen cut along $\\sigma^+ + \\sigma^-$ and glue the two copies of $\\sigma^+$ and $\\sigma^-$.\nAfter the surgery, the two copies of $p$ become a regular and a simple zero, $q^+$ and $q^-$ are glued to become a new simple zero $q$.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.6\\linewidth]{openinglocally}\n \\caption{The opening surgery}\n \\label{opening}\n\\end{figure}\n\nThis surgery is the inverse of the collapsing surgery above: the double zero is replaced by two simple zeros,\nand it can be carried out except for the surgery locus goes through some zero, \nthat is, it has two non-homologous saddle connections with angle $2\\pi$.\nFrom period mapping, such subset has measure zero. We call the subset on which the surgery can be carried out \\emph{permissible set}, \nand denote it by $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$, from the discussion above we have \n\\begin{proposition}\n$\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ is a full measure subset of $\\mathcal{C}^{1,0}_g(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\n\\end{proposition}\n\nBy the surgery of collapsing this map is one-to-one and denote the opening surgery by $F_1$.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.6\\linewidth]{opening}\n \\caption{The resulting surface}\n \\label{opening surface}\n\\end{figure}\n\n\\begin{remark}\nWe can also choose $\\sigma^{\\pm}$ to make the angle between $\\gamma$ and them are $2\\pi$, \nit suffices to make sure the opening operation can obtain an abelian differential and a smooth loop homotopic to $\\gamma$.\n\\end{remark}\n\\subsection{Move zero along closed curve and pinch}\nWe have constructed the mapping $F_1$, \nwhich takes a closed saddle connection $\\gamma$ at a double zero to two saddle connections $\\gamma_1$ and $\\gamma_2$\nsharing same endpoints $p$ and $q$ which are both simple zeros.\nMoreover, the angles between the two saddle connections at the two simple zeros (on both sides) are all $2\\pi$, \nand if we denote by $\\ell_\\omega(\\gamma)=L$, we have $\\ell_{\\omega'}(\\gamma_1)=L$, $\\ell_{\\omega'}(\\gamma_2)=2L$, where $(X',\\omega')=F_1(X,\\omega)$.\nIf we want to collapse $\\gamma_1$ and $\\gamma_2$ simultaneously, we have to make a surgery to move $q$ along $\\gamma_2$ to adjust the length of $\\gamma_2$.\n\n\\subsubsection{Move the zero locally by period mapping.}\n\nFirst we need to choose a special basis of $H^1(X,\\Sigma;\\mathbb{C})$, where $\\Sigma=\\{p,q,p_2,\\cdots,p_{g-1}\\}$\nis the zero set of $(X',\\omega')=F_1(X,\\omega)$.\nFrom Claim \\ref{clm1}, the homology class $[\\gamma]=[\\gamma_1+\\gamma_2]$ is non-separable.\nSo we can choose a basis of $H^1(X,\\mathbb{C})$\n$$(\\alpha_1,\\beta_1,\\cdots,\\alpha_g,\\beta_g),$$\nwhere $\\alpha_1=[\\gamma]$.\n\nNext we choose the relative homology class\n$$(pq,pp_2\\cdots,pp_{g-1}),$$\nwhere $[\\gamma_1]=pq$, and $pp_i$ is freely homotopic to $0$ in $H^1(X,\\mathbb{C})$. \nTogether they compose a basis of $H^1(X,\\Sigma;\\mathbb{C})$, \nand their holonomy gives a period mapping around $(X',\\omega')$ locally.\n\nLet $\\omega'(\\alpha_1)=(x_1,y_1)$, $\\omega'(\\beta_1)=(x_2,y_2)$, $\\omega'(\\alpha_2)=(x_3,y_3)$\n$\\omega'(\\beta_2)=(x_4,y_4)$.\nNow choose a relative cohomology class $\\upsilon \\in H^1(X,\\Sigma;\\mathbb{C})$ ,\nwhich can be considered as a tangent vector in $T_{\\H_g(1,1,\\cdots,2)}(X',\\omega')$ such that\n\\begin{equation}\\label{moving}\n\\upsilon(\\alpha_1)=(-x_1,-y_1), \\upsilon(\\alpha_2)=(-\\frac{y_2x_1}{y_4},\\frac{x_2y_1}{x_4})\n\\end{equation}\nand on the other basis we assign zero to $\\upsilon$.\n\n\\begin{remark}\\label{explain}\nHere $\\upsilon(\\alpha_1)$ is chosen to guarantee that along the curve $(X',\\omega')+t\\upsilon$ in moduli space the zero moves along $\\gamma$, \nand $\\upsilon(\\alpha_2)$ is chosen to guarantee the resulting surface has area $1$, since the area of $(X,\\omega)$ can be written by\n$$\\int_X|\\omega|^2=\\frac{i}{2}\\sum_i(\\int_{\\alpha_i}\\omega \\int_{\\beta_i}\\overline{\\omega}-\\int_{\\beta_i}\\omega \\int_{\\alpha_i}\\overline{\\omega}).$$\n\\end{remark}\n\nConsider the curve in $\\H_g(1,1,2,,\\cdots,2)$: $(X',\\omega')+t\\upsilon, t\\in [0,1]$. \nIf $\\gamma_1$ and $\\gamma_2$ don't degenerate along the curve, then we obtain \n$$F_2:\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}] \\to \\H_g(1^2,2^{g-1})$$\nsuch that $F_2[(X,\\omega)]=F_1[(X,\\omega)]+\\upsilon$. \nFrom the construction above the resulting surface has two saddle connections \nin the relative homology class $\\gamma_1$ and $\\gamma_2$ with equal length.\n\n\\subsubsection{$F_2$ is well-defined almost everywhere}\nWe have defined a mapping if the saddle connections $\\gamma_1$ and $\\gamma_2$ are preserved along the curve $(X',\\omega')+t\\upsilon, t\\in [0,1]$\n on $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})$.\nNext we will see the mapping can be defined for almost every translation surface in $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})$.\n\nFrom above we know the moving surgery can be realized except for some of $\\gamma_1$ or $\\gamma_2$ degenerate along $(X',\\omega')+t\\upsilon, t\\in [0,1]$.\nSuppose $T$ is the first time when the geodesic in $[\\gamma_1]$ can be represented by $pp_i-p_ip_j-\\cdots-p_kq$,\nsince the saddle connection is smooth on $(X',\\omega')+t\\upsilon,t\\in [0,T]$, the corner of $pp_i-p_ip_j-\\cdots-p_kq$ must be $\\pi$. \nBut the holonomy of $pp_i$ is not changed on $(X',\\omega')+t\\upsilon,t\\in [0,T]$\nThis implies there exists some $pp_i$ such that its holonomy on $(X',\\omega')$ has the same direction with $p_0p_1$. \nAnd by the relation of holonomy under $F_1$, this implies $p_0p_i$ has the same direction with the closed curve on $(X,\\omega)$.\nUnder period mapping, on each local chart this is a measure-zero set.\nSo all such $(X,\\omega)$ is a subset of measure zero in $ \\H_g(2^{g-1})$, also in $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\n\n\\subsubsection{Pinching}\nOn the subset of $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ where $F_2$ is well-defined,\nwe can cut along $\\gamma_1+\\gamma_2$ and denote the two copies of $p$ and $q$ by $q^+$, $q^-$, $p^+$, $p^-$, \nthe two copies of $\\gamma_1$ and $\\gamma_2$ by $\\gamma^+_1$, $\\gamma^-_1$, $\\gamma^+_2$, $\\gamma^-_2$.\nNotice that the total angles at $q^+$, $q^-$, $p^+$, $p^-$ are all $2\\pi$, \nwhich means $\\gamma^\\pm_1$ have same direction with $\\gamma^\\pm_2$,\nthen glue $\\gamma^+_1$ with $\\gamma^+_2$, $\\gamma^-_1$ with $\\gamma^-_2$, which make $q^+$, $q^-$, $p^+$, $p^-$ become regular points.\nSince $\\gamma_1+\\gamma_2$ is non-separable, this surgery reduces the genus by one, and obtains a new abelian differential \non which we mark two points $p^+$ and $p^-$ so as to find its inverse. \nConsidering the choice of which zero is selected after applying the inverse map, we finally obtain:\n$$F_3: \\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]\\to M\\times \\H_{g-1}(2^{g-2},0,0)\\times A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]},$$\nwhere $M$ is the combinatorial data of choosing one from $g-1$ zeros.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.6\\linewidth]{movingandcut}\n \\caption{Moving and cut}\n \\label{moving and cut}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.6\\linewidth]{pinching}\n \\caption{Pinching}\n \\label{pinching}\n\\end{figure}\n\n\\subsubsection{$F_3$ is measure-preserving.}\nThis surgery can be inverse except for some directions which has saddle connections, which is a measure zero subset $A_0(X,\\omega)$\nin $A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]}$ for every surface in $\\H_{g-1}(2^{g-2},0,0)$: \nchoose a vector $\\kappa$ in $A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]}$, \nconsider the two rays from $p^+$ and $p^-$ whose holonomy is $\\kappa$, if the rays exist, then cut the surface along the two rays\nto obtain two closed loci and glue them. \nLet \n$$\\tilde{\\H}=\\{(X,\\omega, \\tau): (X,\\omega)\\in \\H_{g-1}(2^{g-2},0,0), \\tau \\in A_0(X,\\omega)\\},$$\nwhich is a measure zero subset of $\\H_{g-1}(2^{g-2},0,0)\\times A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]}$.\nSo we have \n$$F_3( \\mathcal{C}^{1,0}_{g,Per}(2^{g-1}))\\subset [\\H_{g-1}(2^{g-2},0,0)\\times (A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]})]\\setminus \\tilde{\\H}.$$\n\nMoreover, under local chart determined by the basis we choose,\n$$F_3(z_1,\\cdots,z_n)=(z_1,z_i+\\upsilon(z_i)),$$ \nwhose Jacobian has deteminant one and thus is measure-preserving.\n\n\\section{Collapse closed saddle connections simultaneously}\nFrom the construction above we can collapse one closed saddle connection on some $(X,\\omega)\\in \\mathcal{C}^{1,0}_{g}(2^{g-1}))$ and obtain a new surface\n$(X',\\omega')\\in \\H_{g-1}(2^{g-2},0,0)$. if we want to collapse any $k$ closed saddle connections, \nwe have to ensure the locus of surgery for these saddle connections are disjoint.\nFor a given interval $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$,\ndefine the \\emph{exception set} $\\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ to be the subset of \n$\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ \nsuch that on every $(X,\\omega)\\in \\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ there are two closed saddle connections with length in\n$[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ whose surgery loci intersect.\nFirst we will show that the measure of exception set goes to zero as genus goes to infinity.\n\n\\subsection{The measure of exception set}\n\nFor $(X,\\omega)\\in \\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$, there are three situations: \n\\begin{itemize}\n \\item[1] There exist two closed saddle connections on $(X,\\omega)$ with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$\n that do not share a zero. And the loci of opening surgery intersect or the two closed saddle connections intersect.\n \\item[2] There exist two closed saddle connections on $(X,\\omega)$ with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ sharing one zero.\n\\end{itemize}\n\nNote that in the first situation, we can find a curve connecting the two zeros with length no more than $\\frac{2b}{\\sqrt{g}}$.\n\nFix $B\\in \\mathbb{R}^+$, define $\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$ be the set of $(X,\\omega)\\in \\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[0,\\frac{B}{\\sqrt{g}}]$ \nsuch that there exist two closed saddle connections on $(X,\\omega)$\nthat do not share a zero, moreover the lengths of the two closed saddle connections and the distance between the two zeros are less than $\\frac{B}{\\sqrt{g}}$.\nDefine $\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$ be the set\nof $(X,\\omega)\\in \\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[0,\\frac{B}{\\sqrt{g}}]$ such that there exists two closed saddle connections with lengths less than $\\frac{B}{\\sqrt{g}}$ sharing one zero.\nObviously for $B\\geq 2b$, we have \n$$ \\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]\\subset \\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}}) \\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}).$$\n\nWe need to compute the measure of $\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$ and $\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$ when $g\\to \\infty$.\nFirst consider $\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$, we have\n\\begin{proposition}\\label{chain}\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))} = 0.$$\n\\end{proposition}\n\n\\begin{proof}\nFor $(X,\\omega) \\in \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$, \nlet $p$ be a zero on $(X,\\omega)$ and $\\gamma_1, \\gamma_2$ be the closed saddle connections with lengths less than $\\frac{B}{\\sqrt{g}}$ at $p$.\n\nAs above we can collapse $\\gamma_1$ and get a new translation surface in $\\H_{g-1}(2^{g-2},0,0)$. \nAfter the surgery $\\gamma_2$ will become a segment connecting two marked regular points which are from the pinching surgery.\nSo the image of mapping $F_3$ on $\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$ is in \n$\\H_{g-1,\\frac{B}{\\sqrt{g}}}(2^{g-2},0,0)\\times D(\\frac{B}{\\sqrt{g}})$, \nwhere $\\H_{g-1,\\frac{B}{\\sqrt{g}}}(2^{g-2},0,0)$ is the subset that there exists a segment connnecting the two marked points of length less than \n$\\frac{B}{\\sqrt{g}}$ and $D(\\frac{B}{\\sqrt{g}})$ is the disk of radius $\\frac{B}{\\sqrt{g}}$.\nFrom \\cite[Theorem 1.2]{aggarwal2019large}, the Siegel-Veech constant of saddle connections connecting two fix zeros of order $m_1$ and $m_2$ is\n$$c=(m_1+1)(m_2+1)(1+O(\\frac{1}{g})).$$\nSo we have\n$$\\V(\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))\\leq (g-1)\\frac{B^2}{g}\\V(\\H_{g-1,\\frac{B}{\\sqrt{g}}}(2^{g-2},0,0)) \\leq c(g-1)\\frac{B^4}{g^2}\\V(\\H_{g-1}(2^{g-2},0,0)),$$\nwhere the coefficient $g-1$ is the choice of $p$.\nThen by \\ref{Volume}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}\\leq \\lim_{g\\to \\infty}3c\\frac{B^4}{g}=0$$\n\\end{proof}\n\nFor $\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$ we also have\n\\begin{proposition}\\label{intersect}\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))} = 0.$$\n\\end{proposition}\n\n\\begin{proof}\nFor $(X,\\omega) \\in \\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$, let $p_1, p_2$ be the two zeros on $(X,\\omega)$ with distance less than $\\frac{B}{\\sqrt{g}}$ \nand $\\gamma_1, \\gamma_2$ be the two closed saddle connections connecting $p_1, p_2$.\nFor every $k\\in \\mathbb{N}^+$, denote by $\\mathcal{C}_g^{1,k}(\\frac{B}{\\sqrt{g}})$ the subset of $\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$\nthat the geodesic between $p_1$ and $p_2$ is a concatenation of $k$ open saddle connections.\nTheir lengths are less than $\\frac{B}{\\sqrt{g}}$.\nChoose the shortest saddle connection, using the surgery in \\cite{masur2024lengths} we can collapse it and get a new translation surface in \n$\\H_g(3,1,2^{g-3})$. This is because the locus, that is the extension of the shortest saddle connection can not intersect the whole geodesic.\nRepeat the surgery until $p_1$ and $p_2$ are collapsed to one zero and we get a map\n$$F:\\mathcal{C}_g^{1,k}(\\frac{B}{\\sqrt{g}})\\to \\mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}})\\times D^k(\\frac{B}{\\sqrt{g}}),$$\nwhere $\\mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}})$ is the subset of $\\H_g(2+k,1^k,2^{g-2-k})$ such that there are\ntwo closed saddle connections $\\gamma_1, \\gamma_2$ connecting the zero of order $2+k$ with lengths less than $\\frac{B}{\\sqrt{g}}$ .\n\nSimilarly one can collapse $\\gamma_1$ and the other will become a saddle connection connecting two zeros. \nNote that the order is larger than $2$, although the collapsing surgery can also be realized, we need to require the angles \non the two sides of the closed saddle connections collapsed, which decide the order of zeros after collapsing. \nWe will explain this situation in detail in Section \\ref{general case}.\n\nLet $\\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}})$ be the subset of \n$\\mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}})$ such that \nthe two angles on both sides of $\\gamma_1$ are $(2b'+1)\\pi$ and $(2b''+1)\\pi$, where $b'+b''=2+k$.\nThen after collapsing we get a map\n$$F':\\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}}) \\to \\H_{g-1,\\frac{B}{\\sqrt{g}}}(b'-1,b''-1,1^k,2^{g-2-k})\\times D(\\frac{B}{\\sqrt{g}}).$$\n\nSimilar to Proposition \\ref{chain}, we have\n$$\\V(\\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}}))\\leq b'b''\\frac{B^4}{g^2}\\V(\\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k})).$$\n\nSum all $k$ and $(b',b'')$ we have\n$$\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}}))\\leq (g-1)(g-2)\\sum_k \\sum_{(b',b'')}b'b''\\frac{B^4}{g^2}\\V(\\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k}))(\\frac{B^2}{g})^k,$$\nwhere the coefficient $(g-1)(g-2)$ is the choice of $p_1$ and $p_2$.\nThen when $g\\to \\infty$ we have\n$$\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}\\leq\n\\sum_k \\sum_{(b',b'')}b'b''B^4\\frac{\\V(\\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k}))}{\\V(\\H_g(2^{g-1}))}(\\frac{B^2}{g})^k=O(\\frac{1}{g}).$$\n\\end{proof}\n\nCombine the two proposition we have\n\\begin{corollary}\n$$\\frac{\\V(\\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}])}{\\V(\\H_g(2^{g-1}))}=O(\\frac{1}{g}).$$\n\\end{corollary}\n\n\\subsection{Collapsing simultaneously}\nNow fix a positive integer $K$ and $n_1,\\cdots,n_k$ a partition of $K$.\nWe want to compute the limit of the expectation\n$$(L_{g,[a_1,b_1]})_{n_1}\\cdots (L_{g,[a_k,b_k]})_{n_k},$$\nfor which we will collapse $K$ closed saddle connections in order.\n\nDefine $\\hat{\\mathcal{C}}^{1,0}_{g}(2^{g-1})=(X,\\omega,\\Gamma_1,\\cdots,\\Gamma_k)$, \nwhere $(X,\\omega)\\in \\mathcal{C}^{1,0}_{g}(2^{g-1})$, \nand $\\Gamma_i$ is an ordered list of $n_i$ closed saddle connections $(\\gamma_{1,i},\\cdots,\\gamma_{n_i,i})$ \nwith lengths in $[\\frac{a_i}{\\sqrt{g}},\\frac{b_i}{\\sqrt{g}}]$. \nSuppose $a_1\\leq b_1 \\leq a_2\\leq b_2 \\cdots \\leq a_k \\leq b_k$, let $B=b_k$.\nThen we have\n$$\\bigcup_i \\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a_i}{\\sqrt{g}},\\frac{b_i}{\\sqrt{g}}]\\subset \\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}).$$\n\nLet $\\mathcal{C}'_{g,Per}=\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})\\setminus (\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))$,\nwe will adjust the surgery from last section on $\\mathcal{C}'_{g,Per}$ to collapse the closed saddle connections in\n$\\Gamma_1,\\cdots,\\Gamma_k$ in order.\nThe opening and pinching operations are same to the situation of one curve. The only modification is the equation \\ref{moving}:\nby remark \\ref{explain}, we need to choose the moving vector $\\upsilon$ to ensure the area unchanged, \nhence we can choose a basis containing $\\gamma'_1,\\gamma_1\\cdots,\\gamma'_K,\\gamma_K$, \nwhere $\\gamma'_i$ is a saddle connection from opening $\\gamma_i$. \nThen the moving vector $\\upsilon$ can be constructed like equation \\ref{moving}: after defining the value on $\\gamma_i$, \nwe can choose another relative homotopy class and give a value of $\\upsilon$ on it to ensure the area unchanged.\nThis means we open and move the closed saddle connections simultaneously and then pinch them.\n\nFinally we obtain a mapping\n$$\\hat{F}: \\hat{\\mathcal{C}}'_{g,Per}(2^{g-1}) \\to M_K\\cdot \\tilde{\\H}_{g-K}(2^{g-1-K},0^{2K}) \\times \\prod^k_{i=1} (A_{[\\frac{a_i}{\\sqrt{g}},\\frac{b_i}{\\sqrt{g}}]})^{n_i},$$\nwhere $\\hat{\\mathcal{C}}'_{g,Per}(2^{g-1})$ is defined as $\\hat{\\mathcal{C}}^{1,0}_{g}(2^{g-1})$\nand $M_K$ is a combinatorial data consisting of choosing $K$ ordered zeros from $g-1$ zeros to label. As the discussion above, \nwe have \n\\begin{equation*}\n\\begin{aligned}\n&\\int_{\\hat{\\mathcal{C}}'_{g,Per}(2^{g-1})}(L_{g,[a_1,b_1]})_{n_1}\\cdots (L_{g,[a_k,b_k]})_{n_k}d\\mu_{MV}\\\\&=\\V(\\hat{\\mathcal{C}}'_{g}(2^{g-1})))\n\\\\&=|M_K|\\prod^k_{i=1}[\\frac{\\pi(b^2-a^2)}{g}]^{n_i}\\V(\\H_{g-k}(2^{g-1-K}))\\\\\n&\\to \\prod^k_{i=1}[\\pi(b_i^2-a_i^2)]^{n_i}\\V(\\H_{g-K}(2^{g-1-K})), g\\to \\infty,\n\\end{aligned}\n\\end{equation*}\nwhere the limit is because $|M_K|=(g-1)\\cdots (g-K)$ and\n$$\\lim_{g\\to \\infty}\\frac{(g-1)\\cdots (g-K)}{g^{n_1+\\cdots+n_k}}=\\frac{(g-1)\\cdots (g-K)}{g^K}=1.$$\n\nLet $L'_{g,[a_i,b_i]}$ be the restriction of $L_{g,[a_i,b_i]}$ on $\\mathcal{C}'_{g,Per}(2^{g-1})$.\nThen the limit of factorial moment of $(L'_{g,[a_i,b_i]})_i$ is\n$$\\lim_{g\\to \\infty}\\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\\lim_{g\\to \\infty}\\frac{\\V(\\H_{g-K}(2^{g-1-K}))}{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}\\prod^k_{i=1}[\\pi(b_i^2-a_i^2)]^{n_i}.$$\nFrom Proposition \\ref{chain} and \\ref{intersect}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}=1.$$\n\nFrom Theorem \\ref{Volume}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\H_{g-K}(2^{g-1-K}))}{\\V(\\H_{g}(2^{g-1}))}=3^K.$$\nSo we have\n$$\\lim_{g\\to \\infty}\\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\\prod^k_{i=1}[3\\pi(b_i^2-a_i^2)]^{n_i}.$$\nBy theorem \\ref{moment}, $(L'_{g,[a_1,b_1]},\\cdots,L'_{g,[a_k,b_k]})$ converges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\nFrom the property of Poisson distribution, we have\n$$\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)=\\prod_{i=1}^k\\frac{\\lambda_{[a_i,b_i]}^{n_i}e^{-\\lambda_{[a_i,b_i]}}}{n_i!}.$$\nOn the other hand \n$$\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)=\\frac{\\V(\\{(X,\\omega):L'_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})}{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}.$$\nNote that \n\\begin{equation*}\n\\begin{aligned}\n&\\V(\\{(X,\\omega):L_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})-\n\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))\\\\ &\\leq \\V(\\{(X,\\omega):L'_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})\\\\&\\leq\n\\V(\\{(X,\\omega):L_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\}).\n\\end{aligned}\n\\end{equation*}\nFrom Proposition \\ref{chain} and \\ref{intersect}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}=1,\\\\ \\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}=0.$$\nSo\n\\begin{equation*}\n\\begin{aligned}\n&\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k)-\n\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}\\\\ &\\leq \\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}\\\\&\\leq\n\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k).\n\\end{aligned}\n\\end{equation*}\n\nLet $g \\to \\infty$ we have\n$$\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k)=\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k).$$\nSo $(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]})$ also converges jointly in distribution to a vector of random variables \nwith Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\nSince we don't need to deal with simple zeros in the collapsing surgery, \nsamilarly to the proof of theorem \\ref{closed}, we can prove Corollary \\ref{general}.\n\n\\section{General stratum}\\label{general case}\nThis section we consider the stratum $\\H_g(m^{O(g)},1^{2g-2-mO(g)})$, where $m\\geq 3$.\n\nIf a stratum has zeros of order more than $3$, the angles at both sides are not certain. \nBut we can consider some fixed configuration of closed saddle connection.\nFor a configuration $(J,b'_k,b''_k,a'_i,a''_i)$ of closed saddle connection.\nWe have known it suffices to consider multiplicity $1$ without cylinders, then the configuration become $(b',b'')$, where $b'+b''=m$, \nwhich means the angles of the closed saddle connection are $(2b'+1)\\pi$ on one side and $(2b''+1)\\pi$ on the other side.\nNow we choose all such configuration at each $m$-order zeros.\n\nFor $k$ such closed saddle connections, we can use the surgery in Section \\ref{surgery} to collapse them. \nNote that in this situation the opening surgery will replace the zero of order $m$ to two zeros of order $1$ and $m-1$, \nand the angles at both sides of the zero of $m-1$ order are $(2b')\\pi$ and $(2b'')\\pi$. Then after moving and pinching, \nthe simple zero become two regular points and the zero of $m-1$ order become two zeros of order $b'-1$ and $b''-1$.\nFinally we obtain a mapping\n$$\\hat{F}: \\hat{\\mathcal{C}}'_{g,Per}(m^{O(g)},1^{O'(g)})\\to \\tilde{\\H}_{g-K}((b'-1)^K,(b''-1)^K,m^{O(g)-K},1^{O'(g)})\\times \\prod^k_{i=1} (A_{[\\frac{a_i}{\\sqrt{g}},\\frac{b_i}{\\sqrt{g}}]})^{n_i},$$\nwhere $O'(g)=2g-2-mO(g)$.\nAgain we have\n$$\\lim_{g\\to \\infty}\\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\\prod^k_{i=1}(\\frac{(m+1)\\pi(b_i^2-a_i^2)}{b'b''})^{n_i}.$$\n\nAnd similar to the proof of theorem \\ref{closed} we have\n\\begin{theorem}\nFor the stratum $\\H(m^{O(g)},1^{2g-2-mO(g)})$, a configuration $(b',b'')$ and $a,b\\in \\mathbb{R}$,\nlet the random variable\n$$L_{\\mathcal{C},g,[a,b]}:\\H(m^{O(g)},1^{2g-2-mO(g)}) \\to \\mathbb{N}_0$$ \nbe the number of closed saddle connections on $(X,\\omega)$ satisfying the configuration with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nThen for disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{\\mathcal{C},g,[a_1,b_1]},\\cdots,L_{\\mathcal{C},g,[a_k,b_k]}):\\H(m^{O(g)},1^{2g-2-mO(g)}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=\\frac{(m+1)\\pi(b_i^2-a_i^2)}{b'b''}.$$\n\n\\end{theorem}", + "closed": "\\begin{theorem}\\label{closed}\nFor the stratum $\\H(2^{g-1})$ and disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]}):\\H(2^{g-1}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 1963, + "pre_theorem_intro_text": "In 2024, Masur, Rafi and Randecker study the distribution of open saddle connections on a random translation surface in \\cite{masur2024lengths}. \nIn this paper we study the distribution of closed saddle connections.\n\nA translation surface can be denoted by $(X,\\omega)$ where $X$ is a closed Riemann surface of genus $g$ and $\\omega$ is an Abelian differential on $X$.\n$(X,\\omega)$ has a flat structure defined by the conical metric $|\\omega|$, with singularities at the zeros of $\\omega$.\nIf an $|\\omega|$-geodesic of $(X,\\omega)$ connects two zeros (or one zero to itself) of $\\omega$ and has no other zeros in its interior,\nwe call it an open (or closed) saddle connection.\n\nLet $\\H_g$ be the moduli space of unit-area translation surfaces of genus $g$.\n$\\H_g$ has a natural stratification according to the zero order of Abelian differentials.\nParticularly, the stratum consisting of all abelian differentials in $\\H_g$ which have only simple zeros is called principal stratum, \ndenoted by $\\H_g(1^{2g-2})$. For convenience, if a stratum consists of Abelian differentials on which the number of zeros of order $d_i$ is $\\kappa_i$\nwe denote it by \n$$\\H_g(d_1^{\\kappa_1},\\cdots,d_k^{\\kappa_k})$$\nwhere $\\sum^{k}_{i=1}\\kappa_id_i=2g-2$.\n\nTogether with a finite natural measure which is introduced by Masur and Veech denoted by $\\mu_{MV}$, \nevery stratum $\\H_g(\\kappa)$ of $\\H_g$ becomes a probability space\nwith probability measure $\\frac{\\mu_{MV}}{\\operatorname{Vol}{\\H_g(\\kappa)}}$, where $\\operatorname{Vol}{\\H_g(\\kappa)}$ is the measure of total stratum, \nwhich we call the Masur-Veech volume of $\\H_g(\\kappa)$.\n\nGiven $(X,\\omega)\\in \\H_g(1^{2g-2})$ and $a,b\\in \\mathbb{R}^+$,\nlet $N_{g,[a,b]}(X,\\omega)$ be the number of open saddle connections with lengths in the interval $[\\frac{a}{g},\\frac{b}{g}]$ on $(X,\\omega)$.\nThen $N_{g,[a,b]}:\\H_g(1^{2g-2})\\to \\mathbb{N}_0$ becomes a random variable.\nIn \\cite{masur2024lengths}, Masur, Rafi and Randecker proved :", + "context": "In 2024, Masur, Rafi and Randecker study the distribution of open saddle connections on a random translation surface in \\cite{masur2024lengths}. \nIn this paper we study the distribution of closed saddle connections.\n\nA translation surface can be denoted by $(X,\\omega)$ where $X$ is a closed Riemann surface of genus $g$ and $\\omega$ is an Abelian differential on $X$.\n$(X,\\omega)$ has a flat structure defined by the conical metric $|\\omega|$, with singularities at the zeros of $\\omega$.\nIf an $|\\omega|$-geodesic of $(X,\\omega)$ connects two zeros (or one zero to itself) of $\\omega$ and has no other zeros in its interior,\nwe call it an open (or closed) saddle connection.\n\nLet $\\H_g$ be the moduli space of unit-area translation surfaces of genus $g$.\n$\\H_g$ has a natural stratification according to the zero order of Abelian differentials.\nParticularly, the stratum consisting of all abelian differentials in $\\H_g$ which have only simple zeros is called principal stratum, \ndenoted by $\\H_g(1^{2g-2})$. For convenience, if a stratum consists of Abelian differentials on which the number of zeros of order $d_i$ is $\\kappa_i$\nwe denote it by \n$$\\H_g(d_1^{\\kappa_1},\\cdots,d_k^{\\kappa_k})$$\nwhere $\\sum^{k}_{i=1}\\kappa_id_i=2g-2$.\n\nTogether with a finite natural measure which is introduced by Masur and Veech denoted by $\\mu_{MV}$, \nevery stratum $\\H_g(\\kappa)$ of $\\H_g$ becomes a probability space\nwith probability measure $\\frac{\\mu_{MV}}{\\operatorname{Vol}{\\H_g(\\kappa)}}$, where $\\operatorname{Vol}{\\H_g(\\kappa)}$ is the measure of total stratum, \nwhich we call the Masur-Veech volume of $\\H_g(\\kappa)$.\n\nGiven $(X,\\omega)\\in \\H_g(1^{2g-2})$ and $a,b\\in \\mathbb{R}^+$,\nlet $N_{g,[a,b]}(X,\\omega)$ be the number of open saddle connections with lengths in the interval $[\\frac{a}{g},\\frac{b}{g}]$ on $(X,\\omega)$.\nThen $N_{g,[a,b]}:\\H_g(1^{2g-2})\\to \\mathbb{N}_0$ becomes a random variable.\nIn \\cite{masur2024lengths}, Masur, Rafi and Randecker proved :", + "full_context": "In 2024, Masur, Rafi and Randecker study the distribution of open saddle connections on a random translation surface in \\cite{masur2024lengths}. \nIn this paper we study the distribution of closed saddle connections.\n\nA translation surface can be denoted by $(X,\\omega)$ where $X$ is a closed Riemann surface of genus $g$ and $\\omega$ is an Abelian differential on $X$.\n$(X,\\omega)$ has a flat structure defined by the conical metric $|\\omega|$, with singularities at the zeros of $\\omega$.\nIf an $|\\omega|$-geodesic of $(X,\\omega)$ connects two zeros (or one zero to itself) of $\\omega$ and has no other zeros in its interior,\nwe call it an open (or closed) saddle connection.\n\nLet $\\H_g$ be the moduli space of unit-area translation surfaces of genus $g$.\n$\\H_g$ has a natural stratification according to the zero order of Abelian differentials.\nParticularly, the stratum consisting of all abelian differentials in $\\H_g$ which have only simple zeros is called principal stratum, \ndenoted by $\\H_g(1^{2g-2})$. For convenience, if a stratum consists of Abelian differentials on which the number of zeros of order $d_i$ is $\\kappa_i$\nwe denote it by \n$$\\H_g(d_1^{\\kappa_1},\\cdots,d_k^{\\kappa_k})$$\nwhere $\\sum^{k}_{i=1}\\kappa_id_i=2g-2$.\n\nTogether with a finite natural measure which is introduced by Masur and Veech denoted by $\\mu_{MV}$, \nevery stratum $\\H_g(\\kappa)$ of $\\H_g$ becomes a probability space\nwith probability measure $\\frac{\\mu_{MV}}{\\operatorname{Vol}{\\H_g(\\kappa)}}$, where $\\operatorname{Vol}{\\H_g(\\kappa)}$ is the measure of total stratum, \nwhich we call the Masur-Veech volume of $\\H_g(\\kappa)$.\n\nGiven $(X,\\omega)\\in \\H_g(1^{2g-2})$ and $a,b\\in \\mathbb{R}^+$,\nlet $N_{g,[a,b]}(X,\\omega)$ be the number of open saddle connections with lengths in the interval $[\\frac{a}{g},\\frac{b}{g}]$ on $(X,\\omega)$.\nThen $N_{g,[a,b]}:\\H_g(1^{2g-2})\\to \\mathbb{N}_0$ becomes a random variable.\nIn \\cite{masur2024lengths}, Masur, Rafi and Randecker proved :\n\nTogether with a finite natural measure which is introduced by Masur and Veech denoted by $\\mu_{MV}$, \nevery stratum $\\H_g(\\kappa)$ of $\\H_g$ becomes a probability space\nwith probability measure $\\frac{\\mu_{MV}}{\\V{\\H_g(\\kappa)}}$, where $\\V{\\H_g(\\kappa)}$ is the measure of total stratum, \nwhich we call the Masur-Veech volume of $\\H_g(\\kappa)$.\n\nTheir method is based on the relationship between Poisson distribution and its factorial moment \\cite{bollobas2001random}, \nwhich was used first in the work of Mirzakhani–Petri \\cite{mirzakhani2019lengths} to study the distribution of closed hyperbolic geodesics on random hyperbolic surfaces in Teichm\\\"uller space with respect to Weil-Peterson measure.\n\n\\begin{theorem}\\label{closed}\nFor the stratum $\\H(2^{g-1})$ and disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]}):\\H(2^{g-1}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\\end{theorem}\n\\begin{remark}\nWe will expalin the question is trivial for the principal stratum in Section \\ref{claim}: \nthe probability measure of $\\mathcal{C}_g(1^{2g-2})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ is zero when $g\\to \\infty$.\nFor principal stratum, the length interval to be considered should be $[a,b]$.\n\\end{remark}\n\nRecall that a random variable $X:\\Omega \\to \\mathbb{N}_0$ is Poisson distributed with mean $\\lambda$ if\n$$\\mathbb{P}(X=k)=\\frac{\\lambda^k e^{-\\lambda}}{k!}.$$\nThe following theorem from \\cite{bollobas2001random} gives the relationship between factorial moment and distribution for Poisson distribution:\n\\begin{theorem}[The method of moment from \\cite{bollobas2001random}, Theorem 1.23]\\label{moment}\nLet $\\left\\{(\\Omega_i,\\mathbb{P}_i)\\right\\}_{i\\in \\mathbb{N}}$ be a sequence of probability spaces. \nFor $m\\in \\mathbb{N}$, let $N_{1,i},\\cdots,N_{m,i}:\\Omega_i\\to \\mathbb{N}_0$ be random variables for all $i\\in \\mathbb{N}$\nand suppose there exist $\\lambda_1,\\cdots,\\lambda_m\\in(0,\\infty)$ such that:\n$$\\lim_{i\\to \\infty}\\mathbb{E}[(N_{1,i})_{k_1}\\cdots(N_{m,i})_{k_m}]=\\lambda_1^{k_1}\\cdots\\lambda_m^{k_m}$$\nfor all $k_1\\cdots k_m\\in \\mathbb{N}$. Then\n$$\\lim_{i\\to \\infty}\\mathbb{P}[(N_{1,i})=n_1,\\cdots,(N_{m,i})=n_m]=\\prod^k_{j=1}\\frac{\\lambda_j^{n_j} e^{-\\lambda_j}}{n_j!}.$$\nThat is, $(N_{1,i},\\cdots,N_{m,i}):\\Omega_i \\to \\mathbb{N}^m_0$ converges jointly in distribution to a vector which is independently Poisson distributed with means \n$\\lambda_1,\\cdots,\\lambda_m$.\n\nFrom Theorem \\ref{Volume}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\H_{g-K}(2^{g-1-K}))}{\\V(\\H_{g}(2^{g-1}))}=3^K.$$\nSo we have\n$$\\lim_{g\\to \\infty}\\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\\prod^k_{i=1}[3\\pi(b_i^2-a_i^2)]^{n_i}.$$\nBy theorem \\ref{moment}, $(L'_{g,[a_1,b_1]},\\cdots,L'_{g,[a_k,b_k]})$ converges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\nFrom the property of Poisson distribution, we have\n$$\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)=\\prod_{i=1}^k\\frac{\\lambda_{[a_i,b_i]}^{n_i}e^{-\\lambda_{[a_i,b_i]}}}{n_i!}.$$\nOn the other hand \n$$\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)=\\frac{\\V(\\{(X,\\omega):L'_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})}{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}.$$\nNote that \n\\begin{equation*}\n\\begin{aligned}\n&\\V(\\{(X,\\omega):L_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})-\n\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))\\\\ &\\leq \\V(\\{(X,\\omega):L'_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})\\\\&\\leq\n\\V(\\{(X,\\omega):L_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\}).\n\\end{aligned}\n\\end{equation*}\nFrom Proposition \\ref{chain} and \\ref{intersect}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}=1,\\\\ \\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}=0.$$\nSo\n\\begin{equation*}\n\\begin{aligned}\n&\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k)-\n\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}\\\\ &\\leq \\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}\\\\&\\leq\n\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k).\n\\end{aligned}\n\\end{equation*}\n\nLet $g \\to \\infty$ we have\n$$\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k)=\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k).$$\nSo $(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]})$ also converges jointly in distribution to a vector of random variables \nwith Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\nAnd similar to the proof of theorem \\ref{closed} we have\n\\begin{theorem}\nFor the stratum $\\H(m^{O(g)},1^{2g-2-mO(g)})$, a configuration $(b',b'')$ and $a,b\\in \\mathbb{R}$,\nlet the random variable\n$$L_{\\mathcal{C},g,[a,b]}:\\H(m^{O(g)},1^{2g-2-mO(g)}) \\to \\mathbb{N}_0$$ \nbe the number of closed saddle connections on $(X,\\omega)$ satisfying the configuration with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nThen for disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{\\mathcal{C},g,[a_1,b_1]},\\cdots,L_{\\mathcal{C},g,[a_k,b_k]}):\\H(m^{O(g)},1^{2g-2-mO(g)}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=\\frac{(m+1)\\pi(b_i^2-a_i^2)}{b'b''}.$$", + "post_theorem_intro_text_len": 2238, + "post_theorem_intro_text": "Their method is based on the relationship between Poisson distribution and its factorial moment \\cite{bollobas2001random}, \nwhich was used first in the work of Mirzakhani–Petri \\cite{mirzakhani2019lengths} to study the distribution of closed hyperbolic geodesics on random hyperbolic surfaces in Teichm\\\"uller space with respect to Weil-Peterson measure.\n\nAs for translation surface, to use the method, \n\\cite{masur2024lengths} introduces a surgery to collapse the open saddle connections to obtain a new general translation surface in a new stratum.\nThis surgery does not work for closed saddle connections and they proposed the distribution question in the situation of closed saddle connections,\nwhich inspires this work.\nThis paper gives a new surgery to collapse a closed saddle connection, which ensures the same method can be used for closed saddle connections.\n\nLet $L_{g,[a,b]}(X,\\omega)$ be the number of closed saddle connections on $(X,\\omega)$ with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nDenote by $\\mathcal{C}_g(\\kappa)[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ the subset of $\\H_g(\\kappa)$ consisting of translation\nsurfaces on which there exist closed saddle connections\nwith lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nThe main result of this paper is the following theorem \n\n\\begin{theorem}\\label{closed}\nFor the stratum $\\H(2^{g-1})$ and disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]}):\\H(2^{g-1}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\\end{theorem}\n\\begin{remark}\nWe will expalin the question is trivial for the principal stratum in Section \\ref{claim}: \nthe probability measure of $\\mathcal{C}_g(1^{2g-2})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ is zero when $g\\to \\infty$.\nFor principal stratum, the length interval to be considered should be $[a,b]$.\n\\end{remark}\n\n\\begin{corollary}\\label{general}\nLet $\\H_g(2^{O(g)},1^{2g-2-2O(g)})$ be a stratum with $O(g)$ double zeros, then the result in Theorem \\ref{closed} is also true.\n\\end{corollary}", + "sketch": "Their method is based on the relationship between Poisson distribution and its factorial moment \\cite{bollobas2001random}, which was used first in the work of Mirzakhani--Petri \\cite{mirzakhani2019lengths}. For translation surfaces, \\cite{masur2024lengths} introduces a surgery to collapse the open saddle connections to obtain a new general translation surface in a new stratum. \"This surgery does not work for closed saddle connections\"; the paper \"gives a new surgery to collapse a closed saddle connection, which ensures the same method can be used for closed saddle connections.\"", + "expanded_sketch": "Their method is based on the relationship between Poisson distribution and its factorial moment \\cite{bollobas2001random}, which was used first in the work of Mirzakhani--Petri \\cite{mirzakhani2019lengths}. For translation surfaces, \\cite{masur2024lengths} introduces a surgery to collapse the open saddle connections to obtain a new general translation surface in a new stratum. \"This surgery does not work for closed saddle connections\"; the paper \"gives a new surgery to collapse a closed saddle connection, which ensures the same method can be used for closed saddle connections.\"", + "expanded_theorem": "\\label{open}\nLet $[a_1,b_1],\\cdots,[a_k,b_k]$ be $k$ disjoint intervals. Then as $g\\to \\infty$, the vector of random variables\n$$(N_{g,[a_1,b_1]},\\cdots,N_{g,[a_k,b_k]}):\\H_g(1^{2g-2})\\to \\mathbb{N}_0^k$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=8\\pi(b_i^2-a_i^2)$$\nfor $i=1,\\cdots,k$. That is,\n$$\\mathbb{P}(N_{g,[a_1,b_1]}=n_1,\\cdots,N_{g,[a_k,b_k]}=n_k)=\\prod_{i=1}^k\\frac{\\lambda_{[a_i,b_i]}^{n_i}e^{-\\lambda_{[a_i,b_i]}}}{n_i!}$$,", + "theorem_type": [ + "Asymptotic or Limit", + "Universal" + ], + "mcq": { + "question": "Let H_g(1^{2g-2}) denote the principal stratum of unit-area translation surfaces of genus g, consisting of Abelian differentials with 2g-2 simple zeros, equipped with the normalized Masur–Veech probability measure. For (X,ω) in H_g(1^{2g-2}) and an interval [a,b] with a,b>0, let N_{g,[a,b]}(X,ω) be the number of open saddle connections on (X,ω), meaning |ω|-geodesic segments joining two distinct zeros of ω and containing no zero in their interior, whose lengths lie in [a/g,b/g]. If [a_1,b_1],...,[a_k,b_k] are k pairwise disjoint intervals, then as g→∞, which limiting statement holds for the random vector (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]})?", + "correct_choice": { + "label": "A", + "text": "As g→∞, (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means λ_[a_i,b_i] = 8π(b_i^2-a_i^2) for i=1,...,k. Equivalently, for every fixed n_1,...,n_k in N_0, lim_{g→∞} P(N_{g,[a_1,b_1]}=n_1,...,N_{g,[a_k,b_k]}=n_k) = ∏_{i=1}^k [λ_[a_i,b_i]^{n_i} e^{-λ_[a_i,b_i]} / n_i!]." + }, + "choices": [ + { + "label": "B", + "text": "As g→∞, (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means λ_[a_i,b_i] = 8π(b_i-a_i) for i=1,...,k. Equivalently, for every fixed n_1,...,n_k in N_0, lim_{g→∞} P(N_{g,[a_1,b_1]}=n_1,...,N_{g,[a_k,b_k]}=n_k) = ∏_{i=1}^k [λ_[a_i,b_i]^{n_i} e^{-λ_[a_i,b_i]} / n_i!]." + }, + { + "label": "C", + "text": "As g→∞, each individual random variable N_{g,[a_i,b_i]} converges in distribution to a Poisson random variable with mean λ_[a_i,b_i] = 8π(b_i^2-a_i^2) for i=1,...,k." + }, + { + "label": "D", + "text": "As g→∞, (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of Poisson random variables with means λ_[a_i,b_i] = 8π(b_i^2-a_i^2) for i=1,...,k, but in general the limiting coordinates need not be independent. Equivalently, for every fixed n_1,...,n_k in N_0, lim_{g→∞} P(N_{g,[a_1,b_1]}=n_1,...,N_{g,[a_k,b_k]}=n_k) exists and has Poisson marginals with those means." + }, + { + "label": "E", + "text": "As g→∞, (N_{g,[a_1,b_1]},...,N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means λ_[a_i,b_i] = 3π(b_i^2-a_i^2) for i=1,...,k. Equivalently, for every fixed n_1,...,n_k in N_0, lim_{g→∞} P(N_{g,[a_1,b_1]}=n_1,...,N_{g,[a_k,b_k]}=n_k) = ∏_{i=1}^k [λ_[a_i,b_i]^{n_i} e^{-λ_[a_i,b_i]} / n_i!]." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "counting_estimate", + "tampered_component": "quadratic area-scaling in the Poisson mean", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "counting_estimate", + "tampered_component": "joint convergence/independence of counts across disjoint intervals", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "factorial-moment method yields independent joint Poisson law, not merely Poisson marginals", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "mean constant specific to open saddle connections in the principal stratum", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the objects and asks for the limiting law, but it does not explicitly state the correct conclusion or uniquely signal option A. No direct answer leakage is present." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question: it essentially asks for the exact asymptotic statement of a known result. The alternatives do introduce meaningful variations, but the prompt is still a fairly direct request for the theorem’s conclusion." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is required to distinguish the full joint independent Poisson limit from weaker marginal convergence, incorrect dependence claims, and wrong mean formulas/constants. However, solving it is driven more by precise recall/recognition than by substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong and mathematically plausible: one uses the wrong scaling in the mean, one gives a weaker true statement, one removes independence, and one changes the constant. These align well with realistic confusion points." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with no answer leakage and high-quality distractors, but it functions mainly as theorem recognition rather than a deeply generative reasoning task." + } + }, + { + "id": "2511.02790v1", + "paper_link": "http://arxiv.org/abs/2511.02790v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "theorem", + "content": "\\label{Main theorem 1}\n For any $g\\geqslant 3$, the set of base-$g$ Niven numbers is an asymptotic basis of order 3.", + "start_pos": 11081, + "end_pos": 11223, + "label": "Main theorem 1" + }, + "ref_dict": { + "k cond S": "\\begin{equation}\\label{k cond S}\n |k_{i}-\\mu_{K}|\\leq C_{g} \\textrm{ and } k_{1}+k_{2}+k_{3}\\equiv M\\md{g-1},\n\\end{equation}", + "Main theorem S count": "\\begin{theorem}\\label{Main theorem S count} Let $g,K$ and $M$ be integers such that $g\\geq 3$, $K$ is sufficiently large in terms of $g$, and $M\\in (g^{K-1},g^{K}]$. Suppose that $k_{1},k_{2},k_{3}$ satisfy \\cref{k cond S}. Then\n \\[r_{S_{1}+S_{2}+S_{3}}(M)=\\frac{(g-1)M^{2}}{2(2\\pi\\sigma^{2}K)^{3/2}}(1+O_{g}((\\log K)^{4}K^{-1/4})),\\]\nwhere $\\sigma^{2}=(g^{2}-1)/12$.\n\\end{theorem}", + "Main theorem N count": "\\begin{theorem}\\label{Main theorem N count}\nLet $g,K$ and $M$ be integers such that $g\\geq 3$, $K$ is sufficiently large in terms of $g$, and $M\\in (g^{K-1},g^{K}]$. Suppose that $k_{1},k_{2},k_{3}$ fulfil \\cref{k cond S} and \\cref{k cond N}, then\n\\[r_{\\mathcal{N}_{1}+\\mathcal{N}_{2}+\\mathcal{N}_{3}}(M)=\\frac{(g-1)^{2}c_{g}(k_{1},k_{2},k_{3})}{4k_{1}k_{2}k_{3}}r_{S_{1}+S_{2}+S_{3}}(M)+O_{g}(M^{2}K^{-29/6}).\\]\nIn particular,\n \\[r_{\\mathcal{N}_{1}+\\mathcal{N}_{2}+\\mathcal{N}_{3}}(M)=c_{g}(k_{1},k_{2},k_{3})\\frac{M^{2}}{(2\\pi\\sigma^{2})^{3/2}K^{9/2}}+O_{g}(M^{2}(\\log K)^{4}K^{-19/4}),\\]\n where $\\sigma^{2}=(g^{2}-1)/12$.\n\\end{theorem}", + "k cond N": "\\begin{align}\\label{k cond N}\n (k_{i},k_{j})=1 \\textrm{ and }(k_{i},g)=1 \\textrm{ for }i,j=1,2,3 \\textrm{ and } i\\neq j.\n \\end{align}", + "Main theorem 1": "\\begin{theorem}\\label{Main theorem 1}\n For any $g\\geq 3$, the set of base-$g$ Niven numbers is an asymptotic basis of order 3.\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 2025, + "pre_theorem_intro_text": "A central question in additive number theory is to establish whether a given set of integers $\\mathcal{S}$ is an \\emph{asymptotic basis} for the integers, that is, to determine whether there exists a natural number $k$ such that any sufficiently large integer can be written as the sum of $k$ elements of $\\mathcal{S}$. Here, $k$ denotes the \\textit{order} of the basis.\n\nFamously, Lagrange's theorem gives that the squares are a basis of order 4, and Waring's problem, solved by Hilbert, shows that $k$\\textsuperscript{th } powers are also an additive basis. Some interesting variants of Waring's problem consider $k$\\textsuperscript{th } powers of integers which have restrictions on their digits in some base. For example, Pfeiffer and Thuswaldner \\cite{pfeiffer2007waring} show that the $k$\\textsuperscript{th } powers of integers with certain congruence conditions on their sums of digits in different bases is an asymptotic basis. More recently, Green \\cite{green2025waring} established that, given any two digits which are coprime, the integers whose base-$g$ expansions consists of only these digits satisfy Waring's problem. Further references for additive bases coming from sets of integers with digit restrictions are given in the introduction of \\cite{sanna2021additive}.\n\nA \\emph{base-$g$ Niven number} is a natural number that is divisible by its base-$g$ sum of digits. Such integers are also referred to as \\textit{Harshad numbers}. It is shown in \\cite{de2003counting}, and independently in \\cite{mauduit2005distribution}, that the number of base-$g$ Niven numbers less than $x$ is asymptotically $\\eta_{g}x/\\log x$ for some constant $\\eta_g>0$. \n\nIt is conjectured that the set of base-$g$ Niven numbers is an asymptotic basis of order 2. In \\cite{sanna2021additive} Sanna established, conditionally upon a certain generalisation of the Riemann Hypothesis, that the set of base-$g$ Niven numbers is an asymptotic basis with order growing linearly in $g$. Our result is the following unconditional statement.", + "context": "A central question in additive number theory is to establish whether a given set of integers $\\mathcal{S}$ is an \\emph{asymptotic basis} for the integers, that is, to determine whether there exists a natural number $k$ such that any sufficiently large integer can be written as the sum of $k$ elements of $\\mathcal{S}$. Here, $k$ denotes the \\textit{order} of the basis.\n\nFamously, Lagrange's theorem gives that the squares are a basis of order 4, and Waring's problem, solved by Hilbert, shows that $k$\\textsuperscript{th } powers are also an additive basis. Some interesting variants of Waring's problem consider $k$\\textsuperscript{th } powers of integers which have restrictions on their digits in some base. For example, Pfeiffer and Thuswaldner \\cite{pfeiffer2007waring} show that the $k$\\textsuperscript{th } powers of integers with certain congruence conditions on their sums of digits in different bases is an asymptotic basis. More recently, Green \\cite{green2025waring} established that, given any two digits which are coprime, the integers whose base-$g$ expansions consists of only these digits satisfy Waring's problem. Further references for additive bases coming from sets of integers with digit restrictions are given in the introduction of \\cite{sanna2021additive}.\n\nA \\emph{base-$g$ Niven number} is a natural number that is divisible by its base-$g$ sum of digits. Such integers are also referred to as \\textit{Harshad numbers}. It is shown in \\cite{de2003counting}, and independently in \\cite{mauduit2005distribution}, that the number of base-$g$ Niven numbers less than $x$ is asymptotically $\\eta_{g}x/\\log x$ for some constant $\\eta_g>0$.\n\nIt is conjectured that the set of base-$g$ Niven numbers is an asymptotic basis of order 2. In \\cite{sanna2021additive} Sanna established, conditionally upon a certain generalisation of the Riemann Hypothesis, that the set of base-$g$ Niven numbers is an asymptotic basis with order growing linearly in $g$. Our result is the following unconditional statement.", + "full_context": "A central question in additive number theory is to establish whether a given set of integers $\\mathcal{S}$ is an \\emph{asymptotic basis} for the integers, that is, to determine whether there exists a natural number $k$ such that any sufficiently large integer can be written as the sum of $k$ elements of $\\mathcal{S}$. Here, $k$ denotes the \\textit{order} of the basis.\n\nFamously, Lagrange's theorem gives that the squares are a basis of order 4, and Waring's problem, solved by Hilbert, shows that $k$\\textsuperscript{th } powers are also an additive basis. Some interesting variants of Waring's problem consider $k$\\textsuperscript{th } powers of integers which have restrictions on their digits in some base. For example, Pfeiffer and Thuswaldner \\cite{pfeiffer2007waring} show that the $k$\\textsuperscript{th } powers of integers with certain congruence conditions on their sums of digits in different bases is an asymptotic basis. More recently, Green \\cite{green2025waring} established that, given any two digits which are coprime, the integers whose base-$g$ expansions consists of only these digits satisfy Waring's problem. Further references for additive bases coming from sets of integers with digit restrictions are given in the introduction of \\cite{sanna2021additive}.\n\nA \\emph{base-$g$ Niven number} is a natural number that is divisible by its base-$g$ sum of digits. Such integers are also referred to as \\textit{Harshad numbers}. It is shown in \\cite{de2003counting}, and independently in \\cite{mauduit2005distribution}, that the number of base-$g$ Niven numbers less than $x$ is asymptotically $\\eta_{g}x/\\log x$ for some constant $\\eta_g>0$.\n\nIt is conjectured that the set of base-$g$ Niven numbers is an asymptotic basis of order 2. In \\cite{sanna2021additive} Sanna established, conditionally upon a certain generalisation of the Riemann Hypothesis, that the set of base-$g$ Niven numbers is an asymptotic basis with order growing linearly in $g$. Our result is the following unconditional statement.\n\nA \\emph{base-$g$ Niven number} is a natural number that is divisible by its base-$g$ sum of digits. Such integers are also referred to as \\textit{Harshad numbers}. It is shown in \\cite{de2003counting}, and independently in \\cite{mauduit2005distribution}, that the number of base-$g$ Niven numbers less than $x$ is asymptotically $\\eta_{g}x/\\log x$ for some constant $\\eta_g>0$.\n\nTherefore $|\\mathcal{E}_{0}|\\geq |\\mathcal{E}_{y}|\\geq R$. Recall the definition of the tuple $\\mathbf{a}$ from \\cref{a tuple defn}; as $x=0$ we have that \\[\\mathbf{a}=(|\\mathcal{E}_{0}|,0,\\ldots,0,|\\mathcal{E}_{y}|,0,\\ldots,0 ).\\] \nOur aim is to bound $\\Psi(\\mathbf{a};0)$ using \\cref{psi 0 bound} with $m=R$ and $t=y$. To satisfy the assumptions of \\cref{psi 0 bound} we need $R\\ll_{C} |\\mathcal{E}_{0}|^{1/4}$, which follows from the fact that $|\\mathcal{E}_{0}|\\geq (K-|\\mathcal{D}|)/2\\geq (K-LR)/2$. As $LR\\leq CK^{1/4}$ by assumption, we have that $|\\mathcal{E}_{0}|\\gg_{C} K$, and certainly $R\\leq CK^{1/4}$, giving $R\\ll_{C} |\\mathcal{E}_{0}|^{1/4} $. Applying \\cref{psi 0 bound} gives the bound \n\\begin{equation*}\\label{psi cancellation}\n \\Psi(\\mathbf{a};0)\\ll_{C,g} g^{-R}+R^{2}K^{-3/2}.\n\\end{equation*}\nThis gives \\cref{main term cancellation psi}, using that $R\\geq \\tfrac{3}{2}\\log_{g}K$.\n\\end{proof}\n\\subsection{Decoupling the averages over $\\xbd$ and $\\xbee$}\\label{section decoupling}\nIn this section we prove the decoupling result, \\cref{decoupling lemma}. This lemma allows us to replace the condition on the digits of $X$, $s(\\xbd)+s(\\xbee)=\\xi$, with the condition $s(\\xbee)=0$, even as the digits in $\\xbd$ vary.\n\\begin{proof}[Proof of \\cref{decoupling lemma}] Let $(g-1)\\theta$ have centred base-$g$ expansion given by \\cref{theta digit exp}. Dividing through by $(g-1)$ in this expansion gives\n\\[\\theta=\\frac{1}{g-1}\\sum_{j=1}^{w}\\varepsilon_{n_{j}}g^{-n_{j}}+\\frac{\\eta}{g-1},\\]\nwhere $|\\eta|0\\) such that for all \\(m\\ge m_0\\) and \\(q\\ge q_0\\)$\\colon$\n\n\t\t\\begin{enumerate}[label=(\\roman*)]\n\t\t\t\\item all higher cohomology groups vanish,\n\t\t\t\\[\n\t\t\tH^i(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q})=0,\\quad i>0;\n\t\t\t\\]\n\t\t\t\\item consequently,\n\t\t\t\\[\n\t\t\th^0\\!\\big(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q}\\big)\n\t\t\t=\\chi\\!\\big(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q}\\big)\n\t\t\t=\\frac{1}{s}\\!\\int_Y\\! \\mathrm{ch}(E_{k,m})\\,e^{m c_1(A)}\\,\\mathrm{Td}(TY)\n\t\t\t+O(m^{n-1}),\n\t\t\t\\]\n\t\t\twhere \\(s=|\\mathrm{Stab}_{\\mathrm{gen}}|\\);\n\t\t\t\\item the minimal jet order and asymptotic slope at which invariant jet differentials exist depend only on the coarse Kähler class \\([\\omega_A]\\). \n\t\t\tEquivalently,\n\t\t\t\\[\n\t\t\t\\text{$Y$ is GGD-positive}\n\t\t\t\\quad\\Longleftrightarrow\\quad\n\t\t\t\\text{$\\mathcal{X}$ is GGD-positive}.\n\t\t\t\\]\n\t\t\\end{enumerate}\n\t\tThus, orbifold compactification and rigidification neither alter nor shift the GGD threshold.", + "start_pos": 9028, + "end_pos": 10353, + "label": "thm:invariance" + }, + "ref_dict": { + "thm:main-threshold": "\\begin{theorem}\n\t\t\\label{thm:main-threshold}\n\t\tLet $\\pi\\colon \\X\\to Y$ be the coarse moduli map of a compact complex orbifold of complex dimension~$n$, and let $A$ be an ample line bundle on~$Y$ with Kähler form~$\\omega_A$.\n\t\tAssume the curvature hypotheses \\textup{(H1)}–\\textup{(H4)} of \\Cref{assump:curv}.\n\t\tThen$\\colon $\n\t\t\\begin{enumerate}[label=(\\roman*), leftmargin=1.6em]\n\t\t\t\\item The Green--Griffiths--Demailly hyperbolicity threshold depends only on the coarse Kähler class $[\\omega_A]$ and not on the presence of orbifold or stack structure.\n\t\t\t\\item Equivalently,\n\t\t\t\\[\n\t\t\t\\text{$Y$ is GGD--positive at some jet order $k_0$}\n\t\t\t\\quad\\Longleftrightarrow\\quad\n\t\t\t\\text{$\\X$ is GGD--positive at the same order.}\n\t\t\t\\]\n\t\t\t\\item The minimal jet order and asymptotic growth rate of $h^0(E^{\\mathrm{GG}}_{k,m}\\otimes L^{-q})$ remain unchanged under coarse projection, finite quotient, or rigidification.\n\t\t\\end{enumerate}\n\t\\end{theorem}" + }, + "pre_theorem_intro_text_len": 3125, + "pre_theorem_intro_text": "\\label{sec:intro}\n\n\tThe Green--Griffiths--Demailly (GGD) program, initiated by Green and Griffiths~\\cite{GreenGriffiths1979} and developed extensively by Demailly~\\cite{Demailly1997,Demailly2007,Demailly2011}, provides a powerful analytic and cohomological framework for understanding algebraic degeneracy of entire curves in complex projective varieties. \n\tIts central idea is to construct invariant jet differentials along the Demailly--Semple (DS) tower and to evaluate their asymptotic Euler characteristics using curvature positivity and Riemann--Roch theory. \n\tThe minimal jet order \\(k_0\\) and weight slope \\(\\lambda_0\\) for which\n\t\\[\n\tH^0\\!\\big(X,E^{\\mathrm{inv}}_{k,m}\\otimes A^{-q}\\big)\\neq0,\n\t\\qquad q\\simeq \\lambda_0 m,\n\t\\]\n\tquantify the onset of hyperbolicity-type behavior predicted by the Green--Griffiths--Lang conjecture. \n\tThese numerical thresholds depend on the balance between vertical negativity and horizontal positivity in the curvature of the underlying Kähler class.\n\n\tOrbifold and stack generalizations of this program have become increasingly relevant in modern geometry; see Campana–P\\u{a}un~\\cite{CampanaPaun2016}, Borghesi–Tomassini~\\cite{BorghesiTomassini2017}, and Toën–Vezzosi~\\cite{ToenVezzosi2008}. \n\tDeligne--Mumford stacks equipped with orbifold Kähler forms or log pairs, as in Abramovich–Olsson–Vistoli~\\cite{AbramovichOlssonVistoli2008}, naturally arise in moduli theory and arithmetic geometry, where curvature and cohomological tools must be extended to finite quotient groupoids~\\cite{MoerdijkPronk1997,Lerman2008,KeelMori1997,StacksProj04V2}. \n\tHowever, the analytic underpinnings of the GGD framework---Bochner identities, Hörmander-type \\(L^2\\) estimates, and Riemann--Roch asymptotics---were originally formulated for manifolds, not orbifolds. \n\tThis raises the structural question: \n\\begin{question}\nDoes the passage from a smooth variety to a smooth orbifold or stack, with the same coarse Kähler class, alter the positivity thresholds that govern the existence of invariant jet differentials?\n\\end{question}\n\tAt first glance, the answer need not be obvious. \n\tThe orbifold Riemann--Roch theorem of Satake~\\cite{Satake1956} and Kawasaki~\\cite{Kawasaki1979,Kawasaki1981} involves contributions from twisted sectors with denominators determined by isotropy representations. \n\tThus, the asymptotic expansion of the Euler characteristic \n\t\\[\n\t\\chi(\\mathcal{X},E_{k,m}\\otimes \\pi^*A^{-q})\n\t\\]\n\twhere $\\pi\\colon \\mathcal{X}\\to Y$ is the coarse moduli map and $A$ is a line bundle on $Y$,\nmight, in principle, differ from its coarse counterpart on~$Y$.\n\tFurthermore, while the leading term of the Euler characteristic scales by \\(1/s\\), where \\(s=|\\mathrm{Stab}_{\\mathrm{gen}}|\\) is the generic stabilizer order, the actual number of global sections \\(h^0\\) could still depend on higher cohomology groups \\(H^i(\\mathcal{X},E_{k,m}\\otimes L^{-q})\\). \n\tTo establish structural invariance, one must show that these higher cohomology terms are suppressed by curvature positivity in the orbifold setting, ensuring that the growth of \\(h^0\\) and \\(\\chi\\) coincide asymptotically.", + "context": "\\label{sec:intro}\n\nThe Green--Griffiths--Demailly (GGD) program, initiated by Green and Griffiths~\\cite{GreenGriffiths1979} and developed extensively by Demailly~\\cite{Demailly1997,Demailly2007,Demailly2011}, provides a powerful analytic and cohomological framework for understanding algebraic degeneracy of entire curves in complex projective varieties. \n Its central idea is to construct invariant jet differentials along the Demailly--Semple (DS) tower and to evaluate their asymptotic Euler characteristics using curvature positivity and Riemann--Roch theory. \n The minimal jet order \\(k_0\\) and weight slope \\(\\lambda_0\\) for which\n \\[\n H^0\\!\\big(X,E^{\\mathrm{inv}}_{k,m}\\otimes A^{-q}\\big)\\neq0,\n \\qquad q\\simeq \\lambda_0 m,\n \\]\n quantify the onset of hyperbolicity-type behavior predicted by the Green--Griffiths--Lang conjecture. \n These numerical thresholds depend on the balance between vertical negativity and horizontal positivity in the curvature of the underlying Kähler class.\n\nOrbifold and stack generalizations of this program have become increasingly relevant in modern geometry; see Campana–P\\u{a}un~\\cite{CampanaPaun2016}, Borghesi–Tomassini~\\cite{BorghesiTomassini2017}, and Toën–Vezzosi~\\cite{ToenVezzosi2008}. \n Deligne--Mumford stacks equipped with orbifold Kähler forms or log pairs, as in Abramovich–Olsson–Vistoli~\\cite{AbramovichOlssonVistoli2008}, naturally arise in moduli theory and arithmetic geometry, where curvature and cohomological tools must be extended to finite quotient groupoids~\\cite{MoerdijkPronk1997,Lerman2008,KeelMori1997,StacksProj04V2}. \n However, the analytic underpinnings of the GGD framework---Bochner identities, Hörmander-type \\(L^2\\) estimates, and Riemann--Roch asymptotics---were originally formulated for manifolds, not orbifolds. \n This raises the structural question: \n\\begin{question}\nDoes the passage from a smooth variety to a smooth orbifold or stack, with the same coarse Kähler class, alter the positivity thresholds that govern the existence of invariant jet differentials?\n\\end{question}\n At first glance, the answer need not be obvious. \n The orbifold Riemann--Roch theorem of Satake~\\cite{Satake1956} and Kawasaki~\\cite{Kawasaki1979,Kawasaki1981} involves contributions from twisted sectors with denominators determined by isotropy representations. \n Thus, the asymptotic expansion of the Euler characteristic \n \\[\n \\chi(\\mathcal{X},E_{k,m}\\otimes \\pi^*A^{-q})\n \\]\n where $\\pi\\colon \\mathcal{X}\\to Y$ is the coarse moduli map and $A$ is a line bundle on $Y$,\nmight, in principle, differ from its coarse counterpart on~$Y$.\n Furthermore, while the leading term of the Euler characteristic scales by \\(1/s\\), where \\(s=|\\mathrm{Stab}_{\\mathrm{gen}}|\\) is the generic stabilizer order, the actual number of global sections \\(h^0\\) could still depend on higher cohomology groups \\(H^i(\\mathcal{X},E_{k,m}\\otimes L^{-q})\\). \n To establish structural invariance, one must show that these higher cohomology terms are suppressed by curvature positivity in the orbifold setting, ensuring that the growth of \\(h^0\\) and \\(\\chi\\) coincide asymptotically.\n\n\\begin{theorem}\n\t\t\\label{thm:main-threshold}\n\t\tLet $\\pi\\colon \\X\\to Y$ be the coarse moduli map of a compact complex orbifold of complex dimension~$n$, and let $A$ be an ample line bundle on~$Y$ with Kähler form~$\\omega_A$.\n\t\tAssume the curvature hypotheses \\textup{(H1)}–\\textup{(H4)} of \\Cref{assump:curv}.\n\t\tThen$\\colon $\n\t\t\\begin{enumerate}[label=(\\roman*), leftmargin=1.6em]\n\t\t\t\\item The Green--Griffiths--Demailly hyperbolicity threshold depends only on the coarse Kähler class $[\\omega_A]$ and not on the presence of orbifold or stack structure.\n\t\t\t\\item Equivalently,\n\t\t\t\\[\n\t\t\t\\text{$Y$ is GGD--positive at some jet order $k_0$}\n\t\t\t\\quad\\Longleftrightarrow\\quad\n\t\t\t\\text{$\\X$ is GGD--positive at the same order.}\n\t\t\t\\]\n\t\t\t\\item The minimal jet order and asymptotic growth rate of $h^0(E^{\\mathrm{GG}}_{k,m}\\otimes L^{-q})$ remain unchanged under coarse projection, finite quotient, or rigidification.\n\t\t\\end{enumerate}\n\t\\end{theorem}", + "full_context": "\\label{sec:intro}\n\nThe Green--Griffiths--Demailly (GGD) program, initiated by Green and Griffiths~\\cite{GreenGriffiths1979} and developed extensively by Demailly~\\cite{Demailly1997,Demailly2007,Demailly2011}, provides a powerful analytic and cohomological framework for understanding algebraic degeneracy of entire curves in complex projective varieties. \n Its central idea is to construct invariant jet differentials along the Demailly--Semple (DS) tower and to evaluate their asymptotic Euler characteristics using curvature positivity and Riemann--Roch theory. \n The minimal jet order \\(k_0\\) and weight slope \\(\\lambda_0\\) for which\n \\[\n H^0\\!\\big(X,E^{\\mathrm{inv}}_{k,m}\\otimes A^{-q}\\big)\\neq0,\n \\qquad q\\simeq \\lambda_0 m,\n \\]\n quantify the onset of hyperbolicity-type behavior predicted by the Green--Griffiths--Lang conjecture. \n These numerical thresholds depend on the balance between vertical negativity and horizontal positivity in the curvature of the underlying Kähler class.\n\nOrbifold and stack generalizations of this program have become increasingly relevant in modern geometry; see Campana–P\\u{a}un~\\cite{CampanaPaun2016}, Borghesi–Tomassini~\\cite{BorghesiTomassini2017}, and Toën–Vezzosi~\\cite{ToenVezzosi2008}. \n Deligne--Mumford stacks equipped with orbifold Kähler forms or log pairs, as in Abramovich–Olsson–Vistoli~\\cite{AbramovichOlssonVistoli2008}, naturally arise in moduli theory and arithmetic geometry, where curvature and cohomological tools must be extended to finite quotient groupoids~\\cite{MoerdijkPronk1997,Lerman2008,KeelMori1997,StacksProj04V2}. \n However, the analytic underpinnings of the GGD framework---Bochner identities, Hörmander-type \\(L^2\\) estimates, and Riemann--Roch asymptotics---were originally formulated for manifolds, not orbifolds. \n This raises the structural question: \n\\begin{question}\nDoes the passage from a smooth variety to a smooth orbifold or stack, with the same coarse Kähler class, alter the positivity thresholds that govern the existence of invariant jet differentials?\n\\end{question}\n At first glance, the answer need not be obvious. \n The orbifold Riemann--Roch theorem of Satake~\\cite{Satake1956} and Kawasaki~\\cite{Kawasaki1979,Kawasaki1981} involves contributions from twisted sectors with denominators determined by isotropy representations. \n Thus, the asymptotic expansion of the Euler characteristic \n \\[\n \\chi(\\mathcal{X},E_{k,m}\\otimes \\pi^*A^{-q})\n \\]\n where $\\pi\\colon \\mathcal{X}\\to Y$ is the coarse moduli map and $A$ is a line bundle on $Y$,\nmight, in principle, differ from its coarse counterpart on~$Y$.\n Furthermore, while the leading term of the Euler characteristic scales by \\(1/s\\), where \\(s=|\\mathrm{Stab}_{\\mathrm{gen}}|\\) is the generic stabilizer order, the actual number of global sections \\(h^0\\) could still depend on higher cohomology groups \\(H^i(\\mathcal{X},E_{k,m}\\otimes L^{-q})\\). \n To establish structural invariance, one must show that these higher cohomology terms are suppressed by curvature positivity in the orbifold setting, ensuring that the growth of \\(h^0\\) and \\(\\chi\\) coincide asymptotically.\n\n\\begin{theorem}\n\t\t\\label{thm:main-threshold}\n\t\tLet $\\pi\\colon \\X\\to Y$ be the coarse moduli map of a compact complex orbifold of complex dimension~$n$, and let $A$ be an ample line bundle on~$Y$ with Kähler form~$\\omega_A$.\n\t\tAssume the curvature hypotheses \\textup{(H1)}–\\textup{(H4)} of \\Cref{assump:curv}.\n\t\tThen$\\colon $\n\t\t\\begin{enumerate}[label=(\\roman*), leftmargin=1.6em]\n\t\t\t\\item The Green--Griffiths--Demailly hyperbolicity threshold depends only on the coarse Kähler class $[\\omega_A]$ and not on the presence of orbifold or stack structure.\n\t\t\t\\item Equivalently,\n\t\t\t\\[\n\t\t\t\\text{$Y$ is GGD--positive at some jet order $k_0$}\n\t\t\t\\quad\\Longleftrightarrow\\quad\n\t\t\t\\text{$\\X$ is GGD--positive at the same order.}\n\t\t\t\\]\n\t\t\t\\item The minimal jet order and asymptotic growth rate of $h^0(E^{\\mathrm{GG}}_{k,m}\\otimes L^{-q})$ remain unchanged under coarse projection, finite quotient, or rigidification.\n\t\t\\end{enumerate}\n\t\\end{theorem}\n\nOrbifold and stack generalizations of this program have become increasingly relevant in modern geometry; see Campana–P\\u{a}un~\\cite{CampanaPaun2016}, Borghesi–Tomassini~\\cite{BorghesiTomassini2017}, and Toën–Vezzosi~\\cite{ToenVezzosi2008}. \n Deligne--Mumford stacks equipped with orbifold Kähler forms or log pairs, as in Abramovich–Olsson–Vistoli~\\cite{AbramovichOlssonVistoli2008}, naturally arise in moduli theory and arithmetic geometry, where curvature and cohomological tools must be extended to finite quotient groupoids~\\cite{MoerdijkPronk1997,Lerman2008,KeelMori1997,StacksProj04V2}. \n However, the analytic underpinnings of the GGD framework---Bochner identities, Hörmander-type \\(L^2\\) estimates, and Riemann--Roch asymptotics---were originally formulated for manifolds, not orbifolds. \n This raises the structural question: \n\\begin{question}\nDoes the passage from a smooth variety to a smooth orbifold or stack, with the same coarse Kähler class, alter the positivity thresholds that govern the existence of invariant jet differentials?\n\\end{question}\n At first glance, the answer need not be obvious. \n The orbifold Riemann--Roch theorem of Satake~\\cite{Satake1956} and Kawasaki~\\cite{Kawasaki1979,Kawasaki1981} involves contributions from twisted sectors with denominators determined by isotropy representations. \n Thus, the asymptotic expansion of the Euler characteristic \n \\[\n \\chi(\\X,E_{k,m}\\otimes \\pi^*A^{-q})\n \\]\n where $\\pi\\colon \\X\\to Y$ is the coarse moduli map and $A$ is a line bundle on $Y$,\nmight, in principle, differ from its coarse counterpart on~$Y$.\n Furthermore, while the leading term of the Euler characteristic scales by \\(1/s\\), where \\(s=|\\mathrm{Stab}_{\\mathrm{gen}}|\\) is the generic stabilizer order, the actual number of global sections \\(h^0\\) could still depend on higher cohomology groups \\(H^i(\\X,E_{k,m}\\otimes L^{-q})\\). \n To establish structural invariance, one must show that these higher cohomology terms are suppressed by curvature positivity in the orbifold setting, ensuring that the growth of \\(h^0\\) and \\(\\chi\\) coincide asymptotically.\n\nThe proof combines two analytic–cohomological mechanisms. \n First, the curvature–negativity–positivity package shows that vertical negativity of \\(\\ev_0^*A^{-1}\\) on the \\(1\\)-jet bundle produces fiberwise positivity of the tautological bundle \\(\\OO_{\\X_1}(1)\\) via the Chern curvature formula on projectivized bundles; this positivity then propagates along the DS tower, ensuring semipositivity horizontally and strict positivity vertically. \n Second, a chartwise version of the Satake–Kawasaki–Toën Riemann--Roch theorem expresses the Euler characteristic as\n \\[\n \\chi(\\X,E_{k,m}\\otimes L^{-q})\n =\\frac{1}{s}\\int_Y \\mathrm{ch}(E_{k,m})\\,e^{m c_1(A)}\\mathrm{Td}(TY)\n +O(m^{n-1}),\n \\]\n with all twisted-sector terms of order \\(O(m^{n-1})\\). \n Together with an orbifold Kodaira-type vanishing theorem for DS bundles, which ensures that higher cohomology groups vanish for \\(m,q\\gg0\\), the asymptotic growth of \\(h^0\\) and \\(\\chi\\) coincide, yielding the stated invariance.\n\n\\begin{proposition}[Generic stabilizer and normalization of the untwisted integral]\n \\label{prop:generic-stab-normalization}\n Let $\\pi\\colon \\mathcal X\\to Y$ be the coarse moduli map and let $n=\\dim_{\\C}\\mathcal X$. Then$\\colon $\n \\begin{enumerate}[label=(\\roman*)]\n \\item (Existence and constancy) There exists a Zariski open dense sub-orbifold $\\mathcal X^\\circ\\subset \\mathcal X$ on which $|\\Aut(x)|$ is constant. Its common value\n \\[\n s \\; \\coloneq \\; |\\Stab_{\\mathrm{gen}}|\n \\]\n is the \\emph{generic stabilizer order}. The function $x\\mapsto |\\Aut(x)|$ is upper semicontinuous, hence $|\\Aut(x)|\\ge s$ on $\\mathcal X$. \n \\item (Normalization) For any top-degree form $\\alpha$ supported on the untwisted sector,\n \\[\n \\int_{\\mathcal X}\\alpha \\;=\\; \\frac{1}{s}\\int_Y \\pi_*\\alpha.\n \\]\n Equivalently, on $\\mathcal X^\\circ$ the map $\\pi$ is \\'etale of degree $s$, and the identity-sector integral picks up the factor $1/s$.\n \\end{enumerate}\n \\end{proposition}\n\n\\begin{proposition}[Sectorwise asymptotics and slope control]\n \\label{prop:slope-control}\n Let $\\mathcal{X}$ be a connected compact complex orbifold of complex dimension $n$, \n and let $L=\\pi^*A$ be the pullback of an ample line bundle on the coarse space $Y$.\n Then\n \\[\n \\chi(\\mathcal{X},E\\otimes L^{\\otimes m})\n = \\frac{1}{s}\\int_Y \\ch(E)\\,e^{m\\,c_1(A)}\\,\\Td(TY)\n + O(m^{n-1}),\n \\qquad s=|\\Stab_{\\mathrm{gen}}|.\n \\]\n Consequently, the sign of the asymptotic slope\n \\[\n \\mu(L) \\coloneq \\lim_{m\\to\\infty}\\frac{\\chi(\\mathcal{X},E\\otimes L^{\\otimes m})}{m^n/n!} =\\frac{1}{s}\\int_Y \\rk(E)\\,c_1(A)^n\n \\]\n is identical to that of the coarse manifold term. \n Twisted--sector corrections are of strictly lower order \n and therefore cannot alter the positivity or negativity of $\\mu(L)$. \n \\end{proposition}\n\n\\begin{lemma}[Untwisted control of cohomology dimensions]\n \\label{lem:cohomology-control}\n Assume $(L,h)$ is a Hermitian line bundle with Nakano--positive curvature\n and that $L=\\pi^*A$ descends from the coarse space.\n Then for all sufficiently large $m$,\n \\[\n H^q(\\mathcal{X},E\\otimes L^{\\otimes m})=0\\quad\\text{for all }q>0,\n \\]\n and consequently\n \\[\n h^0(\\mathcal{X},E\\otimes L^{\\otimes m})\n =\\chi(\\mathcal{X},E\\otimes L^{\\otimes m})\n =\\frac{1}{s}\\int_Y \\ch(E)\\,e^{m\\,c_1(A)}\\,\\Td(TY)\n +O(m^{n-1}).\n \\]\n Thus the asymptotic growth of global sections is governed entirely by the untwisted (identity) sector.\n \\end{lemma}\n\n\\begin{lemma}[Orbifold HRR asymptotics]\n \\label{lem:eq-HRR}\n Let $\\X$ be a compact complex orbifold with generic stabilizer of order $s$, and let $L=\\pi^*A$ for an ample line bundle $A$ on the coarse space $Y$. \n Let $E^{\\mathrm{inv}}_{k,m}$ denote the invariant jet (or DS) bundle. \n Then, as $m\\to\\infty$,\n \\[\n \\chi\\!\\bigl(\\X,E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q}\\bigr)\n =\\frac{1}{s}\\!\\int_Y\\!\n \\ch(\\underline{E}^{\\mathrm{inv}}_{k,m})\\,e^{-q\\,c_1(A)}\\,\\Td(TY)\n +O(m^{n-1}),\n \\]\n and all twisted-sector terms contribute only $O(m^{n-1})$ due to the fixed-locus dimension drop and the $m$–independence of the Kawasaki denominator.\n \\end{lemma}\n\n\\begin{theorem}[Stack-theoretic GGD degeneracy]\n \\label{thm:stack-ggd-final}\n Under Assumption~\\ref{ass:equiv-setting3}, there exist integers $k\\gg1$, $m\\gg1$, and $q=q(k,m)>0$ such that$\\colon $\n \\begin{enumerate}[label=(\\roman*), leftmargin=1.6em]\n \\item $H^0(\\X,E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q})\\neq 0$;\n \\item the common zero locus \n \\[\n \\mathcal{G}_k\n =\\bigcap_{\\substack{m\\gg1\\\\0\\le q\\le q_0(k)}} \n Z\\!\\big(H^0(\\X,E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q})\\big)\n \\]\n is a proper closed analytic substack $\\mathcal{G}_k\\subsetneq\\X$;\n \\item every nonconstant entire map $f\\colon \\C\\to\\X$ satisfies $f(\\C)\\subset\\mathcal{G}_k$.\n \\end{enumerate}\n \\end{theorem}\n\n\\begin{theorem}\n \\label{thm:main-threshold}\n Let $\\pi\\colon \\X\\to Y$ be the coarse moduli map of a compact complex orbifold of complex dimension~$n$, and let $A$ be an ample line bundle on~$Y$ with Kähler form~$\\omega_A$.\n Assume the curvature hypotheses \\textup{(H1)}–\\textup{(H4)} of \\Cref{assump:curv}.\n Then$\\colon $\n \\begin{enumerate}[label=(\\roman*), leftmargin=1.6em]\n \\item The Green--Griffiths--Demailly hyperbolicity threshold depends only on the coarse Kähler class $[\\omega_A]$ and not on the presence of orbifold or stack structure.\n \\item Equivalently,\n \\[\n \\text{$Y$ is GGD--positive at some jet order $k_0$}\n \\quad\\Longleftrightarrow\\quad\n \\text{$\\X$ is GGD--positive at the same order.}\n \\]\n \\item The minimal jet order and asymptotic growth rate of $h^0(E^{\\mathrm{GG}}_{k,m}\\otimes L^{-q})$ remain unchanged under coarse projection, finite quotient, or rigidification.\n \\end{enumerate}\n \\end{theorem}", + "post_theorem_intro_text_len": 3312, + "post_theorem_intro_text": "The proof combines two analytic–cohomological mechanisms. \n\tFirst, the curvature–negativity–positivity package shows that vertical negativity of \\(\\ev_0^*A^{-1}\\) on the \\(1\\)-jet bundle produces fiberwise positivity of the tautological bundle \\(\\OO_{\\X_1}(1)\\) via the Chern curvature formula on projectivized bundles; this positivity then propagates along the DS tower, ensuring semipositivity horizontally and strict positivity vertically. \n\tSecond, a chartwise version of the Satake–Kawasaki–Toën Riemann--Roch theorem expresses the Euler characteristic as\n\t\\[\n\t\\chi(\\mathcal{X},E_{k,m}\\otimes L^{-q})\n\t=\\frac{1}{s}\\int_Y \\mathrm{ch}(E_{k,m})\\,e^{m c_1(A)}\\mathrm{Td}(TY)\n\t+O(m^{n-1}),\n\t\\]\n\twith all twisted-sector terms of order \\(O(m^{n-1})\\). \n\tTogether with an orbifold Kodaira-type vanishing theorem for DS bundles, which ensures that higher cohomology groups vanish for \\(m,q\\gg0\\), the asymptotic growth of \\(h^0\\) and \\(\\chi\\) coincide, yielding the stated invariance.\n\n\tThis conclusion situates the structural invariance of GGD thresholds at the intersection of analytic and stack-theoretic geometry. \n\tIt refines Demailly’s curvature approach~\\cite{Demailly1997,Demailly2011} and P\\u{a}un’s vector-field method~\\cite{Paun2008}, while connecting Satake–Kawasaki index theory~\\cite{Kawasaki1979,Kawasaki1981} and modern stack Riemann--Roch theorems~\\cite{Toen1999,Vistoli1989}. \n\tBy aligning orbifold HRR asymptotics with curvature positivity, it confirms that orbifold structures preserve, rather than disturb, the hyperbolicity thresholds predicted by the Green--Griffiths--Lang conjecture. \n\tIn particular, higher orbifold Betti numbers or stabilizers may influence lower-order corrections but have no effect on the leading asymptotic behavior that determines the GGD threshold.\n\n\tThe remainder of the paper proceeds as follows. \n\tSection~\\ref{sec:orbifold-hrr-ds} establishes the chartwise Riemann--Roch formula compatible with orbifold descent and applies it to jet bundles on $\\X_k$. \n\tSection~\\ref{sec:stack-ggd} develops the curvature–positivity package on the DS tower, derives \\(L^2\\)-vanishing and Bochner inequalities on orbifold charts, and proves the stack-theoretic GGD degeneracy theorem. \n\tFinally, Section~\\ref{sec:ggd-structural-invariance} combines HRR asymptotics, Kodaira-type vanishing, and slope control to show that the GGD thresholds depend only on the coarse Kähler class and remain invariant under orbifold or stack structures, concluding with examples and applications.\n\n\tThroughout, compact analytic Deligne--Mumford stacks are identified with compact complex orbifolds after rigidification. \n\tIntegration on $\\mathcal{X}$ is normalized by the generic stabilizer order \\(s=|\\mathrm{Stab}_{\\mathrm{gen}}|\\), so that identity-sector integrals correspond to those on \\(Y\\) up to the factor \\(1/s\\). \n\tAll Demailly--Semple and jet constructions are performed chartwise and descend by finite-group equivariance.\n\n\t\\subsection*{Acknowledgements}\nA substantial part of this work was carried out while the second author was visiting Texas State University to give a talk in the topology seminar.\nThe authors thank the host of the seminar Prof. Christine Lee, as well as the department chair and faculty members, for their warm hospitality and stimulating discussions.", + "sketch": "To prove Theorem~\\ref{thm:invariance}, the argument “combines two analytic–cohomological mechanisms.”\n\n1) **Curvature/positivity along the Demailly–Semple (DS) tower.** The “curvature–negativity–positivity package shows that vertical negativity of \\(\\ev_0^*A^{-1}\\) on the \\(1\\)-jet bundle produces fiberwise positivity of the tautological bundle \\(\\OO_{\\X_1}(1)\\) via the Chern curvature formula on projectivized bundles; this positivity then propagates along the DS tower, ensuring semipositivity horizontally and strict positivity vertically.”\n\n2) **Orbifold HRR plus control of twisted sectors.** A “chartwise version of the Satake–Kawasaki–Toën Riemann--Roch theorem” gives\n\\[\n\\chi(\\mathcal{X},E_{k,m}\\otimes L^{-q})\n=\\frac{1}{s}\\int_Y \\mathrm{ch}(E_{k,m})\\,e^{m c_1(A)}\\mathrm{Td}(TY)\n+O(m^{n-1}),\n\\]\n“with all twisted-sector terms of order \\(O(m^{n-1})\\).”\n\n3) **Vanishing to identify \\(h^0\\) with \\(\\chi\\).** “Together with an orbifold Kodaira-type vanishing theorem for DS bundles, which ensures that higher cohomology groups vanish for \\(m,q\\gg0\\), the asymptotic growth of \\(h^0\\) and \\(\\chi\\) coincide,” and this “yield[s] the stated invariance” of the GGD threshold (i.e., dependence only on the coarse Kähler class and invariance under orbifold/stack structures).", + "expanded_sketch": "To prove the main theorem, the argument “combines two analytic–cohomological mechanisms.”\n\n1) **Curvature/positivity along the Demailly–Semple (DS) tower.** The “curvature–negativity–positivity package shows that vertical negativity of \\(\\ev_0^*A^{-1}\\) on the \\(1\\)-jet bundle produces fiberwise positivity of the tautological bundle \\(\\OO_{\\X_1}(1)\\) via the Chern curvature formula on projectivized bundles; this positivity then propagates along the DS tower, ensuring semipositivity horizontally and strict positivity vertically.”\n\n2) **Orbifold HRR plus control of twisted sectors.** A “chartwise version of the Satake–Kawasaki–Toën Riemann--Roch theorem” gives\n\\[\n\\chi(\\mathcal{X},E_{k,m}\\otimes L^{-q})\n=\\frac{1}{s}\\int_Y \\mathrm{ch}(E_{k,m})\\,e^{m c_1(A)}\\mathrm{Td}(TY)\n+O(m^{n-1}),\n\\]\n“with all twisted-sector terms of order \\(O(m^{n-1})\\).”\n\n3) **Vanishing to identify \\(h^0\\) with \\(\\chi\\).** “Together with an orbifold Kodaira-type vanishing theorem for DS bundles, which ensures that higher cohomology groups vanish for \\(m,q\\gg0\\), the asymptotic growth of \\(h^0\\) and \\(\\chi\\) coincide,” and this “yield[s] the stated invariance” of the GGD threshold (i.e., dependence only on the coarse Kähler class and invariance under orbifold/stack structures).", + "expanded_theorem": "[Structural invariance of GGD thresholds {In establishing the main theorem,}{}]\\label{thm:invariance}\n\t\tLet \\(\\pi\\colon \\mathcal{X}\\to Y\\) be the coarse moduli map of a compact smooth analytic Deligne--Mumford stack (orbifold) \\(\\mathcal{X}\\), and let \\(A\\) be an ample line bundle on \\(Y\\) endowed with a smooth positively curved metric. \n\t\tSet \\(L=\\pi^*A\\). \n\t\tThen there exist integers \\(m_0,q_0>0\\) such that for all \\(m\\ge m_0\\) and \\(q\\ge q_0\\)$\\colon$\n\n\t\t\\begin{enumerate}[label=(\\roman*)]\n\t\t\t\\item all higher cohomology groups vanish,\n\t\t\t\\[\n\t\t\tH^i(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q})=0,\\quad i>0;\n\t\t\t\\]\n\t\t\t\\item consequently,\n\t\t\t\\[\n\t\t\th^0\\!\\big(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q}\\big)\n\t\t\t=\\chi\\!\\big(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q}\\big)\n\t\t\t=\\frac{1}{s}\\!\\int_Y\\! \\mathrm{ch}(E_{k,m})\\,e^{m c_1(A)}\\,\\mathrm{Td}(TY)\n\t\t\t+O(m^{n-1}),\n\t\t\t\\]\n\t\t\twhere \\(s=|\\mathrm{Stab}_{\\mathrm{gen}}|\\);\n\t\t\t\\item the minimal jet order and asymptotic slope at which invariant jet differentials exist depend only on the coarse K\u0000ehler class \\([\\omega_A]\\). \n\t\t\tEquivalently,\n\t\t\t\\[\n\t\t\t\\text{$Y$ is GGD-positive}\n\t\t\t\\quad\\Longleftrightarrow\\quad\n\t\t\t\\text{$\\mathcal{X}$ is GGD-positive}.\n\t\t\t\\]\n\t\t\\end{enumerate}\n\t\tThus, orbifold compactification and rigidification neither alter nor shift the GGD threshold.", + "theorem_type": [ + "Existential–Universal", + "Biconditional or Equivalence" + ], + "mcq": { + "question": "Let \\(\\pi\\colon \\mathcal{X}\\to Y\\) be the coarse moduli map of a compact smooth analytic Deligne--Mumford stack (orbifold), let \\(A\\) be an ample line bundle on \\(Y\\) with a smooth positively curved metric, and set \\(L=\\pi^*A\\). Say that a space is GGD-positive if some invariant jet differential bundle \\(E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q}\\) has a nonzero global section for suitable jet order \\(k\\) and integers \\(m,q\\). Which statement is equivalent to saying that the coarse space \\(Y\\) is GGD-positive?", + "correct_choice": { + "label": "A", + "text": "The orbifold/stack \\(\\mathcal{X}\\) is GGD-positive. Equivalently, the minimal jet order and the asymptotic slope at which invariant jet differentials exist depend only on the coarse K\\\"ahler class \\([\\omega_A]\\), so orbifold compactification and rigidification do not alter or shift the GGD threshold." + }, + "choices": [ + { + "label": "B", + "text": "The orbifold/stack \\(\\mathcal{X}\\) is GGD-positive after possibly replacing the jet order by some larger \\(k'\\ge k\\) and rescaling the asymptotic slope by the generic stabilizer factor \\(1/s\\). Equivalently, orbifold compactification preserves existence of invariant jet differentials only up to this stabilizer-dependent shift of the GGD threshold." + }, + { + "label": "C", + "text": "The orbifold/stack \\(\\mathcal{X}\\) is GGD-positive whenever the coarse space \\(Y\\) is GGD-positive." + }, + { + "label": "D", + "text": "The orbifold/stack \\(\\mathcal{X}\\) is GGD-positive if and only if the orbifold Euler characteristic satisfies\n\\[\n\\chi\\!\\big(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q}\\big)>0\n\\]\nfor all sufficiently large \\(m\\) and \\(q\\), since the twisted sectors contribute only the factor \\(1/s\\) and therefore automatically force nonzero global sections without any higher-cohomology vanishing hypothesis." + }, + { + "label": "E", + "text": "The orbifold/stack \\(\\mathcal{X}\\) is GGD-positive if and only if there exist integers \\(m_0,q_0>0\\) such that for all \\(m\\ge m_0\\) and \\(q\\ge q_0\\),\n\\[\nH^i(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q})=0\\quad (i>0),\n\\]\nand\n\\[\nh^0\\!\\big(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q}\\big)\n=\\chi\\!\\big(\\mathcal{X},E^{\\mathrm{inv}}_{k,m}\\otimes L^{-q}\\big)\n=\\frac{1}{s}\\!\\int_Y \\mathrm{ch}(E_{k,m})\\,e^{m c_1(A)}\\,\\mathrm{Td}(TY)+O(m^{n-1}),\n\\]\nso the GGD threshold is characterized exactly by this asymptotic formula." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "finiteness", + "tampered_component": "same jet order and unchanged slope", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped the converse equivalence and threshold invariance", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "finiteness", + "tampered_component": "need for higher-cohomology vanishing to identify h^0 with chi", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "equivalence between positivity and eventual vanishing/asymptotic formula for all large m,q", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly state that GGD-positivity is preserved between the coarse space and the stack. It sets up the objects and definition, but the correct equivalence is not directly revealed." + }, + "TAS": { + "score": 1, + "justification": "The item is close to theorem recall: it asks for a statement equivalent to Y being GGD-positive, and the correct choice is essentially a reformulation of the expected invariance theorem. The extra threshold-language makes it slightly more than a verbatim restatement, but only mildly." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to reject the weaker true implication in C and the technically embellished but false statements B, D, and E. However, the question mainly tests recognition of the precise theorem rather than substantial derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic failure modes: confusing equivalence with one-way implication, inserting a stabilizer-dependent shift, conflating positivity of Euler characteristic with existence of sections, and overstating asymptotic vanishing claims." + }, + "total_score": 6, + "overall_assessment": "A reasonably strong MCQ with good distractors and limited answer leakage, but it leans heavily on theorem recognition and is only mildly non-tautological." + } + }, + { + "id": "2511.03979v1", + "paper_link": "http://arxiv.org/abs/2511.03979v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "\\label{main theorem}\nFor $n>0$, we have\n\\begin{align}\\label{main theorem eqn}\nA(n)=B(n)=C(n+1)=\\frac{1}{2}D(n+1),\n\\end{align}\nwhere $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.", + "start_pos": 6707, + "end_pos": 7145, + "label": "main theorem" + }, + "ref_dict": { + "bijective proofs": "\\label{bijective proofs}\n\n\\subsection{A Bijective proof of $A(n-1)=\\frac{1}{2}D(n)$}\nWe can first remove the non-negative part condition for $D(n)$ as follows:\n\\begin{quote}\nFor $n\\ge 2$, $D(n)$ count", + "bc": "\\label{bc}\n\nWe note that\n\\begin{align}\\label{bn}\n\\sum_{n=1}^\\infty B(n)q^{n+1}&=q\\sum_{n=1}^\\infty\\frac{q^{2n-1}}{(q;q^2)_n} \\notag \\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n},\n\\end{align}\nhere, an", + "cd": "\\begin{align}\n\\sum_{n=1}^\\infty C(n)q^{n}&=\\sum_{n=1}^\\infty\\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\\nonumber\\\\\n&=\\sum_{n=1}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \\label{c1n}\\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n}.\\label{cn}\n\\end{align}\n\nComparing \\eqref{bn} and \\eqref{cn}, we see that $B(n)=C(n+1)$.\n\n\\section{Proof that $C(n)=\\frac{1}{2}D(n)$}\\label{cd}\n\nLet us fix $C(0)=1$. From \\eqref{c1n}, we have\n\\begin{align}\\label{before p19}\n\\sum_{n=0}^\\infty C(n)q^{n}&=\\sum_{n=0}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{n=0}^\\infty\\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\\infty}.\n\\end{align}", + "main theorem": "\\begin{theorem}\\label{main theorem}\nFor $n>0$, we have\n\\begin{align}\\label{main theorem eqn}\nA(n)=B(n)=C(n+1)=\\frac{1}{2}D(n+1),\n\\end{align}\nwhere $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 802, + "pre_theorem_intro_text": "The grandfather of all partition identities is Euler's theorem \\cite{andrews book}, namely,\n\\begin{align*}\nA(n)=B(n),\n\\end{align*}\nwhere $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.\n\nThis theorem contains all the elements that would suggest generalizations, and, over the centuries, generalizations have been found in profusion. The Rogers-Ramanujan identities \\cite[p.~104]{andrews book} and Schur's 1926 theorem \\cite[p.~116]{andrews book} kicked off the twentieth century's contributions, consult Henry Alder's survey \\cite{alder} for an account of some of the results in the late 20th century. \n\nHowever, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.", + "context": "The grandfather of all partition identities is Euler's theorem \\cite{andrews book}, namely,\n\\begin{align*}\nA(n)=B(n),\n\\end{align*}\nwhere $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.\n\nThis theorem contains all the elements that would suggest generalizations, and, over the centuries, generalizations have been found in profusion. The Rogers-Ramanujan identities \\cite[p.~104]{andrews book} and Schur's 1926 theorem \\cite[p.~116]{andrews book} kicked off the twentieth century's contributions, consult Henry Alder's survey \\cite{alder} for an account of some of the results in the late 20th century.\n\nHowever, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.", + "full_context": "The grandfather of all partition identities is Euler's theorem \\cite{andrews book}, namely,\n\\begin{align*}\nA(n)=B(n),\n\\end{align*}\nwhere $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.\n\nThis theorem contains all the elements that would suggest generalizations, and, over the centuries, generalizations have been found in profusion. The Rogers-Ramanujan identities \\cite[p.~104]{andrews book} and Schur's 1926 theorem \\cite[p.~116]{andrews book} kicked off the twentieth century's contributions, consult Henry Alder's survey \\cite{alder} for an account of some of the results in the late 20th century.\n\nHowever, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.\n\n\\subjclass[2020]{Primary 11P84, 05A17}\n \\keywords{Euler's theorem, Partitions, Glaisher's bijection}\n\\maketitle\n\\pagenumbering{arabic}\n\\pagestyle{headings}\n\\begin{abstract}\nEuler's theorem asserts that $A(n)=B(n)$ where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts. In this paper, it is proved that for $n>0$,\n\\begin{align*}\nA(n)=B(n)=C(n+1)=\\frac{1}{2}D(n+1),\n\\end{align*}\nwhere $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.\n\nThe grandfather of all partition identities is Euler's theorem \\cite{andrews book}, namely,\n\\begin{align*}\nA(n)=B(n),\n\\end{align*}\nwhere $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.\n\nHowever, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.\n\nFor example, if $n=6$, the four sets of partitions are following:\n\\begin{center}\n\\begin{tabular}{ p{2cm} p{3.5cm} p{3cm} p{4.1cm} p{4.1cm}}\n\nOn the other hand, we have\n\\begin{align}\n\\sum_{n=1}^\\infty C(n)q^{n}&=\\sum_{n=1}^\\infty\\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\\nonumber\\\\\n&=\\sum_{n=1}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \\label{c1n}\\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n}.\\label{cn}\n\\end{align}\n\nLet us fix $C(0)=1$. From \\eqref{c1n}, we have\n\\begin{align}\\label{before p19}\n\\sum_{n=0}^\\infty C(n)q^{n}&=\\sum_{n=0}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{n=0}^\\infty\\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\\infty}.\n\\end{align}\nWe now employ the following Euler's identity \\cite[p.~19, (2.2.5)]{andrews book}\n\\begin{align}\\label{euler identity}\n\\frac{1}{(t;q)_\\infty}=\\sum_{m=0}^\\infty\\frac{t^m}{(q,q)_m},\n\\end{align}\n(with replacing $q$ by $q^2$ and then letting $t=q^{2n+2}$) in \\eqref{before p19} so as to obtain\n\\begin{align}\\label{series}\n\\sum_{n=0}^\\infty C(n)q^{n}&=(q^2;q^2)_\\infty\\sum_{n=0}^\\infty\\frac{q^{2n}}{(q;q)_{2n}}\\sum_{m=0}^\\infty\\frac{q^{2nm+2m}}{(q^2;q^2)_m}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{m,n=0}^\\infty\\frac{q^{2n+2nm+2m}}{(q;q)_{2n}(q^2;q^2)_m}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{m,n=0}^\\infty\\frac{1}{2}\\left(1+(-1)^n\\right)\\frac{q^{n+nm+2m}}{(q;q)_{n}(q^2;q^2)_m}\\nonumber\\\\\n&=\\frac{1}{2}(q^2;q^2)_\\infty\\sum_{m=0}^\\infty\\frac{q^{2m}}{(q^2;q^2)_m}\\left\\{\\sum_{n=0}^\\infty\\frac{q^{n(m+1)}}{(q;q)_n}+\\sum_{n=0}^\\infty\\frac{(-1)^nq^{n(m+1)}}{(q;q)_n}\\right\\}.\n\\end{align}\nUpon invoking \\eqref{euler identity} twice, once with letting $t=q^{m+1}$ and once with letting $t=-q^{m+1}$, and then substituting both resulting expressions in \\eqref{series}, we conclude that\n\\begin{align}\\label{3.4}\n\\sum_{n=0}^\\infty C(n)q^{n}&=\\frac{1}{2}(q^2;q^2)_\\infty\\sum_{m=0}^\\infty\\frac{q^{2m}}{(q^2;q^2)_m}\\left\\{\\frac{1}{(q^{m+1};q)_\\infty}+\\frac{1}{(-q^{m+1};q)_\\infty}\\right\\}\\nonumber\\\\\n&=\\frac{1}{2}(-q;q)_\\infty\\sum_{m=0}^\\infty\\frac{q^{2m}}{(-q;q)_m}+\\frac{1}{2}(q;q)_\\infty\\sum_{m=0}^\\infty\\frac{q^{2m}}{(q;q)_m}\\nonumber\\\\\n&=\\frac{1}{2}\\sum_{m=0}^\\infty q^{2m}(-q^{m+1};q)_\\infty+\\frac{1}{2}(1-q),\n\\end{align}\nwhere the last step follows upon again using \\eqref{euler identity} with letting $t=q^2$.\n\n\\subsection{A Bijective proof of $A(n-1)=\\frac{1}{2}D(n)$}\nWe can first remove the non-negative part condition for $D(n)$ as follows:\n\\begin{quote}\nFor $n\\ge 2$, $D(n)$ counts the number of partitions of $n$ where only the smallest part can repeat at most twice and all other parts are distinct. \n\\end{quote}\nIn other words, a partition $\\lambda=(\\lambda_1,\\ldots, \\lambda_{\\ell})$ counted by $D(n)$ satisfies\n$$\n\\lambda_1>\\lambda_2> \\cdots > \\lambda_{\\ell-1}\\ge \\lambda_{\\ell} \\text{ if $\\ell>1$} \n$$\nor \n$\\lambda=(\\lambda_1)$ has only one part.\n\nLet $M \\le 2N$ be an even number, which can be written as\n$$\nM=2^k a \\text{ for some odd integer $a$ and some $ k\\ge 1$}.\n$$\nWe apply Glaisher's bijection $\\phi$ to $M$ and obtain $2^k$ parts of size $a$. Note that since\n$\n2^k a \\le 2N,\n$\n$$\na \\le N.\n$$\nAlso, if $M\\le N$, then $M$ can appear only once, so we get all distinct positive powers $2^k$ such that\n\\begin{equation} \n2^k\\le N/a. \\label{power1}\n\\end{equation}\nOn the other hand, if $N< M\\le 2N$, then\n\\begin{equation} \n2^k> N/a, \\label{power2}\n\\end{equation}\nand $M$ can repeat. \nSuppose $M$ appears $f$ many times. Upon applying Glaisher's bijection $\\phi$ to $f$ copies of $M$, we obtain\n$$\n f 2^{k} \\text{ copies of $a$}. \n$$\nBy writing $f$ as a binary expansion, \n$$\nf 2^k =(f_0 \\,2^{0}+f_1 \\, 2^{1} +\\cdots ) 2^k, \n$$\nwhere $f_j$ is either $0$ or $1$ for $j\\ge 0$. By \\eqref{power2}, we see that each summand in the above expression represents a distinct power of $2$ greater than $N/a$, i.e., \n\\begin{equation}\nf_j \\, 2^{j+k} >N/a. \\label{power3}\n\\end{equation}\nIt follows from \\eqref{power1} and \\eqref{power3} that $a$ can appear with any multiplicity greater than $1$. This proves that the resulting partition is counted by $B(n)$.", + "post_theorem_intro_text_len": 1257, + "post_theorem_intro_text": "For example, if $n=6$, the four sets of partitions are following:\n\\begin{center}\n\\begin{tabular}{ p{2cm} p{3.5cm} p{3cm} p{4.1cm} p{4.1cm}}\n\n \\vspace{2mm}$A(6)$\\newline & \\vspace{2mm} $B(6)$ & \\vspace{2mm} $C(7)$& \\vspace{2mm} $D(7)$ \\\\\n\n $6$ \\newline 5+1 \\newline 4+2\\newline 3+2+1 & 5+1 \\newline 3+3 \\newline 3+1+1+1 \\newline 1+1+1+1+1+1 & 6+1\\newline 4+3 \\newline 4+2+1 \\newline 2+2+2+1 & 0+0+7\\newline 0+0+6+1\\newline 0+0+5+2\\newline 0+0+4+3\\newline 0+0+4+2+1\\newline 1+1+5\\newline 1+1+2+3\\newline 2+2+3\n\\end{tabular}\n\\end{center}\n\nWe conclude the introduction with the following remark.\n\\begin{remark}\nIn their recent paper \\cite{ba}, M. El Bachraoui and the first author considered partitions with multiple appearances by the first part. All parts were assumed to be positive. It would be a simple matter to extend the results of that paper to the case of non-negative parts in that this would add $(-q;q)_\\infty$ to the generating functions in question.\n\\end{remark}\n\nThis paper is organised as follows. In section \\ref{bc}, we provide the brief proof that $B(n)=C(n+1)$. In Section \\ref{cd}, we prove that $C(n)=\\frac{1}{2}D(n)$. We also provide bijective proofs our assertions in Theorem \\ref{main theorem} in Section \\ref{bijective proofs}.", + "sketch": "The post-theorem introduction does not give a proof sketch beyond an outline of the paper: in Section~\\ref{bc} they \"provide the brief proof that $B(n)=C(n+1)$\"; in Section~\\ref{cd} they \"prove that $C(n)=\\frac{1}{2}D(n)$\"; and in Section~\\ref{bijective proofs} they \"provide bijective proofs\" of the assertions in Theorem~\\ref{main theorem}.", + "expanded_sketch": "The post-theorem introduction does not give a proof sketch beyond an outline of the paper: next they note that\n\\begin{align}\\label{bn}\n\\sum_{n=1}^\\infty B(n)q^{n+1}&=q\\sum_{n=1}^\\infty\\frac{q^{2n-1}}{(q;q^2)_n} \\notag \\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n},\n\\end{align}\nhere, an\nand also that\n\\begin{align}\n\\sum_{n=1}^\\infty C(n)q^{n}&=\\sum_{n=1}^\\infty\\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\\nonumber\\\\\n&=\\sum_{n=1}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \\label{c1n}\\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n}.\\label{cn}\n\\end{align}\n\nComparing \\eqref{bn} and \\eqref{cn}, we see that $B(n)=C(n+1)$.\n\n\\section{Proof that $C(n)=\\frac{1}{2}D(n)$}\\label{cd}\n\nLet us fix $C(0)=1$. From \\eqref{c1n}, we have\n\\begin{align}\\label{before p19}\n\\sum_{n=0}^\\infty C(n)q^{n}&=\\sum_{n=0}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{n=0}^\\infty\\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\\infty}.\n\\end{align}\nThey then prove that $C(n)=\\frac{1}{2}D(n)$, and finally they provide bijective proofs as follows:\n\\label{bijective proofs}\n\n\\subsection{A Bijective proof of $A(n-1)=\\frac{1}{2}D(n)$}\nWe can first remove the non-negative part condition for $D(n)$ as follows:\n\\begin{quote}\nFor $n\\ge 2$, $D(n)$ count", + "expanded_theorem": "\\label{main theorem}\nFor $n>0$, we have\n\\begin{align}\\label{main theorem eqn}\nA(n)=B(n)=C(n+1)=\\frac{1}{2}D(n+1),\n\\end{align}\nwhere $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.,", + "theorem_type": [ + "Universal", + "Biconditional or Equivalence" + ], + "mcq": { + "question": "For each integer \\(n>0\\), let \\(A(n)\\) be the number of partitions of \\(n\\) into distinct parts, and let \\(B(n)\\) be the number of partitions of \\(n\\) into odd parts. Also, for an integer \\(m\\), let \\(C(m)\\) be the number of partitions of \\(m\\) whose largest part is even and whose parts not exceeding half of the largest part are distinct, and let \\(D(m)\\) be the number of partitions of \\(m\\) into non-negative parts such that the smallest part appears exactly twice and no other part is repeated. Which statement is equivalent to the equality \\(A(n)=B(n)\\) for all \\(n>0\\)?", + "correct_choice": { + "label": "A", + "text": "For every \\(n>0\\), \\(A(n)=B(n)=C(n+1)=\\tfrac12 D(n+1)\\)." + }, + "choices": [ + { + "label": "B", + "text": "For every \\(n>0\\), \\(A(n)=B(n)=C(n)=\\tfrac12 D(n)\\)." + }, + { + "label": "C", + "text": "For every \\(n>0\\), \\(A(n)=B(n)=C(n+1)\\)." + }, + { + "label": "D", + "text": "For every \\(n>0\\), \\(A(n)=B(n)=C(n+1)=D(n+1)\\)." + }, + { + "label": "E", + "text": "For every \\(n>0\\), \\(A(n)=B(n)=C(n+1)=\\tfrac12 D(n)\\)." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "characteristic", + "tampered_component": "index_shift_for_C_and_D", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "characteristic", + "tampered_component": "dropped_the_\\(\\tfrac12 D(n+1)\\)_clause", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "counting_estimate", + "tampered_component": "missing_factor_one_half_in_D_relation", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "characteristic", + "tampered_component": "dependence_on_shifted_argument_of_D", + "template_used": "quantifier_dependence" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the partition functions and asks for an equivalent statement, but it does not explicitly reveal the needed index shift or the factor of 1/2. The correct answer is not stated or strongly hinted at in the wording." + }, + "TAS": { + "score": 1, + "justification": "The item is close to asking for the exact theorem-level equivalence rather than a genuinely new conclusion. It is not a pure restatement of A(n)=B(n), since the C and D relations must be matched correctly, but it still mainly tests recognition of the precise formulation." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish subtle variants involving shifted indices, a missing factor, and a weaker true statement. However, the task is still largely theorem recognition/verification rather than substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and well targeted: off-by-one shifts, omission of the 1/2 factor, and inclusion of a weaker true statement. These reflect realistic mathematical errors and are meaningfully distinct." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with no real answer leakage and strong distractors, but it leans toward recalling the exact theorem statement rather than forcing deep generative reasoning." + } + }, + { + "id": "2511.21628v1", + "paper_link": "http://arxiv.org/abs/2511.21628v1", + "theorems_cnt": 7, + "theorem": { + "env_name": "theorem", + "content": "\\label{t.main}\n\tLet $n,s,\\ell,c$ be positive integers such that $n=2s+c=3s-\\ell$, and $c,\\ell\\in [s-1]$. Then $$e(n, s) = \\max\\big\\{|\\mathcal{P}(s, \\ell)|, |\\mathcal{P}'(\\ell)|, |\\mathcal{Q}(s, \\ell)|, |\\mathcal{W}(s, \\ell)|\\big\\}.$$ Moreover, if $s \\geq 3$, ${\\mathcal F}\\subset 2^{[n]}$ is shifted, has no $s$-matching and $|{\\mathcal F}|=e(n,s)$, then ${\\mathcal F}$ must coincide with one of the families above.", + "start_pos": 76309, + "end_pos": 76727, + "label": "t.main" + }, + "ref_dict": { + "eq008": "\\begin{equation}\\label{eq008} e_k(n,s) = \\max\\bigl\\{|\\aaa_1^{(k)}(n,s-1)|,|\\aaa_k^{(k)}(n,s-1)|\\bigr\\}.\n\\end{equation}", + "thmfk": "\\begin{thm}[\\cite{FK9,FK8}]\\label{thmfk} \\ $e(sm+s-\\ell,s) = |\\pp(s,m,\\ell)|$ holds for\n\\begin{align*} &\\mathrm{(i)}\\ \\ \\ \\ \\ell = 2, \\\\\n &\\mathrm{(ii)}\\ \\ \\ m=1,\\\\\n &\\mathrm{(iii)} \\ \\ s\\ge \\ell m+3\\ell+3.\n\\end{align*}\n\\end{thm}", + "fig2": "\\begin{figure}\n\n\\centering\n\n\\includegraphics[width=0.6\\linewidth]{extremal_families_2.png}\n\n\\caption{Extremal families for small $c,s$. Note that for $c=1, s=5$ there are $3$ different extremal families. For $\\ell = 1$, that is, $s = c + 1$, the family $\\mathcal{P}'$ coincides with $\\mathcal{P}$ (both are equal to ${n \\choose \\geq (m + 1)}$) and thus is formally extremal. When $\\mathcal{P}' \\neq \\mathcal{P}$, for $c \\leq 9$ $\\mathcal{P}'$ cannot be extremal.}\n\n\\label{fig2}\n\n\\end{figure}", + "eq002": "\\begin{align}\\label{eq001} e(sm-1,s) &= \\sum_{t=m}^{sm-1}{sm-1\\choose t},\\\\\n\\label{eq002} e(sm,s) &= {sm-1\\choose m}+\\sum_{t=m+1}^{sm}{sm\\choose t}.\n\\end{align}", + "conj1": "\\begin{gypo}[\\cite{FK9}]\\label{conj1}\nSuppose that $s\\ge 2, m\\ge 1$, and $n = sm+s-\\ell$ for some integer $0<\\ell\\le \\lceil \\frac s2\\rceil$. Then\n\\begin{equation}\\label{eq007} e(sm+s-\\ell,s) = |\\pp(s,m,\\ell)|.\n\\end{equation}\n\\end{gypo}", + "fig1": "\\begin{figure}\n\n\\centering\n\n\\includegraphics[width=0.8\\linewidth]{extremal_families.png}\n\n\\caption{Extremal families. $\\mathcal{W}$ is one of the extremal families for $c = 1, s \\geq 5$.}\n\n\\label{fig1}\n\n\\end{figure}" + }, + "pre_theorem_intro_text_len": 9223, + "pre_theorem_intro_text": "Let $[n] := \\{1,2,\\ldots, n\\}$ and, more generally, $[a,b]=\\{a,a+1,\\ldots, b\\}$. For a set $X$ and an integer $k$, let $2^{X}$, ${X\\choose k}$ and ${X\\choose \\geq k}$ stand for the power set of $X$, the set of its $k$-element subsets and the set of its subsets with size at least $k$, respectively. Any collection of sets is called a {\\it family.} A {\\it matching} is a collection of pairwise disjoint sets. An {\\it $s$-matching} is a matching of size $s$. Given a family ${\\mathcal F},$ its {\\it matching number}\n$\\nu(\\mathcal F)$ is the size of the largest matching in ${\\mathcal F}$.\n\nOne of the classical topics in extremal set theory is the study of {\\it intersecting} families, that is, families with matching number $1$. Erd\\H os, Ko and Rado~\\cite{EKR} showed that the largest intersecting family ${\\mathcal F}\\subset 2^{[n]}$ has size at most $2^{n-1}$, and that for $n\\ge 2k$ the largest intersecting family ${\\mathcal F}\\subset {[n]\\choose k}$ has size ${n-1\\choose k-1}.$ In the several years that followed, Erd\\H os asked for the size of the largest family avoiding an $s$-matching. Let us introduce the following two quantities.\n\\begin{align*}\n e(n,s)&=\\max\\big\\{|{\\mathcal F}|: {\\mathcal F}\\subset 2^{[n]}, \\nu({\\mathcal F})0$). We refer the reader to \\cite{aletal, FK21} for the connections of the Erd\\H os Matching Conjecture and other questions, such as Dirac thresholds and small deviations in probability theory. In \\cite{HLS}, \\cite{K49}, the multi-family variant of the EMC was addressed. In \\cite{FK6}, a Hilton--Milner type stability result for the EMC is obtained.\n\n\\subsection{The non-uniform case}\nThe study of $e(n,s)$ was also initiated by Erd\\H os at around the same time. The behavior of $e(n,s)$ heavily depends on $n\\ ({\\rm mod\\ } s)$. Answering a question of Erd\\H os, Kleitman proved the following theorem. \n\\begin{thm}[Kleitman \\cite{Kl}]\\label{thmkl}\n\\begin{align}\\label{eq001} e(sm-1,s) &= \\sum_{t=m}^{sm-1}{sm-1\\choose t},\\\\\n\\label{eq002} e(sm,s) &= {sm-1\\choose m}+\\sum_{t=m+1}^{sm}{sm\\choose t}.\n\\end{align}\n\\end{thm}\nThe matching example for the first case is the family ${[n]\\choose \\ge m}$ of all subsets of $[n]$ of size at least $m$. It is also not difficult to see that $e(sm,s) = 2e(sm-1,s)$. In general, $e(n+1,s)\\ge 2e(n,s)$ because of the {\\it doubling} construction. Given a family ${\\mathcal F}\\subset 2^{[n]}$ with $\\nu({\\mathcal F})\\min\\big\\{|\\overline{\\mathcal{P'}(s, \\ell)}|, |\\overline{\\mathcal{P}(s, \\ell)}|, |\\overline{\\mathcal{Q}(s, \\ell)}|, |\\overline{\\mathcal{W}(s, \\ell)}|\\big\\}.\n \\end{multline*}\n Moreover,\n \\begin{multline*}\n \\min\\big\\{|\\overline{\\mathcal{P'}(s-1, \\ell-2)}^{(\\leq 3)}|, |\\overline{\\mathcal{P}(s-1, \\ell-2)}^{(\\leq 3)}|, |\\overline{\\mathcal{Q}(s-1, \\ell-2)}^{(\\leq 3)}|, \\\\|\\overline{\\mathcal{W}(s-1, \\ell-2)}^{(\\leq 3)}|\\big\\} >\\min\\big\\{|\\overline{\\mathcal{P'}(s, \\ell)}^{(\\leq 3)}|, |\\overline{\\mathcal{P}(s, \\ell)}^{(\\leq 3)}|, |\\overline{\\mathcal{Q}(s, \\ell)}^{(\\leq 3)}|, |\\overline{\\mathcal{W}(s, \\ell)}^{(\\leq 3)}|\\big\\}.\n \\end{multline*}\n\\label{c.no_siggletons}\n\\end{restatable}\n\nThe case of odd $d$ requires a more careful analysis. In this case, we use the inequality \\eqref{eqd2c} which states that $d(\\mathcal{F}) \\leq 2c$. \n\\begin{lemma} \\label{l.odd_d_to_y2}\n Let $d$ be a positive odd integer, $d \\leq 2c$. If $\\mathcal{F} \\subset 2^{[n]}$ is a shifted family with $d(\\mathcal{F}) = d$, then {\\small \\begin{equation}\\label{eqy22}y(2) \\geq \\min\\Big\\{\\frac{(4\\ell+3c+d-2)(3c-d+1)}{2}, \\frac{(\\ell+3c-\\frac{d-1}{2})(\\ell+3c-\\frac{d+1}2)}{2}\\Big\\}.\\end{equation}} Moreover, equality is achieved only if $\\mathcal{F}^{(2)} = {[2\\ell+d-1] \\choose 2}$ or $\\mathcal{F}^{(2)} = \\{F \\in {n \\choose 2 }: F\\cap[\\ell+\\frac{d-1}{2}] \\neq \\emptyset\\}$.\n\\end{lemma}\n\n\\subsection{$c \\in \\{3, 4\\}$}\n\\begin{lemma} \\label{l.c_eq_3}\n Let $n = 2s+3$ and $\\mathcal{F} \\subset 2^{[n]}$ is a shifted up-set with $\\nu(\\mathcal{F}) < s$ and $\\mathcal{F} \\cap {[n] \\choose 1} = \\emptyset$. Then $|\\mathcal{F}| \\leq \\max(|\\mathcal{P}(s, \\ell)|, |\\mathcal{P}'(s, \\ell)|, |\\mathcal{Q}(s, \\ell)|)$. Moreover, equality is achieved only if $\\mathcal{F}$ is one of the families $\\mathcal{P}(s, \\ell), \\mathcal{P}'(s, \\ell), \\mathcal{Q}(s, \\ell)$.\n\\end{lemma}", + "post_theorem_intro_text_len": 2648, + "post_theorem_intro_text": "\\begin{figure}\n\n\\centering\n\n\\includegraphics[width=0.8\\linewidth]{extremal_families.png}\n\n\\caption{Extremal families. $\\mathcal{W}$ is one of the extremal families for $c = 1, s \\geq 5$.}\n\n\\label{fig1}\n\n\\end{figure}\n\n\\begin{figure}\n\n\\centering\n\n\\includegraphics[width=0.6\\linewidth]{extremal_families_2.png}\n\n\\caption{Extremal families for small $c,s$. Note that for $c=1, s=5$ there are $3$ different extremal families. For $\\ell = 1$, that is, $s = c + 1$, the family $\\mathcal{P}'$ coincides with $\\mathcal{P}$ (both are equal to ${n \\choose \\geq (m + 1)}$) and thus is formally extremal. When $\\mathcal{P}' \\neq \\mathcal{P}$, for $c \\leq 9$ $\\mathcal{P}'$ cannot be extremal.}\n\n\\label{fig2}\n\n\\end{figure}\nOn Figures~\\ref{fig1} and~\\ref{fig2} we show, which families are extremal for different regimes of the parameters $s,c$. For some values we get that three different families are extremal at the same time. \n\nWe define shifted families in the next section. We should note that actually there are rather natural examples of families with no $s$-matching interpolating between $\\mathcal{P}'(s, l)$ and $\\mathcal{Q}(s, l)$ in a somewhat similar way as $\\aaa_i$ interpolate between $\\aaa_0$ and $\\aaa_k$, but, as in the case of the EMC, there is a certain convexity that leads to the fact that it is the endpoints that must be extremal. \n\nIn the proof we will work only with sets of size $3$ or less. Therefore, any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family. We thus get the following theorem about the truncated boolean lattice, confirming a conjecture of Frankl and the first author \\cite{FK9} in our regime of the parameters. \n\n\\begin{theorem} \\label{t.truncated_lattice}\n Let $n,s,\\ell,c$ be positive integers such that $n=2s+c=3s-\\ell$, and $c,\\ell\\in [s-1]$. If $\\mathcal{F} \\subset {[n] \\choose \\leq 3}$ has no $s$-matching, then\n $$|\\mathcal{F}| \\leq \\max\\big\\{|\\mathcal{P}(s, \\ell)^{(\\leq 3)}|, |\\mathcal{P}'(s, \\ell)^{(\\leq 3)}|, |\\mathcal{Q}(s, \\ell)^{(\\leq 3)}|, |\\mathcal{W}(s, \\ell)^{(\\leq 3)}|\\big\\}.$$\n\\end{theorem}\n\nNote that a similar statement about $2$ first layers of boolean lattice is obviously false. Indeed, one of the families $\\aaa_1^{(2)}(n,s-1), \\aaa_2^{(2)}(n,s-1)$ has a larger cardinality than families $\\mathcal{P}(s, \\ell)^{(\\leq 2)}, \\mathcal{P}'(s, \\ell)^{(\\leq 2)}, \\mathcal{Q}(s, \\ell)^{(\\leq 2)}, \\mathcal{W}(s, \\ell)^{(\\leq 2)}$.\n\nIn Section~\\ref{sec2}, we prove several easy facts and make the necessary preparations for the proof of the main theorem. In Section~\\ref{sec3} we describe the strategy of the proof of the main theorem.", + "sketch": "To prove Theorem~\\ref{t.main}, the authors note that they will \"work only with sets of size $3$ or less.\" Hence, \"any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family.\" This reduction leads to an auxiliary result on the truncated Boolean lattice: Theorem~\\ref{t.truncated_lattice}, which bounds $|\\mathcal F\\subset {[n]\\choose \\le 3}|$ with no $s$-matching by the maximum of the truncated versions of the candidate extremal families $\\mathcal P,\\mathcal P',\\mathcal Q,\\mathcal W$. They also mention that while there are \"natural examples\" interpolating between $\\mathcal P'$ and $\\mathcal Q$, \"there is a certain convexity\" implying \"it is the endpoints that must be extremal.\" Finally, they indicate the paper structure: in Section~\\ref{sec2} they \"prove several easy facts and make the necessary preparations\" and in Section~\\ref{sec3} they \"describe the strategy of the proof of the main theorem.\"", + "expanded_sketch": "To prove the main theorem, the authors note that they will \"work only with sets of size $3$ or less.\" Hence, \"any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family.\" This reduction leads to an auxiliary result on the truncated Boolean lattice: Theorem~\\ref{t.truncated_lattice}, which bounds $|\\mathcal F\\subset {[n]\\choose \\le 3}|$ with no $s$-matching by the maximum of the truncated versions of the candidate extremal families $\\mathcal P,\\mathcal P',\\mathcal Q,\\mathcal W$. They also mention that while there are \"natural examples\" interpolating between $\\mathcal P'$ and $\\mathcal Q$, \"there is a certain convexity\" implying \"it is the endpoints that must be extremal.\" Finally, they indicate the paper structure: next they \"prove several easy facts and make the necessary preparations\" and later they \"describe the strategy of the proof of the main theorem.\"", + "expanded_theorem": "\\label{t.main}\n\tLet $n,s,\\ell,c$ be positive integers such that $n=2s+c=3s-\\ell$, and $c,\\ell\\in [s-1]$. Then $$e(n, s) = \\max\\big\\{|\\mathcal{P}(s, \\ell)|, |\\mathcal{P}'(\\ell)|, |\\mathcal{Q}(s, \\ell)|, |\\mathcal{W}(s, \\ell)|\\big\\}.$$ Moreover, if $s \\geq 3$, ${\\mathcal F}\\subset 2^{[n]}$ is shifted, has no $s$-matching and $|{\\mathcal F}|=e(n,s)$, then ${\\mathcal F}$ must coincide with one of the families above.", + "theorem_type": [ + "Classification or Bijection", + "Equality or Bound" + ], + "mcq": { + "question": "Let \\([n]=\\{1,\\dots,n\\}\\), and define\n\\[\ne(n,s)=\\max\\{\\,|\\mathcal F|: \\mathcal F\\subset 2^{[n]},\\ \\nu(\\mathcal F)1$, $C>0$, $a \\in (1,\\infty)$. Let $\\tau_1, \\tau_2 \\in \\mathbb{R}^+$ satisfy\n \\begin{equation*}\n \\tau_1 < \\big(\\frac{2}{(2\\pi)^{m/(m-1)}(ma)^{-1/(m-1)}e}\\big)^{\\frac{m-1}{m}}, \\quad \\quad \\quad \\tau_2< (\\frac{2}{ame})^{\\frac{1}{m}}.\n \\end{equation*}\n Define the set $\\Lambda$ such that: \n \\begin{equation*}\n \\Lambda = \\{(\\pm\\tau_1n^{\\frac{m-1}{m}},\\pm\\tau_2n^{\\frac{1}{m}}),n \\in \\mathbb{N}\\}\n \\end{equation*} \n Then the following statements are equivalent for every $f, h \\in L^2(\\mathbb{R}):$\n \\begin{enumerate}\n \\item $|V_g(f)(\\lambda)| = |V_gh(\\lambda)|$ for every $\\lambda \\in \\Lambda$.\n \\item $f = e^{i \\alpha}h$ for some $\\alpha \\in [0,2\\pi).$\n \\end{enumerate}", + "start_pos": 105912, + "end_pos": 106865, + "label": "MainTheorem" + }, + "ref_dict": { + "MainTheorem": "\\begin{theorem}\\label{MainTheorem}\n Let $g \\in L^2(\\R)$ with Fourier transform satisfies the super-exponential decay condition:\n \\begin{equation*}\n |\\hat{g}(\\xi)| \\leq C e^{-a|\\xi|^m}\n \\end{equation*}\n for some constants $m>1$, $C>0$, $a \\in (1,\\infty)$. Let $\\tau_1, \\tau_2 \\in \\R^+$ satisfy\n \\begin{equation*}\n \\tau_1 < \\big(\\frac{2}{(2\\pi)^{m/(m-1)}(ma)^{-1/(m-1)}e}\\big)^{\\frac{m-1}{m}}, \\quad \\quad \\quad \\tau_2< (\\frac{2}{ame})^{\\frac{1}{m}}.\n \\end{equation*}\n Define the set $\\Lambda$ such that: \n \\begin{equation*}\n \\Lambda = \\{(\\pm\\tau_1n^{\\frac{m-1}{m}},\\pm\\tau_2n^{\\frac{1}{m}}),n \\in \\N\\}\n \\end{equation*} \n Then the following statements are equivalent for every $f, h \\in L^2(\\R):$\n \\begin{enumerate}\n \\item $|V_g(f)(\\lambda)| = |V_gh(\\lambda)|$ for every $\\lambda \\in \\Lambda$.\n \\item $f = e^{i \\alpha}h$ for some $\\alpha \\in [0,2\\pi).$\n \\end{enumerate}\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 34, + "pre_theorem_intro_text": "\\label{sec:1}\n\\input{introduction}", + "context": "\\label{sec:1}\n\\input{introduction}", + "full_context": "\\label{sec:1}\n\\input{introduction}\n\n\\section{Preliminary}\n\\label{sec:2}\nA central principle in Fourier analysis is the connection between the smoothness of a function and the decay rate of its Fourier transform at infinity. The Paley-Wiener theorem and its variants are the primary tools that precisely characterize this relationship, providing the foundation for extending functions into the complex domain based on the behavior of their Fourier transforms. \n\\subsection{Paley-Wiener Type Theorem}\nThe Fourier transform of a Lebesgue integrable function $f$ on $\\R$ is given by \n\\begin{equation*}\n \\mathcal{F}(f):=\\hat{f} (\\xi) = \\int_{\\R} f(t) e^{-2 \\pi i t \\cdot\\xi} dt, \\quad \\xi \\in \\R.\n\\end{equation*}\n\nA standard density argument can be used to extend this definition so that $\\mathcal{F}$ is an isometry on $L^2(\\R)$. The Fourier transform of a function $f \\in L^2(\\R)$ is said to be of \\emph{super-exponential decay} if\n\\begin{equation}\\label{FourierDecay}\n |\\hat{f}(\\xi)| \\lesssim Ce^{-a|\\xi|^m} \n\\end{equation}\nfor some $m>1$, $C>0$ and $a \\in (1,\\infty)$. In this subsection we introduce a slightly modified version of Paley-Wiener theorem. In particular, we show that if the Fourier transform of a function $f$ decays super-exponential, then $f$ can \nbe extended to an entire function whose order and type depend on the parameters $a$ and $m$. To establish this result, we make use of Morera’s Theorem, which we now recall. \n\\begin{mainthm}\\cite[Theorem 5.2]{stein2010complex}\\label{Morera}\n If $\\{ f_n \\}_{n \\in \\N}$ is a sequence of holomorphic functions that converges uniformly to a function $f$ in every compact subset of $\\Omega$, then $f$ is holomorphic in $\\Omega$.\n\\end{mainthm}\n\nSimilarly, fix $z=x+iy$ and consider $z' = x'+iy'\\subset K'$, where $K'$ is compact. Then we have\n\\begin{equation}\\label{fixz}\n \\int_{-\\infty}^{\\infty} |\\overline{g(t-z)}|^2 |e^{2\\pi i z' \\xi}|^2 dt \\leq C\\int_{-\\infty}^{\\infty} e^{-2a|t-x|^{\\rho}+2b|y|^{\\rho}-4\\pi y't}dt.\n\\end{equation}\nSince $z=x+iy$ is fixed, for $z' \\in K'$ it follows that \\eqref{fixz} converges for $z' \\in K$. Following a statement from \\cite[Page 68]{boas2011entire}, the order and type of an entire function is retained under integral, therefore the orders and types for $|V_gf (z,z')|$ can be computed directly using Lemma \\ref{OrderType}. It is noteworthy that our computation for orders is compatible with the result from \\cite[Lemma 3.4]{grohs2024phaseless} when $m = 2$.\n\\end{proof}\n\\section{Main Result}\nDenote the ring of holomorphic functions by $\\mathcal{O}(\\C)$. Consider the collection of entire functions with $\\rho>1$ and $a,b>0$:\n$$\\fG_{a,b,\\rho} = \\{ f \\in \\mathcal{O}(\\C)\\big| |f(x+iy)| \\lesssim e^{-a|x|^{\\rho}+b|y|^{\\rho}} \\}.$$\nIt is straightforward to see that $\\fG_{a,b,\\rho}$ is a linear space. The objective is to establish a uniqueness set so that whenever two functions $f, g \\in \\fG_{a,b,\\rho}$ agree on the set, they agree \\textit{everywhere}. Recent literature shows that such functions cannot be uniquely determined from their values on integer lattices \\cite{alaifari2021phase}. For the purpose of notation, let $U \\subset \\mathcal{O(\\C)}$ denote a linear function space. A set $\\Lambda \\subset \\C$ is called a uniqueness set of $U$ if for any $f \\in U$, it holds that \n$$(f( \\lambda ) = 0 \\; \\text{for all} \\; \\lambda \\in \\Lambda) \\Longrightarrow f = 0 \\; \\text{everywhere on } \\: \\C.$$\nSince zero of any entire function can't be an accumulation point, any open set in $\\C$ is trivially a uniqueness set. Without this property, determining discrete uniqueness sets of $\\fG_{a,b,\\rho}$ contained in $\\R$ is a nontrivial task. We first propose a sufficient condition for such a set.\\\\\n\n\\input{references}\n\\end{document}\n\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000OrderType.tex\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u00000000664\u00000000000\u00000000000\u000000000012657\u000015107436001\u0000012225\u0000 0\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000ustar \u0000root\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000root\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\n\\begin{lemma}\\label{OrderType}\nLet $f \\in L^2(\\mathbb{R})$ be a function whose Fourier transform satisfies $|\\hat{f}(\\xi)| \\le C e^{-a|\\xi|^m}$ for some $m>1$ and $a>0$. Then its analytic continuation $f(z)$ is an entire function of type at most $\\tau = \\frac{m-1}{m}(2\\pi)^{m/(m-1)}(am)^{-1/(m-1)}$ corresponding to maximum possible order $\\rho = \\frac{m}{m-1}$.\n\\end{lemma}\n\\begin{proof}\nThe analytic continuation of the function, $f(z)$, is given by the inverse Fourier transform:\n\\begin{equation*}\n f(z) = \\int_{-\\infty}^\\infty \\hat{f}(\\xi) e^{2\\pi i z \\xi} d\\xi\n\\end{equation*}\nThe Taylor coefficients of $f(z)$ are given by \n\\begin{equation*}\nc_n = \\frac{f^{(n)}(0)}{n!} = \\frac{(2\\pi i)^n}{n!} \\int_{-\\infty}^\\infty \\xi^n \\hat{f}(\\xi) d\\xi\n\\end{equation*}\nWe can now bound the magnitude of the coefficients using the given decay of $\\hat{f}(\\xi)$:\n\\begin{equation*}\n|c_n| \\le \\frac{(2\\pi)^n}{n!} \\int_{-\\infty}^\\infty |\\xi|^n |\\hat{f}(\\xi)| d\\xi \\le \\frac{C(2\\pi)^n}{n!} \\int_{-\\infty}^\\infty |\\xi|^n e^{-a|\\xi|^m} d\\xi\n\\end{equation*}\nThe integral can be evaluated using the Gamma function:\n\\begin{equation*}\n \\int_{-\\infty}^\\infty |\\xi|^n e^{-a|\\xi|^m} d\\xi = \\frac{2}{m a^{(n+1)/m}} \\Gamma\\left(\\frac{n+1}{m}\\right). \n\\end{equation*}\nUsing Stirling's formula for Gamma function, we obtain \n\\begin{equation*}\n \\Gamma(\\frac{n+1}{m}) = \\sqrt{\\frac{2 \\pi m}{n+1}}(\\frac{n+1}{me})^{(n+1)/m}\\big(1+ O(\\frac{m}{n+1})\\big).\n\\end{equation*}\nand \n\\begin{equation*}\n n! \\approx \\sqrt{2 \\pi n}(\\frac{n}{e})^n\n\\end{equation*}\nThen, for some $C' >0$, we derive the following expression:\n \\begin{align}\\label{GeneralTaylorCoefficient}\n |c_n|\n &\\lesssim \\Big((2\\pi)^n \\cdot n^{-\\frac{1}{2}-n}\\cdot e^{n -\\frac{n+1}{m}}\\cdot a^{-\\frac{n+1}{m}}\\cdot m ^{-\\frac{1}{2}-\\frac{n+1}{m}}\\cdot (n+1)^{\\frac{n+1}{m}-\\frac{1}{2}}\\Big)\\cdot\\Big(1 + \\frac{C'm}{n+1}\\Big) \\\\\n &= R(a,n,m)\\cdot n^{-n}\\cdot(n+1)^{\\frac{n+1}{m}}\\cdot\\Big(1 + \\frac{C'm}{n+1}\\Big) \\nonumber\n \\end{align}\nThe term $R(a,n,m)$ collects all remaining exponential and polynomial factors. It follows that:\n\\begin{align*}\n \\rho &= \\limsup_{n\\to\\infty} \\frac{n \\log n}{-\\log|c_n|} \\lesssim \\limsup_{n\\to\\infty} \\frac{n\\log n}{-\\log R(a,m,n) + n\\log n -\\frac{n+1}{m}\\log(n+1)- \\log(1+\\frac{C'm}{n+1})}\\\\\n &\\lesssim \\limsup_{n\\to\\infty} \\frac{1}{\\frac{-\\log R(a,m,n)}{n\\log n} + 1 -\\frac{n+1}{nm}\\frac{\\log(n+1)}{\\log n}- \\frac{\\log(1+\\frac{C'm}{n+1})}{n\\log n}}.\n\\end{align*}\nA series of computations shows that $$\\begin{cases} \\lim_{n\\to \\infty}\\frac{\\log R(a,m,n)}{n\\log n} =0\\\\ \\lim_{n\\to \\infty}\\frac{\\log(1+ \\frac{C'm}{n+1})}{n\\log n}=0\\end{cases}.$$ Therefore, the remaining terms are:\n\\begin{equation*} \n \\rho \\leq \\limsup_{n\\to \\infty}\\frac{1}{1-\\frac{n+1}{mn}\\frac{\\log(n+1)}{\\log n}}= \\frac{1}{1- \\frac{1}{m}} = \\frac{m}{m-1}.\n\\end{equation*}\nLet $\\tilde{\\rho} = \\frac{m}{m-1}$, and choose $A,R >0$ such that \n\\begin{equation}\\label{TypeWRTorder}\n \\max_{|z|= r}|f(z)| < Re^{Ar^{\\tilde{\\rho}}}. \n\\end{equation}\nNow we compute the type $\\tau$ of $f(z)$ with respect to order $\\tilde{\\rho}$, i.e., the greatest lower bound of $A$ in Equation~\\eqref{TypeWRTorder}. We note that", + "post_theorem_intro_text_len": 837, + "post_theorem_intro_text": "\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=0.7\\textwidth]{rhosampling.png}\n \\caption{An illustration of sampling points $\\Lambda$ in time-frequency plane for $g$ with Fourier decay parameter $m=\\frac{3}{2}$ and $a=1$}\n\\end{figure}\n{The rest of the paper is organized as follows: In Section 2 we introduce some necessary preliminaries. Subsequently we prove Theorem~\\ref{MainTheorem} in Section 3. To this end, we first establish a sufficient condition for a discrete set to be a uniqueness set for the relevant class of entire functions (Proposition 1) and use Proposition 2 to show that our result is in a sense \"sharp\". We then use these technical results to prove Theorem~\\ref{MainTheorem}, which provides a concrete discrete sampling set $\\Lambda$ that guarantees the unique recovery of $f$ up to a global phase.", + "sketch": "To prove Theorem~\\ref{MainTheorem}, the paper (in Section 3) proceeds by first establishing “a sufficient condition for a discrete set to be a uniqueness set for the relevant class of entire functions (Proposition 1),” then using “Proposition 2 to show that our result is in a sense \u0018sharp\u0019,” and finally applying “these technical results to prove Theorem~\\ref{MainTheorem},” yielding “a concrete discrete sampling set $\\Lambda$ that guarantees the unique recovery of $f$ up to a global phase.”", + "expanded_sketch": "To prove the main theorem, the paper proceeds by first establishing a sufficient condition for a discrete set to be a uniqueness set for the relevant class of entire functions (Proposition 1), then using Proposition 2 to show that the result is in a sense sharp, and finally applying these technical results to prove the main theorem, yielding a concrete discrete sampling set $\\Lambda$ that guarantees the unique recovery of $f$ up to a global phase.", + "expanded_theorem": "\\label{MainTheorem}\n Let $g \\in L^2(\\mathbb{R})$ with Fourier transform satisfies the super-exponential decay condition:\n \\begin{equation*}\n |\\hat{g}(\\xi)| \\leq C e^{-a|\\xi|^m}\n \\end{equation*}\n for some constants $m>1$, $C>0$, $a \\in (1,\\infty)$. Let $\\tau_1, \\tau_2 \\in \\mathbb{R}^+$ satisfy\n \\begin{equation*}\n \\tau_1 < \\big(\\frac{2}{(2\\pi)^{m/(m-1)}(ma)^{-1/(m-1)}e}\\big)^{\\frac{m-1}{m}}, \\quad \\quad \\quad \\tau_2< (\\frac{2}{ame})^{\\frac{1}{m}}.\n \\end{equation*}\n Define the set $\\Lambda$ such that: \n \\begin{equation*}\n \\Lambda = \\{(\\pm\\tau_1n^{\\frac{m-1}{m}},\\pm\\tau_2n^{\\frac{1}{m}}),n \\in \\mathbb{N}\\}\n \\end{equation*} \n Then the following statements are equivalent for every $f, h \\in L^2(\\mathbb{R}):$\n \\begin{enumerate}\n \\item $|V_g(f)(\\lambda)| = |V_gh(\\lambda)|$ for every $\\lambda \\in \\Lambda$.\n \\item $f = e^{i \\alpha}h$ for some $\\alpha \\in [0,2\\pi).$\n \\end{enumerate}", + "theorem_type": [ + "Biconditional or Equivalence", + "Classification or Bijection" + ], + "mcq": { + "question": "Let the short-time Fourier transform with window \\(g\\) be \\(V_g f(x,\\xi)=\\int_{\\mathbb R} f(t)\\overline{g(t-x)}e^{-2\\pi i\\xi t}\\,dt\\). Suppose \\(g\\in L^2(\\mathbb R)\\) has Fourier transform satisfying\n\\[\n|\\widehat g(\\xi)|\\le C e^{-a|\\xi|^m}\n\\]\nfor some \\(m>1\\), \\(C>0\\), and \\(a>1\\). Let \\(\\tau_1,\\tau_2>0\\) satisfy\n\\[\n\\tau_1<\\Big(\\frac{2}{(2\\pi)^{m/(m-1)}(ma)^{-1/(m-1)}e}\\Big)^{\\frac{m-1}{m}},\n\\qquad\n\\tau_2<\\Big(\\frac{2}{ame}\\Big)^{1/m},\n\\]\nand define\n\\[\n\\Lambda=\\{(\\pm \\tau_1 n^{(m-1)/m},\\, \\pm \\tau_2 n^{1/m}) : n\\in\\mathbb N\\}\\subset \\mathbb R^2,\n\\]\nwith all sign choices allowed. Which statement exactly characterizes the pairs \\(f,h\\in L^2(\\mathbb R)\\) for which\n\\(|V_g f(\\lambda)|=|V_g h(\\lambda)|\\) for every \\(\\lambda\\in\\Lambda\\)?", + "correct_choice": { + "label": "A", + "text": "They are exactly the pairs differing by a global phase: \\(|V_g f(\\lambda)|=|V_g h(\\lambda)|\\) for all \\(\\lambda\\in\\Lambda\\) if and only if \\(f=e^{i\\alpha}h\\) for some \\(\\alpha\\in[0,2\\pi)\\)." + }, + "choices": [ + { + "label": "B", + "text": "They are exactly the pairs related by a time-frequency shift and a global phase: \\(|V_g f(\\lambda)|=|V_g h(\\lambda)|\\) for all \\(\\lambda\\in\\Lambda\\) if and only if there exist \\(x_0,\\xi_0\\in\\mathbb R\\) and \\(\\alpha\\in[0,2\\pi)\\) such that \\(f(t)=e^{i\\alpha}e^{2\\pi i \\xi_0 t}h(t-x_0)\\) a.e." + }, + { + "label": "C", + "text": "If \\(f=e^{i\\alpha}h\\) for some \\(\\alpha\\in[0,2\\pi)\\), then \\(|V_g f(\\lambda)|=|V_g h(\\lambda)|\\) for every \\(\\lambda\\in\\Lambda\\)." + }, + { + "label": "D", + "text": "They are exactly the pairs differing by a global phase provided the equalities hold on one fixed-sign branch of the sampling set, i.e. if and only if \\(|V_g f(\\tau_1 n^{(m-1)/m},\\tau_2 n^{1/m})|=|V_g h(\\tau_1 n^{(m-1)/m},\\tau_2 n^{1/m})|\\) for every \\(n\\in\\mathbb N\\), then \\(f=e^{i\\alpha}h\\) for some \\(\\alpha\\in[0,2\\pi)\\)." + }, + { + "label": "E", + "text": "They are exactly the pairs differing by a global phase for any choice of \\(\\tau_1,\\tau_2>0\\): \\(|V_g f(\\lambda)|=|V_g h(\\lambda)|\\) for all \\(\\lambda\\in\\Lambda\\) if and only if \\(f=e^{i\\alpha}h\\) for some \\(\\alpha\\in[0,2\\pi)\\), without requiring the upper bounds on \\(\\tau_1\\) and \\(\\tau_2\\)." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "uniqueness_conclusion_global_phase_only", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped_only_if_direction", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "case_split", + "tampered_component": "full_four_sign_sampling_set", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "finiteness", + "tampered_component": "threshold_conditions_on_tau1_tau2", + "template_used": "boundary_range" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not state or strongly hint at the correct conclusion; it only gives the hypotheses and asks for the exact characterization." + }, + "TAS": { + "score": 1, + "justification": "This is close to a theorem-recall item: the stem essentially presents the assumptions and asks for the precise conclusion. It is not a pure tautology because the options vary in quantifiers, hypotheses, and equivalence strength." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish the exact iff statement from weaker, stronger, or hypothesis-altered variants, but the task is mainly precise theorem recognition rather than substantial derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and target real failure modes: confusing global phase with time-frequency symmetry, accepting only one implication, dropping the four-sign sampling requirement, or removing the tau bounds." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-identification MCQ with strong distractors and no answer leakage, though it leans more toward exact statement recall than deeper generative reasoning." + } + }, + { + "id": "2511.09264v1", + "paper_link": "http://arxiv.org/abs/2511.09264v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "thm", + "content": "\\label{thm:intro-keller}\n Let $X$ be a quasi-compact separated scheme over a base field $\\mathbb{K}$. Then there exist canonical isomorphisms \n \\begin{align*}\n & HH_\\bullet\\big(\\Perf(X)\\big) \\to HH_\\bullet(X),\\\\\n & HC_\\bullet\\big(\\Perf(X)\\big) \\to HC_\\bullet(X).\n \\end{align*}", + "start_pos": 6552, + "end_pos": 6867, + "label": "thm:intro-keller" + }, + "ref_dict": { + "thm:intro-keller": "\\begin{thm}\\label{thm:intro-keller}\n Let $X$ be a quasi-compact separated scheme over a base field $\\bbK$. Then there exist canonical isomorphisms \n \\begin{align*}\n & HH_\\bullet\\big(\\Perf(X)\\big) \\to HH_\\bullet(X),\\\\\n & HC_\\bullet\\big(\\Perf(X)\\big) \\to HC_\\bullet(X).\n \\end{align*} \n\\end{thm}" + }, + "pre_theorem_intro_text_len": 1036, + "pre_theorem_intro_text": "Let $X$ be a quasi-compact separated scheme over a base field $\\mathbb{K}$. An important set of invariants associated with $X$ is its Hochschild homology $HH_\\bullet(X)$ and cyclic homology $HC_\\bullet(X)$. They are defined as the hyper-cohomology groups of the sheafification of the Hochschild complexes and cyclic chain complexes respectively.\n\nOn the other hand, let $\\Perf(X)$ denote the dg category of perfect complexes over $X$. We emphasize that this is a dg-enhancement of the full subcategory of the bounded derived category $D^b(X)$ consisting of perfect complexes over $X$. For any small $\\mathbb{K}$-linear dg category, we may also define purely algebraically its own Hochschild type invariants. In the case of $\\Perf(X)$, we are naturally lead to the question if $HH_\\bullet\\big(\\Perf(X)\\big)$ (respectively $HC_\\bullet\\big(\\Perf(X)\\big)$) would be naturally isomorphic to the geometrically defined invariants $HH_\\bullet(X)$ (respectively $HC_\\bullet(X)$). In~\\cite{keller1998cyclic}, Keller obtained the following result.", + "context": "Let $X$ be a quasi-compact separated scheme over a base field $\\mathbb{K}$. An important set of invariants associated with $X$ is its Hochschild homology $HH_\\bullet(X)$ and cyclic homology $HC_\\bullet(X)$. They are defined as the hyper-cohomology groups of the sheafification of the Hochschild complexes and cyclic chain complexes respectively.\n\nOn the other hand, let $\\Perf(X)$ denote the dg category of perfect complexes over $X$. We emphasize that this is a dg-enhancement of the full subcategory of the bounded derived category $D^b(X)$ consisting of perfect complexes over $X$. For any small $\\mathbb{K}$-linear dg category, we may also define purely algebraically its own Hochschild type invariants. In the case of $\\Perf(X)$, we are naturally lead to the question if $HH_\\bullet\\big(\\Perf(X)\\big)$ (respectively $HC_\\bullet\\big(\\Perf(X)\\big)$) would be naturally isomorphic to the geometrically defined invariants $HH_\\bullet(X)$ (respectively $HC_\\bullet(X)$). In~\\cite{keller1998cyclic}, Keller obtained the following result.", + "full_context": "Let $X$ be a quasi-compact separated scheme over a base field $\\mathbb{K}$. An important set of invariants associated with $X$ is its Hochschild homology $HH_\\bullet(X)$ and cyclic homology $HC_\\bullet(X)$. They are defined as the hyper-cohomology groups of the sheafification of the Hochschild complexes and cyclic chain complexes respectively.\n\nOn the other hand, let $\\Perf(X)$ denote the dg category of perfect complexes over $X$. We emphasize that this is a dg-enhancement of the full subcategory of the bounded derived category $D^b(X)$ consisting of perfect complexes over $X$. For any small $\\mathbb{K}$-linear dg category, we may also define purely algebraically its own Hochschild type invariants. In the case of $\\Perf(X)$, we are naturally lead to the question if $HH_\\bullet\\big(\\Perf(X)\\big)$ (respectively $HC_\\bullet\\big(\\Perf(X)\\big)$) would be naturally isomorphic to the geometrically defined invariants $HH_\\bullet(X)$ (respectively $HC_\\bullet(X)$). In~\\cite{keller1998cyclic}, Keller obtained the following result.\n\n\\begin{abstract}\nLet $X$ be a quasi-compact separated scheme over a base field. Keller proved a theorem stating that the cyclic homology of $X$ is canonically isomorphic to the cyclic homology of the dg category $\\Perf(X)$ consisting of perfect complexes over $X$. This theorem shows the categorical nature of the cyclic homology. In this note, we generalize Keller's theorem to allow $X$ be defined over a base commutative ring.\n\\end{abstract}\n\nOn the other hand, let $\\Perf(X)$ denote the dg category of perfect complexes over $X$. We emphasize that this is a dg-enhancement of the full subcategory of the bounded derived category $D^b(X)$ consisting of perfect complexes over $X$. For any small $\\bbK$-linear dg category, we may also define purely algebraically its own Hochschild type invariants. In the case of $\\Perf(X)$, we are naturally lead to the question if $HH_\\bullet\\big(\\Perf(X)\\big)$ (respectively $HC_\\bullet\\big(\\Perf(X)\\big)$) would be naturally isomorphic to the geometrically defined invariants $HH_\\bullet(X)$ (respectively $HC_\\bullet(X)$). In~\\cite{keller1998cyclic}, Keller obtained the following result.\n\nIn the study of logrithmic Hochschild homology~\\cite{Olsson,HABLICSEK2026127}, one is often lead to consider Hochschild homology relative to a base space~\\cite{OLSSON2003747}. This serves as at least one of the motivations to obtain a generalization of Keller's theorem above in the relative setting. More precisely, we prove the following\n\n\\begin{thm}\n Let $\\mathfrak{X}$ be a quasi-compact separated scheme over a base commutative ring $R$. Then there are canonical isomorphisms\n \\begin{align*}\n & HH_\\bullet\\big(\\Perf(\\mathfrak{X})/R\\big) \\to HH_\\bullet(\\mathfrak{X}/R),\\\\\n & HC_\\bullet\\big(\\Perf(\\mathfrak{X})/R\\big) \\to HC_\\bullet(\\mathfrak{X}/R).\n \\end{align*}\n\\end{thm}\n\n\\subsection{Conventions and Notations.} We work over a base field $\\mathbb{K}$. For a dg category $\\CC$ over $\\mathbb{K}$, denote by $\\underline{\\CC}$ its homotopy category. This is a category with the same underlying objects as $\\CC$ and with \n\\[ {\\sf Hom}_{\\underline{\\CC}}(X,Y):= H^0\\big(\\hom^\\bullet_\\CC(X,Y)\\big).\\]\nFor a scheme $X$, in Keller's original treatment~\\cite{keller1998cyclic}, he uses the notion of localization pairs, i.e. a pair of dg categories $\\big(\\per(X),{\\sf acyc}(X)\\big)$ formed by the category of perfect complexes over $X$ and its full subcategory of acyclic perfect complexes. By a result of Drinfeld~\\cite{drinfeld2004dg} on the construction of dg quotients of dg categories, we may first take the dg quotient $\\per(X)/{\\sf acyc}(X)$ as our definition of $\\Perf(X)$. Due to this reason, in this paper we shall work exclusively with dg categories instead of localization pairs. In other words, in our setting, we have\n\\[ \\underline{\\Perf(X)} \\cong \\underline{\\per(X)}/\\underline{{\\sf acyc}(X)}.\\]\nThus no further localization is needed when working with $\\Perf(X)$.\n\nLet $X$ be a quasi-compact separated scheme over a base field $\\bbK$. Keller considers a pair of dg categories $\\big(\\per(X),{\\sf acyc}(X)\\big)$ formed by the category of perfect complexes over $X$ and its full subcategory of acyclic perfect complexes. Then he defines the associated mixed complex as\n\\[ M\\big(\\per(X),{\\sf acyc}(X)\\big):={\\sf cone}\\big( M({\\sf acyc}(X))\\to M(\\per(X))\\big),\\]\nusing the canonical inclusion functor. Throughout the paper, we shall denote the pair above by $\\Perf(X):= \\big(\\per(X),{\\sf acyc}(X)\\big)$. By a result of Drinfeld~\\cite{drinfeld2004dg} on the construction of dg quotients of dg categories, this choice of notation should cause no confusion. In other words, we may first take the dg quotient $\\per(X)/{\\sf acyc}(X)$ and then just take its associated mixed complex.\n\n\\begin{thm}\\label{thm:main}\n Let $X$ be a quasi-compact separated scheme over a base field $\\bbK$. Then trace map $\\tau: M\\big(\\Perf(X)\\big)\\to R\\Gamma(X,M\\big(\\cO)^\\sharp)$ defined in Diagram~\\eqref{diagram:tau} is invertible in $\\DD Mix$.\n\\end{thm}\n\n\\begin{thm}\\label{thm:main-relative}\n Let $\\mathfrak{X}$ be a quasi-compact separated scheme over a base commutative ring $R$. Then trace map $\\tau: M\\big(\\Perf(\\mathfrak{X})\\big)\\to R\\Gamma(\\mathfrak{X},M\\big(\\cO)^\\sharp)$ defined in Diagram~\\eqref{diagram:tau} is invertible in $\\DD Mix(R)$. Note that the mixed complex functor $M$ should be understood in the sense of Equation~\\eqref{eq:mixed-ring}.\n\\end{thm}", + "post_theorem_intro_text_len": 2477, + "post_theorem_intro_text": "In the study of logrithmic Hochschild homology~\\cite{Olsson,HABLICSEK2026127}, one is often lead to consider Hochschild homology relative to a base space~\\cite{OLSSON2003747}. This serves as at least one of the motivations to obtain a generalization of Keller's theorem above in the relative setting. More precisely, we prove the following\n\n\\begin{thm}\n Let $\\mathfrak{X}$ be a quasi-compact separated scheme over a base commutative ring $R$. Then there are canonical isomorphisms\n \\begin{align*}\n & HH_\\bullet\\big(\\Perf(\\mathfrak{X})/R\\big) \\to HH_\\bullet(\\mathfrak{X}/R),\\\\\n & HC_\\bullet\\big(\\Perf(\\mathfrak{X})/R\\big) \\to HC_\\bullet(\\mathfrak{X}/R).\n \\end{align*}\n\\end{thm}\n\nNote that in defining the Hochschild invariants above, when the structure sheaf $\\cO_\\mathfrak{X}$ is not flat over $R$, we need to replace it by a flat resolution. This homology theory is sometimes called Shukla-Hochschild homology~\\cite{Shuk}. Similarly, such a replacement is also needed for the dg category $\\Perf(\\mathfrak{X})$. The proof of the relative case follows the same line as that of Theorem~\\ref{thm:intro-keller}. Indeed, in Section~\\ref{sec:review} we briefly sketch Keller's original proof. In Section~\\ref{sec:relative} we deduce our main result by taking care of flatness using semi-free resolutions of dg categories. \n\n\\subsection{Conventions and Notations.} We work over a base field $\\mathbb{K}$. For a dg category $\\mathcal{C}$ over $\\mathbb{K}$, denote by $\\underline{\\mathcal{C}}$ its homotopy category. This is a category with the same underlying objects as $\\mathcal{C}$ and with \n\\[ {\\sf Hom}_{\\underline{\\mathcal{C}}}(X,Y):= H^0\\big(\\hom^\\bullet_\\mathcal{C}(X,Y)\\big).\\]\nFor a scheme $X$, in Keller's original treatment~\\cite{keller1998cyclic}, he uses the notion of localization pairs, i.e. a pair of dg categories $\\big(\\per(X),{\\sf acyc}(X)\\big)$ formed by the category of perfect complexes over $X$ and its full subcategory of acyclic perfect complexes. By a result of Drinfeld~\\cite{drinfeld2004dg} on the construction of dg quotients of dg categories, we may first take the dg quotient $\\per(X)/{\\sf acyc}(X)$ as our definition of $\\Perf(X)$. Due to this reason, in this paper we shall work exclusively with dg categories instead of localization pairs. In other words, in our setting, we have\n\\[ \\underline{\\Perf(X)} \\cong \\underline{\\per(X)}/\\underline{{\\sf acyc}(X)}.\\]\nThus no further localization is needed when working with $\\Perf(X)$.", + "sketch": "The post-theorem introduction does not sketch a proof of Theorem~\\ref{thm:intro-keller} itself, but indicates the strategy for the relative generalization: it says that “The proof of the relative case follows the same line as that of Theorem~\\ref{thm:intro-keller}.” It highlights the extra issue that “when the structure sheaf $\\cO_\\mathfrak{X}$ is not flat over $R$, we need to replace it by a flat resolution” (Shukla-Hochschild homology), and that “such a replacement is also needed for the dg category $\\Perf(\\mathfrak{X})$.” The plan of the argument is then: “in Section~\\ref{sec:review} we briefly sketch Keller's original proof,” and “in Section~\\ref{sec:relative} we deduce our main result by taking care of flatness using semi-free resolutions of dg categories.”", + "expanded_sketch": "The post-theorem introduction does not sketch a proof of the main theorem itself, but indicates the strategy for the relative generalization: it says that “The proof of the relative case follows the same line as that of the main theorem.” It highlights the extra issue that “when the structure sheaf $\\cO_\\mathfrak{X}$ is not flat over $R$, we need to replace it by a flat resolution” (Shukla-Hochschild homology), and that “such a replacement is also needed for the dg category $\\Perf(\\mathfrak{X})$.” The plan of the argument is then: “next we briefly sketch Keller's original proof,” and “later we deduce our main result by taking care of flatness using semi-free resolutions of dg categories.”", + "expanded_theorem": "\\label{thm:intro-keller}\n Let $X$ be a quasi-compact separated scheme over a base field $\\mathbb{K}$. Then there exist canonical isomorphisms \n \\begin{align*}\n & HH_\\bullet\\big(\\Perf(X)\\big) \\to HH_\\bullet(X),\\\\\n & HC_\\bullet\\big(\\Perf(X)\\big) \\to HC_\\bullet(X).\n \\end{align*}", + "theorem_type": [ + "Existence", + "Classification or Bijection" + ], + "mcq": { + "question": "Let $X$ be a quasi-compact separated scheme over a base field $\\mathbb{K}$, and let $\\Perf(X)$ denote the dg category of perfect complexes on $X$. Which invariants are canonically isomorphic to the Hochschild homology $HH_\\bullet(\\Perf(X))$ and the cyclic homology $HC_\\bullet(\\Perf(X))$?", + "correct_choice": { + "label": "A", + "text": "They are canonically isomorphic to the geometrically defined Hochschild and cyclic homology of $X$, respectively; that is, there are canonical isomorphisms $$HH_\\bullet\\big(\\Perf(X)\\big) \\to HH_\\bullet(X), \\qquad HC_\\bullet\\big(\\Perf(X)\\big) \\to HC_\\bullet(X).$$" + }, + "choices": [ + { + "label": "B", + "text": "They are canonically isomorphic to the geometrically defined Hochschild and cyclic homology of $X$ relative to the base field; that is, there are canonical isomorphisms $$HH_\\bullet\\big(\\Perf(X)/\\mathbb{K}\\big) \\to HH_\\bullet(X/\\mathbb{K}), \\qquad HC_\\bullet\\big(\\Perf(X)/\\mathbb{K}\\big) \\to HC_\\bullet(X/\\mathbb{K}).$$" + }, + { + "label": "C", + "text": "At least the Hochschild homology agrees with the geometrically defined Hochschild homology of $X$; namely, there is a canonical isomorphism $$HH_\\bullet\\big(\\Perf(X)\\big) \\to HH_\\bullet(X).$$" + }, + { + "label": "D", + "text": "They are canonically isomorphic after passing to a flat or semi-free resolution of the structure sheaf and of the dg category; in particular, one has canonical isomorphisms only up to such resolutions, rather than directly for $\\Perf(X)$ and $X$ themselves." + }, + { + "label": "E", + "text": "They are canonically isomorphic to the geometrically defined Hochschild and cyclic homology of $X$ for affine quasi-compact separated schemes, and more generally after imposing a flatness hypothesis on $\\mathcal{O}_X$ over $\\mathbb{K}$; namely, under those assumptions one has $$HH_\\bullet\\big(\\Perf(X)\\big) \\to HH_\\bullet(X), \\qquad HC_\\bullet\\big(\\Perf(X)\\big) \\to HC_\\bullet(X).$$" + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "characteristic", + "tampered_component": "absolute_vs_relative_invariants", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "cyclic_homology_conclusion", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "resolution_needed_only_in_relative_nonflat_case", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "no_extra_flatness_or_affineness_hypothesis", + "template_used": "boundary_range" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly state the theorem’s conclusion or name the geometrically defined invariants as the answer. It only asks which invariants match the dg-category homologies, so there is no direct answer leakage." + }, + "TAS": { + "score": 1, + "justification": "This is essentially a theorem-recall question with mild reformulation. The correct option restates the standard comparison result, though the alternatives force attention to scope and hypotheses." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is required to reject subtle variants: relative vs. absolute homology, whether both HH and HC are included, and whether extra flatness/affineness assumptions are needed. Still, it mainly tests precise recall rather than deeper derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically meaningful: confusing absolute with relative invariants, selecting a weaker true statement, adding unnecessary resolution caveats, or imposing extra hypotheses. These reflect common failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it is still fairly close to direct theorem recall rather than a genuinely generative reasoning task." + } + }, + { + "id": "2511.09176v1", + "paper_link": "http://arxiv.org/abs/2511.09176v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "lemma", + "content": "For two points $P,Q\\in k^n$ we have that $$\\dim_k\\ext^1_A(M_P,M_Q)=\\begin{cases}n,\\ P=Q\\\\0,\\ P\\neq Q.\\end{cases}$$", + "start_pos": 9024, + "end_pos": 9164, + "label": null + }, + "ref_dict": {}, + "pre_theorem_intro_text_len": 2190, + "pre_theorem_intro_text": "Real algebraic geometry can be thought of as a generalization of manifolds, where the continuous functions are replaced by polynomials with real coefficients. Application to physics also leads to a necessary generalization to associative algebraic geometry and a generalization of continuous Riemannian metrics, see the book of O.A. Laudal, \\cite{Laudal21}. One of the main problems with this, is that Riemannian metrics are defined over the reals, and the polynomial algebra over the reals, $\\mathbb R[x_1,\\dots,x_n],$ contains more simple modules than $\\mathbb R^n.$ Because the algebraic properties governing the simple modules are better controlled by an algebra over an algebraically closed field, the main result in this text is the construction of a $\\mathbb C$-algebra $A_{\\mathbb R}$ such that $\\simp(A_{\\mathbb R})\\cong\\mathbb R^n.$ Thus the points in $\\mathbb R^n$ is in one-to-one correspondence with the simple $A_{\\mathbb R}$-modules for which $\\aspec(A_{\\mathbb R})$ is a fine moduli, see the book \\cite{S23} or the preprint \\cite{S241}. For any real manifold $M$ we define an associative variety $(\\mathcal M,\\mathcal O^A_L)$ over $\\mathbb C$ such that the points in $M$ is in bijective correspondence with the closed points in $\\mathcal M,$ and such that the charts $U$ in $\\mathcal M$ corresponds to $\\mathcal O^A_L(U)=A_{\\mathbb R}.$\n\nIn short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$\n\nBecause of the discovery of localization in associative rings, \\cite{S252}, we include the general definition of associative schemes. Their purpose is to serve as moduli of algebraic objects which can be put in one-to-one correspondence with modules over associative schemes: That is, we need associative moduli to classify associative objects. This is most easily seen by the following.\nLet $k$ be a field and consider the polynomial algebra in $n\\geq 1$ variables $$A=k[x_1,\\dots,x_n]=k[\\underline x].$$ For a point $P=(p_1,\\dots,p_n)\\in k^n,$ we define the simple $A$-modules $$M(P)=A/(x_1-p_1,\\dots,x_n-p_n).$$", + "context": "Real algebraic geometry can be thought of as a generalization of manifolds, where the continuous functions are replaced by polynomials with real coefficients. Application to physics also leads to a necessary generalization to associative algebraic geometry and a generalization of continuous Riemannian metrics, see the book of O.A. Laudal, \\cite{Laudal21}. One of the main problems with this, is that Riemannian metrics are defined over the reals, and the polynomial algebra over the reals, $\\mathbb R[x_1,\\dots,x_n],$ contains more simple modules than $\\mathbb R^n.$ Because the algebraic properties governing the simple modules are better controlled by an algebra over an algebraically closed field, the main result in this text is the construction of a $\\mathbb C$-algebra $A_{\\mathbb R}$ such that $\\simp(A_{\\mathbb R})\\cong\\mathbb R^n.$ Thus the points in $\\mathbb R^n$ is in one-to-one correspondence with the simple $A_{\\mathbb R}$-modules for which $\\aspec(A_{\\mathbb R})$ is a fine moduli, see the book \\cite{S23} or the preprint \\cite{S241}. For any real manifold $M$ we define an associative variety $(\\mathcal M,\\mathcal O^A_L)$ over $\\mathbb C$ such that the points in $M$ is in bijective correspondence with the closed points in $\\mathcal M,$ and such that the charts $U$ in $\\mathcal M$ corresponds to $\\mathcal O^A_L(U)=A_{\\mathbb R}.$\n\nIn short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$\n\nBecause of the discovery of localization in associative rings, \\cite{S252}, we include the general definition of associative schemes. Their purpose is to serve as moduli of algebraic objects which can be put in one-to-one correspondence with modules over associative schemes: That is, we need associative moduli to classify associative objects. This is most easily seen by the following.\nLet $k$ be a field and consider the polynomial algebra in $n\\geq 1$ variables $$A=k[x_1,\\dots,x_n]=k[\\underline x].$$ For a point $P=(p_1,\\dots,p_n)\\in k^n,$ we define the simple $A$-modules $$M(P)=A/(x_1-p_1,\\dots,x_n-p_n).$$", + "full_context": "Real algebraic geometry can be thought of as a generalization of manifolds, where the continuous functions are replaced by polynomials with real coefficients. Application to physics also leads to a necessary generalization to associative algebraic geometry and a generalization of continuous Riemannian metrics, see the book of O.A. Laudal, \\cite{Laudal21}. One of the main problems with this, is that Riemannian metrics are defined over the reals, and the polynomial algebra over the reals, $\\mathbb R[x_1,\\dots,x_n],$ contains more simple modules than $\\mathbb R^n.$ Because the algebraic properties governing the simple modules are better controlled by an algebra over an algebraically closed field, the main result in this text is the construction of a $\\mathbb C$-algebra $A_{\\mathbb R}$ such that $\\simp(A_{\\mathbb R})\\cong\\mathbb R^n.$ Thus the points in $\\mathbb R^n$ is in one-to-one correspondence with the simple $A_{\\mathbb R}$-modules for which $\\aspec(A_{\\mathbb R})$ is a fine moduli, see the book \\cite{S23} or the preprint \\cite{S241}. For any real manifold $M$ we define an associative variety $(\\mathcal M,\\mathcal O^A_L)$ over $\\mathbb C$ such that the points in $M$ is in bijective correspondence with the closed points in $\\mathcal M,$ and such that the charts $U$ in $\\mathcal M$ corresponds to $\\mathcal O^A_L(U)=A_{\\mathbb R}.$\n\nIn short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$\n\nBecause of the discovery of localization in associative rings, \\cite{S252}, we include the general definition of associative schemes. Their purpose is to serve as moduli of algebraic objects which can be put in one-to-one correspondence with modules over associative schemes: That is, we need associative moduli to classify associative objects. This is most easily seen by the following.\nLet $k$ be a field and consider the polynomial algebra in $n\\geq 1$ variables $$A=k[x_1,\\dots,x_n]=k[\\underline x].$$ For a point $P=(p_1,\\dots,p_n)\\in k^n,$ we define the simple $A$-modules $$M(P)=A/(x_1-p_1,\\dots,x_n-p_n).$$\n\n\\begin{abstract}\nIn the preprint \\cite{S252} we proved that there exists a localizing ring $A_M$ for $A$ an associative ring with unit, and $M=\\oplus_{i=1}^rM_i$ a direct sum of $r\\geq 1$ simple right $A$-modules. For a homomorphism of associative rings $A\\rightarrow B$ we define the contraction of a simple $B$-module to $A.$\nThen we define the set of aprime right $A$-modules $\\aspec A$ to be the set of simple $A$-modules together with contractions of such. When $A$ is commutative, $\\aspec A=\\spec A,$ and we define a topology on $\\aspec A$ such that when $A$ is commutative, this is the Zariski topology. In the preprint \\cite{S251}, we proved that when we have a topology and a localizing subcategory, there exists a sheaf of associative rings $\\mathcal O_X$ on $\\aspec A,$ agreeing with the usual sheaf of rings on $\\spec A.$ In this text, we write out this construction, and we see that we can restrict the sheaf and topology to any subset $V\\subseteq\\aspec A.$ In particular, this \nproves that we can use complex varieties in real algebraic geometry, by restricting in accordance with $\\mathbb R\\subseteq\\mathbb C.$ Thus the theory of schemes over algebraically closed fields and its associative generalization can be applied to real (algebraic) geometry.\n\\end{abstract}\n\nIn short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$\n\n\\begin{proof} From \\cite{S23} we know that $$\\ext^1_A(M_1,M_2)\\simeq\\der_k(A,\\hmm_k(M_1,M_2))/\\inner$$ where $\\hmm_k(M_1,M_2)$ is an $A-A$ bimodule by $a\\phi(m)=\\phi(am), \\phi a(m)=\\phi(m)a.$\n\nWhen $P\\neq Q$ we can consider $M_{\\underline 0}, M_P$ with $p_1\\neq 0.$ \nFor $$\\delta\\in\\der_k(A,\\hmm_k(M_{\\underline 0},M_P)$$ we have $$\\begin{aligned}\\delta(x_ix_1)&=\\delta(x_1x_i)\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)x_1+x_i\\delta(x_1)&=\\delta(x_1)x_i+x_1\\delta(x_i)\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)p_1&=\\delta(x_1)p_i\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)&=\\delta(x_i)\\frac{p_i}{p_1}.\n\\end{aligned}$$\nThis is also true for the inner derivations, proving that $\\dim_k\\inner=1$ so that $\\ext^1_A(M_P,M_Q)=0$ when $P\\neq Q.$\n\\end{proof}\n\n\\begin{proof} We have a homomorphism $f:A\\rightarrow A/\\mathfrak m$ such for every $s\\in A\\setminus\\mathfrak m,$ $f(s)$ is a unit. By the universal property of localization, there exists a unique homomorphism $\\phi:A_\\mathfrak m\\rightarrow A/\\mathfrak m$ such that $\\phi(\\frac{a}{s})=\\iota(a)\\iota(s)^{-1}.$ This homomorphism is clearly surjective, and its kernel is $\\mathfrak m A_\\mathfrak m$ giving the wanted isomorphism.\n\\end{proof}\n\n\\begin{proof} We send the prime ideal $\\mathfrak p\\subset A$ to $A_\\mathfrak p/\\mathfrak p A_\\mathfrak p$ which is $A$-prime by definition. On the other hand, let\n$M$ be an aprime $A$-module, defined by $\\iota_M: A\\rightarrow B$ such that $M$ is a simple $B$-module. Let $\\mathfrak m\\subset B$ be the maximal ideal defining $M$ as a simple $B$-module. Then $\\iota_M^{-1}(\\mathfrak m)$ is a prime ideal in $A.$ That these two operations are inverses to each other follows from the fact that for a maximal ideal in a ring $B$ we have $B/\\mathfrak m\\simeq B_{\\mathfrak m}/\\mathfrak m B_\\mathfrak m$ as proven i Lemma \\ref{anotherloclemma}.\n\\end{proof}\n\n\\begin{definition} Define the subset of $k$-points in $\\mathbb X$ by $$\\tilde{\\mathbb X}(k)=\\{x\\in X\\subseteq\\mathbb X|x\\text{ is simple}\\}\\subseteq\\mathbb X.$$ Then the induced associative subscheme $\\mathbb X(k)$ is called the associative subscheme of $k$-points. \n\\end{definition}\n\n\\begin{proof} Because $k\\subseteq\\Bbbk$ is a sub-algebra, it follows that if $\\phi\\otimes\\id$ is an isomorphism, then $\\dim_k V_1=\\dim_k V_2,$ and that choosing corresponding bases, $$0\\neq\\det(\\phi\\otimes\\id)=\\det\\phi.$$ \n\\end{proof}", + "post_theorem_intro_text_len": 1254, + "post_theorem_intro_text": "\\begin{proof} From \\cite{S23} we know that $$\\ext^1_A(M_1,M_2)\\simeq\\der_k(A,\\hmm_k(M_1,M_2))/\\inner$$ where $\\hmm_k(M_1,M_2)$ is an $A-A$ bimodule by $a\\phi(m)=\\phi(am), \\phi a(m)=\\phi(m)a.$\n\nWhen $P=Q$ we can assume $M_1=M_2=k[\\underline x]/(\\underline x).$ Then any inner derivation is on the form $\\ad_\\phi$ for which $\\ad_\\phi(x_i)=[\\phi,x_i]=0.$ Thus the inner derivations is of dimension zero and $\\ext^1_A(M_1,M_2)\\simeq\\der_k(A,k)$ where \n$\\operatorname d_i(x_i)=1,\\ i=1,\\dots,n$ gives a basis for the derivations.\n\nWhen $P\\neq Q$ we can consider $M_{\\underline 0}, M_P$ with $p_1\\neq 0.$ \nFor $$\\delta\\in\\der_k(A,\\hmm_k(M_{\\underline 0},M_P)$$ we have $$\\begin{aligned}\\delta(x_ix_1)&=\\delta(x_1x_i)\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)x_1+x_i\\delta(x_1)&=\\delta(x_1)x_i+x_1\\delta(x_i)\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)p_1&=\\delta(x_1)p_i\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)&=\\delta(x_i)\\frac{p_i}{p_1}.\n\\end{aligned}$$\nThis is also true for the inner derivations, proving that $\\dim_k\\inner=1$ so that $\\ext^1_A(M_P,M_Q)=0$ when $P\\neq Q.$\n\\end{proof}\n\nIt follows from the lemma that not all finite dimensional simple modules over a noncommutative $k$-algebra can be classified by a finitely generated commutative algebra. See \\cite{S23} for a lot of examples.", + "sketch": "From \\cite{S23}: \\(\\ext^1_A(M_1,M_2)\\simeq \\der_k(A,\\hmm_k(M_1,M_2))/\\inner\\), with \\(\\hmm_k(M_1,M_2)\\) an \\(A\\)-\\(A\\) bimodule via \\(a\\phi(m)=\\phi(am)\\), \\(\\phi a(m)=\\phi(m)a\\).\n\n- Case \\(P=Q\\): assume \\(M_1=M_2=k[\\underline x]/(\\underline x)\\). Any inner derivation is \\(\\ad_\\phi\\) and \\(\\ad_\\phi(x_i)=[\\phi,x_i]=0\\), so \\(\\dim_k\\inner=0\\). Hence \\(\\ext^1_A(M_1,M_2)\\simeq \\der_k(A,k)\\), and the derivations \\(\\operatorname d_i\\) with \\(\\operatorname d_i(x_i)=1\\) for \\(i=1,\\dots,n\\) give a basis, so the dimension is \\(n\\).\n\n- Case \\(P\\neq Q\\): consider \\(M_{\\underline 0}, M_P\\) with \\(p_1\\neq 0\\). For \\(\\delta\\in\\der_k(A,\\hmm_k(M_{\\underline 0},M_P))\\), comparing \\(\\delta(x_ix_1)=\\delta(x_1x_i)\\) yields\n\\[\\delta(x_i)x_1+x_i\\delta(x_1)=\\delta(x_1)x_i+x_1\\delta(x_i)\\Rightarrow \\delta(x_i)p_1=\\delta(x_1)p_i\\Rightarrow \\delta(x_i)=\\delta(x_1)\\frac{p_i}{p_1}.\\]\nThis relation also holds for inner derivations, and it is concluded that \\(\\dim_k\\inner=1\\), so \\(\\ext^1_A(M_P,M_Q)=0\\) when \\(P\\neq Q\\).", + "expanded_sketch": "From \\cite{S23}: \\(\\ext^1_A(M_1,M_2)\\simeq \\der_k(A,\\hmm_k(M_1,M_2))/\\inner\\), with \\(\\hmm_k(M_1,M_2)\\) an \\(A\\)-\\(A\\) bimodule via \\(a\\phi(m)=\\phi(am)\\), \\(\\phi a(m)=\\phi(m)a\\).\n\n- Case \\(P=Q\\): assume \\(M_1=M_2=k[\\underline x]/(\\underline x)\\). Any inner derivation is \\(\\ad_\\phi\\) and \\(\\ad_\\phi(x_i)=[\\phi,x_i]=0\\), so \\(\\dim_k\\inner=0\\). Hence \\(\\ext^1_A(M_1,M_2)\\simeq \\der_k(A,k)\\), and the derivations \\(\\operatorname d_i\\) with \\(\\operatorname d_i(x_i)=1\\) for \\(i=1,\\dots,n\\) give a basis, so the dimension is \\(n\\).\n\n- Case \\(P\\neq Q\\): consider \\(M_{\\underline 0}, M_P\\) with \\(p_1\\neq 0\\). For \\(\\delta\\in\\der_k(A,\\hmm_k(M_{\\underline 0},M_P))\\), comparing \\(\\delta(x_ix_1)=\\delta(x_1x_i)\\) yields\n\\[\\delta(x_i)x_1+x_i\\delta(x_1)=\\delta(x_1)x_i+x_1\\delta(x_i)\\Rightarrow \\delta(x_i)p_1=\\delta(x_1)p_i\\Rightarrow \\delta(x_i)=\\delta(x_1)\\frac{p_i}{p_1}.\\]\nThis relation also holds for inner derivations, and it is concluded that \\(\\dim_k\\inner=1\\), so \\(\\ext^1_A(M_P,M_Q)=0\\) when \\(P\\neq Q\\).", + "expanded_theorem": "For two points $P,Q\\in k^n$ we have that $$\\dim_k\\ext^1_A(M_P,M_Q)=\\begin{cases}n,\\ P=Q\\\\0,\\ P\\neq Q.\\end{cases}$$,", + "theorem_type": [ + "Universal", + "Classification or Bijection" + ], + "mcq": { + "question": "Let $k$ be a field, let $A=k[x_1,\\dots,x_n]$ with $n\\ge 1$, and for each point $P=(p_1,\\dots,p_n)\\in k^n$ define the simple $A$-module\n$$M_P:=A/(x_1-p_1,\\dots,x_n-p_n).$$\nWhich statement correctly describes $\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)$ for two points $P,Q\\in k^n$?", + "correct_choice": { + "label": "A", + "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)=\\begin{cases}n,& P=Q,\\\\0,& P\\ne Q.\\end{cases}$$" + }, + "choices": [ + { + "label": "B", + "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)=\\begin{cases}n,& P=Q,\\\\1,& P\\ne Q.\\end{cases}$$" + }, + { + "label": "C", + "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)=0\\quad\\text{for all }P\\ne Q.$$" + }, + { + "label": "D", + "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)=\\begin{cases}n-1,& P=Q,\\\\0,& P\\ne Q.\\end{cases}$$" + }, + { + "label": "E", + "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)=\\begin{cases}n,& P=Q,\\\\0,& P-Q\\in (k^\times)^n,\\end{cases}$$" + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "quotient_by_inner_in_off-diagonal_case", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "case_split", + "tampered_component": "diagonal_case_value_n", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "number_of_basis_derivations_in_diagonal_case", + "template_used": "boundary_range" + }, + { + "label": "E", + "sketch_hook_type": "case_split", + "tampered_component": "reduction_to_coordinate_with_nonzero_difference", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the modules and asks for the Ext^1-dimension formula, but it does not reveal or strongly hint at the diagonal/off-diagonal values." + }, + "TAS": { + "score": 2, + "justification": "This is not a mere restatement in the stem; the respondent must choose among several competing formulas for both the P=Q and P≠Q cases." + }, + "GPS": { + "score": 2, + "justification": "To identify the correct choice, one must reason about Ext^1 between simples over a polynomial ring, including the vanishing off the diagonal and the n-dimensional tangent-space contribution on the diagonal." + }, + "DQS": { + "score": 2, + "justification": "The distractors are meaningfully different and plausibly reflect common errors: nonzero off-diagonal Ext, forgetting the diagonal case, miscounting the diagonal dimension as n-1, or imposing an irrelevant coordinatewise condition." + }, + "total_score": 8, + "overall_assessment": "A strong MCQ: no answer leakage, non-tautological, and it tests genuine algebraic reasoning with mostly high-quality distractors." + } + }, + { + "id": "2511.22164v2", + "paper_link": "http://arxiv.org/abs/2511.22164v2", + "theorems_cnt": 3, + "theorem": { + "env_name": "introtheorem", + "content": "\\label{thmintro:delocalised-trace-identification-simplified}\n Let $\\cH$ be an Iwahori Hecke algebra of right-angled type $(W,S)$. Let $\\cO \\subseteq W$ be a conjugacy class. Then there is a unique trace $\\vphi_\\cO$ on $\\cH$ that satisfies $\\vphi_\\cO(T_w) = \\mathbb{1}_\\cO(w)$ for every element $w \\in W$ that is of minimal length in its conjugacy class.", + "start_pos": 28108, + "end_pos": 28506, + "label": "thmintro:delocalised-trace-identification-simplified" + }, + "ref_dict": { + "thm:delocalised-trace-identification": "\\begin{theorem}\n \\label{thm:delocalised-trace-identification}\n Let $\\cH$ be a generic Hecke algebra of right-angled type $(W,S)$. For every conjugacy class $\\cO \\subseteq W$, the map $\\vphi_\\cO$ is a trace on $\\cH$. If $(T_{\\cU})_{\\cU}$ denotes the basis of the cocentre of $\\cH$ satisfying $T_{\\cU} = T_w + [\\cH, \\cH]$ for every element of minimal length $w \\in \\cU$, then $\\vphi_\\cO(T_\\cU) = \\delta_{\\cO, \\cU}$.\n\\end{theorem}", + "introthm:Schwartz-algebra-existence": "\\begin{introtheorem}\n \\label{introthm:Schwartz-algebra-existence}\n Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $\\cH$ the complex Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \\in S} \\in \\RR_{> 0}^S$. There is a subalgebra $\\cS(W, S, q) \\subseteq \\Cstarred(W, S, q)$, which\n \\begin{itemize}\n \\item is dense and closed under holomorphic functional calculus,\n \\item and such that for every conjugacy class $\\cO \\subseteq W$, the trace $\\vphi_\\cO$ continuously extends to $\\cS(W, S, q)$.\n \\end{itemize}\n\\end{introtheorem}", + "rem:finite-order-element-cliques": "\\begin{remark}\n \\label{rem:finite-order-element-cliques}\n Let $\\Gamma$ be the commutation graph of a right-angled Coxeter system $(W, S)$. Denote by $\\mathrm{Cliq}(\\Gamma)$ the set of cliques in $\\Gamma$, that is the set of complete subgraphs of $\\Gamma$ (including the empty graph). Identifying vertices of $\\Gamma$ with $S$, we consider the map $C \\mapsto \\prod_{s \\in \\rV(C)} s$. By Remark~\\ref{rem:finite-order-element-conjugacy} every finite order element is conjugate to an element in the image of this map. Further, if $\\prod_{s \\in \\rV(C)} s$ and $\\prod_{s \\in \\rV(D)} s$ are conjugate, they must be equal by Tits solution to the word problem. It follows that $C \\mapsto \\prod_{s \\in \\rV(C)} s$ defines a bijection between $\\mathrm{Cliq}(\\Gamma)$ and the set of conjugacy classes of finite order elements of $W$.\n\\end{remark}", + "thm:right-angled-cocentre-basis": "\\begin{theorem}\n \\label{thm:right-angled-cocentre-basis}\n Let $\\cH$ be a generic Hecke algebra of right-angled type $(W,S)$ over an algebraically closed field $K$ of characteristic different from $2$. Let $(a_s)_{s\\in S}$, $(b_s)_{s\\in S}$ be the deformation parameters of $\\cH$ and assume that $b_s \\neq -\\frac{a_s^2}{4}$ for each $s \\in S$. Then $(T_{\\cO})_{\\cO}$ where $\\cO$ runs through the conjugacy classes of $W$ is a basis of the cocentre of $\\cH$.\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 4937, + "pre_theorem_intro_text": "\\label{sec:introduction}\n\nThe study of delocalised traces forms an important tool in noncommutative geometry and index theory, particularly in the analysis of delocalised $\\Ltwo$-invariants and higher indices associated with manifolds, discrete groups and their operator algebras. These traces, which extend the familiar notion of the canonical trace on group C*-algebras, allow one to distinguish K-theory classes that are indistinguishable by the standard trace. A foundational instance of their use appears in the work of Lott \\cite{lott1999}, who investigated for example delocalised $\\Ltwo$-Betti numbers and delocalised $\\Ltwo$-torsion of hyperbolic manifolds. The theoretical underpinning for the pairing of delocalised traces with K-theory classes in the context of hyperbolic groups was developed by Puschnigg \\cite{puschnigg2010}, who constructed dense subalgebras of group C*-algebras which are at the same time stable under holomorphic functional calculus and host bounded traces that sum coefficients over possibly infinite conjugacy classes. This might be viewed as a natural continuation of the work of Jolissaint \\cite{jolissaint1989-k-theory}, who, inspired by the earlier results of Connes, produced natural smooth subalgebras contained in reduced group C*-algebras, remembering the K-theory of their C*-counterparts. More generally, Puschnigg was able to prove that cyclic homology of the group algebra injects into local cyclic homology of the associated reduced group C*-algebra.\n\nWhile this story is well-developed for group C*-algebras of hyperbolic groups, a parallel and increasingly important direction involves Hecke C*-algebras, which arise as deformations of group algebras associated with Coxeter systems. Historically, spherical and affine Hecke algebras appeared naturally in number theory and representation theory and as such have become a central part of a deep and well-developed theory. They are closely connected to the representation theory of finite groups of Lie type and reductive p-adic groups, respectively. Delorme-Opdam's work in \\cite{delormeopdam2008} considered Hecke C*-algebra of affine types and introduced Schwartz algebras in this context. Notably, by Bernstein's theorem, affine Hecke algebras are finitely generated modules over their centre, so that classical methods related the Fourier transform could be employed. These Hecke-Schwartz algebras of affine type were instrumental in Opdam-Solleveld's approach to determine discrete series representations \\cite{opdamsolleveld2010-discrete-series} and generalised principal series \\cite{delormeopdam2011} of affine Hecke algebras.\n\nHecke operator algebras of indefinite type drew attention only after the introduction of Hecke von Neumann algebras in the context of weighted $\\Ltwo$-cohomology of buildings by Davis-Dymara-Januszkiewicz-Okun \\cite{davisdymarajanusykiewiczokun07} around two decades ago. Specifically right-angled Hecke operator algebras since then gained prominence in the operator algebra community due to their rich structural features, which are at the same time amenable to a variety of established methods from the field. Known features of right-angled Hecke operator algebras by now include explicit, non-trivial factor decomposition \\cite{garncarek16,raumskalski2023} and absence of Cartan or even strong solidity results \\cite{capsers2020,klisse2023-boundaries,borstcasperswasilewski2024} for Hecke-von-Neumann algebras, and the rapid decay property \\cite{caspersklisselarsen2021}, simplicity results \\cite{caspersklisselarsen2021,klisse2023-simplicity} and K-theory calculations \\cite{raumskalski2022} for Hecke C*-algebras. Also compact quantum metric spaces associated with Hecke C*-algebas were very recently introduced and investigated in \\cite{klisseperovic25}. Moreover, Hecke operator algebras encode significant aspects of unitary representation theory, particularly through their role in classifying Iwahori-spherical unitary representations of groups acting strongly transitively on buildings such as certain completed Kac-Moody groups. See e.g.\\ \\cite[Corollary B]{raumskalski2023}.\n\nThe cyclic homology of affine Hecke algebras, was clarified in work Baum-Nistor \\cite{baumnistor2002} in the purely algebraic setup and by Solleveld \\cite{solleveld2009-thesis,solleveld2009-hp} in the operator algebraic setup of Hecke-C*-algebras as well as in the setup of non-commutative geometry provided by Schwartz algebras.\n\nIn this work we consider Hecke algebras of right-angled, hyperbolic type. In this setup, delocalised traces account for all of cyclic cohomology, so that a natural first step is to determine traces on such Hecke algebras. This is achieved in the next theorem, which is also proven in the greater generality of generic Hecke algebras in Theorem~\\ref{thm:delocalised-trace-identification}. For simplicity we state it in a simplified form for Iwahori-Hecke algebras.", + "context": "\\label{sec:introduction}\n\nThe study of delocalised traces forms an important tool in noncommutative geometry and index theory, particularly in the analysis of delocalised $\\Ltwo$-invariants and higher indices associated with manifolds, discrete groups and their operator algebras. These traces, which extend the familiar notion of the canonical trace on group C*-algebras, allow one to distinguish K-theory classes that are indistinguishable by the standard trace. A foundational instance of their use appears in the work of Lott \\cite{lott1999}, who investigated for example delocalised $\\Ltwo$-Betti numbers and delocalised $\\Ltwo$-torsion of hyperbolic manifolds. The theoretical underpinning for the pairing of delocalised traces with K-theory classes in the context of hyperbolic groups was developed by Puschnigg \\cite{puschnigg2010}, who constructed dense subalgebras of group C*-algebras which are at the same time stable under holomorphic functional calculus and host bounded traces that sum coefficients over possibly infinite conjugacy classes. This might be viewed as a natural continuation of the work of Jolissaint \\cite{jolissaint1989-k-theory}, who, inspired by the earlier results of Connes, produced natural smooth subalgebras contained in reduced group C*-algebras, remembering the K-theory of their C*-counterparts. More generally, Puschnigg was able to prove that cyclic homology of the group algebra injects into local cyclic homology of the associated reduced group C*-algebra.\n\nWhile this story is well-developed for group C*-algebras of hyperbolic groups, a parallel and increasingly important direction involves Hecke C*-algebras, which arise as deformations of group algebras associated with Coxeter systems. Historically, spherical and affine Hecke algebras appeared naturally in number theory and representation theory and as such have become a central part of a deep and well-developed theory. They are closely connected to the representation theory of finite groups of Lie type and reductive p-adic groups, respectively. Delorme-Opdam's work in \\cite{delormeopdam2008} considered Hecke C*-algebra of affine types and introduced Schwartz algebras in this context. Notably, by Bernstein's theorem, affine Hecke algebras are finitely generated modules over their centre, so that classical methods related the Fourier transform could be employed. These Hecke-Schwartz algebras of affine type were instrumental in Opdam-Solleveld's approach to determine discrete series representations \\cite{opdamsolleveld2010-discrete-series} and generalised principal series \\cite{delormeopdam2011} of affine Hecke algebras.\n\nHecke operator algebras of indefinite type drew attention only after the introduction of Hecke von Neumann algebras in the context of weighted $\\Ltwo$-cohomology of buildings by Davis-Dymara-Januszkiewicz-Okun \\cite{davisdymarajanusykiewiczokun07} around two decades ago. Specifically right-angled Hecke operator algebras since then gained prominence in the operator algebra community due to their rich structural features, which are at the same time amenable to a variety of established methods from the field. Known features of right-angled Hecke operator algebras by now include explicit, non-trivial factor decomposition \\cite{garncarek16,raumskalski2023} and absence of Cartan or even strong solidity results \\cite{capsers2020,klisse2023-boundaries,borstcasperswasilewski2024} for Hecke-von-Neumann algebras, and the rapid decay property \\cite{caspersklisselarsen2021}, simplicity results \\cite{caspersklisselarsen2021,klisse2023-simplicity} and K-theory calculations \\cite{raumskalski2022} for Hecke C*-algebras. Also compact quantum metric spaces associated with Hecke C*-algebas were very recently introduced and investigated in \\cite{klisseperovic25}. Moreover, Hecke operator algebras encode significant aspects of unitary representation theory, particularly through their role in classifying Iwahori-spherical unitary representations of groups acting strongly transitively on buildings such as certain completed Kac-Moody groups. See e.g.\\ \\cite[Corollary B]{raumskalski2023}.\n\nThe cyclic homology of affine Hecke algebras, was clarified in work Baum-Nistor \\cite{baumnistor2002} in the purely algebraic setup and by Solleveld \\cite{solleveld2009-thesis,solleveld2009-hp} in the operator algebraic setup of Hecke-C*-algebras as well as in the setup of non-commutative geometry provided by Schwartz algebras.\n\nIn this work we consider Hecke algebras of right-angled, hyperbolic type. In this setup, delocalised traces account for all of cyclic cohomology, so that a natural first step is to determine traces on such Hecke algebras. This is achieved in the next theorem, which is also proven in the greater generality of generic Hecke algebras in Theorem~\\ref{thm:delocalised-trace-identification}. For simplicity we state it in a simplified form for Iwahori-Hecke algebras.\n\n\\begin{theorem}\n \\label{thm:delocalised-trace-identification}\n Let $\\cH$ be a generic Hecke algebra of right-angled type $(W,S)$. For every conjugacy class $\\cO \\subseteq W$, the map $\\vphi_\\cO$ is a trace on $\\cH$. If $(T_{\\cU})_{\\cU}$ denotes the basis of the cocentre of $\\cH$ satisfying $T_{\\cU} = T_w + [\\cH, \\cH]$ for every element of minimal length $w \\in \\cU$, then $\\vphi_\\cO(T_\\cU) = \\delta_{\\cO, \\cU}$.\n\\end{theorem}", + "full_context": "\\label{sec:introduction}\n\nThe study of delocalised traces forms an important tool in noncommutative geometry and index theory, particularly in the analysis of delocalised $\\Ltwo$-invariants and higher indices associated with manifolds, discrete groups and their operator algebras. These traces, which extend the familiar notion of the canonical trace on group C*-algebras, allow one to distinguish K-theory classes that are indistinguishable by the standard trace. A foundational instance of their use appears in the work of Lott \\cite{lott1999}, who investigated for example delocalised $\\Ltwo$-Betti numbers and delocalised $\\Ltwo$-torsion of hyperbolic manifolds. The theoretical underpinning for the pairing of delocalised traces with K-theory classes in the context of hyperbolic groups was developed by Puschnigg \\cite{puschnigg2010}, who constructed dense subalgebras of group C*-algebras which are at the same time stable under holomorphic functional calculus and host bounded traces that sum coefficients over possibly infinite conjugacy classes. This might be viewed as a natural continuation of the work of Jolissaint \\cite{jolissaint1989-k-theory}, who, inspired by the earlier results of Connes, produced natural smooth subalgebras contained in reduced group C*-algebras, remembering the K-theory of their C*-counterparts. More generally, Puschnigg was able to prove that cyclic homology of the group algebra injects into local cyclic homology of the associated reduced group C*-algebra.\n\nWhile this story is well-developed for group C*-algebras of hyperbolic groups, a parallel and increasingly important direction involves Hecke C*-algebras, which arise as deformations of group algebras associated with Coxeter systems. Historically, spherical and affine Hecke algebras appeared naturally in number theory and representation theory and as such have become a central part of a deep and well-developed theory. They are closely connected to the representation theory of finite groups of Lie type and reductive p-adic groups, respectively. Delorme-Opdam's work in \\cite{delormeopdam2008} considered Hecke C*-algebra of affine types and introduced Schwartz algebras in this context. Notably, by Bernstein's theorem, affine Hecke algebras are finitely generated modules over their centre, so that classical methods related the Fourier transform could be employed. These Hecke-Schwartz algebras of affine type were instrumental in Opdam-Solleveld's approach to determine discrete series representations \\cite{opdamsolleveld2010-discrete-series} and generalised principal series \\cite{delormeopdam2011} of affine Hecke algebras.\n\nHecke operator algebras of indefinite type drew attention only after the introduction of Hecke von Neumann algebras in the context of weighted $\\Ltwo$-cohomology of buildings by Davis-Dymara-Januszkiewicz-Okun \\cite{davisdymarajanusykiewiczokun07} around two decades ago. Specifically right-angled Hecke operator algebras since then gained prominence in the operator algebra community due to their rich structural features, which are at the same time amenable to a variety of established methods from the field. Known features of right-angled Hecke operator algebras by now include explicit, non-trivial factor decomposition \\cite{garncarek16,raumskalski2023} and absence of Cartan or even strong solidity results \\cite{capsers2020,klisse2023-boundaries,borstcasperswasilewski2024} for Hecke-von-Neumann algebras, and the rapid decay property \\cite{caspersklisselarsen2021}, simplicity results \\cite{caspersklisselarsen2021,klisse2023-simplicity} and K-theory calculations \\cite{raumskalski2022} for Hecke C*-algebras. Also compact quantum metric spaces associated with Hecke C*-algebas were very recently introduced and investigated in \\cite{klisseperovic25}. Moreover, Hecke operator algebras encode significant aspects of unitary representation theory, particularly through their role in classifying Iwahori-spherical unitary representations of groups acting strongly transitively on buildings such as certain completed Kac-Moody groups. See e.g.\\ \\cite[Corollary B]{raumskalski2023}.\n\nThe cyclic homology of affine Hecke algebras, was clarified in work Baum-Nistor \\cite{baumnistor2002} in the purely algebraic setup and by Solleveld \\cite{solleveld2009-thesis,solleveld2009-hp} in the operator algebraic setup of Hecke-C*-algebras as well as in the setup of non-commutative geometry provided by Schwartz algebras.\n\nIn this work we consider Hecke algebras of right-angled, hyperbolic type. In this setup, delocalised traces account for all of cyclic cohomology, so that a natural first step is to determine traces on such Hecke algebras. This is achieved in the next theorem, which is also proven in the greater generality of generic Hecke algebras in Theorem~\\ref{thm:delocalised-trace-identification}. For simplicity we state it in a simplified form for Iwahori-Hecke algebras.\n\n\\begin{theorem}\n \\label{thm:delocalised-trace-identification}\n Let $\\cH$ be a generic Hecke algebra of right-angled type $(W,S)$. For every conjugacy class $\\cO \\subseteq W$, the map $\\vphi_\\cO$ is a trace on $\\cH$. If $(T_{\\cU})_{\\cU}$ denotes the basis of the cocentre of $\\cH$ satisfying $T_{\\cU} = T_w + [\\cH, \\cH]$ for every element of minimal length $w \\in \\cU$, then $\\vphi_\\cO(T_\\cU) = \\delta_{\\cO, \\cU}$.\n\\end{theorem}\n\nThe next theorem extends Puschnigg's construction of Schwartz algebras for hyperbolic groups to the realm of right-angled Hecke algebras.\n\\begin{introtheorem}\n \\label{introthm:Schwartz-algebra-existence}\n Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $\\cH$ the complex Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \\in S} \\in \\RR_{> 0}^S$. There is a subalgebra $\\cS(W, S, q) \\subseteq \\Cstarred(W, S, q)$, which\n \\begin{itemize}\n \\item is dense and closed under holomorphic functional calculus,\n \\item and such that for every conjugacy class $\\cO \\subseteq W$, the trace $\\vphi_\\cO$ continuously extends to $\\cS(W, S, q)$.\n \\end{itemize}\n\\end{introtheorem}\n\nThe proof of \\cite[Theorem 5.3]{henie2014} applies verbatim to show the following result: the collection $(T_\\cO)_{\\cO}$ linearly generates the cocentre of any generic Hecke algebra. We will use the following description of the coefficients appearing in linear combinations of these elements.\n\\begin{lemma}\n \\label{lem:non-minimal-element-cocentre}\n Let $\\cH$ be a generic Hecke algebra of type $(W,S)$. Let $\\cO \\subseteq W$ be a conjugacy class and let $w \\in \\cO$. Then\n \\begin{gather*}\n T_w + [\\cH, \\cH]\n =\n \\sum_\\cK c_{\\cK, w} T_\\cK\n \\end{gather*}\n for certain coefficients $c_{\\cK, w}$, where $\\cK$ runs over all conjugacy classes of elements that are Bruhat subordinate to $w$.\n\\end{lemma}\n\\begin{proof}\n Let $\\ell$ be the minimal length of an element in $\\cO$. We prove the claim by induction on $|w| - \\ell = k \\in \\NN_0$. For $k = 0$ there is nothing to prove.\n\n\\begin{notation}\n \\label{not:delocalised-trace}\n Let $(W, S)$ be a Coxeter system of right-angled type and let $\\cH$ be a generic Hecke algebra of type $(W, S)$ defined over $R$. For a conjugacy class $\\cO \\subseteq W$, we denote by $\\Sigma_\\cO\\colon \\cH \\lra R$ the sum over all coefficients of minimal length elements in $\\cO$, that is\n \\begin{gather*}\n \\Sigma_\\cO(T_w)\n =\n \\begin{cases}\n 1 & \\text{ if $w \\in \\cO$ and $w$ is of minimal length in $\\cO$} \\\\\n 0 & \\text{ otherwise}\\eqstop\n \\end{cases}\n \\end{gather*}\n Combining this with Notation~\\ref{not:cyclic-reduction-map}, we will write $\\vphi_\\cO = \\Sigma_\\cO \\circ \\Phi$.\n\\end{notation}\n\n\\begin{theorem}\n \\label{thm:delocalised-trace-identification}\n Let $\\cH$ be a generic Hecke algebra of right-angled type $(W,S)$. For every conjugacy class $\\cO \\subseteq W$, the map $\\vphi_\\cO$ is a trace on $\\cH$. If $(T_{\\cU})_{\\cU}$ denotes the basis of the cocentre of $\\cH$ satisfying $T_{\\cU} = T_w + [\\cH, \\cH]$ for every element of minimal length $w \\in \\cU$, then $\\vphi_\\cO(T_\\cU) = \\delta_{\\cO, \\cU}$.\n\\end{theorem}\n\\begin{proof}\n In order to show that $\\vphi_\\cO$ is a trace, it suffices to show that for all $w \\in W$ and all $s \\in S$ the equality $\\vphi_\\cO(T_s T_w) = \\vphi_\\cO(T_w T_s)$ holds. We prove this statement by induction on $|w| \\in \\NN_0$ and observe that the case $|w| = 0$ is clear. We distinguish several cases.\n\n\\begin{theorem}\n \\label{thm:delocalised-trace-existence}\n Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and let $\\cH$ be the Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \\in S} \\in \\RR_{> 0}^S$. Let $\\cS'(W, S, q), \\cS(W, S, q) \\subseteq \\Cstarred(W, S, q)$ be smooth subalgebras, such that $\\cS'(W, S, q)$ is an unconditional completion of $\\CC(W, S, q)$ and the natural quasi-derivation\n \\begin{gather*}\n \\Delta\\colon T_w \\mapsto \\sum_{\\substack{w = w_1w_2 \\\\ \\text{ reduced decompositon}}} T_{w_1} \\ot T_{w_2}\n \\end{gather*}\n continuously extends to a map into the unconditional tensor product $\\cS(W, S, q) \\lra \\cS'(W, S, q) \\ot \\cS'(W, S, q)$. Then for every conjugacy class $\\cO \\subseteq W$ the trace $\\vphi_\\cO$ continuously extends to $\\cS(W, S, q)$.\n\\end{theorem}\n\\begin{proof}\n Let us first observe that the map $\\Sigma_\\cO$ summing over the finitely many coefficients of elements of minimal length in $\\cO$ extends to a continuous map on $\\cS'(W, S, q)$, as $\\cS'(W, S, q)$ is an unconditional completion of $\\CC(W, S, q)$. Consider maps $\\gamma, \\rho\\colon W \\lra W$ defined as in Lemma~\\ref{lem:cyclic-reduction-hecke-algebra}. Since $\\Delta$ extends continuously to a map $\\cS(W, S, q) \\lra \\cS'(W, S, q) \\ot \\cS'(W, S, q)$ and since $\\cS'(W, S, q)$ is an unconditional completion of $\\CC(W, S, q)$, the map $\\Delta'\\colon \\cS(W, S, q) \\lra \\cS'(W, S, q) \\ot \\cS'(W, S, q)$ satisfying $\\Delta'(T_w) = T_{\\gamma(w) \\rho(w)} \\ot T_{\\gamma(w)^{-1}}$ is well-defined and continuous. Let $\\Phi\\colon \\CC(W, S, q) \\lra \\CC(W, S, q)$ be the unique linear map satisfying $\\Phi(T_w) = T_{\\gamma(w)^{-1}} T_{\\gamma(w)} T_{\\rho(w)}$. Then we have $\\vphi_\\cO = \\Sigma_\\cO \\circ \\Phi$ as shown in Theorem~\\ref{thm:delocalised-trace-identification}. So a continuous extension of $\\vphi_\\cO$ to $\\cS(W, S, q)$ is obtained by the composition\n \\begin{multline*}\n \\cS(W, S, q) \\stackrel{\\Delta'}{\\lra}\n \\cS'(W, S, q) \\ot \\cS'(W, S, q) \\stackrel{\\text{tensor flip}}{\\lra} \\\\\n \\cS'(W, S, q) \\ot \\cS'(W, S, q) \\stackrel{\\text{multiplication}}{\\lra}\n \\cS'(W, S, q) \\stackrel{\\Sigma_\\cO}{\\lra}\n \\CC\n \\eqcomma\n \\end{multline*}\nThis finishes the proof.\n\\end{proof}\n\nBefore proving Theorem~\\ref{thmintro:pairing}, let us clarify its context and notation beyond what we have done in the introduction already. Recall from \\cite[Theorem A.2.1]{bost1990-principe-oka} that the inclusion of a smooth subalgebra into a unital C*-algebra induces an isomorphism in K-theory. We also recall the K-theory calculations for right-angled Hecke C*-algebras from \\cite[Section 4]{raumskalski2022}. To this end we use the description of conjugacy classes of finite order elements from Remark~\\ref{rem:finite-order-element-cliques}. Denote by $\\mathrm{Cliq}(\\Gamma)$ the set of cliques in the commutation graph $\\Gamma$ of a right-angled Coxeter system $(W, S)$, that is subsets $C \\subseteq S$ such that $st = ts$ for all $s,t \\in C$. For a clique $C \\in \\mathrm{Cliq}(\\Gamma)$, we write $p_C = \\prod_{s \\in C} \\chi_1^s$. Here $\\chi_1^s = \\pi_s(\\chi_1)$ denotes the image of the projection $\\chi_1 \\in \\CC[\\ZZ/2]$ associated with the trivial representation under the linear map extending the group homomorphism $\\pi_s\\colon \\ZZ/2 \\lra \\CC(W, S, q)^\\times$ which satisfies $\\pi_s(1) = \\frac{2}{q_s^{1/2} + q_s^{-1/2}}T_s + \\frac{q_s^{-1/2} - q_s^{1/2}}{q_s^{1/2} + q_s^{-1/2}}$. Then for every deformation parameter $q \\in \\RR_{> 0}^S$ the map $\\mathrm{Cliq}(\\Gamma) \\lra \\rK_0(\\Cstarred(W, S, q))\\colon C \\mapsto [p_C]$ induces an isomorphism $\\ZZ^{\\mathrm{Cliq}(\\Gamma)} \\lra \\rK_0(\\Cstarred(W, S, q))$.\n\\begin{proof}[Proof of Theorem~\\ref{thmintro:pairing}]\n As described above, for $D \\in \\mathrm{Cliq}(\\Gamma)$ we have $p_D = \\prod_{s \\in D} \\chi_1^s$ where $\\chi_1^s$ is the projection associated with the trivial representation of the unitary representation $\\ZZ/2 \\mapsto \\CC(W, S, q)$ mapping $1$ to $\\frac{2}{q_s^{1/2} + q_s^{-1/2}}T_s + \\frac{q_s^{-1/2} - q_s^{1/2}}{q_s^{1/2} + q_s^{-1/2}}$. We have $\\chi_1^s = \\mathbb{1}_{(0,\\infty)}(T_s) = \\frac{1}{q_s^{1/2} + q_s^{-1/2}} (T_s + q_s^{-1/2}) =\\frac{1}{q_s^{1/2} + q_s^{-1/2}} T_s + \\frac{1}{1 + q_s}$. Writing $T_D = \\prod_{s \\in D} T_s$, we find that\n \\begin{gather*}\n p_D = \\sum_{C \\subseteq D} \\bigl ( \\prod_{s \\in C} \\frac{1}{q_s^{1/2} + q_s^{-1/2}} \\bigr ) \\bigl ( \\prod_{s \\in D \\setminus C} \\frac{1}{1 + q_s} \\bigr ) T_C\n \\eqstop\n \\end{gather*}\n Since all elements in the support of $p_D$ have minimal length and are pairwise non-conjugate, we find that\n \\begin{gather*}\n \\vphi_C(p_D)\n =\n \\Sigma_C(p_D)\n =\n \\begin{cases}\n 0 & C \\nsubseteq D \\eqcomma \\\\\n \\prod_{s \\in C} \\frac{1}{q_s^{1/2} + q_s^{-1/2}} \\prod_{s \\in D \\setminus C} \\frac{1}{1 + q_s} & C \\subseteq D \\eqstop\n \\end{cases}\n \\end{gather*}\n Now the claimed formula for $\\vphi_C(p_D)$ follows.", + "post_theorem_intro_text_len": 5773, + "post_theorem_intro_text": "We mention that in order to prove this result, we establish in Theorem~\\ref{thm:right-angled-cocentre-basis} that the cocentre of Iwahori Hecke algebra of right-angled type has a basis indexed by representatives of conjugacy classes. The problem of describing the cocentre of Hecke algebras is well-known in the context of character theory. See e.g. \\cite{geckpfeiffer2000-characters} for spherical Coxeter types and \\cite{henie2014} for affine Coxeter types. The recent work \\cite{chen2025-centralizers} explicitly states folklore conjectures about the cocentre for indefinite Coxeter types.\n\nThe next theorem extends Puschnigg's construction of Schwartz algebras for hyperbolic groups to the realm of right-angled Hecke algebras.\n\\begin{introtheorem}\n \\label{introthm:Schwartz-algebra-existence}\n Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $\\cH$ the complex Iwahori-Hecke algebra of type $(W,S)$ with deformation parameter $(q_s)_{s \\in S} \\in \\RR_{> 0}^S$. There is a subalgebra $\\cS(W, S, q) \\subseteq \\Cstarred(W, S, q)$, which\n \\begin{itemize}\n \\item is dense and closed under holomorphic functional calculus,\n \\item and such that for every conjugacy class $\\cO \\subseteq W$, the trace $\\vphi_\\cO$ continuously extends to $\\cS(W, S, q)$.\n \\end{itemize}\n\\end{introtheorem}\n\nFrom the point of view of operator algebras, one of the main purposes of delocalised traces is to determine K-theory classes via their pairing. The K-theory of right-angled Hecke C*-algebras was calculated in \\cite{raumskalski2022}. We briefly recall that for every clique $C$ in the commutation graph of a right-angled Coxeter system there is a projection $p_C$, such that the classes $([p_C])_C$ freely generate K-theory. In Remark~\\ref{rem:finite-order-element-cliques}, we observe that these cliques are in natural bijection with finite order elements of a right-angled Coxeter system. We obtain the following result, which determines the pairing of K-theory and delocalised traces.\n\\begin{introtheorem}\n \\label{thmintro:pairing}\n Let $(W,S)$ be a right-angled, hyperbolic Coxeter system and $(q_s)_{s \\in S} \\in \\RR_{> 0}^S$ a deformation parameter. Denote by $\\mathrm{Cliq}(\\Gamma)$ the set of cliques of the commutation graph $\\Gamma$ of $(W, S)$, identified with the set of conjugacy classes of finite order elements in $W$. For a clique $C \\in \\mathrm{Cliq}(\\Gamma)$ let $p_C$ be the associated projection and let $\\vphi_C$ be the delocalised trace associated with $C$. Then the pairing between K-theory of $\\Cstarred(W, S, q)$ and delocalised traces of finite order elements is determined by the formula\n \\begin{gather*}\n \\vphi_C(p_D) =\n \\begin{cases}\n 0 & C \\nsubseteq D\\eqcomma \\\\\n \\prod_{s \\in C} \\frac{1}{q_s^{1/2} + q_s^{-1/2}} \\prod_{s \\in D \\setminus C} \\frac{1}{1 + q_s} & C \\subseteq D \\eqstop\n \\end{cases}\n \\end{gather*}\n In particular, the pairing $\\rK_0(\\Cstarred(W,S,q)) \\times \\RR^{\\mathrm{Cliq(\\Gamma)}} \\lra \\RR$ is non-degenerate.\n\\end{introtheorem}\n\nAs mentioned above, in the last section of \\cite{puschnigg2010} Puschnigg used his main results to obtain certain decompositions of cyclic cohomology for his smooth algebras and the group C*-algebras of hyperbolic groups, close in spirit to the computation of cyclic homology of group rings due to Burghelea \\cite{burghelea1985, burghelea2023-erratum-for-1985}. This line of research was later continued, using the Jolissaint's rapid decay subalgebras, for example in \\cite{jiogleramsey14} and in \\cite{engel20}. Our results in principle open the way to obtaining similar decompositions of cyclic cohomology in the context of Hecke algebras, such as the Fr{\\'e}chet algebra $\\cS(W, S, q)$ appearing in Theorem \\ref{introthm:Schwartz-algebra-existence}. However, the setting of right-angled hyperbolic Coxeter types remains too narrow to make such an endeavour worthwhile at present, since higher cyclic cohomology is expected to vanish. Further work in this direction, would chime with the recent progress in understanding noncommutative geometric properties of the Iwahori-Hecke algebras such as in the work \\cite{klisseperovic25} already mentioned.\n\nThis paper is organised as follows: after this introduction, in the preliminaries we describe background material and fix notation, focusing on Coxeter groups and Hecke algebras. Then in the third section, we investigate the combinatorics of the multiplication in right-angled Hecke algebras, which plays a major role in this work. In the fourth section, we adapt work of Blackadar-Cuntz and Puschnigg on differential seminorms and quasi-derivations to the needed generality. In the fifth section, we use this framework in order to exhibit specific smooth subalgebras of the Hecke algebras under consideration, to which the natural quasi-derivation extends. In the sixth section, we describe traces on right-angled Hecke algebras, and in that last section combine this with the previous results on smooth subalgebras, to obtain delocalised traces on Hecke-Schwartz algebras and calculate their pairing with K-theory.\n\n\\subsection*{Acknowledgements}\n\nSP was partially supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 677120-INDEX) and the Knut and Alice Wallenberg Foundation (grant number KAW 2020.0252).\n\nSR was partially supported by the Swedish Research Council (grant number 2018-04243) and the German Research Foundation (DFG project no. 550184791).\n\nPN was supported by the National Science Center Grant Maestro-13 UMO-2021/42/A/ST1/00306.\n\nThe authors would like to thank IMPAN and its B{\\c e}dlewo Conference Center as well as the University to Potsdam for their hospitality during respective visits.", + "sketch": "To prove Theorem~\\ref{thmintro:delocalised-trace-identification-simplified}, the introduction says that the authors first establish in Theorem~\\ref{thm:right-angled-cocentre-basis} that “the cocentre of Iwahori Hecke algebra of right-angled type has a basis indexed by representatives of conjugacy classes.”", + "expanded_sketch": "No expanded sketch found.", + "expanded_theorem": "\\label{thmintro:delocalised-trace-identification-simplified}\n Let $\\cH$ be an Iwahori Hecke algebra of right-angled type $(W,S)$. Let $\\cO \\subseteq W$ be a conjugacy class. Then there is a unique trace $\\vphi_\\cO$ on $\\cH$ that satisfies $\\vphi_\\cO(T_w) = \\mathbb{1}_\\cO(w)$ for every element $w \\in W$ that is of minimal length in its conjugacy class.", + "theorem_type": [ + "Uniqueness", + "Existence" + ], + "mcq": { + "question": "Let \\(\\mathcal H\\) be the Iwahori Hecke algebra associated with a right-angled Coxeter system \\((W,S)\\), and let \\(\\{T_w\\}_{w\\in W}\\) denote its standard basis indexed by \\(W\\). Let \\(\\mathcal O\\subseteq W\\) be a conjugacy class. A trace on \\(\\mathcal H\\) means a linear functional \\(\\varphi:\\mathcal H\\to\\mathbb C\\) such that \\(\\varphi(xy)=\\varphi(yx)\\) for all \\(x,y\\in\\mathcal H\\). For \\(w\\in W\\), say that \\(w\\) is of minimal length in its conjugacy class if its Coxeter length is minimal among all elements conjugate to \\(w\\). Which existence statement holds under these assumptions?", + "correct_choice": { + "label": "A", + "text": "There exists a unique trace \\(\\varphi_{\\mathcal O}:\\mathcal H\\to\\mathbb C\\) such that for every \\(w\\in W\\) that is of minimal length in its conjugacy class, \\(\\varphi_{\\mathcal O}(T_w)=\\mathbb 1_{\\mathcal O}(w)\\), where \\(\\mathbb 1_{\\mathcal O}(w)=1\\) if \\(w\\in\\mathcal O\\) and \\(0\\) otherwise." + }, + "choices": [ + { + "label": "B", + "text": "There exists a unique trace \\(\\varphi_{\\mathcal O}:\\mathcal H\\to\\mathbb C\\) such that for every \\(w\\in W\\), \\(\\varphi_{\\mathcal O}(T_w)=\\mathbb 1_{\\mathcal O}(w)\\), where \\(\\mathbb 1_{\\mathcal O}(w)=1\\) if \\(w\\in\\mathcal O\\) and \\(0\\) otherwise." + }, + { + "label": "C", + "text": "There exists a trace \\(\\varphi_{\\mathcal O}:\\mathcal H\\to\\mathbb C\\) such that for every \\(w\\in W\\) that is of minimal length in its conjugacy class, \\(\\varphi_{\\mathcal O}(T_w)=\\mathbb 1_{\\mathcal O}(w)\\), where \\(\\mathbb 1_{\\mathcal O}(w)=1\\) if \\(w\\in\\mathcal O\\) and \\(0\\) otherwise." + }, + { + "label": "D", + "text": "For each conjugacy class \\(\\mathcal O\\subseteq W\\), there exists a unique linear functional \\(\\varphi_{\\mathcal O}:\\mathcal H\\to\\mathbb C\\) such that for every \\(w\\in W\\) that is of minimal length in its conjugacy class, \\(\\varphi_{\\mathcal O}(T_w)=\\mathbb 1_{\\mathcal O}(w)\\)." + }, + { + "label": "E", + "text": "There exists a trace \\(\\varphi:\\mathcal H\\to\\mathbb C\\) such that for every conjugacy class \\(\\mathcal O\\subseteq W\\) and every \\(w\\in W\\) that is of minimal length in its conjugacy class, \\(\\varphi(T_w)=\\mathbb 1_{\\mathcal O}(w)\\), where \\(\\mathbb 1_{\\mathcal O}(w)=1\\) if \\(w\\in\\mathcal O\\) and \\(0\\) otherwise." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "minimal-length restriction", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "uniqueness of the trace", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "trace property replaced by mere linearity", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "dependence on the conjugacy class parameter \\(\\mathcal O\\)", + "template_used": "quantifier_dependence" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives only background definitions and asks which existence statement is valid; it does not explicitly state the theorem or directly reveal the correct option." + }, + "TAS": { + "score": 1, + "justification": "The item is very close to a theorem-recall question: the correct choice is essentially the precise theorem statement, though the alternatives introduce meaningful variations in quantifiers, uniqueness, and the trace condition." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish the exact valid statement from stronger or weaker variants, but the task is mainly precise theorem recognition rather than substantial mathematical generation or derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically targeted: they test common confusions about minimal-length restriction, uniqueness, dependence on the conjugacy class parameter, and trace versus arbitrary linear functional." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it remains largely a near-restatement/recall item rather than a high-generative-reasoning question." + } + }, + { + "id": "2511.16782v1", + "paper_link": "http://arxiv.org/abs/2511.16782v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "thm", + "content": "\\label{mainthm}\n Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.", + "start_pos": 35776, + "end_pos": 35919, + "label": "mainthm" + }, + "ref_dict": { + "mainthm": "\\begin{thm}\\label{mainthm}\n Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.\n\\end{thm}", + "discuss": "\\begin{thm}\\label{mainthm}\n Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.\n\\end{thm}\n\n\\par In slightly more detail, \ngiven a boundaryless surface with negative Euler characteristic\nand a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\\colon S \\to S$, one can construct the veering triangulation of the mapping torus of $f$, whose\nflow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.\n\n\\subsection{Connections to literature}\\label{discuss}\n\n\\par Work of Penner-Papadopoulos \\cite[Theorem 4.1]{PP} is sometimes cited as producing $f$-invariant generic train tracks with irreducible transition matrix for a pseudo-Anosov $f$, however it was pointed out to us by Chris Leininger that this construction seems to only produce train tracks which are $f^n$-invariant with irreducible transition matrix for some $n$ possibly greater than one.\n\\par In the setting of \\cite[Theorem 4.1]{PP},\nthe support map $\\sigma$ produced for the $f$-invariant train track $\\tau$ is not required to map switches to switches, rather the transition matrix is defined by choosing test points in the interiors of branches and counting the number of times the image of each branch hits each test point. Since switches don't have to map to switches, a branch $b$ of $\\tau$ could e.g. be mapped by $\\sigma \\circ f$ over itself and a small piece of an adjacent branch. Then the transition matrix for $f$ will record that $b$ only maps over itself. However, for some $n$, $(\\sigma \\circ f)^n(b)$ will be mapped over all of the adjacent branch, so the transition matrix for $(\\sigma \\circ f)^n$ will record that $b$ maps over itself and the adjacent branch. Hence, it's possible that $f^n(\\tau) \\prec \\tau$ has irreducible transition matrix while $f(\\tau) \\prec \\tau$ does not.\n\n\\par Returning to discussion of the present work, we note that much of the theory of train tracks assumes that the tracks considered are \\textit{\\textbf{generic}}, i.e. trivalent, since this simplification can often be made without loss of generality. In particular, the train tracks associated to veering triangulations are always generic. However, if the flow graph for a pseudo-Anosov $f$ is not irreducible, \\Cref{mainthm} necessarily produces train tracks which are not generic. \nThis motivates the following question:\n\n\\begin{qn}\n For all $f\\colon S \\to S$ pseudo-Anosov, does there exist a \\textit{generic} invariant train track for $f$ with irreducible transition matrix?\n\\end{qn}\n\n\\par \\textbf{Acknowledgements.} I would like to thank Sam Taylor for his generous guidance and feedback. I would also like to thank Chi Cheuk Tsang for helpful conversations and comments on a draft of this paper.\n\n\\section{Background}\\label{background}\n\n\\par A \\textit{\\textbf{strongly connected component}} of a directed graph $G$ is a maximal subset $C$ of the vertices of $G$ such that for any two vertices in $C$, there is a path in $G$ from the first vertex to the second.\nIf $G$ has only one strongly connected component, it is called \\textit{\\textbf{strongly connected}}.\nA nonnegative square matrix $A$ is called \\textit{\\textbf{irreducible}} if for every pair of indices $i$, $j$ there is an $n>0$ such that the $(i,j)$-th entry of $A^n$ is positive. $A$ is irreducible if and only if it is the adjacency matrix of a strongly connected graph.\n\\par\n Given an orientable finite-type surface $S$ without boundary and negative Euler characteristic, a homeomorphism $f\\colon S \\to S$ is called \\textit{\\textbf{pseudo-Anosov}} if there is a pair of transverse measured singular foliations $(\\Lambda^u,\\mu_u)$ and $(\\Lambda^s,\\mu_s)$, called the \\textit{\\textbf{unstable}} and \\textit{\\textbf{stable}} foliations, and a constant $\\lambda > 1$, called the \\textit{\\textbf{stretch factor}}, such that $f(\\Lambda^u)=\\Lambda^u$, $f(\\Lambda^s)=\\Lambda^s$ and $f(\\mu^u) = \\lambda \\mu^u$, $f(\\mu^s) = \\lambda^{-1} \\mu^s$. No nontrivial power of a pseudo-Anosov map fixes the homotopy class of any essential closed curve in $S$. A pseudo-Anosov map is a diffeomorphism except at the singular points of $\\Lambda^u$.\n\n \\par The \\textit{\\textbf{mapping torus of $f$}} is the 3-manifold\n $$ M_f = \\faktor{(S \\times [0,1])}{ (f(p),0) \\sim (p,1)} $$ which fibers over the circle with fiber $S$ and monodromy $f$.\n\n \\par If all of the singularities of the foliations are at the punctures of $S$, we say $f$ is \\textit{\\textbf{fully punctured}}. We will study general pseudo-Anosovs by deleting the singularities of the foliations to obtain a new surface $S^\\circ$ on which the restriction of $f$ is fully punctured.\nFor full background on pseudo-Anosov homeomorphisms, see Fathi--Laudenbach--Poénaru \\cite{FLP}.\n\n\\subsection{Train tracks}\\label{tts}\n\\par A \\textit{\\textbf{train track}} $\\tau$ is a closed 1-complex embedded in a surface $S$ with a ``smoothing\" at each vertex so that $\\tau$ has a well-defined tangent space at each vertex, and $S \\setminus \\tau$ contains no nullgons, unpunctured monogons or unpunctured bigons. See Penner-Harer \\cite{PH}. The vertices of $\\tau$ are called \\textit{\\textbf{switches}} and the edges are called \\textit{\\textbf{branches}}. If the switches of $\\tau$ all have degree three, $\\tau$ is called \\textit{\\textbf{generic}}.\n\n\\par Given two train tracks $\\tau_1$ and $\\tau_2$ on $S$, $\\tau_2$ is said to \\textit{\\textbf{carry}} $\\tau_1$ (or $\\tau_1$ \\textit{\\textbf{is carried by}} $\\tau_2$), denoted $\\tau_1 \\prec \\tau_2$, if there is a $C^1$ map $\\sigma\\colon S \\to S$ called the \\textit{\\textbf{support map}} \nsuch that\n \\begin{enumerate}\n \\item $\\sigma$ is homotopic to the identity,\n \\item $\\sigma(\\tau_1) \\subseteq \\tau_2$,\n \\item for all points $x$ in $\\tau_1$,\n $d_x \\sigma_{|\\tau_1}\\colon T_x \\tau_1 \\to T_{h(x)}\\tau_2$ is an isomorphism of tangent spaces,\n \\end{enumerate} \nNote that carrying is transitive: if $\\tau_1 \\prec \\tau_2$ and $\\tau_2 \\prec \\tau_3$, then\n$\\tau_1 \\prec \\tau_3$. \nWe also write $\\tau_1 \\prec_\\sigma \\tau_2$ to indicate that $\\sigma$ is the support map for $\\tau_1 \\prec \\tau_2$. A carrying map $\\sigma$ for $\\tau_1 \\prec \\tau_2$ is called \\textit{\\textbf{combinatorial}} if for every switch $v$ in $\\tau_1$, $\\sigma(v)$ is a switch in $\\tau_2$.\n \\par\n Given $\\tau_1$ with branches $\\{b^1_j\\}_{j=1\\ldots n}$ and $\\tau_2$ with branches $\\{b^2_i\\}_{i=1\\ldots m}$ such that $\\tau_1$ is carried by $\\tau_2$ with combinatorial support map $\\sigma$,\n the \\textit{\\textbf{transition matrix}} is the $m \\times n$ matrix $T = [t_{i,j}]$ where \n $t_{i,j}$ is the number of times the image of $b^1_j$ under $\\sigma$ passes over $b^2_i$.\n\\par\n A \\textit{\\textbf{train path}} is a $C^1$ immersion $p\\colon [0,1] \\to \\tau$ so that the endpoints $\\{0,1\\} = \\partial[0,1]$ are mapped to switches. We will sometimes conflate $p$ with its image in $\\tau$. For $t \\in (0,1)$ such that $p(t)$ is a switch, the sub-paths $p_0 = p_{|[0,t]}\\colon [0,t] \\to \\tau$, $p_1 = p_{|[t,1]}\\colon [t,1] \\to \\tau$ are also train paths (after reparametrizing). Similarly, if $p_0$ and $p_1$ are train paths such that $p_0(1)=p_1(0)$ and the differentials $d_1 p_0(\\partial_t)$ and $d_0 p_1(\\partial_t)$ are either both positive or both negative. The concatenation $(p_0 p_1)\\colon [0,2] \\to \\tau$ is a train path after reparametrizing, hence any train path can be written as $(p_0\\ldots p_{n-1})$, where $p_i\\colon\\left[\\frac{i}{n},\\frac{i+1}{n}\\right] \\to \\tau$ has image consisting of a single branch. If $p$ is injective we say the train path is \\textit{\\textbf{embedded}}.\n\\par\n A \\textit{\\textbf{fold}} is a particular combinatorial support map $\\phi$ for $\\tau_1 \\prec \\tau_2$ which induces a bijection between the branches of $\\tau_1$ and $\\tau_2$\n except on a particular connected subgraph where it is defined according to either of the two pictures in \\Cref{fig:fold}. The three branches of $\\tau_1$ which are mapped over the small branch $e'$ in the domain of \\Cref{fig:fold} are said to \\textit{\\textbf{fold to}} $e'$. \n If there is a $\\tau_1 \\prec_\\phi \\tau_2$\n we say $\\tau_1$ \\textit{\\textbf{folds to}} $\\tau_2$, denoted $\\tau_1 \\leftharpoonup \\tau_2$. If $\\tau_1$ folds to $\\tau_2$, $\\tau_1$ is carried by $\\tau_2$. If $\\tau_1 \\prec \\tau_2$ with support map given by a sequence of folds we also say $\\tau_1$ folds to $\\tau_2$.\n \\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.66\\textwidth]{foldgood.pdf}\n \\caption{The two possible pictures of a fold. The fold on the left is a left fold and the one on the right is a right fold. Colors indicate the image of each branch on the top after folding.}\n \\label{fig:fold}\n\\end{figure}\n \\par\n A train track $\\tau$ is an \\textit{\\textbf{invariant}} track for a map $f: S \\to S$ with support map $\\sigma$ if $f(\\tau)$ is carried by $\\tau$ with support map $\\sigma$.\n If the transition matrix for $f(\\tau) \\prec_\\sigma \\tau$ is irreducible, we also say $\\tau$ is \\textit{\\textbf{irreducible}}.\nBy smoothly isotoping $\\tau$ if necessary, we may assume it is disjoint from the singularities of $\\Lambda^u$. This will be convenient later when we remove the singularities of $\\Lambda^u$. The following theorem due to Agol is the starting point of the construction of the veering triangulation of the mapping torus of a pseudo-Anosov $f$:\n\n\\begin{thm}[\\protect{\\cite[Theorem 3.5]{A}}]\\label{ttfs} Given $f\\colon S \\to S$ pseudo-Anosov, there exists a generic invariant train track $\\tau$ for $f$ such that $f(\\tau)$ folds to $\\tau$, i.e. there exist a finite sequence of train tracks \n$$f(\\tau) = \\tau_0 \\leftharpoonup \\tau_1 \\leftharpoonup \\ldots \\leftharpoonup \\tau_n = \\tau,$$ \nwhere $\\tau_i$ is carried by $\\tau_{i+1}$ with support map given by a single fold.\n\\end{thm}" + }, + "pre_theorem_intro_text_len": 2101, + "pre_theorem_intro_text": "\\label{intro}\nGiven a pseudo-Anosov homeomorphism $f$, an invariant train track provides a combinatorial way of recording how curves and other objects behave under iteration of $f$.\nThe existence of invariant train tracks with irreducible transition matrix\nallows the application of Perron-Frobenius theory to the study of the properties of these maps. \nFor example, when an invariant train track is irreducible it can be used to build a Markov partition for the map. \nThe existence of irreducible invariant train tracks is often taken for granted \n(See e.g. \\cite{FR}, \\cite{F}, \\cite{J}, \\cite{EF}, \\cite{M}, \\cite{Pe}) \nbut to our knowledge no complete proof is currently in the literature. See \\Cref{discuss} for more details.\n\n\\par The veering triangulation associated to $f$ provides a unified way to study both the topology of its mapping torus and the dynamics of its invariant train tracks.\nOne important datum attached to a veering triangulation is its \\textit{flow graph}, which encodes a Markov partition of a pseudo-Anosov map or flow. Flow graphs have been used by Landry--Minsky--Taylor used to count periodic points of pseudo-Anosov maps, and to define polynomial invariants of pseudo-Anosov flows without perfect fits \\cite{LMT1}.\nWhen the flow graph associated to the veering triangulation of a pseudo-Anosov mapping torus is strongly connected, the triangulation can be used to obtain invariant train tracks with \nirreducible transition matrices. \n\\par For general veering triangulations, Agol--Tsang have characterized the strongly connected components of the flow graph, which consist of one ``reduced component\" from which there are paths to every vertex, and some number of ``infinitesimal cycles\" which do not have any paths back to the rest of the graph \\cite{AT}. \nBy studying the interaction between these infinitesimal components and the invariant train tracks arising from the triangulation, we are able to produce new train tracks where the obstructions to irreducibility coming from the infinitesimal components have been bypassed. This allows us to prove the following:", + "context": "\\label{intro}\nGiven a pseudo-Anosov homeomorphism $f$, an invariant train track provides a combinatorial way of recording how curves and other objects behave under iteration of $f$.\nThe existence of invariant train tracks with irreducible transition matrix\nallows the application of Perron-Frobenius theory to the study of the properties of these maps. \nFor example, when an invariant train track is irreducible it can be used to build a Markov partition for the map. \nThe existence of irreducible invariant train tracks is often taken for granted \n(See e.g. \\cite{FR}, \\cite{F}, \\cite{J}, \\cite{EF}, \\cite{M}, \\cite{Pe}) \nbut to our knowledge no complete proof is currently in the literature. See \\Cref{discuss} for more details.\n\n\\par The veering triangulation associated to $f$ provides a unified way to study both the topology of its mapping torus and the dynamics of its invariant train tracks.\nOne important datum attached to a veering triangulation is its \\textit{flow graph}, which encodes a Markov partition of a pseudo-Anosov map or flow. Flow graphs have been used by Landry--Minsky--Taylor used to count periodic points of pseudo-Anosov maps, and to define polynomial invariants of pseudo-Anosov flows without perfect fits \\cite{LMT1}.\nWhen the flow graph associated to the veering triangulation of a pseudo-Anosov mapping torus is strongly connected, the triangulation can be used to obtain invariant train tracks with \nirreducible transition matrices. \n\\par For general veering triangulations, Agol--Tsang have characterized the strongly connected components of the flow graph, which consist of one ``reduced component\" from which there are paths to every vertex, and some number of ``infinitesimal cycles\" which do not have any paths back to the rest of the graph \\cite{AT}. \nBy studying the interaction between these infinitesimal components and the invariant train tracks arising from the triangulation, we are able to produce new train tracks where the obstructions to irreducibility coming from the infinitesimal components have been bypassed. This allows us to prove the following:", + "full_context": "\\label{intro}\nGiven a pseudo-Anosov homeomorphism $f$, an invariant train track provides a combinatorial way of recording how curves and other objects behave under iteration of $f$.\nThe existence of invariant train tracks with irreducible transition matrix\nallows the application of Perron-Frobenius theory to the study of the properties of these maps. \nFor example, when an invariant train track is irreducible it can be used to build a Markov partition for the map. \nThe existence of irreducible invariant train tracks is often taken for granted \n(See e.g. \\cite{FR}, \\cite{F}, \\cite{J}, \\cite{EF}, \\cite{M}, \\cite{Pe}) \nbut to our knowledge no complete proof is currently in the literature. See \\Cref{discuss} for more details.\n\n\\par The veering triangulation associated to $f$ provides a unified way to study both the topology of its mapping torus and the dynamics of its invariant train tracks.\nOne important datum attached to a veering triangulation is its \\textit{flow graph}, which encodes a Markov partition of a pseudo-Anosov map or flow. Flow graphs have been used by Landry--Minsky--Taylor used to count periodic points of pseudo-Anosov maps, and to define polynomial invariants of pseudo-Anosov flows without perfect fits \\cite{LMT1}.\nWhen the flow graph associated to the veering triangulation of a pseudo-Anosov mapping torus is strongly connected, the triangulation can be used to obtain invariant train tracks with \nirreducible transition matrices. \n\\par For general veering triangulations, Agol--Tsang have characterized the strongly connected components of the flow graph, which consist of one ``reduced component\" from which there are paths to every vertex, and some number of ``infinitesimal cycles\" which do not have any paths back to the rest of the graph \\cite{AT}. \nBy studying the interaction between these infinitesimal components and the invariant train tracks arising from the triangulation, we are able to produce new train tracks where the obstructions to irreducibility coming from the infinitesimal components have been bypassed. This allows us to prove the following:\n\n\\par In slightly more detail, \ngiven a boundaryless surface with negative Euler characteristic\nand a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\\colon S \\to S$, one can construct the veering triangulation of the mapping torus of $f$, whose\nflow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.\n\n\\begin{qn}\n For all $f\\colon S \\to S$ pseudo-Anosov, does there exist a \\textit{generic} invariant train track for $f$ with irreducible transition matrix?\n\\end{qn}\n\n\\par Given two train tracks $\\tau_1$ and $\\tau_2$ on $S$, $\\tau_2$ is said to \\textit{\\textbf{carry}} $\\tau_1$ (or $\\tau_1$ \\textit{\\textbf{is carried by}} $\\tau_2$), denoted $\\tau_1 \\prec \\tau_2$, if there is a $C^1$ map $\\sigma\\colon S \\to S$ called the \\textit{\\textbf{support map}} \nsuch that\n \\begin{enumerate}\n \\item $\\sigma$ is homotopic to the identity,\n \\item $\\sigma(\\tau_1) \\subseteq \\tau_2$,\n \\item for all points $x$ in $\\tau_1$,\n $d_x \\sigma_{|\\tau_1}\\colon T_x \\tau_1 \\to T_{h(x)}\\tau_2$ is an isomorphism of tangent spaces,\n \\end{enumerate} \nNote that carrying is transitive: if $\\tau_1 \\prec \\tau_2$ and $\\tau_2 \\prec \\tau_3$, then\n$\\tau_1 \\prec \\tau_3$. \nWe also write $\\tau_1 \\prec_\\sigma \\tau_2$ to indicate that $\\sigma$ is the support map for $\\tau_1 \\prec \\tau_2$. A carrying map $\\sigma$ for $\\tau_1 \\prec \\tau_2$ is called \\textit{\\textbf{combinatorial}} if for every switch $v$ in $\\tau_1$, $\\sigma(v)$ is a switch in $\\tau_2$.\n \\par\n Given $\\tau_1$ with branches $\\{b^1_j\\}_{j=1\\ldots n}$ and $\\tau_2$ with branches $\\{b^2_i\\}_{i=1\\ldots m}$ such that $\\tau_1$ is carried by $\\tau_2$ with combinatorial support map $\\sigma$,\n the \\textit{\\textbf{transition matrix}} is the $m \\times n$ matrix $T = [t_{i,j}]$ where \n $t_{i,j}$ is the number of times the image of $b^1_j$ under $\\sigma$ passes over $b^2_i$.\n\\par\n A \\textit{\\textbf{train path}} is a $C^1$ immersion $p\\colon [0,1] \\to \\tau$ so that the endpoints $\\{0,1\\} = \\partial[0,1]$ are mapped to switches. We will sometimes conflate $p$ with its image in $\\tau$. For $t \\in (0,1)$ such that $p(t)$ is a switch, the sub-paths $p_0 = p_{|[0,t]}\\colon [0,t] \\to \\tau$, $p_1 = p_{|[t,1]}\\colon [t,1] \\to \\tau$ are also train paths (after reparametrizing). Similarly, if $p_0$ and $p_1$ are train paths such that $p_0(1)=p_1(0)$ and the differentials $d_1 p_0(\\partial_t)$ and $d_0 p_1(\\partial_t)$ are either both positive or both negative. The concatenation $(p_0 p_1)\\colon [0,2] \\to \\tau$ is a train path after reparametrizing, hence any train path can be written as $(p_0\\ldots p_{n-1})$, where $p_i\\colon\\left[\\frac{i}{n},\\frac{i+1}{n}\\right] \\to \\tau$ has image consisting of a single branch. If $p$ is injective we say the train path is \\textit{\\textbf{embedded}}.\n\\par\n A \\textit{\\textbf{fold}} is a particular combinatorial support map $\\phi$ for $\\tau_1 \\prec \\tau_2$ which induces a bijection between the branches of $\\tau_1$ and $\\tau_2$\n except on a particular connected subgraph where it is defined according to either of the two pictures in \\Cref{fig:fold}. The three branches of $\\tau_1$ which are mapped over the small branch $e'$ in the domain of \\Cref{fig:fold} are said to \\textit{\\textbf{fold to}} $e'$. \n If there is a $\\tau_1 \\prec_\\phi \\tau_2$\n we say $\\tau_1$ \\textit{\\textbf{folds to}} $\\tau_2$, denoted $\\tau_1 \\leftharpoonup \\tau_2$. If $\\tau_1$ folds to $\\tau_2$, $\\tau_1$ is carried by $\\tau_2$. If $\\tau_1 \\prec \\tau_2$ with support map given by a sequence of folds we also say $\\tau_1$ folds to $\\tau_2$.\n \\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.66\\textwidth]{foldgood.pdf}\n \\caption{The two possible pictures of a fold. The fold on the left is a left fold and the one on the right is a right fold. Colors indicate the image of each branch on the top after folding.}\n \\label{fig:fold}\n\\end{figure}\n \\par\n A train track $\\tau$ is an \\textit{\\textbf{invariant}} track for a map $f: S \\to S$ with support map $\\sigma$ if $f(\\tau)$ is carried by $\\tau$ with support map $\\sigma$.\n If the transition matrix for $f(\\tau) \\prec_\\sigma \\tau$ is irreducible, we also say $\\tau$ is \\textit{\\textbf{irreducible}}.\nBy smoothly isotoping $\\tau$ if necessary, we may assume it is disjoint from the singularities of $\\Lambda^u$. This will be convenient later when we remove the singularities of $\\Lambda^u$. The following theorem due to Agol is the starting point of the construction of the veering triangulation of the mapping torus of a pseudo-Anosov $f$:\n\n\\par Note that the referenced theorem is stated in terms of \\textit{splits} rather than folds, which are the combinatorial inverse of folds. \n Additionally, the referenced theorem is stronger than what we state here because the splitting sequence is shown to be canonical up to ``commuting maximal splits\": at each step in the sequence, the train track is split at a branch carrying maximal measure of the invariant lamination, and the splits of maximal branches commute if there is not a unique branch carrying maximal measure.\n This theorem motivates the following definition:\n\\begin{defn}\n An invariant train track $\\tau$ for a pseudo-Anosov map $f$ is called \\textit{\\textbf{veering}} if it \n is generic and $\\tau$ can be obtained from $f(\\tau)$ by a sequence of folds, in which case there is a sequence of train tracks \n $$f(\\tau) = \\tau_0 \\leftharpoonup \\tau_1 \\leftharpoonup \\ldots \\leftharpoonup \\tau_n = \\tau$$ \n where $\\tau_i$ is carried by $\\tau_{i+1}$ with support map consisting of a single fold.\n\\end{defn}\n\\par In \\Cref{clvts} we will see that a veering train track as defined here is sufficient to produce the veering triangulation of the fully punctured mapping torus of $f$. While veering train tracks are not necessarily irreducible, they will be the beginning of our construction of irreducible tracks.\n\\par\n Given two train tracks $\\tau_1$, $\\tau_2$ on $S$, a (surface) \\textit{\\textbf{train track map}} is a map $t: S \\to S$ so that $t(\\tau_1)$ is contained in $\\tau_2$, and the restriction of $t$ to $\\tau_1$ is \n such that for any train path $p\\colon [0,1] \\to \\tau_1$, $t \\circ p\\colon [0,1] \\to \\tau_2$ is also a train path. In other words, a train track map is like a combinatorial support map except that it does not have to be homotopic to the identity map. \n \\par The following lemma will be useful for producing support maps:\n\\begin{lemma}\\label{supplemma}\n Let $t\\colon S \\to S$ be a train track map taking $\\tau_1$ to $\\tau_2$, and $g\\colon S \\to S$ be a diffeomorphism.\n If $t$ is homotopic to $g$, then $g(\\tau_1) \\prec \\tau_2$ with support map given by $t \\circ g^{-1}$.\n\\end{lemma}\n\\begin{proof}\n Since $t$ and $g$ are homotopic, $t \\circ g^{-1}$ is homotopic to the identity. Since $t(\\tau_1) \\subseteq \\tau_2$, $(t \\circ g^{-1})(g(\\tau_1)) = t(\\tau_1) \\subseteq \\tau_2$. Let $p \\in g(\\tau_1)$. Since $g$ is a diffeomorphism, the restriction of the differential \n $$d_p g^{-1}_{|g(\\tau_1)}\\colon T_p g(\\tau_1) \\to T_{g^{-1}(p)} \\tau_1$$\n is an isomorphism. Since $t$ is a train track map, $d_{g^{-1}(p)} t_{| \\tau_1}$ is an isomorphism, so the composition\n $$d_p (t \\circ g^{-1})_{|g(\\tau_1)} = (d_{g^{-1}(p)} t_{| \\tau_1}) \\circ (d_p g^{-1}_{|g(\\tau_1)})\\colon T_p g(\\tau_1) \\to T_{(t \\circ g^{-1})(p)} \\tau_2$$\n is also an isomorphism.\n\\end{proof}", + "post_theorem_intro_text_len": 3015, + "post_theorem_intro_text": "\\par In slightly more detail, \ngiven a boundaryless surface with negative Euler characteristic\nand a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\\colon S \\to S$, one can construct the veering triangulation of the mapping torus of $f$, whose\nflow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.\n\n\\subsection{Connections to literature}\\label{discuss}\n\n\\par Work of Penner-Papadopoulos \\cite[Theorem 4.1]{PP} is sometimes cited as producing $f$-invariant generic train tracks with irreducible transition matrix for a pseudo-Anosov $f$, however it was pointed out to us by Chris Leininger that this construction seems to only produce train tracks which are $f^n$-invariant with irreducible transition matrix for some $n$ possibly greater than one.\n\\par In the setting of \\cite[Theorem 4.1]{PP},\nthe support map $\\sigma$ produced for the $f$-invariant train track $\\tau$ is not required to map switches to switches, rather the transition matrix is defined by choosing test points in the interiors of branches and counting the number of times the image of each branch hits each test point. Since switches don't have to map to switches, a branch $b$ of $\\tau$ could e.g. be mapped by $\\sigma \\circ f$ over itself and a small piece of an adjacent branch. Then the transition matrix for $f$ will record that $b$ only maps over itself. However, for some $n$, $(\\sigma \\circ f)^n(b)$ will be mapped over all of the adjacent branch, so the transition matrix for $(\\sigma \\circ f)^n$ will record that $b$ maps over itself and the adjacent branch. Hence, it's possible that $f^n(\\tau) \\prec \\tau$ has irreducible transition matrix while $f(\\tau) \\prec \\tau$ does not.\n\n\\par Returning to discussion of the present work, we note that much of the theory of train tracks assumes that the tracks considered are \\textit{\\textbf{generic}}, i.e. trivalent, since this simplification can often be made without loss of generality. In particular, the train tracks associated to veering triangulations are always generic. However, if the flow graph for a pseudo-Anosov $f$ is not irreducible, \\Cref{mainthm} necessarily produces train tracks which are not generic. \nThis motivates the following question:\n\n\\begin{qn}\n For all $f\\colon S \\to S$ pseudo-Anosov, does there exist a \\textit{generic} invariant train track for $f$ with irreducible transition matrix?\n\\end{qn}\n\n\\par \\textbf{Acknowledgements.} I would like to thank Sam Taylor for his generous guidance and feedback. I would also like to thank Chi Cheuk Tsang for helpful conversations and comments on a draft of this paper.", + "sketch": "Given a boundaryless surface with negative Euler characteristic and a train track folding (splitting) sequence for a pseudo-Anosov $f\\colon S\\to S$, one constructs the veering triangulation of the mapping torus of $f$, whose flow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, one shows that the only obstructions to irreducibility are branches dual to infinitesimal cycles of the flow graph, and characterizes the local structure of veering tracks near these branches. Using this local picture, one modifies the track by contracting the obstructing branches, and recovers a support map for the modified train track with irreducible transition matrix.", + "expanded_sketch": "Given a boundaryless surface with negative Euler characteristic and a train track folding (splitting) sequence for a pseudo-Anosov $f\\colon S\\to S$, one constructs the veering triangulation of the mapping torus of $f$, whose flow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, one shows that the only obstructions to irreducibility are branches dual to infinitesimal cycles of the flow graph, and characterizes the local structure of veering tracks near these branches. Using this local picture, one modifies the track by contracting the obstructing branches, and recovers a support map for the modified train track with irreducible transition matrix.", + "expanded_theorem": "\\label{mainthm}\n Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.,", + "theorem_type": [ + "Universal", + "Existence" + ], + "mcq": { + "question": "Let \\(f\\colon S\\to S\\) be a pseudo-Anosov homeomorphism. A train track \\(\\tau\\) on \\(S\\) is called invariant for \\(f\\) if \\(f(\\tau)\\) is carried by \\(\\tau\\) via a combinatorial support map. If the branches of \\(f(\\tau)\\) are \\(\\{b^1_j\\}\\) and the branches of \\(\\tau\\) are \\(\\{b^2_i\\}\\), the associated transition matrix is the nonnegative matrix \\(T=[t_{i,j}]\\), where \\(t_{i,j}\\) is the number of times the image of \\(b^1_j\\) under the support map passes over \\(b^2_i\\). A nonnegative matrix is irreducible if for every \\(i,j\\) there exists \\(n\\ge 1\\) such that \\((T^n)_{i,j}>0\\). Under these assumptions, which existence statement holds?", + "correct_choice": { + "label": "A", + "text": "For every pseudo-Anosov homeomorphism \\(f\\colon S\\to S\\), there exists an invariant train track \\(\\tau\\) for \\(f\\) such that the transition matrix associated to \\(f(\\tau)\\prec \\tau\\) is irreducible." + }, + "choices": [ + { + "label": "B", + "text": "For every pseudo-Anosov homeomorphism \\(f\\colon S\\to S\\), every invariant train track \\(\\tau\\) for \\(f\\) has the property that the transition matrix associated to \\(f(\\tau)\\prec \\tau\\) is irreducible." + }, + { + "label": "C", + "text": "For every pseudo-Anosov homeomorphism \\(f\\colon S\\to S\\), there exists an invariant train track \\(\\tau\\) for \\(f\\) such that the transition matrix associated to \\(f(\\tau)\\prec \\tau\\) is nonzero." + }, + { + "label": "D", + "text": "For every pseudo-Anosov homeomorphism \\(f\\colon S\\to S\\), there exists an invariant train track \\(\\tau\\) for \\(f\\) such that for every pair of branches \\(b^2_i,b^2_j\\) of \\(\\tau\\), one has \\(t_{i,j}>0\\) in the transition matrix associated to \\(f(\\tau)\\prec \\tau\\)." + }, + { + "label": "E", + "text": "For every pseudo-Anosov homeomorphism \\(f\\colon S\\to S\\), there exists an invariant train track \\(\\tau\\) for \\(f\\) such that each strongly connected component of the transition matrix associated to \\(f(\\tau)\\prec \\tau\\) is irreducible." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "existential choice of modified track after contracting obstructing branches", + "template_used": "quantifier_dependence" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "irreducibility requirement dropped to mere existence of a nontrivial transition matrix", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "counting_estimate", + "tampered_component": "irreducible replaced by strictly positive one-step transition matrix", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "case_split", + "tampered_component": "infinitesimal-cycle obstruction bypassed only after contracting branches, not for the original component decomposition", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives definitions of invariant train tracks and irreducibility, but it does not explicitly state the target existence theorem or uniquely point to choice A. The correct answer is not leaked, though the setup clearly narrows the topic." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question: after introducing the relevant definitions, it asks which existence statement holds. It is not a verbatim restatement, because the alternatives vary in quantifier strength and matrix properties, but it still largely tests recognition of a standard result." + }, + "GPS": { + "score": 1, + "justification": "There is some reasoning pressure in separating 'there exists' from 'for every,' and irreducible from merely nonzero or strictly positive. However, for a student who knows the theorem, the answer is mainly direct identification rather than substantial derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target natural failure modes: overstrengthening to every invariant track (B), weakening to nonzero (C), confusing irreducibility with positivity in one step (D), and using a subtle component-wise condition (E). They are distinct and well aligned with common misunderstandings." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with strong distractors and little answer leakage, but it leans toward theorem recognition rather than deep generative reasoning." + } + }, + { + "id": "2511.15021v1", + "paper_link": "http://arxiv.org/abs/2511.15021v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "Theorem", + "content": "\\label{thm:uniqueness within periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ be a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$ satisfying \\eqref{eq:A compatible}. Then there exists a unique periodic solution (up to addition of constants) to \\eqref{eq:periodic equation ma} with quadratic component $P$.", + "start_pos": 14162, + "end_pos": 14620, + "label": "thm:uniqueness within periodic" + }, + "ref_dict": { + "def:phi homogenization": "\\begin{Definition}\\label{def:phi homogenization}\nFix $\\lambda = \\lambda(n) > 0$ sufficiently small such that $\\sigma = C_1(n)\\lambda$ satisfies $4^{n+1}\\sigma < c(n)$, where $C_1(n)$ is the one in Lemma \\ref{lem:covering Lemma} and $c(n)$ is the one in Proposition \\ref{prop:basic}. For $\\Lambda\\ge \\max\\{\\Lambda_0,\\Lambda_1,\\Lambda_2\\}$, and any nonnegative functions $v \\in C(\\Omega)$, we define its $\\phi$-homogenization $M_{\\phi,\\Lambda}v(x):=M_{\\phi,\\lambda,\\Lambda}v(x)$ in $\\frac{7}{8}\\Omega$ through the following maximal averaging type operator:\n\\begin{equation}\\label{eq:phi homogenization}\n\\begin{split}\nM_{\\phi,\\Lambda}v(x)= \\sup\\left\\{ t\\in\\R:\\; \\exists \\ \\check{h}(x) t\\} \\cap S_h (x) )}{\\mu(S_h (x))} > \\lambda \\right\\}.\n\\end{split}\n\\end{equation} \n\\end{Definition}", + "eq:periodic equation ma": "\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\R^n,\n\\end{equation}", + "eq:A compatible": "\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation}", + "eq:periodic solution": "\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}", + "thm:solutions are periodic": "\\begin{Theorem}\\label{thm:solutions are periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $u$ be a convex solution of \\eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \\ref{def:periodic solution}.\n\\end{Theorem}", + "thm:uniqueness within periodic": "\\begin{Theorem}\\label{thm:uniqueness within periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ be a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$ satisfying \\eqref{eq:A compatible}. Then there exists a unique periodic solution (up to addition of constants) to \\eqref{eq:periodic equation ma} with quadratic component $P$. \n\\end{Theorem}", + "thm:harnack": "\\begin{Theorem}\\label{thm:harnack}\nLet $\\Omega$ and $\\widetilde{\\Omega}$ be open convex subsets of $\\mathbb{R}^n$ satisfying $B_1\\subset \\Omega \\subset B_n, B_1\\subset \\widetilde{\\Omega} \\subset B_n$, $n\\geq 2$, and let $\\phi\\in C^2(\\overline{\\Omega})$ and $\\tilde{\\phi}\\in C^2(\\overline{\\widetilde{\\Omega}})$ be convex functions satisfying, \n\\[\n\\mu_\\phi =\\mu \\quad \\text{in } \\Omega, \\quad \\phi=0 \\quad \\text{on } \\partial \\Omega,\n\\]\n\\[ \n\\mu_{\\tilde{\\phi}} =\\mu \\quad \\text{in } \\widetilde{\\Omega}, \\quad \\tilde{\\phi}=0 \\quad \\text{on } \\partial \\widetilde{\\Omega},\n\\] \nwhere $\\mu$ is a periodic Borel measure defined on $\\R^n$ satisfying \\eqref{eq:period gamma}, \\eqref{eq:normalized assumption}, and \\eqref{eq:small period assumption}. Assume that $v\\in C^2(\\Omega) \\cap C^2(\\widetilde{\\Omega})$ with $v\\geq 0$ satisfies\n\\[\nL_{\\phi}v \\leq 0 \\quad \\text{in } \\Omega, \\quad L_{\\tilde{\\phi}}v \\geq 0 \\quad \\text{in }\\widetilde{\\Omega}.\n\\] \n\nThen there exist positive constants $\\beta(n)$, $\\Lambda_9(n)$, and $C(n)$ such that if $\\Lambda>\\Lambda_9(n)$, then either \n\\[\n\\sup_{B_{1/4}}v\\leq C\\inf_{B_{1/4}}v,\n\\]\nor for $\\kappa =\\kappa(B_1)\\geq \\Lambda^2$ that\n\\[\n\\sup_{B_{1/2}} v \\geq e^{\\beta \\kappa^{\\frac{1}{6}}}\\sup_{B_{1/4}}v.\n\\] \nConsequently, we have\n\\[\n\\sup_{B_{1/4}}v \\leq C\\inf_{B_{1/4}}v+e^{-\\beta \\kappa^{\\frac{1}{6}}}\\sup_{B_{1/2}} v.\n\\]\n\\end{Theorem}", + "eq:periodic f": "\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}", + "def:periodic solution": "\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic with respect to $\\mathbb{Z}^n$ and $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\R^n$ and $c\\in\\R$. \n\\end{Definition}", + "thm:harnack super": "\\begin{Theorem}\\label{thm:harnack super} \nAssume that $v \\in C^2(\\Omega)$ is a nonnegative supersolution of \\eqref{eq:linearized ma phi} in $\\Omega$.\nThen there exist positive constants $\\Lambda_6(n)$, $\\epsilon(n) $ and $C(n) $ such that if \n\\[\n\\inf_{B_{1/4}} v \\leq 1,\n\\]\nthen for $\\Lambda\\ge \\Lambda_6(n)$, we have\n\\begin{equation}\\label{eq:harnack super} \n\\mu(\\{M_{\\phi,\\Lambda}v > t\\} \\cap B_{1/2})< Ct^{-\\epsilon} .\n\\end{equation} \n\\end{Theorem}", + "prop:semiconcave": "\\begin{Proposition}\\label{prop:semiconcave}\nLet $u$ be a convex solution to \\eqref{eq:periodic equation ma} with $\\mu$ being a $\\mathbb{Z}^n/8n$-periodic measure (where $\\mathbb{Z}^n := \\Gamma_{\\{e_1,\\cdots,e_n\\}}$) satisfying $\\mu(\\mathbb{Z}^n) = 1$. \nUnder the nondegeneracy condition\n\\begin{equation}\\label{eq:nondegenerate}\n\\Delta^2_{i} u(0):=\\Delta^2_{e_i} u(0) > 0 \\quad \\text{for all } 1 \\leq i \\leq n,\n\\end{equation}\nthere exists a unique compatible matrix $A$ satisfying:\n\\[\n\\sup_{ \\R^n}\\Delta_z^2 u=z^{\\top}Az ,\\quad \\forall z \\in \\mathbb{Z}^n.\n\\]\n\\end{Proposition}", + "thm:harnack sub": "\\begin{Theorem}\\label{thm:harnack sub} \nAssume that $v \\in C^2(\\Omega)$ with $v \\geq 0$ is a subsolution of \\eqref{eq:linearized ma phi} in $\\Omega$.\nThen, there exist positive constants $\\Lambda_7(n)$ and $\\beta(n)$ such that for any $\\epsilon > 0$, there is a constant $C_2(n, \\epsilon)$ with the following property: If $\\Lambda\\ge \\Lambda_7(n)$ and\n\\[\n\\min\\{\\mu(\\{v > t\\} \\cap B_{1/2}), \\mu(\\{M_{\\phi,\\Lambda}v > t\\} \\cap B_{1/2}) \\} \\leq t^{-\\epsilon} \\quad \\forall t>0, \n\\]\nthen either\n\\[\n\\sup_{B_{1/4} } v \\leq C_2(n,\\epsilon), \n\\] \nor for $\\kappa =\\kappa(B_1)\\geq \\Lambda^2$ that\n\\[\n\\sup_{B_{1/2} } v \\geq e^{\\beta \\kappa^{\\frac{1}{3}} } \\sup_{B_{1/4} }v.\n\\] \n\\end{Theorem}" + }, + "pre_theorem_intro_text_len": 1825, + "pre_theorem_intro_text": "In this paper, we study convex solutions to the Monge-Amp\\`ere equation\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\mathbb{R}^n,\n\\end{equation}\nwhere $\\mu\\not\\equiv 0$ is a nonnegative locally finite Borel measure on $\\mathbb{R}^n$ and is periodic in $n$ linearly independent directions. By exploiting the affine invariance of the equation, we may, without loss of generality, restrict our analysis to the case where $\\mu $ is periodic with respect to the integer lattice $\\mathbb{Z}^n := \\{(k_1,\\dots,k_n) \\mid k_i \\in \\mathbb{Z}\\} \\subset \\mathbb{R}^n$, i.e., for any Borel set $E \\subset \\mathbb{R}^n$, \n\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}\n\nLet $\\mathcal{S}_+^{n\\times n}$ denote the set of positive definite symmetric $n\\times n$ matrices.\n\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic with respect to $\\mathbb{Z}^n$ and $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\mathbb{R}^n$ and $c\\in\\mathbb{R}$. \n\\end{Definition}\n\nNote that if $u$ is a periodic solution to \\eqref{eq:periodic equation ma} and is expressed as \\eqref{eq:periodic solution}, then the quadratic component $P(x)$ always satisfies the compatibility condition\n\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation} \nwhere $\\mathbb{T}^n=\\mathbb{R}^n/\\mathbb{Z}^n$.\n\nOur first result is the uniqueness of periodic solutions.", + "context": "In this paper, we study convex solutions to the Monge-Amp\\`ere equation\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\mathbb{R}^n,\n\\end{equation}\nwhere $\\mu\\not\\equiv 0$ is a nonnegative locally finite Borel measure on $\\mathbb{R}^n$ and is periodic in $n$ linearly independent directions. By exploiting the affine invariance of the equation, we may, without loss of generality, restrict our analysis to the case where $\\mu $ is periodic with respect to the integer lattice $\\mathbb{Z}^n := \\{(k_1,\\dots,k_n) \\mid k_i \\in \\mathbb{Z}\\} \\subset \\mathbb{R}^n$, i.e., for any Borel set $E \\subset \\mathbb{R}^n$, \n\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}\n\nLet $\\mathcal{S}_+^{n\\times n}$ denote the set of positive definite symmetric $n\\times n$ matrices.\n\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic with respect to $\\mathbb{Z}^n$ and $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\mathbb{R}^n$ and $c\\in\\mathbb{R}$. \n\\end{Definition}\n\nNote that if $u$ is a periodic solution to \\eqref{eq:periodic equation ma} and is expressed as \\eqref{eq:periodic solution}, then the quadratic component $P(x)$ always satisfies the compatibility condition\n\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation} \nwhere $\\mathbb{T}^n=\\mathbb{R}^n/\\mathbb{Z}^n$.\n\nOur first result is the uniqueness of periodic solutions.\n\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\R^n,\n\\end{equation}", + "full_context": "In this paper, we study convex solutions to the Monge-Amp\\`ere equation\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\mathbb{R}^n,\n\\end{equation}\nwhere $\\mu\\not\\equiv 0$ is a nonnegative locally finite Borel measure on $\\mathbb{R}^n$ and is periodic in $n$ linearly independent directions. By exploiting the affine invariance of the equation, we may, without loss of generality, restrict our analysis to the case where $\\mu $ is periodic with respect to the integer lattice $\\mathbb{Z}^n := \\{(k_1,\\dots,k_n) \\mid k_i \\in \\mathbb{Z}\\} \\subset \\mathbb{R}^n$, i.e., for any Borel set $E \\subset \\mathbb{R}^n$, \n\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}\n\nLet $\\mathcal{S}_+^{n\\times n}$ denote the set of positive definite symmetric $n\\times n$ matrices.\n\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic with respect to $\\mathbb{Z}^n$ and $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\mathbb{R}^n$ and $c\\in\\mathbb{R}$. \n\\end{Definition}\n\nNote that if $u$ is a periodic solution to \\eqref{eq:periodic equation ma} and is expressed as \\eqref{eq:periodic solution}, then the quadratic component $P(x)$ always satisfies the compatibility condition\n\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation} \nwhere $\\mathbb{T}^n=\\mathbb{R}^n/\\mathbb{Z}^n$.\n\nOur first result is the uniqueness of periodic solutions.\n\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\R^n,\n\\end{equation}\n\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite periodic Borel measure on $\\R^n$. We show that any convex solution to the Monge-Amp\\`ere equation\n\\[ \n\\det D^2 u = \\mu \\quad \\text{in } \\R^n\n\\]\nadmits a unique decomposition (up to addition of constants) as the sum of a quadratic polynomial and a periodic function. This result extends,\nin full generality, the earlier works for the case $\\mu=f(x)\\,\\ud x$: when $\\log f \\in C^\\alpha$, it was established by Caffarelli and Li; and\nwhen $\\log f$ is merely bounded, it was proved by Li and Lu. Our result thus answers a question raised by Li and Lu. A key ingredient in the proof is a new dichotomous Harnack-type inequality for linearized Monge-Amp\\`ere equations with nonnegative periodic measures.\n\nLet $\\mathcal{S}_+^{n\\times n}$ denote the set of positive definite symmetric $n\\times n$ matrices.\n\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic with respect to $\\mathbb{Z}^n$ and $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\R^n$ and $c\\in\\R$. \n\\end{Definition}\n\nOur first result is the uniqueness of periodic solutions.\n\nThe existence of periodic solutions follows easily from Theorem 2.1 in Li \\cite{li1990existence} and its proof. For the uniqueness, when $\\mu=f(x)\\,\\ud x$ is a positive measure, the case $\\log f \\in C^\\alpha$ was proved by Li \\cite{li1990existence}, and the case $\\log f\\in L^\\infty$ was shown by Li-Lu \\cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.\n\n\\begin{Theorem}\\label{thm:solutions are periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $u$ be a convex solution of \\eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \\ref{def:periodic solution}.\n\\end{Theorem}\n\n\\begin{Proposition}\\label{prop:semiconcave}\nLet $u$ be a convex solution to \\eqref{eq:periodic equation ma} with $\\mu$ being a $\\mathbb{Z}^n/8n$-periodic measure (where $\\mathbb{Z}^n := \\Gamma_{\\{e_1,\\cdots,e_n\\}}$) satisfying $\\mu(\\mathbb{Z}^n) = 1$. \nUnder the nondegeneracy condition\n\\begin{equation}\\label{eq:nondegenerate}\n\\Delta^2_{i} u(0):=\\Delta^2_{e_i} u(0) > 0 \\quad \\text{for all } 1 \\leq i \\leq n,\n\\end{equation}\nthere exists a unique compatible matrix $A$ satisfying:\n\\[\n\\sup_{ \\R^n}\\Delta_z^2 u=z^{\\top}Az ,\\quad \\forall z \\in \\mathbb{Z}^n.\n\\]\n\\end{Proposition} \n\\begin{proof}\nWithout loss of generality, we may assume $u(0) = 0$ and $u \\geq 0$. \nLet us define $S_h := \\{x \\in \\mathbb{R}^n : u(x) \\leq h\\}$, and select $\\Lambda_u>0$ such that the section $S_{\\Lambda_u}$ satisfies $\\kappa_{\\mathbb{Z}^n}(S_{\\Lambda_u}) \\geq \\Lambda_0^2$. \nLet us now normalize $S_h$ such that $B_1(0) \\subset T_h^{-1} S_h \\subset B_n(0)$ and consider the normalized function\n\\[\nu_{h}(x) =\\frac{u(T_h x)}{(\\det T_h)^{\\frac{2}{n}}} \\quad \\text{with } \\det D^2 u_h= \\mu_h=\\mu \\circ T_h .\n\\] \nDue to the periodicity of $\\mu$ and large $h$, we know that \n\\[\nc(n)h^{\\frac{n}{2}} \\leq \\det T_h \\leq C(n)h^{\\frac{n}{2}},\n\\] \nand thus, $c(n)h^{\\frac{n}{2}} \\leq |S_h| \\leq C(n)h^{\\frac{n}{2}}$. Applying Proposition \\ref{prop:basic} to $\\phi=u_h-h(\\det T_h)^{-\\frac{2}{n}}$, we obtain that for sufficiently large $h$, \n\\[\nc(n) (h/\\Lambda_u)^{\\frac{1}{3}} S_{\\Lambda_u} \\subset S_h \\subset C(n) (h/\\Lambda_u)^{\\frac{2}{3}} S_{\\Lambda_u},\n\\] \nand consequently, $B_{c(n) h^{\\frac{1}{4}}}\\subset S_h\\subset B_{C(n) h^{\\frac{3}{4}}}$ for all large $h$. That is, for sufficiently large $h$, we now have\n\\[\n\\quad c(n)h^{\\frac{1}{4}} I\\leq T_h \\leq C(n)h^{\\frac{3}{4}} I.\n\\]\nThe transformation law \\eqref{eq:transformation law} then yields the following estimates at corresponding points \n\\begin{equation}\\label{eq:transformh}\nc(n)h^{-\\frac{1}{2}}\\Delta^2_{e_i} u \\leq \\Delta^2_{T_h^{-1} e_i} u_h \\leq C(n)h^{\\frac{1}{2}}\\Delta^2_{e_i} u.\n\\end{equation}\n\n\\textbf{Step 4.} Suppose for some sequence $h \\to \\infty$, the rescaled functions $\\tilde{u}_h$ locally converge to a quadratic form $P_A(x) = \\frac{1}{2}x^{\\top}A x$ for $A \\in \\mathcal{S}_+^{n\\times n}$ with $\\det A = 1$. \nFor each fixed direction $i$, we denote\n\\[\n\\alpha=\\sup_{\\R^n}\\Delta^2_{e_i} u,\\quad \\beta= e_i^{\\top} A e_i.\n\\] \nNote that the estimate \\eqref{eq:w21 estimate} continues to hold for $\\tilde{u}_{h}$, from which we derive $\\alpha \\geq \\beta$.\nWe claim\n\\begin{equation}\\label{eq:w2infty sup}\n\\alpha=\\beta.\n\\end{equation}\nAssume by contradiction that $\\alpha = \\beta + 4s$ for some $s > 0$.\nThrough rescaling from infinity, we may assume without loss of generality that there exist points $x_h \\in B_{1/8}$ satisfying\n\\[\n\\alpha- \\Delta^2_{h^{\\frac{1}{2}}e_i} \\tilde{u}_{h}(x_h)= \\inf_{B_{1/8}} (\\alpha- \\Delta^2_{h^{\\frac{1}{2}} e_i} \\tilde{u}_{h} ) \\leq a_h,\n\\]\nwhere $a_h \\to 0 $ as $h \\to \\infty$. Define the normalized function $v = (\\alpha - \\Delta^2_{h^{\\frac{1}{2}}e_i} \\tilde{u}_{h})/a_h$ and set $\\delta = s/[C(\\beta + 2s)]$. Applying Lemma \\ref{lem:Theorem 7.3.1 modify} yields with $\\phi=\\tilde{u}_h-1$ and $\\Omega=S_{1,h}=\\{\\tilde{u}_h \\leq 1\\} $, we obtain that\n\\[\n\\tilde{\\mu}_h ( \\{v\\leq M_1(n,\\delta) \\} \\cap S_{1,h} ) \\geq (1-\\delta) \\tilde{\\mu}_h(S_{1,h} ),\n\\]\nwhere $\\tilde{\\mu}_h=\\tilde{f}_{h}\\,\\ud x$. For sufficiently small $a_h$, we consequently obtain\n\\[\n\\tilde{\\mu}_h\\left(\\left\\{\\Delta^2_{h^{\\frac{1}{2}} e_i} \\tilde{u}_h>\\beta +2 s\\right\\} \\cap S_{1,h}\\right)= \\tilde{\\mu}_h\\left(\\left\\{v <\\frac{2s}{a_h}\\right\\} \\cap S_{1,h}\\right)\\geq (1-\\delta) \\tilde{\\mu}_h (S_{1,h}).\n\\]\nThis leads to the integral lower bound\n\\[\n\\int_{S_{1,h}} \\Delta^2_{h^{\\frac{1}{2}} e_i} \\tilde{u}_h \\ud \\tilde{\\mu}_h \\geq (1-\\delta) (\\beta+2s) \\tilde{\\mu}_h (S_{1,h})\\geq \n(\\beta+s) \\tilde{\\mu}_h (S_{1,h}).\n\\]\nThis contradicts the uniform estimate \\eqref{eq:w21 estimate} after performing a suitable rescaling.\n\n\\begin{proof}[Proof of Theorems \\ref{thm:uniqueness within periodic} and \\ref{thm:solutions are periodic}] \nLet $u$ be a global solution of \\eqref{eq:periodic equation ma}. \nClearly, $u$ cannot be linear on any ray segment. Indeed, if it were linear on a ray, after subtracting its supporting hyperplane on this ray and by using the convexity of $u$, $u$ would be bounded within any cylinder having that ray as its axis. This would imply that the associated measure $\\mu$ is finite on any strictly smaller cylinder, contradicting the assumptions that $\\mu$ is periodic and $\\mu \\not\\equiv 0$. Therefore, there exists a sufficiently large integer $R$ such that $S_1(0) \\subset B_R(0)$.\nWe now rescale $u$ by defining $u(8nRx)/64n^2R^2$. For simplicity, we continue to denote this rescaled function as $u$. Under this scaling, one can verify that condition \\eqref{eq:nondegenerate} in Proposition \\ref{prop:semiconcave} is satisfied. Hence, by Proposition \\ref{prop:semiconcave}, there exists a compatible matrix $A \\in \\mathcal{S}_+^{n \\times n}$ such that \n\\[\n\\sup_{ \\R^n}\\Delta_z^2 u=z^{\\top}Az,\\quad \\forall z \\in \\mathbb{Z}^n.\n\\]\nLet $w$ be a periodic solution to \\eqref{eq:periodic equation ma} in the sense of Definition \\ref{def:periodic solution} with quadratic part $\\frac{1}{2}z^{\\top} A z $. Then $\\Delta_z^2 w=z^{\\top}Az$. Let \n\\[\nv(x)=u(x)-w(x).\n\\]\nBy subtracting a suitable linear function, we now assume that $v(0)=0$ and \n\\[\nv(e_i)=v(-e_i),\\quad 1\\leq i\\leq n.\n\\]\nNoting that $\\sup_{\\R^n} \\Delta_z^2 v = \\sup_{\\R^n} \\Delta_z^2 (u - w) = 0$, we then find that\n\\[\nv(\\pm ke_i)\\leq 0 \\quad \\text{for all } k\\in \\mathbb{Z},\\ 1\\leq i\\leq n.\n\\]", + "post_theorem_intro_text_len": 5293, + "post_theorem_intro_text": "The existence of periodic solutions follows easily from Theorem 2.1 in Li \\cite{li1990existence} and its proof. For the uniqueness, when $\\mu=f(x)\\,\\mathrm{d} x$ is a positive measure, the case $\\log f \\in C^\\alpha$ was proved by Li \\cite{li1990existence}, and the case $\\log f\\in L^\\infty$ was shown by Li-Lu \\cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.\n\nOur second result is the classification below, which extends the main results of Caffarelli-Li \\cite{caffarelli2004liouville} and Li-Lu \\cite{li2019monge} to the full generality. In particular, it provides a complete answer to Question 1.1 in \\cite{li2019monge}. \n\n\\begin{Theorem}\\label{thm:solutions are periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $u$ be a convex solution of \\eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \\ref{def:periodic solution}.\n\\end{Theorem}\n\nCaffarelli-Li \\cite{caffarelli2004liouville} first proved Theorem \\ref{thm:solutions are periodic} for positive measures $\\mu=f(x)\\,\\mathrm{d} x$ with $\\log f \\in C^\\alpha$, and conjectured the extension to merely bounded $\\log f$. This conjecture was resolved by Li-Lu \\cite{li2019monge}, who further asked whether the result holds for the nonnegative case $f \\geq 0$ with $f \\not\\equiv 0$. Theorem \\ref{thm:solutions are periodic} affirmatively answers this question -- and indeed establishes it in full generality.\n\n\\begin{Remark} \nWhen $\\mu \\equiv 0$, equation \\eqref{eq:periodic equation ma} reduces to the homogeneous Monge-Amp\\`ere equation $\\det D^2u = 0$. The solutions are precisely those functions that are linear along certain directions (see, e.g., Caffarelli-Nirenberg-Spruck \\cite{caffarelli1986dirichlet}). \nIn particular, writing $x = (x',x_n) \\in \\mathbb{R}^{n-1} \\times \\mathbb{R}$, any convex function of the form $u(x) = w(x')$ is a solution, and such solutions are not necessarily periodic.\n\\end{Remark}\n\nSince $\\mu$ is assumed to be nonnegative, equation \\eqref{eq:periodic equation ma} represents a possibly degenerate Monge-Amp\\`ere equation, for which many arguments in \\cite{caffarelli2004liouville,li2019monge} do not apply. The principal challenge stems from the behavior of $u$ on small sections: the associated measure $\\mu$ may not satisfy the doubling condition. This failure implies that sections lose the engulfing property, consequently invalidating the Vitali or Besicovitch covering lemma which is essential for the Calder\\'on-Zygmund decomposition on sections. \nTherefore, we cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\\'errez \\cite{caffarelli1997properties} for subsolutions or supersolutions of linearized Monge-Amp\\`ere equations. \n\nNonetheless, by exploiting the equation's periodic structure, we derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator (see Definition \\ref{def:phi homogenization}). This constitutes one of the key innovations of the present work. For supersolutions, we address the challenge by excluding contributions from small sections. Theorem \\ref{thm:harnack super} establishes decay estimates for homogenized level-sets of supersolutions within our framework. For subsolutions, for which standard $L^{\\infty}$ bounds fail, Theorem \\ref{thm:harnack sub} establishes an alternative dichotomy regarding their growth behavior: either the $L^{\\infty}$ bound holds or the $L^{\\infty}$ norm grows exponentially. Finally, we obtain in Theorem \\ref{thm:harnack} a Harnack inequality with a quantitatively controlled error term.\n\nAfter overcoming this critical difficulty arising from the degeneracy of equation \\eqref{eq:periodic equation ma}, the rest proof focuses on analyzing the second-order difference quotient $\\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart. As observed in \\cite{caffarelli2004liouville}, these quantities naturally arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\\`ere equations. This is another point where the periodicity of the measure plays a crucial role. We then establish the quadratic behavior of solutions near infinity, extending the arguments in \\cite{caffarelli2004liouville,li2019monge} for positive measures $\\mu=f(x)\\mathrm{d} x$ to the degenerate case considered here. In this step, we must also overcome additional difficulties due to the degeneracy of\n \\eqref{eq:periodic equation ma}. Consequently, when Theorem \\ref{thm:harnack} is applied to the difference of two solutions, exponential growth is precluded. Hence, the classical Harnack inequality holds for this difference, and our theorems would follow. \n\nThis paper is organized as follows. In Section \\ref{sec:harnack}, we study convex functions whose Monge-Amp\\`ere measure has small period, establishing a dichotomous Harnack type inequality for linearized Monge-Amp\\`ere equations. Section \\ref{sec:semiconvavity} investigates the uniqueness of compatible quadratic component of each solution in the sense of Proposition \\ref{prop:semiconcave}. Finally, in Section \\ref{sec:main theorem}, we prove Theorems \\ref{thm:uniqueness within periodic} and \\ref{thm:solutions are periodic}.", + "sketch": "The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \\cite{li1990existence} and its proof,” while Theorem~\\ref{thm:uniqueness within periodic} “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including Theorem~\\ref{thm:uniqueness within periodic}) in the nonnegative/degenerate setting: “the associated measure $\\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\\'errez.”\n\nTo overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” For supersolutions they do so “by excluding contributions from small sections,” obtaining “decay estimates for homogenized level-sets” (Theorem~\\ref{thm:harnack super}); for subsolutions (where “standard $L^{\\infty}$ bounds fail”), they prove a “dichotomy” (Theorem~\\ref{thm:harnack sub}): “either the $L^{\\infty}$ bound holds or the $L^{\\infty}$ norm grows exponentially”; combining these yields “a Harnack inequality with a quantitatively controlled error term” (Theorem~\\ref{thm:harnack}).\n\n“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \\cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying Theorem~\\ref{thm:harnack} “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and “our theorems would follow.”", + "expanded_sketch": "The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \\cite{li1990existence} and its proof,” while in establishing the main theorem it “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including the main theorem) in the nonnegative/degenerate setting: “the associated measure $\\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\\'errez.”\n\nTo overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” We first use the following theorem for supersolutions:\n\\begin{Theorem}\\label{thm:harnack super} \nAssume that $v \\in C^2(\\Omega)$ is a nonnegative supersolution of \\eqref{eq:linearized ma phi} in $\\Omega$.\nThen there exist positive constants $\\Lambda_6(n)$, $\\epsilon(n) $ and $C(n) $ such that if \n\\[\n\\inf_{B_{1/4}} v \\leq 1,\n\\]\nthen for $\\Lambda\\ge \\Lambda_6(n)$, we have\n\\begin{equation}\\label{eq:harnack super} \n\\mu(\\{M_{\\phi,\\Lambda}v > t\\} \\cap B_{1/2})< Ct^{-\\epsilon} .\n\\end{equation} \n\\end{Theorem}\nFor subsolutions (where “standard $L^{\\infty}$ bounds fail”), they prove the following “dichotomy”:\n\\begin{Theorem}\\label{thm:harnack sub} \nAssume that $v \\in C^2(\\Omega)$ with $v \\geq 0$ is a subsolution of \\eqref{eq:linearized ma phi} in $\\Omega$.\nThen, there exist positive constants $\\Lambda_7(n)$ and $\\beta(n)$ such that for any $\\epsilon > 0$, there is a constant $C_2(n, \\epsilon)$ with the following property: If $\\Lambda\\ge \\Lambda_7(n)$ and\n\\[\n\\min\\{\\mu(\\{v > t\\} \\cap B_{1/2}), \\mu(\\{M_{\\phi,\\Lambda}v > t\\} \\cap B_{1/2}) \\} \\leq t^{-\\epsilon} \\quad \\forall t>0, \n\\]\nthen either\n\\[\n\\sup_{B_{1/4} } v \\leq C_2(n,\\epsilon), \n\\] \nor for $\\kappa =\\kappa(B_1)\\geq \\Lambda^2$ that\n\\[\n\\sup_{B_{1/2} } v \\geq e^{\\beta \\kappa^{\\frac{1}{3}} } \\sup_{B_{1/4} }v.\n\\] \n\\end{Theorem}\nCombining these yields “a Harnack inequality with a quantitatively controlled error term,” namely the following theorem:\n\\begin{Theorem}\\label{thm:harnack}\nLet $\\Omega$ and $\\widetilde{\\Omega}$ be open convex subsets of $\\mathbb{R}^n$ satisfying $B_1\\subset \\Omega \\subset B_n, B_1\\subset \\widetilde{\\Omega} \\subset B_n$, $n\\geq 2$, and let $\\phi\\in C^2(\\overline{\\Omega})$ and $\\tilde{\\phi}\\in C^2(\\overline{\\widetilde{\\Omega}})$ be convex functions satisfying, \n\\[\n\\mu_\\phi =\\mu \\quad \\text{in } \\Omega, \\quad \\phi=0 \\quad \\text{on } \\partial \\Omega,\n\\]\n\\[ \n\\mu_{\\tilde{\\phi}} =\\mu \\quad \\text{in } \\widetilde{\\Omega}, \\quad \\tilde{\\phi}=0 \\quad \\text{on } \\partial \\widetilde{\\Omega},\n\\] \nwhere $\\mu$ is a periodic Borel measure defined on $\\R^n$ satisfying \\eqref{eq:period gamma}, \\eqref{eq:normalized assumption}, and \\eqref{eq:small period assumption}. Assume that $v\\in C^2(\\Omega) \\cap C^2(\\widetilde{\\Omega})$ with $v\\geq 0$ satisfies\n\\[\nL_{\\phi}v \\leq 0 \\quad \\text{in } \\Omega, \\quad L_{\\tilde{\\phi}}v \\geq 0 \\quad \\text{in }\\widetilde{\\Omega}.\n\\] \n\nThen there exist positive constants $\\beta(n)$, $\\Lambda_9(n)$, and $C(n)$ such that if $\\Lambda>\\Lambda_9(n)$, then either \n\\[\n\\sup_{B_{1/4}}v\\leq C\\inf_{B_{1/4}}v,\n\\]\nor for $\\kappa =\\kappa(B_1)\\geq \\Lambda^2$ that\n\\[\n\\sup_{B_{1/2}} v \\geq e^{\\beta \\kappa^{\\frac{1}{6}}}\\sup_{B_{1/4}}v.\n\\] \nConsequently, we have\n\\[\n\\sup_{B_{1/4}}v \\leq C\\inf_{B_{1/4}}v+e^{-\\beta \\kappa^{\\frac{1}{6}}}\\sup_{B_{1/2}} v.\n\\]\n\\end{Theorem}\n\n“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \\cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying the preceding theorem (the Harnack inequality with controlled error term) “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and this completes the proof strategy for the main theorem.", + "expanded_theorem": "\\label{thm:uniqueness within periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying\n\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}\nand let $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ be a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$ satisfying\n\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation}\nThen, to prove the main theorem, we show that there exists a unique periodic solution (up to addition of constants) to\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\R^n,\n\\end{equation}\nwith quadratic component $P$.", + "theorem_type": [ + "Uniqueness", + "Existence" + ], + "mcq": { + "question": "Let $\\mu\\not\\equiv 0$ be a nonnegative locally finite Borel measure on $\\mathbb{R}^n$ that is $\\mathbb{Z}^n$-periodic in the sense that for every Borel set $E\\subset \\mathbb{R}^n$ and every $z\\in \\mathbb{Z}^n$,\\n\\[\\n\\mu(E+z)=\\mu(E).\\n\\]\\nLet\\n\\[\\nP(x)=\\tfrac12 x^\\top A x+b\\cdot x+c,\\n\\]\\nwhere $A\\in \\mathcal S_+^{n\\times n}$ is positive definite symmetric, $b\\in \\mathbb{R}^n$, $c\\in \\mathbb{R}$, and\\n\\[\\n\\det A=\\mu(\\mathbb{T}^n),\\qquad \\mathbb{T}^n=\\mathbb{R}^n/\\mathbb{Z}^n.\\n\\]\\nA periodic solution of\\n\\[\\n\\det D^2u=\\mu\\quad\\text{in }\\mathbb{R}^n\\n\\]\\nmeans a convex function $u\\in C(\\mathbb{R}^n)$ satisfying the equation in the Alexandrov sense and admitting a decomposition $u(x)=v(x)+P(x)$ with $v$ $\\mathbb{Z}^n$-periodic. Under these assumptions, which statement holds?", + "correct_choice": { + "label": "A", + "text": "There exists a periodic solution $u$ of $\\det D^2u=\\mu$ in $\\mathbb{R}^n$ with quadratic component $P$, and it is unique up to addition of constants; equivalently, if $u_1$ and $u_2$ are periodic solutions with quadratic component $P$, then $u_1-u_2$ is constant on $\\mathbb{R}^n$." + }, + "choices": [ + { + "label": "B", + "text": "There exists a periodic solution $u$ of $\\det D^2u=\\mu$ in $\\mathbb{R}^n$ with quadratic component $P$, and it is unique without any ambiguity; equivalently, if $u_1$ and $u_2$ are periodic solutions with quadratic component $P$, then $u_1=u_2$ on $\\mathbb{R}^n$." + }, + { + "label": "C", + "text": "There exists at most one periodic solution $u$ of $\\det D^2u=\\mu$ in $\\mathbb{R}^n$ with quadratic component $P$ up to addition of constants; equivalently, if $u_1$ and $u_2$ are periodic solutions with quadratic component $P$, then $u_1-u_2$ is constant on $\\mathbb{R}^n$." + }, + { + "label": "D", + "text": "For every quadratic function $P(x)=\\tfrac12 x^\\top A x+b\\cdot x+c$ with $A\\in \\mathcal S_+^{n\\times n}$ satisfying $\\det A\\ge \\mu(\\mathbb{T}^n)$, there exists a periodic solution $u$ of $\\det D^2u=\\mu$ in $\\mathbb{R}^n$ with quadratic component $P$, unique up to addition of constants." + }, + { + "label": "E", + "text": "There exists a periodic solution $u$ of $\\det D^2u=\\mu$ in $\\mathbb{R}^n$ with quadratic component $P$, and for any two periodic solutions $u_1,u_2$ with quadratic components having the same matrix part $A$, the difference $u_1-u_2$ is an affine function on $\\mathbb{R}^n$." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "constant-shift ambiguity in comparing two solutions", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped existence while retaining uniqueness up to constants", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "exact compatibility condition $\\det A=\\mu(\\mathbb{T}^n)$", + "template_used": "boundary_range" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "normalization removing linear parts before applying Harnack to the difference", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives the hypotheses and asks for the valid conclusion, but it does not explicitly reveal the correct option. There is no direct wording that singles out existence plus uniqueness up to constants." + }, + "TAS": { + "score": 1, + "justification": "The item is quite close to a theorem-recall question: the assumptions are laid out in theorem style and the correct answer is essentially the theorem’s conclusion. However, the alternatives introduce meaningful nearby variants, so it is not a pure verbatim restatement." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish exact uniqueness up to constants from strict uniqueness, mere uniqueness without existence, and an incorrect compatibility condition. Still, the task is driven more by precise theorem recognition than by substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong and mathematically targeted: they test common mistakes about constant-shift ambiguity, omission of existence, weakening/altering the determinant compatibility condition, and overgeneralizing the difference of two solutions." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with little answer leakage and strong distractors, but it leans heavily on theorem recall rather than deeper generative reasoning." + } + }, + { + "id": "2511.07937v1", + "paper_link": "http://arxiv.org/abs/2511.07937v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "thm", + "content": "[Main Result]\\label{thm:main}\n{\\em (1)} The $h$-polynomial $h_R(t)$ of any cyclotomic standard graded $\\Bbbk$-algebra $R$ is of type CI if $h_R(1) \\in \\{1,4,6\\}$ or $h_R(1)$ is prime. \n\n\\noindent\n{\\em (2)} There exists a cyclotomic standard graded $\\Bbbk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1)=n$ if $n$ is a non-prime number at least $8$.", + "start_pos": 10277, + "end_pos": 10677, + "label": "thm:main" + }, + "ref_dict": { + "prop:ci": "\\begin{prop}[{cf.~\\cite[Corollary~3.4]{S78}}]\\label{prop:ci}\nLet $R$ be a standard graded $\\kk$-algebra. \nIf $R$ is a complete intersection, then the $h$-polynomial of $R$ is of type~CI, that is, $h_R(t)=\\prod_{i=1}^r\\Psi_{m_i}(t)$ for some non-negative integer $r$ and positive integers $m_1, \\dots, m_r$.\n\\end{prop}", + "thm:main": "\\begin{thm}[Main Result]\\label{thm:main}\n{\\em (1)} The $h$-polynomial $h_R(t)$ of any cyclotomic standard graded $\\kk$-algebra $R$ is of type CI if $h_R(1) \\in \\{1,4,6\\}$ or $h_R(1)$ is prime. \n\n\\noindent\n{\\em (2)} There exists a cyclotomic standard graded $\\kk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1)=n$ if $n$ is a non-prime number at least $8$.\n\\end{thm}" + }, + "pre_theorem_intro_text_len": 4524, + "pre_theorem_intro_text": "Throughout this paper, let $\\Bbbk$ denote a field. We say that $R$ is a \\emph{standard graded} $\\Bbbk$-algebra \nif $R = \\bigoplus_{n \\ge 0} R_n$ is a commutative graded $\\Bbbk$-algebra with $R_0 = \\Bbbk$ and finitely generated in degree one.\n(A standard graded $\\Bbbk$-algebra is also called a \\emph{homogeneous} $\\Bbbk$-algebra in other literature.)\n\n\\subsection{Background}\nThe Hilbert series of a (not necessarily commutative) graded $\\Bbbk$-algebra is one of the most fundamental invariants in the study of graded $\\Bbbk$-algebras.\nIt provides a concise analytic expression encoding the dimensions of the homogeneous components and thus captures the growth and structural features of the $\\Bbbk$-algebra as a $\\Bbbk$-vector space. \nIt is a classical and well-known fact that if a standard graded $\\Bbbk$-algebra is Cohen-Macaulay, then the numerator polynomial of its Hilbert series, called the \\textit{$h$-polynomial}, has all positive coefficients.\nThis fact is one of the cornerstones in graded commutative algebra.\nFurthermore, several algebraic properties can be effectively characterized in terms of the Hilbert series. \nA prominent theorem is Stanley's characterization of the Gorenstein property via the symmetry of the Hilbert series (\\cite{S78}), \nwhich has inspired subsequent studies establishing necessary conditions for generalized notions of Gorensteinness, \nsuch as almost Gorensteinness (\\cite{H}), nearly Gorensteinness (\\cite{M}), and Artin-Schelter Gorensteinness (\\cite{JZ}). \n\nIt is also known that when a standard graded $\\Bbbk$-algebra is a complete intersection, its Hilbert series takes a particularly simple form.\nIn fact, for a positive integer $m$, let \\[\\Psi_m(t)=1+t+\\cdots+t^{m-1}.\\]\nThen the $h$-polynomial of a standard graded complete intersection is of the form of a product of a finite collection of $\\Psi_m(t)$'s (see Proposition~\\ref{prop:ci}). \nIn addition, the Hilbert series also serves as an essential tool in investigating noncommutative graded complete intersections (see \\cite{KKZ, KKZ2}).\n\n\\subsection{Cyclotomic standard graded $\\Bbbk$-algebras and polynomials of type CI}\n\nIn recent years, increasing attention has been directed to the finer analytic behavior of the Hilbert series of standard graded $\\Bbbk$-algebras, particularly the roots of their $h$-polynomials. \nThese roots encode subtle structural information and sometimes admit striking combinatorial interpretations \n(see, e.g., the survey \\cite{Br} for combinatorial consequences of the real-rootedness of the numerator polynomials of generating series). \n\nMotivated by this perspective, we focus on the following notion for standard graded $\\Bbbk$-algebras:\n\n\\begin{dfn}[cf. \\cite{BD, KKZ}]\nA standard graded $\\Bbbk$-algebra $R$ is called \\textit{cyclotomic} if all roots of its $h$-polynomial $h_R(t)$ lie on the unit circle in the complex plane. \n\\end{dfn}\n\nMonic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \\textit{Kronecker polynomials}.\nIt is known that every such polynomial can be expressed as a product of cyclotomic polynomials.\nNote that the $h$-polynomials of cyclotomic $\\Bbbk$-algebras are \\textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials. \nWe say that a polynomial is \\textit{of type CI} if it can be written as a product of $\\Psi_m(t)$'s. \nIn particular, the $h$-polynomials of standard graded complete intersections are of type CI. \nHowever, the converse does not hold in general; see, for example, \\cite[Example~3.7]{S78}. \nBorzi and D'Al\\`i showed that the converse does hold when $R$ is Koszul~\\cite[Theorem~3.9]{BD}. \nMoreover, it was proved that a cyclotomic polynomial $\\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\\Bbbk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\\cite[Theorem~4.3]{BD}.\n\nDespite these results, in general, one cannot expect the ring-theoretic structure of $R$ to be determined by the cyclotomicity of its $h$-polynomial. \nThis naturally leads to the following question:\n\\begin{q}\nWhich polynomials can occur as the $h$-polynomials of cyclotomic standard graded $\\Bbbk$-algebras, other than those of type CI? \n\\end{q}\nFor example, \\cite[Theorem~4.3]{BD} indicates that a single cyclotomic polynomial $\\Phi_m(t)$ is rarely equal to the $h$-polynomial of a standard graded $\\Bbbk$-algebra.\n\n\\subsection{Main Results}\n\nThe aim of this paper is to make a contribution to this question.\nOur main result is stated as follows.", + "context": "\\subsection{Background}\nThe Hilbert series of a (not necessarily commutative) graded $\\Bbbk$-algebra is one of the most fundamental invariants in the study of graded $\\Bbbk$-algebras.\nIt provides a concise analytic expression encoding the dimensions of the homogeneous components and thus captures the growth and structural features of the $\\Bbbk$-algebra as a $\\Bbbk$-vector space. \nIt is a classical and well-known fact that if a standard graded $\\Bbbk$-algebra is Cohen-Macaulay, then the numerator polynomial of its Hilbert series, called the \\textit{$h$-polynomial}, has all positive coefficients.\nThis fact is one of the cornerstones in graded commutative algebra.\nFurthermore, several algebraic properties can be effectively characterized in terms of the Hilbert series. \nA prominent theorem is Stanley's characterization of the Gorenstein property via the symmetry of the Hilbert series (\\cite{S78}), \nwhich has inspired subsequent studies establishing necessary conditions for generalized notions of Gorensteinness, \nsuch as almost Gorensteinness (\\cite{H}), nearly Gorensteinness (\\cite{M}), and Artin-Schelter Gorensteinness (\\cite{JZ}).\n\n\\begin{dfn}[cf. \\cite{BD, KKZ}]\nA standard graded $\\Bbbk$-algebra $R$ is called \\textit{cyclotomic} if all roots of its $h$-polynomial $h_R(t)$ lie on the unit circle in the complex plane. \n\\end{dfn}\n\nMonic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \\textit{Kronecker polynomials}.\nIt is known that every such polynomial can be expressed as a product of cyclotomic polynomials.\nNote that the $h$-polynomials of cyclotomic $\\Bbbk$-algebras are \\textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials. \nWe say that a polynomial is \\textit{of type CI} if it can be written as a product of $\\Psi_m(t)$'s. \nIn particular, the $h$-polynomials of standard graded complete intersections are of type CI. \nHowever, the converse does not hold in general; see, for example, \\cite[Example~3.7]{S78}. \nBorzi and D'Al\\`i showed that the converse does hold when $R$ is Koszul~\\cite[Theorem~3.9]{BD}. \nMoreover, it was proved that a cyclotomic polynomial $\\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\\Bbbk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\\cite[Theorem~4.3]{BD}.\n\nDespite these results, in general, one cannot expect the ring-theoretic structure of $R$ to be determined by the cyclotomicity of its $h$-polynomial. \nThis naturally leads to the following question:\n\\begin{q}\nWhich polynomials can occur as the $h$-polynomials of cyclotomic standard graded $\\Bbbk$-algebras, other than those of type CI? \n\\end{q}\nFor example, \\cite[Theorem~4.3]{BD} indicates that a single cyclotomic polynomial $\\Phi_m(t)$ is rarely equal to the $h$-polynomial of a standard graded $\\Bbbk$-algebra.\n\n\\subsection{Main Results}\n\nThe aim of this paper is to make a contribution to this question.\nOur main result is stated as follows.", + "full_context": "\\subsection{Background}\nThe Hilbert series of a (not necessarily commutative) graded $\\Bbbk$-algebra is one of the most fundamental invariants in the study of graded $\\Bbbk$-algebras.\nIt provides a concise analytic expression encoding the dimensions of the homogeneous components and thus captures the growth and structural features of the $\\Bbbk$-algebra as a $\\Bbbk$-vector space. \nIt is a classical and well-known fact that if a standard graded $\\Bbbk$-algebra is Cohen-Macaulay, then the numerator polynomial of its Hilbert series, called the \\textit{$h$-polynomial}, has all positive coefficients.\nThis fact is one of the cornerstones in graded commutative algebra.\nFurthermore, several algebraic properties can be effectively characterized in terms of the Hilbert series. \nA prominent theorem is Stanley's characterization of the Gorenstein property via the symmetry of the Hilbert series (\\cite{S78}), \nwhich has inspired subsequent studies establishing necessary conditions for generalized notions of Gorensteinness, \nsuch as almost Gorensteinness (\\cite{H}), nearly Gorensteinness (\\cite{M}), and Artin-Schelter Gorensteinness (\\cite{JZ}).\n\n\\begin{dfn}[cf. \\cite{BD, KKZ}]\nA standard graded $\\Bbbk$-algebra $R$ is called \\textit{cyclotomic} if all roots of its $h$-polynomial $h_R(t)$ lie on the unit circle in the complex plane. \n\\end{dfn}\n\nMonic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \\textit{Kronecker polynomials}.\nIt is known that every such polynomial can be expressed as a product of cyclotomic polynomials.\nNote that the $h$-polynomials of cyclotomic $\\Bbbk$-algebras are \\textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials. \nWe say that a polynomial is \\textit{of type CI} if it can be written as a product of $\\Psi_m(t)$'s. \nIn particular, the $h$-polynomials of standard graded complete intersections are of type CI. \nHowever, the converse does not hold in general; see, for example, \\cite[Example~3.7]{S78}. \nBorzi and D'Al\\`i showed that the converse does hold when $R$ is Koszul~\\cite[Theorem~3.9]{BD}. \nMoreover, it was proved that a cyclotomic polynomial $\\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\\Bbbk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\\cite[Theorem~4.3]{BD}.\n\nDespite these results, in general, one cannot expect the ring-theoretic structure of $R$ to be determined by the cyclotomicity of its $h$-polynomial. \nThis naturally leads to the following question:\n\\begin{q}\nWhich polynomials can occur as the $h$-polynomials of cyclotomic standard graded $\\Bbbk$-algebras, other than those of type CI? \n\\end{q}\nFor example, \\cite[Theorem~4.3]{BD} indicates that a single cyclotomic polynomial $\\Phi_m(t)$ is rarely equal to the $h$-polynomial of a standard graded $\\Bbbk$-algebra.\n\n\\subsection{Main Results}\n\nThe aim of this paper is to make a contribution to this question.\nOur main result is stated as follows.\n\n\\begin{abstract}\nWe call a standard graded commutative $\\kk$-algebra \\textit{cyclotomic} if its $h$-polynomial has all its roots on the unit circle in the complex plane. \nComplete intersections provide typical examples of cyclotomic algebras, \nsince the $h$-polynomial of any standard graded complete intersection is a product of polynomials of the form $1 + t + \\cdots + t^{m-1}$. \nWe refer to such polynomials as being \\textit{of type CI}. \nA natural question is whether there exists a cyclotomic standard graded $\\kk$-algebra whose $h$-polynomial is not of type CI.\nIn this paper, we give a partial answer to this question. \nWe show that the $h$-polynomial $h_R(t)$ of a cyclotomic standard graded $\\kk$-algebra $R$ is of type CI whenever $h_R(1) \\in \\{1, 4, 6\\}$ or $h_R(1)$ is prime. \nOn the other hand, if $n \\ge 8$ and $n$ is not prime, then there exists a cyclotomic standard graded $\\kk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1) = n$.\n\\end{abstract}\n\nMonic polynomials with integer coefficients whose roots all lie on the unit circle are sometimes called \\textit{Kronecker polynomials}.\nIt is known that every such polynomial can be expressed as a product of cyclotomic polynomials.\nNote that the $h$-polynomials of cyclotomic $\\kk$-algebras are \\textit{not necessarily cyclotomic polynomials themselves}, but rather products of cyclotomic polynomials. \nWe say that a polynomial is \\textit{of type CI} if it can be written as a product of $\\Psi_m(t)$'s. \nIn particular, the $h$-polynomials of standard graded complete intersections are of type CI. \nHowever, the converse does not hold in general; see, for example, \\cite[Example~3.7]{S78}. \nBorzi and D'Al\\`i showed that the converse does hold when $R$ is Koszul~\\cite[Theorem~3.9]{BD}. \nMoreover, it was proved that a cyclotomic polynomial $\\Phi_m(t)$ coincides with $h_R(t)$ of some standard graded $\\kk$-algebra $R$ if and only if $m$ is prime and $R$ is a hypersurface~\\cite[Theorem~4.3]{BD}.\n\nThe aim of this paper is to make a contribution to this question.\nOur main result is stated as follows.\n\nNote that $h_R(1)$ is nothing but the multiplicity of $R$ (see \\cite[Proposition 4.1.9]{BH}), which is one of the important invariants of graded modules.\n\n\\section{Proof of Theorem~\\ref{thm:main} (2)}\\label{sec:(2)}\nThe goal of this section is to give a proof of Theorem~\\ref{thm:main} (2). \nSpecifically, for any non-prime $n$ with $n \\geq 8$, we construct a cyclotomic standard graded $\\kk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$. \nSuch $\\kk$-algebras are realized as Stanley-Reisner rings of certain simplicial complexes.\n\nWe prove the following theorem, which implies Theorem~\\ref{thm:main} (2). \n\\begin{thm}\\label{thm:p=3}\n{\\em (1)} Let $q \\ge 5$ be an odd integer.\nThen there exists a cyclotomic standard graded $\\kk$-algebra $R$ such that $h_R(t)$ is not of type CI with $h_R(1)=2q$.\n\n\\noindent\n{\\em (3)} Let $5 \\le p \\le q$ be odd integers.\nThen there exists a cyclotomic standard graded $\\kk$-algebra $R$ such that $h_R(t)$ is not of type CI with $h_R(1)=pq$. \n\\end{thm}\n\nWe postpone the proof of this theorem. So far, assume Theorem~\\ref{thm:p=3} holds. Then we can prove the desired result. In fact, \nlet $R$ be a cyclotomic standard graded $\\kk$-algebra whose $h$-polynomial is not of type CI. \nThen, for any positive integer $a$, the standard graded $\\kk$-algebra $R'=R[x]/(x^a)$ has the $h$-polynomial $h_{R'}(t)$ which is not of type CI and $h_{R'}(1)=a\\cdot h_R(1)$. \n(In fact, we have $h_{R'}(t)=h_R(t)\\Psi_a(t)$.)\n\n\\begin{thm}[Main Result]\\label{thm:main}\n{\\em (1)} The $h$-polynomial $h_R(t)$ of any cyclotomic standard graded $\\kk$-algebra $R$ is of type CI if $h_R(1) \\in \\{1,4,6\\}$ or $h_R(1)$ is prime. \n\n\\noindent\n{\\em (2)} There exists a cyclotomic standard graded $\\kk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1)=n$ if $n$ is a non-prime number at least $8$.\n\\end{thm}", + "post_theorem_intro_text_len": 714, + "post_theorem_intro_text": "Note that $h_R(1)$ is nothing but the multiplicity of $R$ (see \\cite[Proposition 4.1.9]{BH}), which is one of the important invariants of graded modules.\n\n\\subsection{Structure of the paper}\nWe briefly describe the structure of this paper. \nIn Section~\\ref{sec:pre}, we fix our notation (Hilbert series, $h$-polynomials, simplicial complexes, etc.), \nreview fundamental notions (cyclotomic polynomials, Stanley-Reisner theory, and related topics), \nand recall classical results, namely Macaulay's theorem and the $g$-theorem, which will be used in the proof of Theorem~\\ref{thm:main}. \nSections~\\ref{sec:(1)} and~\\ref{sec:(2)} are devoted to the proofs of parts~(1) and~(2) of Theorem~\\ref{thm:main}, respectively.", + "sketch": "The post-theorem introduction gives only a high-level roadmap: it says that Section~\\ref{sec:pre} recalls classical results—\"namely Macaulay's theorem and the $g$-theorem\"—\"which will be used in the proof of Theorem~\\ref{thm:main},\" and that Sections~\\ref{sec:(1)} and~\\ref{sec:(2)} contain the proofs of parts~(1) and~(2) of Theorem~\\ref{thm:main}, respectively.", + "expanded_sketch": "The post-theorem introduction gives only a high-level roadmap: it says that next it recalls classical results—\"namely Macaulay's theorem and the $g$-theorem\"—\"which will be used in establishing the main theorem,\" and that later it contains the proofs of parts~(1) and~(2) of the main theorem, respectively.", + "expanded_theorem": "[Main Result]\\label{thm:main}\n{\\em (1)} The $h$-polynomial $h_R(t)$ of any cyclotomic standard graded $\\Bbbk$-algebra $R$ is of type CI if $h_R(1) \\in \\{1,4,6\\}$ or $h_R(1)$ is prime. \n\n\\noindent\n{\\em (2)} There exists a cyclotomic standard graded $\\Bbbk$-algebra $R$ whose $h$-polynomial $h_R(t)$ is not of type CI and satisfies $h_R(1)=n$ if $n$ is a non-prime number at least $8$., ", + "theorem_type": [ + "Universal", + "Existence" + ], + "mcq": { + "question": "Let $R$ be a standard graded commutative $\\Bbbk$-algebra with Hilbert series of the form $H_R(t)=\\dfrac{h_R(t)}{(1-t)^d}$, where $h_R(t)$ is its $h$-polynomial. Call $R$ cyclotomic if all roots of $h_R(t)$ lie on the unit circle in $\\mathbb{C}$. A polynomial is said to be of type CI if it can be written as a product of polynomials $\\Psi_m(t)=1+t+\\cdots+t^{m-1}$. Which statement holds for such algebras and $h$-polynomials?", + "correct_choice": { + "label": "A", + "text": "For every cyclotomic standard graded $\\Bbbk$-algebra $R$, if $h_R(1)\\in\\{1,4,6\\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI. Moreover, for every non-prime integer $n\\ge 8$, there exists a cyclotomic standard graded $\\Bbbk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$." + }, + "choices": [ + { + "label": "B", + "text": "For every cyclotomic standard graded $\\Bbbk$-algebra $R$, if $h_R(1)\\in\\{1,4,6,8\\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI. Moreover, for every non-prime integer $n\\ge 9$, there exists a cyclotomic standard graded $\\Bbbk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$." + }, + { + "label": "C", + "text": "For every cyclotomic standard graded $\\Bbbk$-algebra $R$, if $h_R(1)\\in\\{1,4,6\\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI." + }, + { + "label": "D", + "text": "For every cyclotomic standard graded $\\Bbbk$-algebra $R$, if $h_R(1)\\in\\{1,4,6\\}$ or $h_R(1)$ is prime, then $h_R(t)$ is of type CI. Moreover, there exists a non-prime integer $N\\ge 8$ such that for every non-prime integer $n\\ge N$, there exists a cyclotomic standard graded $\\Bbbk$-algebra $R$ with $h_R(t)$ not of type CI and $h_R(1)=n$." + }, + { + "label": "E", + "text": "For every cyclotomic standard graded $\\Bbbk$-algebra $R$, if $h_R(1)\\in\\{1,4,6\\}$ or $h_R(1)$ is composite, then $h_R(t)$ is of type CI. Moreover, for every prime integer $n\\ge 8$, there exists a cyclotomic standard graded $\\Bbbk$-algebra $R$ such that $h_R(t)$ is not of type CI and $h_R(1)=n$." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "B" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "sharp exceptional values and threshold $n\\ge 8$", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped the existence clause in part (2)", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "uniform bound replaced exact range of all non-prime $n\\ge 8$", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "prime/non-prime dichotomy in both directions", + "template_used": "property_confusion" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem only defines the terms and asks which statement is true; it does not reveal the specific exceptional values, quantifiers, or existence claim appearing in the correct option." + }, + "TAS": { + "score": 1, + "justification": "The item is largely theorem-recognition: the correct answer is essentially the exact statement of a result. However, it is not a pure tautology because the options introduce meaningful competing variants with altered thresholds and quantifiers." + }, + "GPS": { + "score": 1, + "justification": "Selecting the best answer requires moderate reasoning or careful recall, especially to track the sharp set {1,4,6}, the prime condition, and the exact existence clause for every non-prime n >= 8. Still, it does not require constructing a proof or applying the theorem in a new setting." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong: they are close to the target statement, differ in mathematically meaningful ways, and reflect common failure modes such as weakening the conclusion, shifting thresholds, or confusing prime/composite cases." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with little answer leakage and strong distractors, but it mainly tests precise theorem recall/comparison rather than deeper generative mathematical reasoning." + } + }, + { + "id": "2511.06484v1", + "paper_link": "http://arxiv.org/abs/2511.06484v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm_main}\nLet $M$ be a closed topological manifold of dimension $2n$ and let $b = \\dim H^2(M,\\mathbb{C})$.\n\nThen there exist non-zero elements $e_1,\\ldots,e_q \\in H^2(M,\\mathbb{C})$ with $q \\leq b+1$ such that if $X$ is a smooth complex projective variety of dimension $n$ with underlying topological space $M$ and $f\\colon X \\to Y$ is a divisorial contraction to a point with exceptional divisor $E$, then there exists $i \\in \\{1, \\dots, q\\}$ such that $[E] = e_i$.", + "start_pos": 4807, + "end_pos": 5309, + "label": "thm_main" + }, + "ref_dict": { + "thm:W_F": "\\begin{theorem}\\label{thm:W_F}\nLet $F \\in \\C[x_0, x_1, \\ldots,x_b]$ be a form of degree $n \\ge 3$. \n\\begin{enumerate}\n\\item If $F$ is honest, then $W_F \\cap \\{F \\ne 0\\}$ is a finite set, and more precisely, \n$$\n|W_F \\cap \\{F \\ne 0\\} | \\le b+1.\n$$\n\n\\item If $F$ is non-degenerate, then $W_F$ is a finite set.\n\\end{enumerate}\n\\end{theorem}", + "s:intersection": "\\label{s:intersection}\n\nLet $M$ be a closed (i.e.\\ compact and without boundary) oriented topological manifold of dimension $d=2n$. \nConsider the symmetric \n$n$-multilinear map defined by the cup prod", + "s_preliminaries": "\\begin{theorem}\\label{thm_main}\nLet $M$ be a closed topological manifold of dimension $2n$ and let $b = \\dim H^2(M,\\mathbb{C})$.\n\nThen there exist non-zero elements $e_1,\\ldots,e_q \\in H^2(M,\\mathbb{C})$ with $q \\leq b+1$ such that if $X$ is a smooth complex projective variety of dimension $n$ with underlying topological space $M$ and $f\\colon X \\to Y$ is a divisorial contraction to a point with exceptional divisor $E$, then there exists $i \\in \\{1, \\dots, q\\}$ such that $[E] = e_i$.\n\\end{theorem}\n\nThis result provides a topological bound on the possible exceptional divisors arising from some of the steps of the Minimal Model Program, establishing a new link between topology and birational geometry. It is a consequence of Theorem \\ref{thm:W_F}, where we prove the finiteness of rank one points of $\\mathcal H_F$ where $F$ is a non-degenerate $n$-form, a result of independent interest. \n\n\\medskip \n\nThe structure of the paper is as follows. In Section \\ref{s_preliminaries}, we introduce the necessary background on tensors and forms. Sections \\ref{s_low} and \\ref{s:intersection} are devoted to the study of the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$ with $n\\ge 3$. We show, in particular, that, up to a non-zero scalar, there are at most finitely many classes $e\\in H^2(M,\\mathbb C)$ such that \n$e^n\\neq 0$ and the rank of the Hessian of $F$ at $e$ is one.\n(cf. Theorem \\ref{thm:W_F}). Section \\ref{s_blowups} establishes our main theorem by examining the rank of the associated form in the presence of a divisorial contraction. In Section \\ref{s_volume}, we present some related open problems. \n\n\\medskip\n\n\\textbf{Acknowledgements:} The first and third authors are members of the GNSAGA - Istituto Nazionale di Alta Matematica.\nThe first author was also partially supported by the PRIN ``Vartietà reali e complesse: geometria, topologia e analisi armonica\". \nThe second author is partially supported by a Simons collaboration grant and would like to thank the National Center for Theoretical Sciences in Taiwan and Professor Jungkai Chen for\ntheir hospitality, where some of the work for this paper was completed.\nThe third author is partially supported by the PRIN2020 research grant ``2020KKWT53”.\nThe authors would also like to thank the referee for careful reading the paper and for several useful comments.\n\n\\section{Preliminaries}\\label{s_preliminaries}\nWe work over the field of complex numbers $\\mathbb C$. A \\emph{form} over $\\mathbb C$ is a homogeneous polynomial $F\\in \\mathbb C[x_0,\\ldots,x_b]$.\n\n\\subsection{Tensors} Let $a_1,\\dots,a_n$ be positive integers. A \\emph{tensor} of type $a_1\\times \\ldots \\times a_n$ is a multilinear map \n$$T\\colon \\mathbb C^{a_1}\\times \\ldots \\times \\mathbb C^{a_n}\\to \\mathbb C.$$\nFor any positive integer $a$, denote $[a]:=\\{ 1,2,\\ldots, a \\}$. Then a tensor $T$ of type $a_1\\times \\ldots \\times a_n$ is determined uniquely by a function \n$$\\tilde T\\colon [a_1]\\times \\cdots \\times [a_n] \\to \\mathbb{C}.$$\nWe will refer to $\\tilde T$ as the \\emph{hypermatrix associated to $T$}. \nGiven positive integers \n$a_j^{'} \\le a_j$ for $1 \\le j \\le n,$ and strictly increasing functions $f_j\\colon [a_j^{'}] \\to [a_j]$,\nwe define a \\emph{sub-tensor} $T'$ of type $a'_1\\times\\ldots\\times a'_n$ so that if $\\tilde T'$ is the hypermatrix associated to $T'$ then\n$$\\tilde T'=\\tilde T(f_1,\\dots,f_n)\\colon [a'_1]\\times \\cdots \\times [a'_n] \\to \\mathbb{C}.$$\nNote that if $v_i\\in \\mathbb C^{a_i}$ for $i=1,\\dots,n$, then there exists a natural tensor of type $a_1\\times \\ldots \\times a_n$ defined by \n$v_1\\otimes \\ldots \\otimes v_n$.\n\n\\begin{definition}\nA nonzero tensor $T$ of type $a_1\\times \\ldots \\times a_n$ has {\\it rank one} if there are nonzero vectors $v_i \\in \\mathbb{C}^{a_i}$ such that $T=v_1 \\otimes \\cdots \\otimes v_n $. We define the\n{\\it rank} of a nonzero tensor $T$ ($=\\rk(T)$) to be the minimum positive integer $r$\nsuch that there exist $r$ tensors $T_1, \\cdots , T_r$ of rank one with \n\n$$T = T_1 + \\cdots + T_r .$$\n\\end{definition}\n\nNote that the rank of any sub-tensor of $T$ cannot be larger than the rank of $T$. \n\\begin{definition}\\label{d_cubic}\nA \\emph{cubic tensor} $T$ is a tensor of type $d^{\\times n} =d \\times \\cdots \\times d$. Let $A$ be the hypermatrix associated to $T$. \nWe denote by $\\det A$ the \\emph{hyperdeterminant} of $A$ (see \\cite[Chapter 14]{GKZ} for the definition and some of its main properties). In particular, $\\det A$ is a polynomial in the entries of $A$. \n\n\\end{definition}\n\nThe following theorem is useful to compute the rank of a tensor:\n\\begin{theorem}{\\cite[Thm. 3.1.1.1]{Landsberg-book}} \\label{thm:slices}\nLet $A_1,\\dots,A_n$ be $l\\times m$ matrices and consider the tensor $A=[A_1,\\ldots,A_n]$ of type $l\\times m\\times n$. \n\nThen $\\rk(A)$ is equal to the minimum number of rank one matrices needed to span a vector space containing $\\langle A_1,\\ldots, A_n \\rangle$.\n\\end{theorem}", + "s_low": "\\begin{proof}\nLet $r=\\rk(A)$ and let $B_1,\\ldots, B_r$ be rank one matrices such that \n$$\\langle A_0,\\ldots, A_q\\rangle\\subset \\langle B_1,\\ldots, B_r\\rangle,$$\nas in Theorem \\ref{thm:slices}. \nThus, for any $j=0,\\ldots,q$, we may write\n$$\nA_j= \\sum_{i=1}^r \\lambda_{ij}B_i\n$$\nfor $\\lambda_{ij} \\in \\C$ for $i=1,\\ldots,r$ and $j=0,\\dots,q$. Since $A_1,\\ldots, A_q$ are linearly independent, after possibly reordering the $B_i$'s, we may assume that the $q\\times q$-matrix \n$$L=(\\lambda_{ij})_{i,j=1,\\ldots,q}$$\n is of maximal rank $q$. Thus, the linear system \n$$L \\cdot X =(-\\lambda_{10},-\\lambda_{20},\\ldots,-\\lambda_{q0})^t$$ \nadmits a solution $(\\mu_1,\\ldots,\\mu_q)\\in \\mathbb C^q$. \nIt follows that \n $$A_0 + \\sum_{i=1}^q \\mu_i A_i=\\sum_{j=q+1}^{r} s_j B_j$$ for some $s_{q+1},\\ldots,s_r \\in \\C$.\n Since the rank of $\\sum_{j=q+1}^{r} s_j B_j$ is at most $r-q$, our assumption implies that $r-q \\ge t$ and the claim follows. \n\\end{proof}\n\n\\section{Points of low rank}\\label{s_low}\n\n\\begin{definition}\n\tGiven a form $F \\in \\C[x_0,\\ldots,x_b]$ of degree $n \\ge 3$ we consider the tensor \nof type $(b+1)^{\\times (n-1)}$\n\t\tdefined by the hypermatrix $\\mathcal H_F$ of the $(n-1)$-th order derivatives, whose entries are linear forms in $x_0, \\ldots, x_b$. \n\tWe say that $F$ is \\emph{honest} if for any non-zero element $v \\in \\mathbb C^{b+1}$, we have $\\mathcal H_F(v) \\ne 0$. \n We say that $F$ is \\emph{non-degenerate} if $ \\det \\mathcal H_F(v) \\ne 0$ for some \n non-zero $v \\in \\mathbb C^{b+1}$\n (note that this definition is consistent with \\cite[Page 7929]{CT18} but not with \\cite[14.1.A]{GKZ}).\n\\end{definition}\n\nNote that the locus $p \\in \\mathbb P^b$ such that $\\mathcal H_F(p) = 0$ is well defined. By \\cite[Remark 6.3.5]{Bocci-Chiantini}, the rank $\\rk \\mathcal H_F(p)$ is also well-defined.\n\n\\begin{example}\\label{ex:degenerate}\n\tConsider the form \n\t$$\n\tF(x_0,\\ldots,x_4)=\\frac{x_0x_1^2}{2} + x_1x_3x_4 + \\frac{x_2x_3^2}{2}.\n\t$$ \n\tThen\n\n\t$$\n\t\\mathcal H_F=\\begin{pmatrix}\n\t0 & x_1 & 0 & 0 & 0 \\\\\n\tx_1 & x_0 & 0 & x_4 & x_3 \\\\\n\t0 & 0 & 0 & x_3 & 0 \\\\ \n\t0 & x_4 & x_3 & x_2 & x_1 \\\\\n\t0 & x_3 & 0 & x_1 & 0 \n\t\\end{pmatrix}\n\t$$\nThus, the form $F$ is honest, but it is degenerate since $\\det \\mathcal H_F$ is identically zero. \n\\end{example}\t\n\nVaguely speaking, an honest form can be characterised by the fact that it depends on all the variables. More precisely, we have:\n\\begin{lemma}\\label{lem:H_F(p)=0}\n\tLet $F \\in \\C[x_0,\\ldots,x_b]$ be a form of degree $n$ and let \n $p:=[1,0,\\ldots,0] \\in \\mathbb P^b$. \n\n Then $\\mathcal H_F(p) = 0$ if and only if $x_0$ does not appear in the expression of $F$. In particular, a non-degenerate form is honest.\n\\end{lemma}\t\n\n\\begin{proof}\n\tAssume first that $\\mathcal H_F(p) = 0$. By the Euler formula for homogeneous polynomials we have that\n\t$$\n\t(n-k)\\partial_{i_1 \\ldots i_k}F(x_0,\\ldots,x_b)= \\sum_{\\ell=0}^b x_\\ell \\partial_{\\ell, i_1 \\ldots i_k}F(x_0, \\ldots, x_b)\n\t$$\n\tfor any $k=0,\\ldots, n-1$ and $i_j \\in \\{0,\\ldots, b\\}$. \n Proceeding by induction and using the fact that $\\mathcal H_F(p) = 0$, we get that $\\partial_{i_1 \\ldots i_k}F(p)=0$ for any $k=0,\\ldots, n-1$ and $i_j \\in \\{0,\\ldots, b\\}$. Thus, $x_0$ does not appear in $F(x_0, \\ldots, x_n)$.\n\tThe converse is a simple computation. \t\t\n\n\\medskip\n\nAssume now that $F(x_0, \\ldots, x_n)$ is a non-degenerate form. If by contradiction there exists $p\\in \\mathbb P^b$ such that $\\mathcal H_F(p) = 0$ then, up to a change of coordinates, we may assume $p=[1,0,\\ldots,0]$. This means that $F$ does not depends on $x_0$ and so the face of $\\mathcal H_F(p)=0$ corresponding to $\\partial_0$ is trivial. By \\cite[Corollary XIV.1.5(d)]{GKZ}, it follows that $\\det \\mathcal H_F$ is identically zero, which is a contradiction.\n\\end{proof}", + "s_blowups": "\\begin{pmatrix}\n0 & x_1 \\\\\nx_1 & x_0 \\\\\n\\end{pmatrix}.\n$$\nBy \\cite[Proposition XIV.1.7]{GKZ}, it follows that $F_X$ is degenerate. \n\\end{example}\n\n\\section{Blow-ups}\\label{s_blowups}\n\n\\begin{lemma}\\label{lem:blowup}\nLet $X$ be a smooth projective variety of dimension $n$ and let $f\\colon Y \\to X$ be the blow-up along a smooth subvariety $Z$ of $X$ of dimension $k < n$. \nLet $[E], \\beta_1:=f^*\\gamma_1, \\ldots, \\beta_b:=f^*\\gamma_b$ be a basis of $H^2(Y,\\Q)$ where $E$ is the exceptional divisor and $\\gamma_1,\\ldots,\\gamma_b$ is a basis of $H^2(X,\\Q)$.\t\nThen, with respect to this basis, we may write\n$$\nF_Y(x_0,\\ldots,x_n)=ax_0^n +\\sum_{i=1}^{n-k} x_0^{n-i}R_i + F_X(x_1, \\ldots, x_b)\n$$\nwhere $a=E^n$ and $R_i \\in \\mathbb Q[x_1,\\ldots,x_b]$ is a form of degree $i$, for $i=1,\\dots,n-k$. \n\n\\end{lemma}\n\\begin{proof}\nThis follows easily from the projection formula. \n\\end{proof}\n\n\\begin{proposition}\\label{prop:blowuprank}\nLet $X$ be a smooth K\\\"ahler manifold of dimension $n$ and let $f\\colon Y \\to X$ be the blow-up along a closed submanifold $Z$ of $X$ of dimension $k \\le 2$. Let $p=[E] \\in H^2(Y,\\C)$ be the class of the exceptional divisor.\n\\begin{enumerate}\n\\item \\label{k=0} If $n\\ge 2$ and $k=0$ then $\\rk \\mathcal H_{F_Y}(p)=1$.\n\\item \\label{k=1} If $n\\ge 3$ and $k=1$ then $\\rk \\mathcal H_{F_Y}(p) \\ge 2$. \n\\item \\label{k=2} If $n\\ge 4$ and $k=2$ then, after a base change, we may write \n$$\nF_Y(x_0,\\ldots,x_b)=ax_0^n + x_0^{n-1}L(x_1,\\ldots,x_b) + x_0^{n-2}Q(x_1,\\ldots,x_b) + F_X(x_1, \\ldots, x_b)\n$$\nwhere $a \\in \\mathbb Q$, $L$ is a linear form and $Q$ is a quardric form of rank $q$ for some positive integer $q$ such that $\\rk \\mathcal H_{F_Y}(p) \\ge 2q$.\n\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nWe use the same notation as in Lemma \\ref{lem:blowup}.\n\nWe first prove \\eqref{k=0}. We have \n$$\nF_Y(x_0,\\ldots,x_n)=ax_0^n + F_X(x_1, \\ldots, x_b)\n$$\nwhere $a=E^n \\ne 0$. Hence $\\rk \\mathcal H_{F_Y}(p)=1$.\nThus, \\eqref{k=0} follows. \n\\medskip \n\nWe now prove \\eqref{k=1}. We have \n$$\nF_Y(x_0,\\ldots,x_n)=ax_0^n+ x_0^{n-1}\\cdot \\left(\\sum_{i=1}^{b}\\lambda_ix_i \\right ) + F_X(x_1, \\ldots, x_b )\n$$\nwhere $a=E^n$ and $\\lambda_i=Z \\cdot \\gamma_i$, for $i=1,\\dots,b$. After taking a base change that fixes $(1,0,\\ldots,0)$, we may assume $\\lambda_i=0$ for any $i=2, \\ldots ,b$, i.e.\n\n$$\nF_Y(x_0,\\ldots,x_n)=ax_0^n+ \\lambda_1 x_0^{n-1}x_1 + F_X(x_1, \\ldots, x_b ),\n$$\nwhere $\\lambda_1 \\ne 0$ because we are blowing up a curve $Z$ in a K\\\"ahler manifold and so there exists at least one class in $H^2(X,\\mathbb Z)$ with non-zero intersection with $Z$. Thus, it is easy to check that the subtensor given by \n$$(\\partial_0^{n-3}\\partial_i\\partial_j)_{i,j=0,\\dots,b}$$ \nhas rank two, which implies that $\\rk \\mathcal H_{F_Y}(p) \\ge 2$. \n\n\\medskip \n\nWe finally prove \\eqref{k=2}. Let $j\\colon Z \\hookrightarrow X$ be the inclusion. \nLet $V \\subset H^2(Z, \\C)$ be the subspace generated by $j^*\\gamma_1, \\ldots, j^*\\gamma_b$ and let $q$ be the rank of the quadratic form $Q$ obtained restrincting to $V$ the quadratic form of $H^2(Z,\\mathbb C)$.\n Note that $q \\ge 1$ because a K\\\"ahler class of $X$ restricts to a K\\\"ahler class on $Z$. \nAfter a base change that diagonalises $Q$, we can write\n$$\nF=F_Y(x_0,\\ldots,x_b)=\\frac{ax_0^n}{n!} + \\frac{x_0^{n-1}}{(n-1)!}(c_1x_1+\\ldots +c_bx_b) + \\frac{x_0^{n-2}}{2(n-2)!}(x_1^2+\\ldots+ x_q^2) + F_X(x_1, \\ldots, x_b)\n$$\nfor some $c_1,\\dots,c_b \\in \\C$. \n\nConsider the $(b+1)\\times (b+1) \\times (b+1)$ subtensor $A$ of $\\mathcal H_{F}(p)$ given by the slices $A_0, \\ldots, A_b$ where \n $$\nA_h= ( \\partial^{n-4}_0 \\partial_h \\partial_i \\partial_j F(p))_{i,j=0,\\dots,b}\n $$\n for $h=0,\\dots,b$. \n\nIt is enough to prove that $\\rk A \\ge 2q$, as it immediately implies that $\\rk \\mathcal H_{F}(p) \\ge 2q$.\nWe have\n$$\nA_0 = \n\\begin{pmatrix}\n1 & c_1 & \\ldots & c_q & c_{q+1} & \\ldots & c_b \\\\\nc_1 & 1 & 0 & \\ldots & \\ldots & \\dots & 0 \\\\\n\\vdots & 0 & \\ddots & 0 & \\cdots & \\cdots & \\vdots \\\\\nc_q & 0 & \\ldots & 1 & 0 & \\ldots & 0 \\\\\nc_{q+1} &\\vdots & \\vdots &\\vdots & 0 & \\ldots & 0 \\\\\n\\vdots&\\vdots & \\vdots &\\vdots & \\vdots & \\ddots & \\vdots \\\\\nc_b & \\cdots & \\cdots & \\cdots & 0 & \\cdots & 0 \\\\\n\\end{pmatrix}", + "s_volume": "\\label{s_volume}\n\nThe goal of this section is to present some open problems on the relationship between birational invariants and topological invariants of a smooth complex projective variety. \n\n\\sub" + }, + "pre_theorem_intro_text_len": 1620, + "pre_theorem_intro_text": "Let $X$ be a compact K\\\"ahler manifold of dimension $n$. One of the fundamental objects associated with $X$ is the natural form $F_X$ on $H^2(X,\\mathbb{Z})$ of degree $n$, induced by the cup product structure. This form serves as a topological invariant of $X$ and plays a crucial role in understanding the intersection theory of higher-dimensional algebraic varieties.\n\nFor $n=2$, the quadratic form $F_X$ is central to the theory of algebraic surfaces, yielding results in classification theory and the study of their moduli spaces. When $n=3$, the study of the cubic form $F_X$ was initiated in \\cite{OV95}, leading to numerous applications in algebraic geometry, differential geometry, and dynamical systems. Recent advancements have further deepened our understanding of $F_X$ and its role in various geometric problems \\cite{KW, BCT16, CT18, ST19, ST20, Wilson21, Wilson21b}.\n\nDespite these advances, relatively little is known about the structure and applications of $F_X$ in dimensions $n \\geq 4$. The goal of this paper is to initiate a study of $F_X$ in higher dimensions, exploring its fundamental properties and potential applications to birational geometry and topology. In particular, we examine how the rank of the hypermatrix $\\mathcal H_{F_X}$ of the $(n-1)$-th derivatives of $F_X$ influence the behavior of divisorial contractions (see Section \\ref{s:intersection} for the definition of $\\mathcal H_{F_X}$ and some of its basic properties).\n\n\\medskip \n\nOur main result establishes the existence of a finite set of cohomology classes that control the exceptional divisors of divisorial contractions:", + "context": "Let $X$ be a compact K\\\"ahler manifold of dimension $n$. One of the fundamental objects associated with $X$ is the natural form $F_X$ on $H^2(X,\\mathbb{Z})$ of degree $n$, induced by the cup product structure. This form serves as a topological invariant of $X$ and plays a crucial role in understanding the intersection theory of higher-dimensional algebraic varieties.\n\nFor $n=2$, the quadratic form $F_X$ is central to the theory of algebraic surfaces, yielding results in classification theory and the study of their moduli spaces. When $n=3$, the study of the cubic form $F_X$ was initiated in \\cite{OV95}, leading to numerous applications in algebraic geometry, differential geometry, and dynamical systems. Recent advancements have further deepened our understanding of $F_X$ and its role in various geometric problems \\cite{KW, BCT16, CT18, ST19, ST20, Wilson21, Wilson21b}.\n\nDespite these advances, relatively little is known about the structure and applications of $F_X$ in dimensions $n \\geq 4$. The goal of this paper is to initiate a study of $F_X$ in higher dimensions, exploring its fundamental properties and potential applications to birational geometry and topology. In particular, we examine how the rank of the hypermatrix $\\mathcal H_{F_X}$ of the $(n-1)$-th derivatives of $F_X$ influence the behavior of divisorial contractions (see Section \\ref{s:intersection} for the definition of $\\mathcal H_{F_X}$ and some of its basic properties).\n\n\\medskip\n\nOur main result establishes the existence of a finite set of cohomology classes that control the exceptional divisors of divisorial contractions:\n\n\\label{s:intersection}\n\nLet $M$ be a closed (i.e.\\ compact and without boundary) oriented topological manifold of dimension $d=2n$. \nConsider the symmetric \n$n$-multilinear map defined by the cup prod", + "full_context": "Let $X$ be a compact K\\\"ahler manifold of dimension $n$. One of the fundamental objects associated with $X$ is the natural form $F_X$ on $H^2(X,\\mathbb{Z})$ of degree $n$, induced by the cup product structure. This form serves as a topological invariant of $X$ and plays a crucial role in understanding the intersection theory of higher-dimensional algebraic varieties.\n\nFor $n=2$, the quadratic form $F_X$ is central to the theory of algebraic surfaces, yielding results in classification theory and the study of their moduli spaces. When $n=3$, the study of the cubic form $F_X$ was initiated in \\cite{OV95}, leading to numerous applications in algebraic geometry, differential geometry, and dynamical systems. Recent advancements have further deepened our understanding of $F_X$ and its role in various geometric problems \\cite{KW, BCT16, CT18, ST19, ST20, Wilson21, Wilson21b}.\n\nDespite these advances, relatively little is known about the structure and applications of $F_X$ in dimensions $n \\geq 4$. The goal of this paper is to initiate a study of $F_X$ in higher dimensions, exploring its fundamental properties and potential applications to birational geometry and topology. In particular, we examine how the rank of the hypermatrix $\\mathcal H_{F_X}$ of the $(n-1)$-th derivatives of $F_X$ influence the behavior of divisorial contractions (see Section \\ref{s:intersection} for the definition of $\\mathcal H_{F_X}$ and some of its basic properties).\n\n\\medskip\n\nOur main result establishes the existence of a finite set of cohomology classes that control the exceptional divisors of divisorial contractions:\n\n\\label{s:intersection}\n\nLet $M$ be a closed (i.e.\\ compact and without boundary) oriented topological manifold of dimension $d=2n$. \nConsider the symmetric \n$n$-multilinear map defined by the cup prod\n\nOur main result establishes the existence of a finite set of cohomology classes that control the exceptional divisors of divisorial contractions:\n\nThis result provides a topological bound on the possible exceptional divisors arising from some of the steps of the Minimal Model Program, establishing a new link between topology and birational geometry. It is a consequence of Theorem \\ref{thm:W_F}, where we prove the finiteness of rank one points of $\\mathcal H_F$ where $F$ is a non-degenerate $n$-form, a result of independent interest.\n\nThe structure of the paper is as follows. In Section \\ref{s_preliminaries}, we introduce the necessary background on tensors and forms. Sections \\ref{s_low} and \\ref{s:intersection} are devoted to the study of the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$ with $n\\ge 3$. We show, in particular, that, up to a non-zero scalar, there are at most finitely many classes $e\\in H^2(M,\\mathbb C)$ such that \n$e^n\\neq 0$ and the rank of the Hessian of $F$ at $e$ is one.\n(cf. Theorem \\ref{thm:W_F}). Section \\ref{s_blowups} establishes our main theorem by examining the rank of the associated form in the presence of a divisorial contraction. In Section \\ref{s_volume}, we present some related open problems.\n\n\\subsection{Tensors} Let $a_1,\\dots,a_n$ be positive integers. A \\emph{tensor} of type $a_1\\times \\ldots \\times a_n$ is a multilinear map \n$$T\\colon \\mathbb C^{a_1}\\times \\ldots \\times \\mathbb C^{a_n}\\to \\mathbb C.$$\nFor any positive integer $a$, denote $[a]:=\\{ 1,2,\\ldots, a \\}$. Then a tensor $T$ of type $a_1\\times \\ldots \\times a_n$ is determined uniquely by a function \n$$\\tilde T\\colon [a_1]\\times \\cdots \\times [a_n] \\to \\mathbb{C}.$$\nWe will refer to $\\tilde T$ as the \\emph{hypermatrix associated to $T$}. \nGiven positive integers \n$a_j^{'} \\le a_j$ for $1 \\le j \\le n,$ and strictly increasing functions $f_j\\colon [a_j^{'}] \\to [a_j]$,\nwe define a \\emph{sub-tensor} $T'$ of type $a'_1\\times\\ldots\\times a'_n$ so that if $\\tilde T'$ is the hypermatrix associated to $T'$ then\n$$\\tilde T'=\\tilde T(f_1,\\dots,f_n)\\colon [a'_1]\\times \\cdots \\times [a'_n] \\to \\mathbb{C}.$$\nNote that if $v_i\\in \\mathbb C^{a_i}$ for $i=1,\\dots,n$, then there exists a natural tensor of type $a_1\\times \\ldots \\times a_n$ defined by \n$v_1\\otimes \\ldots \\otimes v_n$.\n\n\\begin{lemma}\\label{lem:blowup}\nLet $X$ be a smooth projective variety of dimension $n$ and let $f\\colon Y \\to X$ be the blow-up along a smooth subvariety $Z$ of $X$ of dimension $k < n$. \nLet $[E], \\beta_1:=f^*\\gamma_1, \\ldots, \\beta_b:=f^*\\gamma_b$ be a basis of $H^2(Y,\\Q)$ where $E$ is the exceptional divisor and $\\gamma_1,\\ldots,\\gamma_b$ is a basis of $H^2(X,\\Q)$. \nThen, with respect to this basis, we may write\n$$\nF_Y(x_0,\\ldots,x_n)=ax_0^n +\\sum_{i=1}^{n-k} x_0^{n-i}R_i + F_X(x_1, \\ldots, x_b)\n$$\nwhere $a=E^n$ and $R_i \\in \\mathbb Q[x_1,\\ldots,x_b]$ is a form of degree $i$, for $i=1,\\dots,n-k$.\n\n\\begin{proposition}\\label{prop:blowuprank}\nLet $X$ be a smooth K\\\"ahler manifold of dimension $n$ and let $f\\colon Y \\to X$ be the blow-up along a closed submanifold $Z$ of $X$ of dimension $k \\le 2$. Let $p=[E] \\in H^2(Y,\\C)$ be the class of the exceptional divisor.\n\\begin{enumerate}\n\\item \\label{k=0} If $n\\ge 2$ and $k=0$ then $\\rk \\mathcal H_{F_Y}(p)=1$.\n\\item \\label{k=1} If $n\\ge 3$ and $k=1$ then $\\rk \\mathcal H_{F_Y}(p) \\ge 2$. \n\\item \\label{k=2} If $n\\ge 4$ and $k=2$ then, after a base change, we may write \n$$\nF_Y(x_0,\\ldots,x_b)=ax_0^n + x_0^{n-1}L(x_1,\\ldots,x_b) + x_0^{n-2}Q(x_1,\\ldots,x_b) + F_X(x_1, \\ldots, x_b)\n$$\nwhere $a \\in \\mathbb Q$, $L$ is a linear form and $Q$ is a quardric form of rank $q$ for some positive integer $q$ such that $\\rk \\mathcal H_{F_Y}(p) \\ge 2q$.\n\n\\begin{example}\nLet $X$ be a complex projective manifold of dimension $n$ and $b_2(X)=1$. Let $f\\colon Y \\to X$ be the blow-up along a smooth curve $Z$ of $X$. Let $p=[E] \\in H^2(Y,\\C)$ be the class of the exceptional divisor. Then we can write \n$$\nF_Y(x_0,\\ldots,x_n)=\\frac{a}{n!}x_0^n+ \\frac{x_0^{n-1}x_1}{(n-1)!} + x_1^n,\n$$ \nfor some $a \\in \\mathbb Q$. If $n=3$, then \n$$\n\\mathcal H_{F_Y}(p)= \n\\begin{pmatrix}\na & 1 \\\\\n1 & 0 \\\\\n\\end{pmatrix}\n$$\nhas rank 2. If $n=4$, then \n$$\n\\mathcal H_{F_Y}(p)= \n\\left[ \\begin{pmatrix}\na & 1 \\\\\n1 & 0 \\\\\n\\end{pmatrix},\n\\begin{pmatrix}\n1 & 0 \\\\\n0 & 0 \\\\\n\\end{pmatrix}\n\\right]\n$$\nhas rank at least 3 by Lemma \\ref{lem:trick}. On the other hand, 3 is the maximal rank for a $2\\times 2 \\times 2$ tensor (see, for instance, \\cite[Section 3]{Kruskal}). Thus, $\\mathcal H_{F_Y}(p)$ has rank $3$.\n\\end{example}\n\n\\begin{question}\\label{q:volume}\nLet $M$ be a closed topological manifold of dimension $2n$. Is there a constant $C$, depending only on $M$, such that for any smooth complex projective variety $X$ whose underlying topological space is $M$, we have\n\\[ \\vol(X) \\le C? \\]\n\\end{question}", + "post_theorem_intro_text_len": 1885, + "post_theorem_intro_text": "This result provides a topological bound on the possible exceptional divisors arising from some of the steps of the Minimal Model Program, establishing a new link between topology and birational geometry. It is a consequence of Theorem \\ref{thm:W_F}, where we prove the finiteness of rank one points of $\\mathcal H_F$ where $F$ is a non-degenerate $n$-form, a result of independent interest. \n\n\\medskip \n\nThe structure of the paper is as follows. In Section \\ref{s_preliminaries}, we introduce the necessary background on tensors and forms. Sections \\ref{s_low} and \\ref{s:intersection} are devoted to the study of the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$ with $n\\ge 3$. We show, in particular, that, up to a non-zero scalar, there are at most finitely many classes $e\\in H^2(M,\\mathbb C)$ such that \n$e^n\\neq 0$ and the rank of the Hessian of $F$ at $e$ is one.\n(cf. Theorem \\ref{thm:W_F}). Section \\ref{s_blowups} establishes our main theorem by examining the rank of the associated form in the presence of a divisorial contraction. In Section \\ref{s_volume}, we present some related open problems. \n\n\\medskip\n\n\\textbf{Acknowledgements:} The first and third authors are members of the GNSAGA - Istituto Nazionale di Alta Matematica.\nThe first author was also partially supported by the PRIN ``Vartietà reali e complesse: geometria, topologia e analisi armonica\". \nThe second author is partially supported by a Simons collaboration grant and would like to thank the National Center for Theoretical Sciences in Taiwan and Professor Jungkai Chen for\ntheir hospitality, where some of the work for this paper was completed.\nThe third author is partially supported by the PRIN2020 research grant ``2020KKWT53”.\nThe authors would also like to thank the referee for careful reading the paper and for several useful comments.", + "sketch": "The proof of Theorem~\\ref{thm_main} is obtained by reducing it to a finiteness statement about certain cohomology classes detected by the intersection form. Concretely, the paper studies “the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$,” and shows that “up to a non-zero scalar, there are at most finitely many classes $e\\in H^2(M,\\mathbb C)$ such that $e^n\\neq 0$ and the rank of the Hessian of $F$ at $e$ is one” (cf. Theorem~\\ref{thm:W_F}), which itself comes from “the finiteness of rank one points of $\\mathcal H_F$ where $F$ is a non-degenerate $n$-form.” Then Section~\\ref{s_blowups} “establishes our main theorem by examining the rank of the associated form in the presence of a divisorial contraction,” relating the exceptional divisor class $[E]$ of a contraction $f\\colon X\\to Y$ to one of those finitely many rank-one Hessian classes, yielding the finite list $e_1,\\ldots,e_q$.", + "expanded_sketch": "The proof of Theorem~\\ref{thm_main} is obtained by reducing it to a finiteness statement about certain cohomology classes detected by the intersection form. Concretely, the paper studies “the tensor arising from the intersection form on a closed topological manifold $M$ of dimension $2n$,” and shows that “up to a non-zero scalar, there are at most finitely many classes $e\\in H^2(M,\\mathbb C)$ such that $e^n\\neq 0$ and the rank of the Hessian of $F$ at $e$ is one.” We first record the precise finiteness statement:\n\n\\begin{theorem}\\label{thm:W_F}\nLet $F \\in \\C[x_0, x_1, \\ldots,x_b]$ be a form of degree $n \\ge 3$. \n\\begin{enumerate}\n\\item If $F$ is honest, then $W_F \\cap \\{F \\ne 0\\}$ is a finite set, and more precisely, \n$$\n|W_F \\cap \\{F \\ne 0\\} | \\le b+1.\n$$\n\n\\item If $F$ is non-degenerate, then $W_F$ is a finite set.\n\\end{enumerate}\n\\end{theorem}\n\nThis itself comes from “the finiteness of rank one points of $\\mathcal H_F$ where $F$ is a non-degenerate $n$-form.” Next we study blow-ups and use them to connect exceptional divisor classes to these finitely many rank-one Hessian classes. In particular, we use the following structural description of the intersection form under blow-up:\n\n\\begin{lemma}\\label{lem:blowup}\nLet $X$ be a smooth projective variety of dimension $n$ and let $f\\colon Y \\to X$ be the blow-up along a smooth subvariety $Z$ of $X$ of dimension $k < n$. \nLet $[E], \\beta_1:=f^*\\gamma_1, \\ldots, \\beta_b:=f^*\\gamma_b$ be a basis of $H^2(Y,\\Q)$ where $E$ is the exceptional divisor and $\\gamma_1,\\ldots,\\gamma_b$ is a basis of $H^2(X,\\Q)$.\t\nThen, with respect to this basis, we may write\n$$\nF_Y(x_0,\\ldots,x_n)=ax_0^n +\\sum_{i=1}^{n-k} x_0^{n-i}R_i + F_X(x_1, \\ldots, x_b)\n$$\nwhere $a=E^n$ and $R_i \\in \\mathbb Q[x_1,\\ldots,x_b]$ is a form of degree $i$, for $i=1,\\dots,n-k$. \n\n\\end{lemma}\n\nWe then control the rank of the Hessian at the exceptional divisor class $p=[E]$ in low-dimensional centers of blow-up via the following result:\n\n\\begin{proposition}\\label{prop:blowuprank}\nLet $X$ be a smooth K\\\"ahler manifold of dimension $n$ and let $f\\colon Y \\to X$ be the blow-up along a closed submanifold $Z$ of $X$ of dimension $k \\le 2$. Let $p=[E] \\in H^2(Y,\\C)$ be the class of the exceptional divisor.\n\\begin{enumerate}\n\\item \\label{k=0} If $n\\ge 2$ and $k=0$ then $\\rk \\mathcal H_{F_Y}(p)=1$.\n\\item \\label{k=1} If $n\\ge 3$ and $k=1$ then $\\rk \\mathcal H_{F_Y}(p) \\ge 2$. \n\\item \\label{k=2} If $n\\ge 4$ and $k=2$ then, after a base change, we may write \n$$\nF_Y(x_0,\\ldots,x_b)=ax_0^n + x_0^{n-1}L(x_1,\\ldots,x_b) + x_0^{n-2}Q(x_1,\\ldots,x_b) + F_X(x_1, \\ldots, x_b)\n$$\nwhere $a \\in \\mathbb Q$, $L$ is a linear form and $Q$ is a quardric form of rank $q$ for some positive integer $q$ such that $\\rk \\mathcal H_{F_Y}(p) \\ge 2q$.\n\n\\end{enumerate}\n\\end{proposition}\n\nIn establishing the main theorem, we apply these blow-up computations to examine “the rank of the associated form in the presence of a divisorial contraction,” relating the exceptional divisor class $[E]$ of a contraction $f\\colon X\\to Y$ to one of the finitely many rank-one Hessian classes provided by Theorem~\\ref{thm:W_F}, yielding the finite list $e_1,\\ldots,e_q$.", + "expanded_theorem": "\\label{thm_main}\nLet $M$ be a closed topological manifold of dimension $2n$ and let $b = \\dim H^2(M,\\mathbb{C})$.\n\nThen there exist non-zero elements $e_1,\\ldots,e_q \\in H^2(M,\\mathbb{C})$ with $q \\leq b+1$ such that if $X$ is a smooth complex projective variety of dimension $n$ with underlying topological space $M$ and $f\\colon X \\to Y$ is a divisorial contraction to a point with exceptional divisor $E$, then there exists $i \\in \\{1, \\dots, q\\}$ such that $[E] = e_i$.", + "theorem_type": [ + "Existential–Universal", + "Existence" + ], + "mcq": { + "question": "Let $M$ be a closed (compact and without boundary) topological manifold of real dimension $2n$, and set $b=\\dim_{\\mathbb C} H^2(M,\\mathbb C)$. For any smooth complex projective variety $X$ of complex dimension $n$ whose underlying topological space is $M$, identify $H^2(X,\\mathbb C)$ with $H^2(M,\\mathbb C)$. If $f\\colon X\\to Y$ is a divisorial contraction to a point, let $E$ denote its exceptional divisor and let $[E]\\in H^2(M,\\mathbb C)$ be its cohomology class under this identification. Which existence statement holds?", + "correct_choice": { + "label": "A", + "text": "There exist nonzero classes $e_1,\\ldots,e_q\\in H^2(M,\\mathbb C)$ with $q\\le b+1$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\\colon X\\to Y$ to a point with exceptional divisor $E$, there is an index $i\\in\\{1,\\ldots,q\\}$ for which $[E]=e_i$." + }, + "choices": [ + { + "label": "B", + "text": "There exist nonzero classes $e_1,\\ldots,e_q\\in H^2(M,\\mathbb C)$ with $q\\le b+1$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\\colon X\\to Y$ with exceptional divisor $E$, there is an index $i\\in\\{1,\\ldots,q\\}$ for which $[E]=e_i$." + }, + { + "label": "C", + "text": "There exist nonzero classes $e_1,\\ldots,e_q\\in H^2(M,\\mathbb C)$ for some finite $q$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\\colon X\\to Y$ to a point with exceptional divisor $E$, there is an index $i\\in\\{1,\\ldots,q\\}$ for which $[E]=e_i$." + }, + { + "label": "D", + "text": "For every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, there exist nonzero classes $e_1,\\ldots,e_q\\in H^2(M,\\mathbb C)$ with $q\\le b+1$ such that for every divisorial contraction $f\\colon X\\to Y$ to a point with exceptional divisor $E$, there is an index $i\\in\\{1,\\ldots,q\\}$ for which $[E]=e_i$." + }, + { + "label": "E", + "text": "There exist nonzero classes $e_1,\\ldots,e_q\\in H^2(M,\\mathbb C)$ with $q\\le b+1$ such that for every smooth complex projective variety $X$ of dimension $n$ whose underlying topological space is $M$, and every divisorial contraction $f\\colon X\\to Y$ to a point with exceptional divisor $E$, there is an index $i\\in\\{1,\\ldots,q\\}$ for which $[E]$ is proportional to $e_i$ by a nonzero scalar." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "case_split", + "tampered_component": "restriction_to_contractions_to_a_point", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "finiteness", + "tampered_component": "explicit_bound_q_le_b+1", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "finiteness", + "tampered_component": "uniformity_in_X", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "finiteness", + "tampered_component": "equality_of_exceptional_class_vs_projective_class", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem sets up notation and context but does not explicitly or implicitly reveal which of the listed existence statements is correct. The correct choice is not singled out by wording in the prompt." + }, + "TAS": { + "score": 1, + "justification": "This is very close to a theorem-statement recognition item: the task is essentially to identify the exact valid formulation among nearby variants. It is not a pure verbatim restatement because the options differ in meaningful quantifiers and strength, but it remains only mildly non-tautological." + }, + "GPS": { + "score": 1, + "justification": "The question requires some careful reasoning about uniformity in X, the restriction to contractions to a point, exact equality versus proportionality, and the bound q ≤ b+1. However, it mainly tests precise statement discrimination rather than deeper mathematical generation or derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically targeted: one removes the 'to a point' restriction, one weakens the bound, one alters quantifier uniformity, and one replaces equality by proportionality. These are distinct and reflect common theorem-statement confusion modes." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it primarily tests precise recall/parsing of a statement rather than deeper generative reasoning." + } + }, + { + "id": "2511.06487v2", + "paper_link": "http://arxiv.org/abs/2511.06487v2", + "theorems_cnt": 2, + "theorem": { + "env_name": "thm", + "content": "\\label{thm:sos}\n\t\tFor $f\\in\\cA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\mathcal{K}$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\mathcal{B}(\\mathcal{K})^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\mathbb N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\mathbb C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}\n\n\t\t\\noindent If $\\mathcal{H}$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[label=\\textup{(\\roman*)}, resume]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:iv}\n\t\t\tThere exists $r\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}\n\n\t\t\\end{enumerate}", + "start_pos": 14614, + "end_pos": 15554, + "label": "thm:sos" + }, + "ref_dict": { + "ssec:readersguide": "\\begin{remark}[What's new?]\\rm\n\t\t\\label{rem:whatsnew}\n\t\tThe passage to operator coefficients necessitates several novel\n\t\tresults and constructions that we expect to be of independent interest. At a high level, the proofs of Theorem~\\ref{thm:sos} and Theorem~\\ref{thm:usos} still follow the now standard paradigm for establishing sum of squares (sos) representations (factorizations).\n\t\tNamely, the Hahn-Banach theorem produces a separating linear functional $\\varphi$, and then a Gelfand-Naimark-Segal (GNS) construction\n\t\tbased on $\\varphi$ ultimately produces a tuple $Y$. \n\t\tHere we \n\t\troughly follow the outline of \\cite{MP05}. \n\n\t\tA key construction is that of a tuple of self-adjoint operators $A$\n\t\tbased upon the left regular representation on Fock space; see Section~\\ref{sec:fock}.\n\t\tWe then show that, up to a universal constant, for a sum of squares polynomial $p,$ \n\t\tthe norm\n\t\tof $p(A)$ bounds the norm of any non-commutative psd Gram matrix $G$ that represents $p$; see Proposition~\\ref{prop:bounded}. \n\t\tThis uniform bound is the main input in Section~\\ref{sec:top} for proving that the cone $\\cC_d$ of sums of squares is closed in the product ultraweak topology on the coefficients. A separate approximation argument then replaces an ultraweak continuous separating functional by a WOT continuous one and hence yields closedness of $\\cC_d$ in the product WOT. This two-step interplay between the ultraweak and WOT topologies enables an application of the Hahn-Banach separation theorem.\n\n\t\tOn the GNS side, we introduce a new argument that exploits the WOT to associate to a\n\t\tseparating linear functional a finite-rank psd noncommutative representing Hankel matrix and,\n\t\ton its range, construct the desired tuple~$Y$; see Section~\\ref{sec:gns}. For the unitary result,\n\t\tTheorem~\\ref{thm:usos}, we additionally modify the construction of Section~\\ref{sec:fock} to produce a canonical\n\t\ttuple of unitary operators from the left-regular representation and adapt the GNS procedure\n\t\tto obtain a unitary tuple together with a representing vector that realizes the separating\n\t\tfunctional as a vector state; see Subsection~\\ref{ssec:uGNS}.\\qed\n\t\\end{remark}\n\n\t\\subsection{Reader's guide}\\label{ssec:readersguide}\n\n\tThe paper is structured as follows. The convex cone of sums of squares (making an appearance in \\eqref{eq:sosthm} of Theorem~\\ref{thm:sos}) is introduced and characterized in the next Section~\\ref{sec:sos}. In Section~\\ref{sec:fock} we define creation operators $L_i$ on the full Fock $\\cF_\\vg^2$ space and their symmetrized analogs $A_i$. How they pertain to the sum of squares statement at hand is explored in Subsection~\\ref{ssec:coeff}, where evaluations at $A$ are used to extract coefficients of a polynomial. In Section~\\ref{sec:top} we \n\tintroduce a suitable topology on $\\cA_\\ad$ and\n\tcollect all the necessary topological properties needed in the sequel. With respect to this topology, the convex cone of sums of squares is closed, see Proposition~\\ref{prop:closedcone}. The fact that the cone is closed allows for an application of the Hahn--Banach Separation Theorem, which is then followed by an appropriate version of the GNS construction, carried out in Section~\\ref{sec:gns}; see Proposition~\\ref{prop:GNS}. \n\tThen Theorem~\\ref{thm:sos} is proved in Section~\\ref{sec:proof} and\n\tTheorem~\\ref{thm:usos} is proved in Section~\\ref{sec:uproof}.\n\n\t\\subsection*{Acknowledgment} The authors thank Dr. Matthias Schötz for carefully reading an earlier version of this manuscript and for pointing out gaps in the proof of Proposition~\\ref{prop:closedcone}. \n\n\t\\section{Convex Cone of (Sums of) Squares}\\label{sec:sos}\n\n\tIn this section a key player in the proof of Theorem~\\ref{thm:sos}, the convex cone of sums of squares of polynomials, is introduced and\n\tstudied. The main result in this section is Proposition~\\ref{prop:closed under addition} (see also Remark \\ref{rem:soscone}), which gives a bound on the number of sums of squares needed to write a polynomial as a sum of squares. \n\n\t\\begin{lemma}\\label{lem:psd}\n\t\tIf $T: \\cH^{n} \\to \\cH^{n}$ be a psd linear map, then there exist linear maps $R_{i}: \\cH \\to \\cH^{n},$ $i = 1, \\dots, n,$ such that $T = \\sum_{i=1}^{n} R_{i}R_{i}^{*}.$ Moreover, if $\\cH$ is infinite-dimensional, then $T = R R^{*}$ for some $R : \\cH \\to \\cH^{n}.$\n\t\\end{lemma} \n\t\\begin{proof}\n\n\t\tSince $T$ is psd, there exists a linear map $\\tilde{R}: \\cH^{n} \\to \\cH^{n}$ such that $T = \\tilde{R} \\tilde{R}^{*}.$ Write \n\t\t\\[\n\t\t\\tilde{R} \\ =\\ \\begin{bmatrix}\n\t\t\tR_{1}, \\dots, R_{n}\n\t\t\\end{bmatrix}\n\t\t\\]\n\t\twith respect to the orthogonal decomposition $\\cH^{n} = \\cH \\oplus \\dots \\oplus \\cH.$ The first part of the lemma follows by noting that each $R_{i}$ is a map from $\\cH$ into $\\cH^{n}.$ For the moreover part, let $U: \\cH \\to \\cH^{n}$ be any unitary, and set $R = \\tilde{R} U.$\n\t\\end{proof}\n\n\tIndex \\df{$\\mathbb{\\cH}^{N(d)}$} and \\df{$\\cA_{d}^{N(d)}$} (the algebraic direct sum of $\\cA_{d}$ with itself $N(d)$ times) by $\\la x \\ra_{d}.$ \n\tLet $V_{d} \\in \\cA_{d}^{N}$ denote the \\df{Veronese column vector} whose $w\\in \\la x \\ra_{d}$ entry is $w$\n\t(adopting the usual convention of viewing $w$ as the the $\\cB(\\cH)$-valued polynomial $I_\\cH \\, w$). For instance, \n\tif $\\vg=2$ and $d=2$, then \\index{$V_d$} \n\t\\[\n\tV_2\\ =\\ \\text{col} \\begin{pmatrix}\n\t\t1 & x_1 & x_2 & x_1^2 & x_1 x_2 & x_2 x_1 & x_2^2\n\t\\end{pmatrix}.\n\t\\]\n\tLet \\df{$\\cC_d$} denote the \\df{cone of sums of squares} of polynomials of degree at most $d,$\n\t\\begin{equation}\\label{eq:sosdef}\n\t\t\\cC_{d} \\ :=\\ \\left\\{ \\sum\\limits_{i=1}^{N(d)} r_{i}^{*}r_{i}: \\quad r_{i}\\in \\cA_{d}, \\quad i=1, \\dots, N(d) \\right\\}\n\t\t\\subseteq \\cA_{2d}.\n\t\\end{equation}\n\tGiven $r\\in \\cA_d,$ the column vector $R$ with $w$ entry $R_{w}^{*}$ \n\tis called the \\df{coefficient vector} of $r$ since $r = R^{*} V_{d}.$ In particular, \n\t\\[\n\tr^* r \\ =\\ V_d^* RR^* V_d\n\t\\]\n\tso that $r^*r$ has a representation as $V_d^* \\SG V_d$ for a psd matrix $\\SG.$\n\n\t\\begin{prop} \\label{prop:closed under addition}\n\t\tA polynomial $p\\in \\cA_{2d}$ is in $\\cC_d$ if and only if there is a psd block matrix $\\SG$ such that\n\t\t\\begin{equation}\n\t\t\t\\label{e:gram-rep}\n\t\t\tp \\ =\\ V_d^* \\SG V_d. \n\t\t\\end{equation}\n\t\tIn fact, if $p=V_d^* \\SG V_d,$ then factoring $\\SG=\\sum_{j=1}^{N(d)} R_jR_j^*$ with $R_j:\\cH \\to \\oplus_{w\\in \\la x\\ra_d}\\cH$\n\t\tas in Lemma~\\ref{lem:psd}, \n\t\tsetting $r_j = R_{j}^{*} V_{d} $\n\t\tgives,\n\t\t\\[\n\t\tp \\ =\\ \\sum_{j=1}^{N(d)} r_j^* r_j.\n\t\t\\]\n\n\t\tIn particular, the set $\\cC_d$ is a (convex) cone. \t\\end{prop} \n\n\tWe call any psd block matrix $\\SG$ satisfying equation~\\eqref{e:gram-rep} a \\df{Gram representation} for $p.$\n\n\t\\begin{proof}\n\t\tGiven a sum of squares $p = \\sum_{i=1}^{N(d)} r_{i}^{*}r_{i},$ writing $r_j=R_j^{*} V_d$ gives \n\t\t$p = \\sum_{i=1}^{N(d)} V_{d}^{*} R_{i} R_{i}^{*} V_{d},$ where $R_{i}$ is the coefficient vector corresponding to the polynomial $r_{i}.$ It follows that $p = V_{d}^{*} \\SG V_{d},$ where $\\SG = \\sum_{i=1}^{N(d)} R_{i} R_{i}^{*}.$ In particular, $\\SG : \\cH^{N(d)} \\to \\cH^{N(d)}$ is a psd linear map. \n\n\t\tConversely, suppose there is a psd linear map $\\SG:\\oplus_{w\\in \\la x\\ra_d} \\cH \\to \\oplus_{w\\in \\la x \\ra_d}\\cH$ such that $p = V_d^* \\SG V_d.$ \n\t\tBy Lemma~\\ref{lem:psd}, there exist $R_j: \\cH\\to \\oplus_{w\\in \\la x \\ra_d}\\cH$ such that\n\t\t$\\SG=\\sum_{j=1}^{N(d)} R_j R_j^*.$ Setting \n\t\t$r_j =R_j^* V_d$, one obtains $p=\\sum_{j=1}^{N(d)} r_j^*r_j.$ \n\n\t\tBy what has already been proved,\n\t\tif $p,q \\in \\cC_{d},$ then there exist (psd) Gram representations $p = V^{*} \\SG_{p} V$ and $ q = V^{*} \\SG_{q} V.$\n\t\tNow $p + q = V_{d}^{*} (\\SG_{p} + \\SG _{q}) V_{d} .$ Since $\\SG_{p} + \\SG_{q} : \\cH^{N(d)} \\to \\cH^{N(d)}$ is a psd linear map, \n\t\twhat has already been proved shows \n\t\t$p + q \\in \\cC_{d}.$ \n\t\\end{proof}\n\n\t\\begin{corollary}\\label{cor:soscone}\n\t\tLetting $V_{d} \\in \\cA_{d}^{N(d)}$ denote the Veronese column vector,\n\t\tthe convex cone of sums of squares of degree at most $2d$ is\n\t\t\\[\n\t\t\\cC_{d} = \\left\\{ V_{d}^{*} \\SG V_{d}: \\quad \\SG = [\\SG_{v,w}]_{v,w\\in \\la x \\ra_{d}} \\in \\cB(\\cH)^{N(d) \\times N(d)}, \\quad \\SG \\succeq 0 \\right\\}.\n\t\t\\]\n\t\\end{corollary}\n\n\t\\begin{remark}\\rm\\label{rem:soscone}\n\t\tIt follows from the preceding discussion that the convex cone $\\cC_{d}$ takes the form \n\t\t\\[ \n\t\t\\cC_{d} \\ =\\ \\{r^{*}r : \\quad r \\in \\cA_{d}\\}\n\t\t\\]\n\t\twhen the Hilbert space $\\cH$ is infinite-dimensional.\n\t\t\\qed\n\t\\end{remark}", + "prop:bounded": "\\begin{prop} \\label{prop:bounded}\n\t\tIf $p \\in \\cC_{d},$ then the set \n\t\t\\[\n\t\t\\Gamma_p \\ =\\ \\{\\, \\SG \\in \\cB(\\cH)^{N(d)\\times N(d)} \\ : \\quad \\SG \\succeq 0, \\quad V_d^* \\SG V_d = p \\,\\}\n\t\t\\]\n\t\tis norm bounded $($with respect to the operator norm on $\\cB(\\cH^{N(d)})$$)$. More precisely, \n\t\tthere exists a constant $\\mu_{d}$ $($depending only on $d$ and $\\vg$ and not on $p$$)$ such that,\n\t\tfor all $\\SG\\in\\Gamma_p,$ \n\t\t\\[ \n\t\t\\|\\SG\\| \\ \\leq\\ \\mu_{d} \\; \\|p(A)\\| ,\n\t\t\\]\n\t\twhere the tuple $A=A^{(\\ad)}$ is defined in \\eqref{eq:symcreate}\n\t\tfor any $\\ad\\geq 2d$. \n\t\\end{prop}", + "thm:usos": "\\begin{thm}\\label{thm:usos}\n\t\tFor $f\\in\\ccA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:usos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of unitary operators $U = (U_{1}, \\dots, U_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of unitary matrices \n\t\t\t$U = (U_{1}, \\dots, U_{\\vg})\\in M_n(\\C)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N_{\\rm red}(d)}\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\\label{eq:usosthm}\n\t\t\t\tf\\ =\\ \\sum_{i=1}^{N_{\\rm red}(d)} r_i^*r_i.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\tIf $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item[\\rm (iv)] \\label{i:usos:iv}\n\t\t\tThere exists $r\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\n\t\t\t\tf\\ =\\ r^*r.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\\end{thm}", + "prop:GNS": "\\begin{prop}\\label{prop:GNS}\n\t\tIf $\\varphi:\\mathcal A_{2d+2}\\to\\mathbb C$ is a continuous linear functional such that\n\t\t\\[\n\t\t\\varphi(p^ *p)\\ \\ge\\ 0 \t\t\\]\n\t\tfor all $p\\in \\cA_{d+1},$ \n\t\tthen there exist a finite-dimensional Hilbert space $\\cE$, a self-adjoint $g$-tuple $Y=(Y_1,\\dots,Y_\\vg)$ on $\\cE$, and a vector $\\gamma\\in \\cH\\otimes \\cE$ such that\n\t\t\\[\n\t\t\\varphi(q^ *p)\\ =\\ \\big\\langle p(Y)\\gamma,\\ q(Y)\\gamma\\big\\rangle_{\\cH\\otimes \\cE} \\qquad\\text{for all }p \\in \\cA_{d+1},\\ q\\in\\mathcal A_{d}. \n\t\t\\]\n\t\tTherefore, for all $p\\in\\mathcal A_{2d+1}$,\n\t\t\\[\n\t\t\\varphi(p)\\ =\\ \\langle p(Y)\\gamma,\\ \\gamma\\rangle. \n\t\t\\]\n\t\\end{prop}", + "ssec:uGNS": "\\label{ssec:uGNS}\n\tThe only other point that needs attention is the proof of a suitable GNS construction as in Proposition~\\ref{prop:GNS}.\n\tSince we cannot rely on non-cancellation of the highest orde", + "prop:closedcone": "\\begin{prop}\\label{prop:closedcone}\n\t\tThe convex cone $\\cC_{d}$ is closed in $\\cA_{\\ad}$.\n\t\\end{prop}", + "i:sos:ii": "\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}", + "ssec:coeff": "\\begin{proof}\n\t\tRecall that we have endowed $\\la x \\ra$ with graded lexicographic order. If $\\ad\\ge |v|$ and $v>w$, then \n\t\t\\[\n\t\t\\langle A^{w} \\Omega, e_{v} \\rangle \\ =\\ 0 \n\t\t\\]\n\t\tby Lemma~\\ref{lem:faithful}. Hence, $\\ME_\\ad$ is upper triangular.\n\t\tMoreover, each diagonal entry is $1$ by Lemma~\\ref{lem:faithful}. Thus, $\\ME_{\\ad}$ is invertible.\n\t\\end{proof}\n\n\t\\subsection{Extraction formula for coefficients}\\label{ssec:coeff}\n\tLet $q = \\sum Q_{w} w \\in \\cA_{\\ad}.$ For $v\\in \\la x\\ra_{\\ad}$ \tdefine the linear functional\n\t$\\Omega_{v} : \\cB(\\Ftgd) \\to \\mathbb{C}$ \tby\n\t\\[\n\t\\Omega_{{v}}(T) = \\langle T \\Omega, e_{v} \\rangle. \n\t\\]\n\tThe operator coefficients \\(Q_v\\) are obtained from $q(A)$ by solving the linear system\n\t\\begin{align*} \n\t\tZ_{v}(q) \\ :=&\\ (\\mathrm{id}_{\\cB(\\cH)} \\otimes \\Omega_{v}) q(A) \\ =\\ \\sum\\limits_{w} Q_{w} \\otimes \\Omega_{v} (A^{w}) \\\\\n\t\t\\ =&\\ \\sum\\limits_{w} \\langle A^{w} \\Omega , e_{v} \\rangle \\, Q_{w} \\ =\\ \\sum\\limits_{w} [\\ME_{\\ad}]_{v,w} Q_{w},\n\t\\end{align*}\n\twhere $[\\ME_{\\ad}]_{v,w}$ is the $(v,w)$ entry of the matrix $\\ME_{\\ad}.$ In short, \\index{$[\\ME_d]$}\n\t\\begin{equation}\\label{eq:coeff2}\n\t\tZ(q) \\ =\\ \\ME_{\\ad} Q,\n\t\\end{equation}\n\twhere $Z(q)$ and $Q$ are column vectors with $Z_{v}(q)$ and $Q_{v}$ as the $v^{\\rm th}$ entry of $Z$ and $Q,$ respectively. \n\tSince, by Lemma~\\ref{lem:invertible M}, $\\ME_d$ is invertible, \n\t\\begin{equation} \\label{eq:coeff}\n\t\tQ \\ =\\ \\ME_{\\ad}^{-1} Z(q).\n\t\\end{equation} \n\t{We refer to $\\ME$ as the \\df{extraction matrix}, and equation~\\eqref{eq:coeff} as the \\df{extraction formula} for the coefficients of $q.$ Note \n\t\tthat this formula depends only upon $q(A);$ that is, the coefficients of $q$ are determined uniquely\n\t\tby $q(A).$}\n\n\tIt follows from equation~\\eqref{eq:coeff} that there exists a positive constant $\\lambda_{\\ad}$ (independent of $q$) such that \n\t\\begin{equation} \\label{eq:Coeff bound}\n\t\t\\|Q_{w}\\| \\ \\leq\\ \\lambda_{\\ad} \\, \\|q(A)\\| \\quad \\text{ for all $w \\in \\la x\\ra_{\\ad}$.}\n\t\\end{equation}\n\n\t\\begin{prop} \\label{prop:bounded}\n\t\tIf $p \\in \\cC_{d},$ then the set \n\t\t\\[\n\t\t\\Gamma_p \\ =\\ \\{\\, \\SG \\in \\cB(\\cH)^{N(d)\\times N(d)} \\ : \\quad \\SG \\succeq 0, \\quad V_d^* \\SG V_d = p \\,\\}\n\t\t\\]\n\t\tis norm bounded $($with respect to the operator norm on $\\cB(\\cH^{N(d)})$$)$. More precisely, \n\t\tthere exists a constant $\\mu_{d}$ $($depending only on $d$ and $\\vg$ and not on $p$$)$ such that,\n\t\tfor all $\\SG\\in\\Gamma_p,$ \n\t\t\\[ \n\t\t\\|\\SG\\| \\ \\leq\\ \\mu_{d} \\; \\|p(A)\\| ,\n\t\t\\]\n\t\twhere the tuple $A=A^{(\\ad)}$ is defined in \\eqref{eq:symcreate}\n\t\tfor any $\\ad\\geq 2d$. \n\t\\end{prop}\n\n\t\\begin{proof}\n\t\tFix $p \\in \\cC_{d}$ and $\\SG \\in \\Gamma_p.$ Thus $p=V_d^* \\SG V_d.$ By\n\t\tProposition~\\ref{prop:closed under addition}, there exists $Q_j: \\cH\\to \\oplus_{w\\in\\la x\\ra_d} \\cH$ such that\n\t\t\\begin{equation}\n\t\t\t\\label{e:bound:1}\n\t\t\tp \\ =\\ \\sum\\limits_{j=1}^{N(d)} q_{j}^{*}q_{j} \\ =\\ V_d^* \\left [ \\sum\\limits_{j=1}^{N(d)} Q_{j} Q_{j}^* \\right ] V_d,\n\t\t\\end{equation}\n\t\twhere\n\t\t\\[\n\t\tq_j \\ =\\ Q_j^* V_d = \\sum_{w\\in\\la x\\ra_d} Q_{j,w} w.\n\t\t\\]\n\t\tBy equation~\\eqref{eq:Coeff bound}, for $v\\in\\la x \\ra_d,$\n\t\t\\[\n\t\t\\| Q_{j,v}\\| \\ \\le\\ \\lambda_d\\; \\|q_j(A)\\|.\n\t\t\\]\n\t\tFrom equation~\\eqref{e:bound:1}, \n\t\t\\[\n\t\t\\|q_j(A)\\|^2 \\ =\\ \\|q_j(A)^* q_j(A)\\| \\ \\le\\ \\|p(A)\\|.\n\t\t\\]\n\t\tThus, again using equation~\\eqref{e:bound:1},\n\t\t\\[\n\t\t\\sum_{u,v\\in \\la x \\ra_d} \\|\\SG_{u,v}\\| \\ \\le\\ \\sum_{u,v\\in \\la x \\ra_d} \\, \\sum_{j=1}^{N(d)} \\|Q_{j,u}Q_{j,v}^*\\| \n\t\t\\ \\le\\ N(d)^3 \\lambda_d^{2} \\; \\|p(A)\\|.\n\t\t\\]\n\t\tIt follows that \n\t\t$\\|\\SG\\|\\le \\mu_d \\, \\|p(A)\\|$ for $\\mu_d = N(d)^3 \\lambda_d^{2}.$\n\t\\end{proof}", + "eq:poly": "\\begin{equation}\\label{eq:poly}\n\t\tp \\ =\\ \\sum_{w\\in\\la x\\ra}^{\\rm finite} P_{w} w,\n\t\\end{equation}", + "eq:sosthm": "\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}", + "ssec:mainresults": "\\begin{equation}\n\t\t\\label{d:evalPunitary}\n\t\t\\sum_{u\\in\\freeg}^{\\rm finite} P_u u \\ \\mapsto\\ \\sum_u P_u^* u^{-1}.\n\t\\end{equation}\n\tIf $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so \n\t\\[\n\tp(X)^* \\ =\\ \\left ( \\sum P_w \\otimes X^w\\right)^* \\ =\\ \\sum P_w^* \\otimes X^{w^*} \\ =\\ p^*(X)\n\t\\]\n\tfor all $p\\in\\ccA$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\\ccA$ and we let,\n\tfor positive integers $d,$ \\index{$\\ccA$} \\index{$\\ccA_d$}\n\t\\[\n\t\\ccA_d \\ =\\ \\left\\{ \n\t\\sum_{\\substack{u\\in\\freeg\\\\ |u|\\le d}} P_u u \\, :\\, P_u \\in \\cB(\\cH)\n\t\\right\\}.\n\t\\]\n\tThe number of words in $\\freeg$ of length $\\le d$, $(\\freeg)_d$, is denoted by \\df{$N_{\\rm red}(d)$} and equals\n\t\\[\n\tN_{\\rm red}(d)\n\t\\ =\\ \n\t1+\\sum_{k=1}^{d} 2\\vg\\,(2\\vg-1)^{k-1}\n\t\\ =\\ \\frac{\\vg (2\\vg-1)^{d}-1}{\\vg-1}.\n\t\\]\n\n\t\\subsection{Main results}\\label{ssec:mainresults}\n\tWe are now ready to state our main results. The first is an operator-valued version of the classical \tsum of squares theorem of Helton \\cite{Hel02} and McCullough \\cite{McC01}, Theorem~\\ref{thm:sos}. The second, Theorem~\\ref{thm:usos},\n\tis a factorization result for positive operator-valued trigonometric polynomials extending a long list of \n\tresults pertaining to scalar-valued noncommutative trigonometric polynomials \\cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \\succeq 0$ means that the operator $T$ is positive semidefinite (psd).\n\n\t\\begin{thm}\\label{thm:sos}\n\t\tFor $f\\in\\cA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}", + "i:sos:iii": "\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}", + "ssec:notation": "\\label{ssec:notation}\n\tFix a positive integer $\\vg$. Let \n\t\\df{$\\la x \\ra$} denote the free monoid on the $\\vg$ letters of the alphabet $x = \\{ x_{1}, \\dots, x_{\\vg}\\}.$ \n\tIts multiplicative identity", + "eq:1sosthm": "\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}", + "i:sos:i": "\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}", + "rem:soscone": "\\begin{remark}\\rm\\label{rem:soscone}\n\t\tIt follows from the preceding discussion that the convex cone $\\cC_{d}$ takes the form \n\t\t\\[ \n\t\t\\cC_{d} \\ =\\ \\{r^{*}r : \\quad r \\in \\cA_{d}\\}\n\t\t\\]\n\t\twhen the Hilbert space $\\cH$ is infinite-dimensional.\n\t\t\\qed\n\t\\end{remark}", + "thm:sos": "\\begin{thm}\\label{thm:sos}\n\t\tFor $f\\in\\cA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}\n\n\t\t\\noindent If $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[label=\\textup{(\\roman*)}, resume]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:iv}\n\t\t\tThere exists $r\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}\n\n\t\t\\end{enumerate} \n\t\\end{thm}" + }, + "pre_theorem_intro_text_len": 6629, + "pre_theorem_intro_text": "Positivity and factorization lie at the heart of real algebraic geometry and operator theory. In the commutative setting, positivity certificates via sums of squares (sos) trace back to \n\tHilbert's $17^{\\rm th}$ problem in 1900; for classical results and modern treatments see \\cite{BCR98,Mar08,Sc24}.\\looseness=-1\n\n\tIn the 21st century, motivated by developments in linear systems theory \\cite{SIG98,dOHMP09}, quantum physics \\cite{brunner}, and free probability \\cite{MS17}, the free (noncommutative) counterpart has evolved into a broad program within noncommutative function theory \\cite{KVV14,MS11,AM15,BMV16,PTD22}. This framework encompasses noncommutative factorizations and noncommutative Positivstellens\\\"atze. Early landmarks include Helton’s theorem that (scalar) positive noncommutative polynomials are sums of squares \\cite{Hel02} and McCullough’s factorization theory for noncommutative polynomials \\cite{McC01}; see also \\cite{HM04,HMP04,Po95,JM12,JMS21} and the references therein for further developments.\n\n\tThis paper establishes operator-valued analogs of these factorization theorems: every positive operator-valued noncommutative polynomial $p$ admits a single-square factorization $p=r^{*}r$, with an analogous result for operator-valued noncommutative trigonometric polynomials (elements of the free group algebra).\n\n\tBeyond the noncommutative positivity literature, our results resonate with classical and modern operator factorization themes, including canonical/state-space factorizations of Bart--Gohberg--Kaashoek and collaborators \\cite{BGK79,BGKR10}, and the operator Fejér--Riesz and multivariable outer factorization lines \\cite{DR10,DW05,GW05}. While our focus is the free (noncommutative) polynomial and free group contexts, the methods developed, such as the WOT-closure mechanism via Fock-space evaluations and the finite-rank Hankel realization, are of independent interest and may be useful in adjacent problems within free analysis and operator theory.\n\n\t\\subsection*{Guide to the introduction.}\n\tNotation is introduced in Subsection~\\ref{ssec:notation}. The main results are stated and their proofs outlined in Subsection~\\ref{ssec:mainresults}, while Subsection~\\ref{ssec:readersguide} provides a roadmap for the remainder of the paper.\n\n\t\\subsection{Notation}\\label{ssec:notation}\n\tFix a positive integer $\\vg$. Let \n\t\\df{$\\langle x \\rangle$} denote the free monoid on the $\\vg$ letters of the alphabet $x = \\{ x_{1}, \\dots, x_{\\vg}\\}.$ \n\tIts multiplicative identity is the empty word \n\t\\df{$\\varnothing$}. \n\tWe endow $\\langle x\\rangle$ with the \\df{graded lexicographic order}.\n\tThe length of a word $w \\in \\langle x \\rangle$ is denoted by $|w|.$ The\n\tset of all elements (words) of $\\langle x \\rangle$ of length (or degree) at most $d$\n\tis denoted \\df{$\\langle x \\ra_{d}$}.\n\tThe cardinality of $\\langle x\\ra_d$ is \\index{$N(d)$}\n\t\\[\n\tN(d) \\ =\\ \\sum_{i=0}^d \\vg^i \\ =\\ \\frac{\\vg^{d+1}-1}{\\vg-1}.\n\t\\]\n\n\tLet \\df{$\\mathcal{H}$} be a fixed complex Hilbert space.\n\tLet \\df{$\\mathcal{B}(\\mathcal{H})$} be the space of all bounded linear operators on $\\mathcal{H}$, and \n\tlet \n\t\\df{$\\mathcal{A} $} to be the free semigroup $\\mathcal{B}(\\mathcal{H})$-algebra on $x,$\n\ti.e., $\\mathcal{A}=\\mathcal{B}(\\mathcal{H})\\langle x\\rangle$. An element $p$ of $\\mathcal{A}$ takes the form, \n\t\\begin{equation}\\label{eq:poly}\n\t\tp \\ =\\ \\sum_{w\\in\\langle x\\rangle}^{\\rm finite} P_{w} w,\n\t\\end{equation}\n\twhere\n\t$P_{w} \\in \\mathcal{B}(\\mathcal{H}),$ and is referred to as an (operator-valued) \n\t\\df{polynomial} in $x.$ \n\tLet \\df{$\\cA_{d}$} denote the elements from $\\mathcal{A}$ of degree at most $d$.\n\n\tEquip $\\mathcal{A}$ with the involution \\df{$^*$}: on letters, $x_{j}^{*} = x_{j},$ on a word $w = x_{i_1} \\cdots x_{i_n} \\in \\langle x \\rangle,$ \n\t\\[\n\tw^{*} \\ =\\ x_{i_n} \\cdots x_{i_1};\n\t\\]\n\tand, on a polynomial $p$ as in \\eqref{eq:poly}, \\index{$p^*$}\n\t\\[\n\tp^{*} \\ =\\ \\sum P_{w}^{*} w^{*},\n\t\\]\n\twhere $P_{w}^{*}$ is the adjoint of the operator $P_{w}$ in $\\mathcal{B}(\\mathcal{H}).$\n\n\tLet $\\mathcal{K}$ be a Hilbert space and $X = (X_{1}, \\dots, X_{\\vg})$ be a tuple of operators from $\\mathcal{B}(\\mathcal{K}).$ The \n\t\\df{evaluation} of $p$ at $X$ is defined as \n\t\\[\n\tp(X) \\ =\\ \\sum P_{w} \\otimes X^{w},\n\t\\]\n\twhere $X^{w} = X_{i_1} \\dots X_{i_n}$ for $w = x_{i_1} \\dots x_{i_n}.$ In general, $p(X)^{*}$ (the adjoint of $p(X)$) and $p^{*}(X)$ are not the same. They are the same if $X$ is a tuple of self-adjoint operators. \n\n\t\\subsubsection{Trigonometric polynomials}\n\n\tWe will also be interested in evaluating noncommutative polynomials in tuples of unitaries on Hilbert space. An appropriate setting to consider these is the group algebra of the free group $\\mathbb F_{\\vg}$ on the $\\vg$ letters $x_i$, $i=1,\\ldots,\\vg$. \n\tElements of $\\mathbb F_{\\vg}$ are (reduced) words in the alphabet $x_i,x_i^{-1}$.\n\n\tLet $\\mathscr{A}$ be the algebra $\\mathcal{B}(\\mathcal{H})[\\mathbb F_{\\vg}] = \\mathcal{B}(\\mathcal{H})\\otimes\\mathbb C[\\mathbb F_{\\vg}]$. Its elements are called \\df{trigonometric polynomials}, and $\\mathscr{A}$\n\tis\n\tendowed with the involution\n\t\\begin{equation}\n\t\t\\label{d:evalPunitary}\n\t\t\\sum_{u\\in\\mathbb F_{\\vg}}^{\\rm finite} P_u u \\ \\mapsto\\ \\sum_u P_u^* u^{-1}.\n\t\\end{equation}\n\tIf $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so \n\t\\[\n\tp(X)^* \\ =\\ \\left ( \\sum P_w \\otimes X^w\\right)^* \\ =\\ \\sum P_w^* \\otimes X^{w^*} \\ =\\ p^*(X)\n\t\\]\n\tfor all $p\\in\\mathscr{A}$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\\mathscr{A}$ and we let,\n\tfor positive integers $d,$ \\index{$\\mathscr{A}$} \\index{$\\ccA_d$}\n\t\\[\n\t\\ccA_d \\ =\\ \\left\\{ \n\t\\sum_{\\substack{u\\in\\mathbb F_{\\vg}\\\\ |u|\\le d}} P_u u \\, :\\, P_u \\in \\mathcal{B}(\\mathcal{H})\n\t\\right\\}.\n\t\\]\n\tThe number of words in $\\mathbb F_{\\vg}$ of length $\\le d$, $(\\mathbb F_{\\vg})_d$, is denoted by \\df{$N_{\\rm red}(d)$} and equals\n\t\\[\n\tN_{\\rm red}(d)\n\t\\ =\\ \n\t1+\\sum_{k=1}^{d} 2\\vg\\,(2\\vg-1)^{k-1}\n\t\\ =\\ \\frac{\\vg (2\\vg-1)^{d}-1}{\\vg-1}.\n\t\\]\n\n\t\\subsection{Main results}\\label{ssec:mainresults}\n\tWe are now ready to state our main results. The first is an operator-valued version of the classical \tsum of squares theorem of Helton \\cite{Hel02} and McCullough \\cite{McC01}, Theorem~\\ref{thm:sos}. The second, Theorem~\\ref{thm:usos},\n\tis a factorization result for positive operator-valued trigonometric polynomials extending a long list of \n\tresults pertaining to scalar-valued noncommutative trigonometric polynomials \\cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \\succeq 0$ means that the operator $T$ is positive semidefinite (psd).", + "context": "\\subsection{Notation}\\label{ssec:notation}\n Fix a positive integer $\\vg$. Let \n \\df{$\\langle x \\rangle$} denote the free monoid on the $\\vg$ letters of the alphabet $x = \\{ x_{1}, \\dots, x_{\\vg}\\}.$ \n Its multiplicative identity is the empty word \n \\df{$\\varnothing$}. \n We endow $\\langle x\\rangle$ with the \\df{graded lexicographic order}.\n The length of a word $w \\in \\langle x \\rangle$ is denoted by $|w|.$ The\n set of all elements (words) of $\\langle x \\rangle$ of length (or degree) at most $d$\n is denoted \\df{$\\langle x \\ra_{d}$}.\n The cardinality of $\\langle x\\ra_d$ is \\index{$N(d)$}\n \\[\n N(d) \\ =\\ \\sum_{i=0}^d \\vg^i \\ =\\ \\frac{\\vg^{d+1}-1}{\\vg-1}.\n \\]\n\nLet \\df{$\\mathcal{H}$} be a fixed complex Hilbert space.\n Let \\df{$\\mathcal{B}(\\mathcal{H})$} be the space of all bounded linear operators on $\\mathcal{H}$, and \n let \n \\df{$\\mathcal{A} $} to be the free semigroup $\\mathcal{B}(\\mathcal{H})$-algebra on $x,$\n i.e., $\\mathcal{A}=\\mathcal{B}(\\mathcal{H})\\langle x\\rangle$. An element $p$ of $\\mathcal{A}$ takes the form, \n \\begin{equation}\\label{eq:poly}\n p \\ =\\ \\sum_{w\\in\\langle x\\rangle}^{\\rm finite} P_{w} w,\n \\end{equation}\n where\n $P_{w} \\in \\mathcal{B}(\\mathcal{H}),$ and is referred to as an (operator-valued) \n \\df{polynomial} in $x.$ \n Let \\df{$\\cA_{d}$} denote the elements from $\\mathcal{A}$ of degree at most $d$.\n\nLet $\\mathcal{K}$ be a Hilbert space and $X = (X_{1}, \\dots, X_{\\vg})$ be a tuple of operators from $\\mathcal{B}(\\mathcal{K}).$ The \n \\df{evaluation} of $p$ at $X$ is defined as \n \\[\n p(X) \\ =\\ \\sum P_{w} \\otimes X^{w},\n \\]\n where $X^{w} = X_{i_1} \\dots X_{i_n}$ for $w = x_{i_1} \\dots x_{i_n}.$ In general, $p(X)^{*}$ (the adjoint of $p(X)$) and $p^{*}(X)$ are not the same. They are the same if $X$ is a tuple of self-adjoint operators.\n\nWe will also be interested in evaluating noncommutative polynomials in tuples of unitaries on Hilbert space. An appropriate setting to consider these is the group algebra of the free group $\\mathbb F_{\\vg}$ on the $\\vg$ letters $x_i$, $i=1,\\ldots,\\vg$. \n Elements of $\\mathbb F_{\\vg}$ are (reduced) words in the alphabet $x_i,x_i^{-1}$.\n\nLet $\\mathscr{A}$ be the algebra $\\mathcal{B}(\\mathcal{H})[\\mathbb F_{\\vg}] = \\mathcal{B}(\\mathcal{H})\\otimes\\mathbb C[\\mathbb F_{\\vg}]$. Its elements are called \\df{trigonometric polynomials}, and $\\mathscr{A}$\n is\n endowed with the involution\n \\begin{equation}\n \\label{d:evalPunitary}\n \\sum_{u\\in\\mathbb F_{\\vg}}^{\\rm finite} P_u u \\ \\mapsto\\ \\sum_u P_u^* u^{-1}.\n \\end{equation}\n If $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so \n \\[\n p(X)^* \\ =\\ \\left ( \\sum P_w \\otimes X^w\\right)^* \\ =\\ \\sum P_w^* \\otimes X^{w^*} \\ =\\ p^*(X)\n \\]\n for all $p\\in\\mathscr{A}$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\\mathscr{A}$ and we let,\n for positive integers $d,$ \\index{$\\mathscr{A}$} \\index{$\\ccA_d$}\n \\[\n \\ccA_d \\ =\\ \\left\\{ \n \\sum_{\\substack{u\\in\\mathbb F_{\\vg}\\\\ |u|\\le d}} P_u u \\, :\\, P_u \\in \\mathcal{B}(\\mathcal{H})\n \\right\\}.\n \\]\n The number of words in $\\mathbb F_{\\vg}$ of length $\\le d$, $(\\mathbb F_{\\vg})_d$, is denoted by \\df{$N_{\\rm red}(d)$} and equals\n \\[\n N_{\\rm red}(d)\n \\ =\\ \n 1+\\sum_{k=1}^{d} 2\\vg\\,(2\\vg-1)^{k-1}\n \\ =\\ \\frac{\\vg (2\\vg-1)^{d}-1}{\\vg-1}.\n \\]\n\n\\subsection{Main results}\\label{ssec:mainresults}\n We are now ready to state our main results. The first is an operator-valued version of the classical sum of squares theorem of Helton \\cite{Hel02} and McCullough \\cite{McC01}, Theorem~\\ref{thm:sos}. The second, Theorem~\\ref{thm:usos},\n is a factorization result for positive operator-valued trigonometric polynomials extending a long list of \n results pertaining to scalar-valued noncommutative trigonometric polynomials \\cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \\succeq 0$ means that the operator $T$ is positive semidefinite (psd).\n\n\\begin{thm}\\label{thm:sos}\n\t\tFor $f\\in\\cA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}\n\n\t\t\\noindent If $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[label=\\textup{(\\roman*)}, resume]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:iv}\n\t\t\tThere exists $r\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}\n\n\t\t\\end{enumerate} \n\t\\end{thm}\n\n\\begin{thm}\\label{thm:usos}\n\t\tFor $f\\in\\ccA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:usos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of unitary operators $U = (U_{1}, \\dots, U_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of unitary matrices \n\t\t\t$U = (U_{1}, \\dots, U_{\\vg})\\in M_n(\\C)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N_{\\rm red}(d)}\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\\label{eq:usosthm}\n\t\t\t\tf\\ =\\ \\sum_{i=1}^{N_{\\rm red}(d)} r_i^*r_i.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\tIf $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item[\\rm (iv)] \\label{i:usos:iv}\n\t\t\tThere exists $r\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\n\t\t\t\tf\\ =\\ r^*r.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\\end{thm}", + "full_context": "\\subsection{Notation}\\label{ssec:notation}\n Fix a positive integer $\\vg$. Let \n \\df{$\\langle x \\rangle$} denote the free monoid on the $\\vg$ letters of the alphabet $x = \\{ x_{1}, \\dots, x_{\\vg}\\}.$ \n Its multiplicative identity is the empty word \n \\df{$\\varnothing$}. \n We endow $\\langle x\\rangle$ with the \\df{graded lexicographic order}.\n The length of a word $w \\in \\langle x \\rangle$ is denoted by $|w|.$ The\n set of all elements (words) of $\\langle x \\rangle$ of length (or degree) at most $d$\n is denoted \\df{$\\langle x \\ra_{d}$}.\n The cardinality of $\\langle x\\ra_d$ is \\index{$N(d)$}\n \\[\n N(d) \\ =\\ \\sum_{i=0}^d \\vg^i \\ =\\ \\frac{\\vg^{d+1}-1}{\\vg-1}.\n \\]\n\nLet \\df{$\\mathcal{H}$} be a fixed complex Hilbert space.\n Let \\df{$\\mathcal{B}(\\mathcal{H})$} be the space of all bounded linear operators on $\\mathcal{H}$, and \n let \n \\df{$\\mathcal{A} $} to be the free semigroup $\\mathcal{B}(\\mathcal{H})$-algebra on $x,$\n i.e., $\\mathcal{A}=\\mathcal{B}(\\mathcal{H})\\langle x\\rangle$. An element $p$ of $\\mathcal{A}$ takes the form, \n \\begin{equation}\\label{eq:poly}\n p \\ =\\ \\sum_{w\\in\\langle x\\rangle}^{\\rm finite} P_{w} w,\n \\end{equation}\n where\n $P_{w} \\in \\mathcal{B}(\\mathcal{H}),$ and is referred to as an (operator-valued) \n \\df{polynomial} in $x.$ \n Let \\df{$\\cA_{d}$} denote the elements from $\\mathcal{A}$ of degree at most $d$.\n\nLet $\\mathcal{K}$ be a Hilbert space and $X = (X_{1}, \\dots, X_{\\vg})$ be a tuple of operators from $\\mathcal{B}(\\mathcal{K}).$ The \n \\df{evaluation} of $p$ at $X$ is defined as \n \\[\n p(X) \\ =\\ \\sum P_{w} \\otimes X^{w},\n \\]\n where $X^{w} = X_{i_1} \\dots X_{i_n}$ for $w = x_{i_1} \\dots x_{i_n}.$ In general, $p(X)^{*}$ (the adjoint of $p(X)$) and $p^{*}(X)$ are not the same. They are the same if $X$ is a tuple of self-adjoint operators.\n\nWe will also be interested in evaluating noncommutative polynomials in tuples of unitaries on Hilbert space. An appropriate setting to consider these is the group algebra of the free group $\\mathbb F_{\\vg}$ on the $\\vg$ letters $x_i$, $i=1,\\ldots,\\vg$. \n Elements of $\\mathbb F_{\\vg}$ are (reduced) words in the alphabet $x_i,x_i^{-1}$.\n\nLet $\\mathscr{A}$ be the algebra $\\mathcal{B}(\\mathcal{H})[\\mathbb F_{\\vg}] = \\mathcal{B}(\\mathcal{H})\\otimes\\mathbb C[\\mathbb F_{\\vg}]$. Its elements are called \\df{trigonometric polynomials}, and $\\mathscr{A}$\n is\n endowed with the involution\n \\begin{equation}\n \\label{d:evalPunitary}\n \\sum_{u\\in\\mathbb F_{\\vg}}^{\\rm finite} P_u u \\ \\mapsto\\ \\sum_u P_u^* u^{-1}.\n \\end{equation}\n If $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so \n \\[\n p(X)^* \\ =\\ \\left ( \\sum P_w \\otimes X^w\\right)^* \\ =\\ \\sum P_w^* \\otimes X^{w^*} \\ =\\ p^*(X)\n \\]\n for all $p\\in\\mathscr{A}$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\\mathscr{A}$ and we let,\n for positive integers $d,$ \\index{$\\mathscr{A}$} \\index{$\\ccA_d$}\n \\[\n \\ccA_d \\ =\\ \\left\\{ \n \\sum_{\\substack{u\\in\\mathbb F_{\\vg}\\\\ |u|\\le d}} P_u u \\, :\\, P_u \\in \\mathcal{B}(\\mathcal{H})\n \\right\\}.\n \\]\n The number of words in $\\mathbb F_{\\vg}$ of length $\\le d$, $(\\mathbb F_{\\vg})_d$, is denoted by \\df{$N_{\\rm red}(d)$} and equals\n \\[\n N_{\\rm red}(d)\n \\ =\\ \n 1+\\sum_{k=1}^{d} 2\\vg\\,(2\\vg-1)^{k-1}\n \\ =\\ \\frac{\\vg (2\\vg-1)^{d}-1}{\\vg-1}.\n \\]\n\n\\subsection{Main results}\\label{ssec:mainresults}\n We are now ready to state our main results. The first is an operator-valued version of the classical sum of squares theorem of Helton \\cite{Hel02} and McCullough \\cite{McC01}, Theorem~\\ref{thm:sos}. The second, Theorem~\\ref{thm:usos},\n is a factorization result for positive operator-valued trigonometric polynomials extending a long list of \n results pertaining to scalar-valued noncommutative trigonometric polynomials \\cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \\succeq 0$ means that the operator $T$ is positive semidefinite (psd).\n\n\\begin{thm}\\label{thm:sos}\n\t\tFor $f\\in\\cA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}\n\n\t\t\\noindent If $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[label=\\textup{(\\roman*)}, resume]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:iv}\n\t\t\tThere exists $r\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}\n\n\t\t\\end{enumerate} \n\t\\end{thm}\n\n\\begin{thm}\\label{thm:usos}\n\t\tFor $f\\in\\ccA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:usos:i}\n\t\t\tFor any Hilbert space $\\cK$ and any tuple of unitary operators $U = (U_{1}, \\dots, U_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:ii}\n\t\t\tFor any $n\\in\\N$ and any tuple of unitary matrices \n\t\t\t$U = (U_{1}, \\dots, U_{\\vg})\\in M_n(\\C)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N_{\\rm red}(d)}\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\\label{eq:usosthm}\n\t\t\t\tf\\ =\\ \\sum_{i=1}^{N_{\\rm red}(d)} r_i^*r_i.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\tIf $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item[\\rm (iv)] \\label{i:usos:iv}\n\t\t\tThere exists $r\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\n\t\t\t\tf\\ =\\ r^*r.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\\end{thm}\n\nLet $\\ccA$ be the algebra $\\cB(\\cH)[\\freeg] = \\cB(\\cH)\\otimes\\C[\\freeg]$. Its elements are called \\df{trigonometric polynomials}, and $\\ccA$\n is\n endowed with the involution\n \\begin{equation}\n \\label{d:evalPunitary}\n \\sum_{u\\in\\freeg}^{\\rm finite} P_u u \\ \\mapsto\\ \\sum_u P_u^* u^{-1}.\n \\end{equation}\n If $X$ is a tuple of unitary operators, then $(X^w)^* = X^{w^{-1}}$ and so \n \\[\n p(X)^* \\ =\\ \\left ( \\sum P_w \\otimes X^w\\right)^* \\ =\\ \\sum P_w^* \\otimes X^{w^*} \\ =\\ p^*(X)\n \\]\n for all $p\\in\\ccA$. The notions of length of a word, degree of a polynomial, etc.~extend naturally to $\\ccA$ and we let,\n for positive integers $d,$ \\index{$\\ccA$} \\index{$\\ccA_d$}\n \\[\n \\ccA_d \\ =\\ \\left\\{ \n \\sum_{\\substack{u\\in\\freeg\\\\ |u|\\le d}} P_u u \\, :\\, P_u \\in \\cB(\\cH)\n \\right\\}.\n \\]\n The number of words in $\\freeg$ of length $\\le d$, $(\\freeg)_d$, is denoted by \\df{$N_{\\rm red}(d)$} and equals\n \\[\n N_{\\rm red}(d)\n \\ =\\ \n 1+\\sum_{k=1}^{d} 2\\vg\\,(2\\vg-1)^{k-1}\n \\ =\\ \\frac{\\vg (2\\vg-1)^{d}-1}{\\vg-1}.\n \\]\n\n\\subsection{Main results}\\label{ssec:mainresults}\n We are now ready to state our main results. The first is an operator-valued version of the classical sum of squares theorem of Helton \\cite{Hel02} and McCullough \\cite{McC01}, Theorem~\\ref{thm:sos}. The second, Theorem~\\ref{thm:usos},\n is a factorization result for positive operator-valued trigonometric polynomials extending a long list of \n results pertaining to scalar-valued noncommutative trigonometric polynomials \\cite{McC01,HMP04,BT07,NT13,KVV17,Oz13}. For a bounded operator $T$, the notation $T \\succeq 0$ means that the operator $T$ is positive semidefinite (psd).\n\n\\noindent If $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n \\begin{enumerate}[label=\\textup{(\\roman*)}, resume]\\itemsep=5pt\n \\item \\label{i:sos:iv}\n There exists $r\\in\\cA_d$ s.t.\n \\begin{equation}\\label{eq:1sosthm}\n f \\ =\\ r^*r.\n \\end{equation}\n\n\\begin{remark} \\rm\n Several remarks related to Theorem~\\ref{thm:sos} are in order. \\begin{enumerate}[\\rm(a)]\\itemsep=5pt\n \\item Item~\\ref{i:sos:iii} can also be phrased as a factorization result. \n Letting $r=\\text{col} \\begin{pmatrix} r_1 & \\cdots & r_{N(d)}\\end{pmatrix} \\in \n \\cA^{N(d)}=\n \\cB(\\cH,\\cH^{N(d)})\\la x\\ra$, \\eqref{eq:sosthm} simply states\n \\[\n f\\ =\\ r^*r.\n \\]\n We refer to \\cite{BGK79,BGKR10,DW05,GW05,DR10} and the references therein for an in depth investigation of factorization.\n \\item\n That item~\\ref{i:sos:iii} implies \n item~\\ref{i:sos:i} implies item~\\ref{i:sos:ii} is trivial. The main content of Theorem~\\ref{thm:sos} is that item~\\ref{i:sos:ii}\n implies item~\\ref{i:sos:iii}. A routine argument shows the equivalence between \\eqref{eq:sosthm} and \\eqref{eq:1sosthm} in the infinite-dimensional case, see Remark~\\ref{rem:soscone}.\n \\item\n Our proof yields no bound on the size $n$ of matrices needed in item~\\ref{i:sos:ii}. \n \\item\n From Theorem~\\ref{thm:sos} one can easily \n deduce its version for free non-self-adjoint variables $z,z^*$ via the usual identification\n $z_j\\mapsto \\real{z_j}= \\frac{z_j+z_j^*}2$ and hence\n $z_j^*\\mapsto \\imag{z_j}= \\frac{z_j-z_j^*}{2i}$.\n \\qed\n \\end{enumerate}\n \\end{remark} The following result is the unitary version of Theorem~\\ref{thm:sos}.\n\n\\begin{thm}\\label{thm:usos}\n For $f\\in\\ccA_{2d}$ the following are equivalent:\n \\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n \\item \\label{i:usos:i}\n For any Hilbert space $\\cK$ and any tuple of unitary operators $U = (U_{1}, \\dots, U_{\\vg})\\in\\cB(\\cK)^\\vg$, $f(U)\\succeq0$;\n \\item \\label{i:usos:ii}\n For any $n\\in\\N$ and any tuple of unitary matrices \n $U = (U_{1}, \\dots, U_{\\vg})\\in M_n(\\C)^\\vg$, $f(U)\\succeq0$;\n \\item \\label{i:usos:iii}\n There exist $r_1,\\ldots,r_{N_{\\rm red}(d)}\\in\\ccA_d$ s.t.\n \\begin{equation*}\\label{eq:usosthm}\n f\\ =\\ \\sum_{i=1}^{N_{\\rm red}(d)} r_i^*r_i.\n \\end{equation*}\n \\end{enumerate}\n If $\\cH$ is infinite-dimensional, then the above statements are also equivalent to\n \\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n \\item[\\rm (iv)] \\label{i:usos:iv}\n There exists $r\\in\\ccA_d$ s.t.\n \\begin{equation*}\n f\\ =\\ r^*r.\n \\end{equation*}\n \\end{enumerate}\n \\end{thm}\n\nIndex \\df{$\\mathbb{\\cH}^{N(d)}$} and \\df{$\\cA_{d}^{N(d)}$} (the algebraic direct sum of $\\cA_{d}$ with itself $N(d)$ times) by $\\la x \\ra_{d}.$ \n Let $V_{d} \\in \\cA_{d}^{N}$ denote the \\df{Veronese column vector} whose $w\\in \\la x \\ra_{d}$ entry is $w$\n (adopting the usual convention of viewing $w$ as the the $\\cB(\\cH)$-valued polynomial $I_\\cH \\, w$). For instance, \n if $\\vg=2$ and $d=2$, then \\index{$V_d$} \n \\[\n V_2\\ =\\ \\text{col} \\begin{pmatrix}\n 1 & x_1 & x_2 & x_1^2 & x_1 x_2 & x_2 x_1 & x_2^2\n \\end{pmatrix}.\n \\]\n Let \\df{$\\cC_d$} denote the \\df{cone of sums of squares} of polynomials of degree at most $d,$\n \\begin{equation}\\label{eq:sosdef}\n \\cC_{d} \\ :=\\ \\left\\{ \\sum\\limits_{i=1}^{N(d)} r_{i}^{*}r_{i}: \\quad r_{i}\\in \\cA_{d}, \\quad i=1, \\dots, N(d) \\right\\}\n \\subseteq \\cA_{2d}.\n \\end{equation}\n Given $r\\in \\cA_d,$ the column vector $R$ with $w$ entry $R_{w}^{*}$ \n is called the \\df{coefficient vector} of $r$ since $r = R^{*} V_{d}.$ In particular, \n \\[\n r^* r \\ =\\ V_d^* RR^* V_d\n \\]\n so that $r^*r$ has a representation as $V_d^* \\SG V_d$ for a psd matrix $\\SG.$\n\n\\begin{lemma}\\label{lem:uweak2WOT}\n If $\\varphi:\\mathcal A_{2d}\\to\\mathbb C$ be an ultraweak continuous linear functional that \n separates the cone $\\cC_{d}$ from a fixed polynomial $p$ in $\\cA_{2d},$ that is, \n \\[\n \\varphi(r^{*}r) \\geq 0 \\quad \\text{for all $r \\in \\cA_{d}$} \\quad \\text{and} \\quad \\varphi(p+p^{*}) <0,\n \\]\n then there exists a WOT continuous linear functionl $\\tilde{\\varphi} : \\cA_{2d} \\to \\mathbb{C}$ such that \n \\[\n \\tilde{\\varphi}(r^{*}r) \\geq 0 \\quad \\text{for all $r \\in \\cA_{d}$} \\quad \\text{and} \\quad \\tilde{\\varphi}(p+p^{*}) <0.\n \\]\n \\end{lemma}\n \\begin{proof}\n Since $\\varphi$ is ultraweak continuous, there exist trace class operators $S_{w}$ $(w \\in \\la x\\ra_{2d})$ in $\\cB(\\cH)$ such that \n \\[\n \\varphi(q) = \\sum\\limits_{w \\in \\la x\\ra_{2d}} {\\rm Tr}\\, (S_{w} Q_{w}),\n \\]\n where $q = \\sum_{w \\in \\la x \\ra_{2d}} Q_{w} w.$ For $r,r^\\prime \\in \\cA_{d}$, \\[\n \\varphi(r^{*}r^\\prime) \\ =\\ \\sum\\limits_{u,v\\in \\la x \\ra_{d}} {\\rm Tr}\\, (S_{u^{*}v} R_{u}^{*}R^\\prime_{v}),\n \\]\n where $r = \\sum_{u \\in \\la x \\ra_{d}} R_{u} u$\n and $r^\\prime = \\sum_{v \\in \\la x \\ra_{d}} R^\\prime_{v} v.$ Denote by $S$ the $N(d) \\times N(d)$ block operator matrix whose $(u,v)$ entry is $S_{v^{*}u}.$\n\n\\begin{prop}\\label{prop:GNS}\n If $\\varphi:\\mathcal A_{2d+2}\\to\\mathbb C$ is a continuous linear functional such that\n \\[\n \\varphi(p^ *p)\\ \\ge\\ 0 \\]\n for all $p\\in \\cA_{d+1},$ \n then there exist a finite-dimensional Hilbert space $\\cE$, a self-adjoint $g$-tuple $Y=(Y_1,\\dots,Y_\\vg)$ on $\\cE$, and a vector $\\gamma\\in \\cH\\otimes \\cE$ such that\n \\[\n \\varphi(q^ *p)\\ =\\ \\big\\langle p(Y)\\gamma,\\ q(Y)\\gamma\\big\\rangle_{\\cH\\otimes \\cE} \\qquad\\text{for all }p \\in \\cA_{d+1},\\ q\\in\\mathcal A_{d}. \n \\]\n Therefore, for all $p\\in\\mathcal A_{2d+1}$,\n \\[\n \\varphi(p)\\ =\\ \\langle p(Y)\\gamma,\\ \\gamma\\rangle. \n \\]\n \\end{prop}", + "post_theorem_intro_text_len": 5987, + "post_theorem_intro_text": "\\begin{remark} \\rm\n\t\tSeveral remarks related to Theorem~\\ref{thm:sos} are in order. \t\t\\begin{enumerate}[\\rm(a)]\\itemsep=5pt\n\t\t\t\\item Item~\\ref{i:sos:iii} can also be phrased as a factorization result. \n\t\t\tLetting $r=\\text{col} \\begin{pmatrix} r_1 & \\cdots & r_{N(d)}\\end{pmatrix} \\in \n\t\t\t\\mathcal{A}^{N(d)}=\n\t\t\t\\mathcal{B}(\\mathcal{H},\\mathcal{H}^{N(d)})\\langle x\\rangle$, \\eqref{eq:sosthm} simply states\n\t\t\t\\[\n\t\t\tf\\ =\\ r^*r.\n\t\t\t\\]\n\t\t\tWe refer to \\cite{BGK79,BGKR10,DW05,GW05,DR10} and the references therein for an in depth investigation of factorization.\n\t\t\t\\item\n\t\t\tThat item~\\ref{i:sos:iii} implies \n\t\t\titem~\\ref{i:sos:i} implies item~\\ref{i:sos:ii} is trivial. The main content of Theorem~\\ref{thm:sos} is that item~\\ref{i:sos:ii}\n\t\t\timplies item~\\ref{i:sos:iii}. A routine argument shows the equivalence between \\eqref{eq:sosthm} and \\eqref{eq:1sosthm} in the infinite-dimensional case, see Remark~\\ref{rem:soscone}.\n\t\t\t\\item\n\t\t\tOur proof yields no bound on the size $n$ of matrices needed in item~\\ref{i:sos:ii}. \n\t\t\t\\item\n\t\t\tFrom Theorem~\\ref{thm:sos} one can easily \n\t\t\tdeduce its version for free non-self-adjoint variables $z,z^*$ via the usual identification\n\t\t\t$z_j\\mapsto \\operatorname{real}{z_j}= \\frac{z_j+z_j^*}2$ and hence\n\t\t\t$z_j^*\\mapsto \\operatorname{imag}{z_j}= \\frac{z_j-z_j^*}{2i}$.\n\t\t\t\\qed\n\t\t\\end{enumerate}\n\t\\end{remark} The following result is the unitary version of Theorem~\\ref{thm:sos}.\n\n\t\\begin{thm}\\label{thm:usos}\n\t\tFor $f\\in\\ccA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:usos:i}\n\t\t\tFor any Hilbert space $\\mathcal{K}$ and any tuple of unitary operators $U = (U_{1}, \\dots, U_{\\vg})\\in\\mathcal{B}(\\mathcal{K})^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:ii}\n\t\t\tFor any $n\\in\\mathbb N$ and any tuple of unitary matrices \n\t\t\t$U = (U_{1}, \\dots, U_{\\vg})\\in M_n(\\mathbb C)^\\vg$, $f(U)\\succeq0$;\n\t\t\t\\item \\label{i:usos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N_{\\rm red}(d)}\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\\label{eq:usosthm}\n\t\t\t\tf\\ =\\ \\sum_{i=1}^{N_{\\rm red}(d)} r_i^*r_i.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\tIf $\\mathcal{H}$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item[\\rm (iv)] \\label{i:usos:iv}\n\t\t\tThere exists $r\\in\\ccA_d$ s.t.\n\t\t\t\\begin{equation*}\n\t\t\t\tf\\ =\\ r^*r.\n\t\t\t\\end{equation*}\n\t\t\\end{enumerate}\n\t\\end{thm}\n\n\t\\begin{remark}[What's new?]\\rm\n\t\t\\label{rem:whatsnew}\n\t\tThe passage to operator coefficients necessitates several novel\n\t\tresults and constructions that we expect to be of independent interest. At a high level, the proofs of Theorem~\\ref{thm:sos} and Theorem~\\ref{thm:usos} still follow the now standard paradigm for establishing sum of squares (sos) representations (factorizations).\n\t\tNamely, the Hahn-Banach theorem produces a separating linear functional $\\varphi$, and then a Gelfand-Naimark-Segal (GNS) construction\n\t\tbased on $\\varphi$ ultimately produces a tuple $Y$. \n\t\tHere we \n\t\troughly follow the outline of \\cite{MP05}. \n\n\t\tA key construction is that of a tuple of self-adjoint operators $A$\n\t\tbased upon the left regular representation on Fock space; see Section~\\ref{sec:fock}.\n\t\tWe then show that, up to a universal constant, for a sum of squares polynomial $p,$ \n\t\tthe norm\n\t\tof $p(A)$ bounds the norm of any non-commutative psd Gram matrix $G$ that represents $p$; see Proposition~\\ref{prop:bounded}. \n\t\tThis uniform bound is the main input in Section~\\ref{sec:top} for proving that the cone $\\cC_d$ of sums of squares is closed in the product ultraweak topology on the coefficients. A separate approximation argument then replaces an ultraweak continuous separating functional by a WOT continuous one and hence yields closedness of $\\cC_d$ in the product WOT. This two-step interplay between the ultraweak and WOT topologies enables an application of the Hahn-Banach separation theorem.\n\n\t\tOn the GNS side, we introduce a new argument that exploits the WOT to associate to a\n\t\tseparating linear functional a finite-rank psd noncommutative representing Hankel matrix and,\n\t\ton its range, construct the desired tuple~$Y$; see Section~\\ref{sec:gns}. For the unitary result,\n\t\tTheorem~\\ref{thm:usos}, we additionally modify the construction of Section~\\ref{sec:fock} to produce a canonical\n\t\ttuple of unitary operators from the left-regular representation and adapt the GNS procedure\n\t\tto obtain a unitary tuple together with a representing vector that realizes the separating\n\t\tfunctional as a vector state; see Subsection~\\ref{ssec:uGNS}.\\qed\n\t\\end{remark}\n\n\t\\subsection{Reader's guide}\\label{ssec:readersguide}\n\n\tThe paper is structured as follows. The convex cone of sums of squares (making an appearance in \\eqref{eq:sosthm} of Theorem~\\ref{thm:sos}) is introduced and characterized in the next Section~\\ref{sec:sos}. In Section~\\ref{sec:fock} we define creation operators $L_i$ on the full Fock $\\cF_\\vg^2$ space and their symmetrized analogs $A_i$. How they pertain to the sum of squares statement at hand is explored in Subsection~\\ref{ssec:coeff}, where evaluations at $A$ are used to extract coefficients of a polynomial. In Section~\\ref{sec:top} we \n\tintroduce a suitable topology on $\\cA_\\lcal{d}$ and\n\tcollect all the necessary topological properties needed in the sequel. With respect to this topology, the convex cone of sums of squares is closed, see Proposition~\\ref{prop:closedcone}. The fact that the cone is closed allows for an application of the Hahn--Banach Separation Theorem, which is then followed by an appropriate version of the GNS construction, carried out in Section~\\ref{sec:gns}; see Proposition~\\ref{prop:GNS}. \n\tThen Theorem~\\ref{thm:sos} is proved in Section~\\ref{sec:proof} and\n\tTheorem~\\ref{thm:usos} is proved in Section~\\ref{sec:uproof}.\n\n\t\\subsection*{Acknowledgment} The authors thank Dr. Matthias Schötz for carefully reading an earlier version of this manuscript and for pointing out gaps in the proof of Proposition~\\ref{prop:closedcone}.", + "sketch": "The post-theorem discussion says the nontrivial direction in Theorem~\\ref{thm:sos} is item~\\ref{i:sos:ii} $\\Rightarrow$ item~\\ref{i:sos:iii} (while item~\\ref{i:sos:iii} $\\Rightarrow$ item~\\ref{i:sos:i} $\\Rightarrow$ item~\\ref{i:sos:ii} is “trivial”). It then outlines a “standard paradigm” proof (following \\cite{MP05}): apply the Hahn–Banach theorem to obtain a separating linear functional $\\varphi$, and then use a Gelfand–Naimark–Segal (GNS) construction based on $\\varphi$ to produce a tuple $Y$.\n\nKey intermediate steps highlighted are:\n\\begin{itemize}\n\\item Construct a tuple of self-adjoint operators $A$ from the left regular representation on Fock space (Section~\\ref{sec:fock}).\n\\item Prove a uniform bound: “up to a universal constant,” for an sos polynomial $p$, $\\|p(A)\\|$ bounds the norm of any psd noncommutative Gram matrix $G$ representing $p$ (Proposition~\\ref{prop:bounded}).\n\\item Use this bound (Section~\\ref{sec:top}) to show the sos cone $\\cC_d$ is closed in the product ultraweak topology; then use “a separate approximation argument” to replace an ultraweak-continuous separating functional by a WOT-continuous one, yielding closedness in the product WOT. This “two-step interplay between the ultraweak and WOT topologies” enables applying Hahn–Banach separation.\n\\item On the GNS side (Section~\\ref{sec:gns}), “exploit the WOT to associate to a separating linear functional a finite-rank psd noncommutative representing Hankel matrix and, on its range, construct the desired tuple~$Y$.”\n\\end{itemize}\nIt also notes that in the infinite-dimensional case “a routine argument shows the equivalence between \\eqref{eq:sosthm} and \\eqref{eq:1sosthm}.”", + "expanded_sketch": "The post-theorem discussion says the nontrivial direction in the main theorem is item~\\ref{i:sos:ii} $\\Rightarrow$ item~\\ref{i:sos:iii} (while item~\\ref{i:sos:iii} $\\Rightarrow$ item~\\ref{i:sos:i} $\\Rightarrow$ item~\\ref{i:sos:ii} is “trivial”). It then outlines a “standard paradigm” proof (following \\cite{MP05}): apply the Hahn–Banach theorem to obtain a separating linear functional $\\varphi$, and then use a Gelfand–Naimark–Segal (GNS) construction based on $\\varphi$ to produce a tuple $Y$.\n\nKey intermediate steps highlighted are:\n\\begin{itemize}\n\\item Construct a tuple of self-adjoint operators $A$ from the left regular representation on Fock space (Section~\\ref{sec:fock}).\n\\item Prove a uniform bound: “up to a universal constant,” for an sos polynomial $p$, $\\|p(A)\\|$ bounds the norm of any psd noncommutative Gram matrix $G$ representing $p$. We first record the needed proposition.\n\n\\begin{prop} \\label{prop:bounded}\n\t\tIf $p \\in \\cC_{d},$ then the set \n\t\t\\[\n\t\t\\Gamma_p \\ =\\ \\{\\, \\SG \\in \\cB(\\cH)^{N(d)\\times N(d)} \\ : \\quad \\SG \\succeq 0, \\quad V_d^* \\SG V_d = p \\,\\}\n\t\t\\]\n\t\tis norm bounded $($with respect to the operator norm on $\\cB(\\cH^{N(d)})$$)$. More precisely, \n\t\tthere exists a constant $\\mu_{d}$ $($depending only on $d$ and $\\vg$ and not on $p$$)$ such that,\n\t\tfor all $\\SG\\in\\Gamma_p,$ \n\t\t\\[ \n\t\t\\|\\SG\\| \\ \\leq\\ \\mu_{d} \\; \\|p(A)\\| ,\n\t\t\\]\n\t\twhere the tuple $A=A^{(\\ad)}$ is defined in \\eqref{eq:symcreate}\n\t\tfor any $\\ad\\geq 2d$. \n\t\\end{prop}\n\n\\item Use this bound (Section~\\ref{sec:top}) to show the sos cone $\\cC_d$ is closed in the product ultraweak topology; then use “a separate approximation argument” to replace an ultraweak-continuous separating functional by a WOT-continuous one, yielding closedness in the product WOT. This “two-step interplay between the ultraweak and WOT topologies” enables applying Hahn–Banach separation.\n\\item On the GNS side (Section~\\ref{sec:gns}), “exploit the WOT to associate to a separating linear functional a finite-rank psd noncommutative representing Hankel matrix and, on its range, construct the desired tuple~$Y$.”\n\\end{itemize}\nIt also notes that in the infinite-dimensional case “a routine argument shows the equivalence between\n\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\nand\n\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}\n.”", + "expanded_theorem": "\\label{thm:sos}\n\t\tFor $f\\in\\cA_{2d}$ the following are equivalent:\n\t\t\\begin{enumerate}[\\rm(i)]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:i}\n\t\t\tFor any Hilbert space $\\mathcal{K}$ and any tuple of self-adjoint operators $Y = (Y_{1}, \\dots, Y_{\\vg})\\in\\mathcal{B}(\\mathcal{K})^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:ii}\n\t\t\tFor any $n\\in\\mathbb N$ and any tuple of self-adjoint matrices \n\t\t\t$Y = (Y_{1}, \\dots, Y_{\\vg})\\in M_n(\\mathbb C)^\\vg$, $f(Y)\\succeq0$;\n\t\t\t\\item \\label{i:sos:iii}\n\t\t\tThere exist $r_1,\\ldots,r_{N(d)}\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:sosthm}\n\t\t\t\tf \\ =\\ \\sum_{i=1}^{N(d)} r_i^*r_i.\n\t\t\t\\end{equation}\n\t\t\\end{enumerate}\n\n\t\t\\noindent If $\\mathcal{H}$ is infinite-dimensional, then the above statements are also equivalent to\n\t\t\\begin{enumerate}[label=\\textup{(\\roman*)}, resume]\\itemsep=5pt\n\t\t\t\\item \\label{i:sos:iv}\n\t\t\tThere exists $r\\in\\cA_d$ s.t.\n\t\t\t\\begin{equation}\\label{eq:1sosthm}\n\t\t\t\tf \\ =\\ r^*r.\n\t\t\t\\end{equation}\n\n\t\t\\end{enumerate}", + "theorem_type": [ + "Biconditional or Equivalence", + "Existence" + ], + "mcq": { + "question": "Fix a positive integer $\\vg$ and a complex Hilbert space $\\mathcal H$. Let $\\mathcal A=\\mathcal B(\\mathcal H)\\langle x_1,\\dots,x_{\\vg}\\rangle$ be the algebra of operator-valued noncommutative polynomials in free variables $x_1,\\dots,x_{\\vg}$, and let $\\mathcal A_d$ denote the polynomials of degree at most $d$. For a Hilbert space $\\mathcal K$ and a self-adjoint tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in \\mathcal B(\\mathcal K)^{\\vg}$, $f(Y)$ denotes the evaluation of $f$ at $Y$. Also set $N(d)=\\sum_{i=0}^d \\vg^i$. Given $f\\in \\mathcal A_{2d}$, which existence statement holds precisely under the positivity assumption that $f(Y)\\succeq 0$ for every Hilbert space $\\mathcal K$ and every self-adjoint tuple $Y\\in\\mathcal B(\\mathcal K)^{\\vg}$ (equivalently, for every $n\\in\\mathbb N$ and every self-adjoint tuple $Y\\in M_n(\\mathbb C)^{\\vg}$)?", + "correct_choice": { + "label": "A", + "text": "For $f\\in\\mathcal A_{2d}$, the following are equivalent: (i) for every Hilbert space $\\mathcal K$ and every self-adjoint tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in\\mathcal B(\\mathcal K)^{\\vg}$, one has $f(Y)\\succeq 0$; (ii) for every $n\\in\\mathbb N$ and every self-adjoint tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in M_n(\\mathbb C)^{\\vg}$, one has $f(Y)\\succeq 0$; (iii) there exist $r_1,\\dots,r_{N(d)}\\in\\mathcal A_d$ such that $f=\\sum_{i=1}^{N(d)} r_i^*r_i$. If $\\mathcal H$ is infinite-dimensional, these are also equivalent to (iv) there exists a single $r\\in\\mathcal A_d$ such that $f=r^*r$." + }, + "choices": [ + { + "label": "B", + "text": "For $f\\in\\mathcal A_{2d}$, the following are equivalent: (i) for every Hilbert space $\\mathcal K$ and every self-adjoint tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in\\mathcal B(\\mathcal K)^{\\vg}$, one has $f(Y)\\succeq 0$; (ii) for every $n\\in\\mathbb N$ and every self-adjoint tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in M_n(\\mathbb C)^{\\vg}$, one has $f(Y)\\succeq 0$; (iii) there exist $r_1,\\dots,r_{N(2d)}\\in\\mathcal A_d$ such that $f=\\sum_{i=1}^{N(2d)} r_i^*r_i$. If $\\mathcal H$ is infinite-dimensional, these are also equivalent to (iv) there exists a single $r\\in\\mathcal A_d$ such that $f=r^*r$." + }, + { + "label": "C", + "text": "If there exist $r_1,\\dots,r_{N(d)}\\in\\mathcal A_d$ such that $f=\\sum_{i=1}^{N(d)} r_i^*r_i$, then for every Hilbert space $\\mathcal K$ and every self-adjoint tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in\\mathcal B(\\mathcal K)^{\\vg}$, one has $f(Y)\\succeq 0$. Moreover, if $\\mathcal H$ is infinite-dimensional and $f=r^*r$ for some $r\\in\\mathcal A_d$, then the same positivity conclusion holds." + }, + { + "label": "D", + "text": "For $f\\in\\mathcal A_{2d}$, the following are equivalent: (i) for every Hilbert space $\\mathcal K$ and every tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in\\mathcal B(\\mathcal K)^{\\vg}$, one has $f(Y)\\succeq 0$; (ii) for every $n\\in\\mathbb N$ and every tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in M_n(\\mathbb C)^{\\vg}$, one has $f(Y)\\succeq 0$; (iii) there exist $r_1,\\dots,r_{N(d)}\\in\\mathcal A_d$ such that $f=\\sum_{i=1}^{N(d)} r_i^*r_i$. If $\\mathcal H$ is infinite-dimensional, these are also equivalent to (iv) there exists a single $r\\in\\mathcal A_d$ such that $f=r^*r$." + }, + { + "label": "E", + "text": "For $f\\in\\mathcal A_{2d}$, the following are equivalent: (i) for every Hilbert space $\\mathcal K$ and every self-adjoint tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in\\mathcal B(\\mathcal K)^{\\vg}$, one has $f(Y)\\succeq 0$; (ii) for every $n\\le N(d)$ and every self-adjoint tuple $Y=(Y_1,\\dots,Y_{\\vg})\\in M_n(\\mathbb C)^{\\vg}$, one has $f(Y)\\succeq 0$; (iii) there exist $r_1,\\dots,r_{N(d)}\\in\\mathcal A_d$ such that $f=\\sum_{i=1}^{N(d)} r_i^*r_i$. If $\\mathcal H$ is infinite-dimensional, these are also equivalent to (iv) there exists a single $r\\in\\mathcal A_d$ such that $f=r^*r$." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "counting_estimate", + "tampered_component": "sharp Gram-size bound $N(d)$", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped converse/equivalence, keeping only sos-implies-positivity", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "characteristic", + "tampered_component": "self-adjointness hypothesis in evaluation", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "finiteness", + "tampered_component": "quantification over all matrix sizes", + "template_used": "quantifier_dependence" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly reveal the correct option. It states the positivity hypothesis and asks for the precise corresponding existence/equivalence statement, but the exact sum-of-squares form, sharp bound N(d), and infinite-dimensional single-factor refinement are not given away." + }, + "TAS": { + "score": 1, + "justification": "This is largely a theorem-identification item: the correct answer is essentially the exact Positivstellensatz statement with small perturbations in the alternatives. It is not a pure verbatim restatement, since the student must distinguish nearby variants, but it still tests recall of a known theorem more than independent conclusion-drawing." + }, + "GPS": { + "score": 1, + "justification": "Moderate reasoning is needed to reject plausible variants: N(d) versus N(2d), self-adjoint versus arbitrary tuples, all matrix sizes versus only n ≤ N(d), and equivalence versus a one-way implication. Still, the task is mainly precision matching of a theorem statement rather than deeper generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong and mathematically meaningful. They target realistic failure modes: weakening an equivalence, changing the sharp Gram bound, dropping self-adjointness, and incorrectly restricting matrix sizes. They are distinct, plausible, and well-aligned with the content." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with high-quality distractors and little answer leakage, but it mainly assesses precise recall/recognition of a theorem rather than substantial generative reasoning." + } + }, + { + "id": "2511.03863v1", + "paper_link": "http://arxiv.org/abs/2511.03863v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:main}\n There is a polynomial time algorithm that, given a matching covered graph $G$, finds a lattice basis for $\\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$.", + "start_pos": 14398, + "end_pos": 14622, + "label": "thm:main" + }, + "ref_dict": {}, + "pre_theorem_intro_text_len": 1429, + "pre_theorem_intro_text": "Consider a graph $G=(V,E)$ with an even number of vertices. A \\emph{perfect matching} is a set of edges $M\\subseteq E$ incident with each vertex exactly once. $G$ is \\emph{matching covered} if every edge of $G$ belongs to some perfect matching.\nThe theory of matching covered graphs is a rich area of combinatorics with pioneering results due to Edmonds, Seymour, and Lov\\'asz. An object of particular interest is the \\emph{perfect matching lattice}, defined as the set of integral linear combinations of the incidence vectors of perfect matchings: \\begin{equation*}\n \\mathcal L(G): = \\lat(\\mathcal{M})= \\{\\sum_{M \\in \\mathcal{M}}k_M \\mathbf{1}_M: k_M \\in \\mathbb{Z}\\},\n\\end{equation*} where $\\mathcal{M}$ denotes the set of all perfect matchings of $G$.\nLov\\'asz \\cite{lovaszPM} gave a characterization of this lattice based on a structural decomposition of the graph into blocks called \\emph{bricks} and \\emph{braces}, with the Petersen graph being the only obstruction for this lattice to contain all the integral points in the linear hull of the perfect matchings. \nA natural question is whether the generating set of this lattice, i.e. the perfect matchings of $G$, contains a basis for $\\mathcal L(G)$. This question was answered positively by de Carvalho, Lucchesi, and Murty \\cite{dimension} who proved it through ear-decompositions of a graph. In this paper, we present an algorithmic analogue of this result, mainly:", + "context": "Consider a graph $G=(V,E)$ with an even number of vertices. A \\emph{perfect matching} is a set of edges $M\\subseteq E$ incident with each vertex exactly once. $G$ is \\emph{matching covered} if every edge of $G$ belongs to some perfect matching.\nThe theory of matching covered graphs is a rich area of combinatorics with pioneering results due to Edmonds, Seymour, and Lov\\'asz. An object of particular interest is the \\emph{perfect matching lattice}, defined as the set of integral linear combinations of the incidence vectors of perfect matchings: \\begin{equation*}\n \\mathcal L(G): = \\lat(\\mathcal{M})= \\{\\sum_{M \\in \\mathcal{M}}k_M \\mathbf{1}_M: k_M \\in \\mathbb{Z}\\},\n\\end{equation*} where $\\mathcal{M}$ denotes the set of all perfect matchings of $G$.\nLov\\'asz \\cite{lovaszPM} gave a characterization of this lattice based on a structural decomposition of the graph into blocks called \\emph{bricks} and \\emph{braces}, with the Petersen graph being the only obstruction for this lattice to contain all the integral points in the linear hull of the perfect matchings. \nA natural question is whether the generating set of this lattice, i.e. the perfect matchings of $G$, contains a basis for $\\mathcal L(G)$. This question was answered positively by de Carvalho, Lucchesi, and Murty \\cite{dimension} who proved it through ear-decompositions of a graph. In this paper, we present an algorithmic analogue of this result, mainly:", + "full_context": "Consider a graph $G=(V,E)$ with an even number of vertices. A \\emph{perfect matching} is a set of edges $M\\subseteq E$ incident with each vertex exactly once. $G$ is \\emph{matching covered} if every edge of $G$ belongs to some perfect matching.\nThe theory of matching covered graphs is a rich area of combinatorics with pioneering results due to Edmonds, Seymour, and Lov\\'asz. An object of particular interest is the \\emph{perfect matching lattice}, defined as the set of integral linear combinations of the incidence vectors of perfect matchings: \\begin{equation*}\n \\mathcal L(G): = \\lat(\\mathcal{M})= \\{\\sum_{M \\in \\mathcal{M}}k_M \\mathbf{1}_M: k_M \\in \\mathbb{Z}\\},\n\\end{equation*} where $\\mathcal{M}$ denotes the set of all perfect matchings of $G$.\nLov\\'asz \\cite{lovaszPM} gave a characterization of this lattice based on a structural decomposition of the graph into blocks called \\emph{bricks} and \\emph{braces}, with the Petersen graph being the only obstruction for this lattice to contain all the integral points in the linear hull of the perfect matchings. \nA natural question is whether the generating set of this lattice, i.e. the perfect matchings of $G$, contains a basis for $\\mathcal L(G)$. This question was answered positively by de Carvalho, Lucchesi, and Murty \\cite{dimension} who proved it through ear-decompositions of a graph. In this paper, we present an algorithmic analogue of this result, mainly:\n\n\\section{Introduction}\n\nThe main idea of the paper is to utilize odd cut decomposition methods to reduce our graph to a number of graphs whose perfect matching polytope coincides with its \\emph{bipartite relaxation}:\\begin{equation}\\label{eq:pm-bipartite}\n P(G):=\\left\\{x\\in \\R^A_{+}: x(\\delta(v))= 1 \\forall v\\in V \\right\\}.\n\\end{equation} \nMatching covered graphs for which $PM(G)=P(G)$ are called \\emph{Birkhoff von Neumann (BvN)} graphs. All bipartite graphs and some non-bipartite graphs, in particular certain types of bricks (to be defined later) are BvN. Our algorithm for finding a basis for $\\zL(G)$ either finds a set of $\\dim(P(G))$ linearly independent integral vertices of $P(G)$ or finds a facet of $PM(G)$induced by an odd cut constraint $x(C)\\geq 1$. We then use this facet to further decompose the graph into smaller ones until eventually the algorithm successfully finds a basis.\n\nSince we are not interested in edges that do not belong to any perfect matching, it is convenient to delete from $G$ all edges that are not in any perfect matching, thus making the graph matching covered.\nThe number of bricks in a tight cut decomposition of a matching covered graph affects the dimension of its perfect matching polytope. In particular, the dimension of the perfect matching polytope is \\begin{equation}\\label{eq:pm-dimension}\n \\dim(PM(G)) = |E'|-|V|+1-b,\n\\end{equation} where $b$ denotes the number of bricks in a tight cut decomposition of $G$ and $E'$ is the set of edges that belong to a perfect matching \\cite{edmonds1982brick}, with $E=E'$ for matching covered $G$. Furthermore, the dimension of the perfect matching lattice and its linear hull is \\begin{equation}\\label{eq:pm-lat-dimension}\n \\dim (\\lat(G)) = |E'|-|V|+2-b.\n\\end{equation}\nWe will call \\emph{near-bricks} all matching covered graphs $G$ with exactly one brick.\nThe following fact due to Lov\\'asz connects the linear hull of perfect matchings with $\\zL(G)$: \\begin{theorem}[Theorem $6.3$ in \\cite{lovaszPM}]\\label{thm:lovasz}\n In a matching covered graph $G=(V,E)$, let $\\mathcal{M}$ be the set of perfect matchings. Then any $x \\in \\lin(\\mathcal{M})\\cap \\Z^E$ satisfies $2x\\in \\zL(G)$. Furthermore, if $G$ has no Petersen bricks then $\\lin(\\mathcal{M})\\cap \\Z^E=\\zL(G)$. \n\\end{theorem}\n\n\\begin{theorem}\\label{thm:algo-correct}\n The result of Algorithm \\ref{algorithm:basis-bvn} applied to a matching covered BvN graph is a set $\\zB$ of linearly independent indicator vectors of perfect matchings that forms an integral basis for $\\zL(G)$.\n\\end{theorem}\n\\begin{proof}\n Because $G$ is BvN it implies that any face of $P(G)$ has integral vertices. Therefore, all $y(e)$ found as a corner solution to \\ref{LP2_match} will indeed be incidence vectors of some perfect matchings of $G$ (in fact, of $G[E_t]$). \n It is clear that the final set $\\zB$ will contain exactly $\\dim(PM(G))$ perfect matchings with the property that each one uses an edge not used in the previous ones. This means that the elements of $\\zB$ are linearly independent and moreover, create an integral basis for $\\lin(\\zB)$.\n\\end{proof}\n\nIf, given any brick $G$, the above process terminates in $|E|-|V|+1$ steps and all the output vectors are integral, then they correspond to a basis for $\\zL(G)$.\nTherefore, the only algorithm obstruction in the non-BvN case is \\ref{LP2_match} having a fractional solution $x^*$. This means that $x^*$ is a fractional vertex of $P(G)$ itself, which gives a certificate that $G$ is non-BvN.\nAs $x^*$ has to be half-integral, its support consists of vertex-disjoint edges of weight $1$ and odd cycles of weight $1/2$. Picking any of the odd cycles, we can consider the odd cut $C'$ whose shore is exactly the vertices of this cycle. Inevitably, we have $x^*(C')=0$.\nIn \\Cref{sec:sep-find}, we describe a combinatorial algorithm to find an odd cut $C$ such that the inequality $x(C)\\geq 1$ is violated by $x^*$ with more specific properties:\n\\begin{theorem}\\label{thm:find-robust}\n Let $G$ be a brick that is non-BvN. Let $x^*$ be a fractional vertex of $P(G)$. There is a polynomial time algorithm that finds a separating cut $C$ of $G$ such that $x^*(G)<1$ and both $C$-contractions of $G$ are near-bricks whose brick is not the Petersen brick.\n\\end{theorem}\n\nAs in Section \\ref{subsec:non-bvn-detect}, consider a non-BvN brick $G$ for which the BvN algorithm fails. Theorem \\ref{thm:find-robust} guarantees the existence of a separating cut $C$ both of whose contractions, call them $G_1$ and $G_2$, have exactly one brick that is not the Petersen brick.\nSince neither $G_1$ nor $G_2$ contain the Petersen graph, we can construct lattice bases $\\zB_1$ and $\\zB_2$ for the corresponding graphs. Since both contractions do not contain Petersen bricks, $\\lin(\\zB_i)\\cap \\Z^{E_i} = \\zL(G_i)$ and hence by \\Cref{thm:basis-comb} the set $\\zB:=\\zB_1 \\odot \\zB_2$ will be an integral basis for its linear hull. However, all elements of $\\zB$ satisfy $x^T \\mathbf{1}_C=1$, so to make it a basis for $\\zL(G)$ we need to add at least one perfect matching $M$ with $|M\\cap C|>1$.\nWe use the following fact shown in \\cite{camposlucchesi}.\n\\begin{theorem}\\label{thm:char-three}\n In a non-BvN brick that is not Petersen, every separating cut has a perfect matching intersecting it three times.\n\\end{theorem}\nSuch a perfect matching $M^*$ can be found in polynomial time by considering all possible intersections $M^*\\cap C$ of size three. Finally, we can use $M^*$ to increment the basis. We can now prove that such a set $\\zB \\cup \\{\\mathbf{1}_{M^*}\\}$ will be a basis for $\\zL(G)$.\n\\begin{theorem}\n Let $G=(V,E)$ be a matching covered graph and let $C$ be a non-tight separating cut such that both $C$-contractions are near-bricks whose brick is not the Petersen brick. Let $M^*$ be a perfect matching with $|M^*\\cap C|=3$. Then $\\zB \\cup \\{\\mathbf{1}_{M^*}\\}$ is an integral basis for $\\zL(G)$ consisting of perfect matchings. \n\\end{theorem}\n\\begin{proof}\n By Theorem \\ref{thm:fdi-charact}, such $C$ defines a facet of $PM(G)$. Hence, using part $(iii)$ of Theorem \\ref{thm:basis-comb}, $|\\zB|=\\dim(\\zL(G))-1$. Thus, adding $M^*$ to $\\zB$ makes it a basis for the linear hull of $\\zL(G)$, so it suffices to prove that it will be an integral basis.\n Indeed, consider any $y \\in \\zL(G)$. We also know it is in the linear hull of $\\zB\\cup\\{\\mathbf{1}_{M^*}\\}$. Therefore, we can write it as follows \\begin{equation}\\label{eq:linear-comb}\n y = \\sum_{M\\in \\mathcal{M}} \\alpha_M \\mathbf{1}_M = \\sum_{z^i\\in \\zB} \\beta_i z^i+\\beta_* \\mathbf{1}_{M^*},\n \\end{equation} where all $\\alpha_M$ are integers and the goal is to prove that $\\beta_i$ are all integer.\n The first equality of (\\ref{eq:linear-comb}) when multiplied by $\\textbf{1}_{E}$ gives \\begin{equation*}\n \\frac{1}{n/2} \\textbf{1}^T_{E} y = \\frac{1}{n/2}\\sum_{M\\in \\mathcal{M}} \\alpha_M \\textbf{1}^T_{E}\\mathbf{1}_M=\\sum_{M\\in \\mathcal{M}} \\alpha_M \\in \\Z.\n \\end{equation*} Similarly, multiplying it by $\\textbf{1}_C$ gives \\begin{equation*}\n \\textbf{1}^T_{C} y = \\sum_{M\\in \\mathcal{M}} \\alpha_M \\textbf{1}^T_{C}\\mathbf{1}_M\\equiv \\sum_{M\\in \\mathcal{M}} \\alpha_M \\mod 2.\n \\end{equation*} Applying the same idea to the last expression of equality (\\ref{eq:linear-comb}) we obtain \\begin{equation*}\n \\frac{1}{n/2} \\textbf{1}^T_{E} y =\\sum_{\\mathbf{z}^i \\in \\zB} \\beta_i + \\beta_*, \\textbf{1}^T_{C} y = \\sum_{\\mathbf{z}^i \\in \\zB} \\beta_i + 3\\beta_*.\n \\end{equation*} Therefore, $\\beta_*=\\frac{1}{2} (\\textbf{1}^T_{C} y-\\frac{1}{n/2} \\textbf{1}^T_{E} y)$, which is integral by the above.\n Finally, consider $y-\\beta_* \\mathbf{1}_{M^*} \\in \\lin(\\zB)$. Since we already showed that $\\zB$ is an integral basis for its linear hull, all the remaining coefficients $\\beta_i$ are also integer.\n\\end{proof}", + "post_theorem_intro_text_len": 3066, + "post_theorem_intro_text": "The main contribution of this paper is a new approach to basis construction for the perfect matching lattice based on polyhedral theory. In addition to the algorithmic result, we identify a class of graphs where a lattice basis can be obtained by a simple inductive process. Moreover, we connect the combinatorial and polyhedral properties of the perfect matchings, bridging our approach and structural methods of de Carvalho, Lucchesi, and Murty. More broadly, our polyhedral techniques offer new tools that may prove valuable in tackling longstanding open problems in matching theory. \nTo begin with, the \\emph{perfect matching polytope} $PM(G)$ is the convex hull of its perfect matching incidence vectors and is described by the following set of inequalities (Edmonds, \\cite{edmonds1965maximum}):\n\\begin{equation}\\label{eq:pm-general}\n PM(G)=\\left\\{x\\in \\mathbb{R}^A_{+}: \\begin{array}{cc}\n x(\\delta(v))= 1 & \\forall v\\in V\\\\\n x(C)\\geq 1 & \\forall C \\textmd{ odd cut}\n\\end{array}\\right\\}\n\\end{equation}\nHere, a \\emph{cut} $C = \\delta(U)$ for some $U\\subseteq V$ is the set of edges with exactly one endpoint in $U$; the sets $U$ and its complement $V\\backslash U$ are the \\emph{shores} of $C$. A cut is \\emph{odd} if both of its shores have an odd cardinality. \n\nThe main idea of the paper is to utilize odd cut decomposition methods to reduce our graph to a number of graphs whose perfect matching polytope coincides with its \\emph{bipartite relaxation}:\\begin{equation}\\label{eq:pm-bipartite}\n P(G):=\\left\\{x\\in \\mathbb{R}^A_{+}: x(\\delta(v))= 1 \\forall v\\in V \\right\\}.\n\\end{equation} \nMatching covered graphs for which $PM(G)=P(G)$ are called \\emph{Birkhoff von Neumann (BvN)} graphs. All bipartite graphs and some non-bipartite graphs, in particular certain types of bricks (to be defined later) are BvN. Our algorithm for finding a basis for $\\mathcal L(G)$ either finds a set of $\\dim(P(G))$ linearly independent integral vertices of $P(G)$ or finds a facet of $PM(G)$induced by an odd cut constraint $x(C)\\geq 1$. We then use this facet to further decompose the graph into smaller ones until eventually the algorithm successfully finds a basis.\n\nThe paper is organized as follows. In Section \\ref{sec:prelim} we discuss a way to decompose the graph into simple pieces and show how to reduce the basis construction for $\\mathcal L(G)$ to these pieces. In Section \\ref{sec:bvn} we describe a basis construction algorithm for \\emph{Birkhoff von Neumann (BvN)} graphs based on polyhedral techniques and discuss what happens when applying this algorithm to a non-BvN graph. In Section \\ref{sec:put-together} we show how to decompose non-BvN graphs into BvN ones using a specific type of odd cuts in a process similar to the tight cut decomposition. In Section \\ref{sec:sep-find} we provide an algorithm to find an appropriate cut to be used in the above mentioned decomposition. Finally, in Appendix \\ref{sec:classical} we draw connections between our polyhedral view of the basis and the classical approach based on \\emph{ear decompositions}.", + "sketch": "To prove Theorem~\\ref{thm:main}, the paper’s approach is “based on polyhedral theory” via the perfect matching polytope $PM(G)$ (Edmonds’ description \\eqref{eq:pm-general}) and its “bipartite relaxation” $P(G)$ in \\eqref{eq:pm-bipartite}. The “main idea … is to utilize odd cut decomposition methods to reduce our graph to a number of graphs whose perfect matching polytope coincides with its bipartite relaxation,” i.e., to reduce to Birkhoff--von Neumann (BvN) graphs where $PM(G)=P(G)$. The algorithm for a basis of $\\mathcal L(G)$ then “either finds a set of $\\dim(P(G))$ linearly independent integral vertices of $P(G)$ or finds a facet of $PM(G)$ induced by an odd cut constraint $x(C)\\ge 1$”; in the latter case, it “use[s] this facet to further decompose the graph into smaller ones until eventually the algorithm successfully finds a basis.” The overall proof/argument is organized by: reducing basis construction to decomposed “simple pieces” (Section~\\ref{sec:prelim}); giving a basis construction algorithm for BvN graphs and analyzing applying it to non-BvN graphs (Section~\\ref{sec:bvn}); decomposing non-BvN graphs into BvN ones “using a specific type of odd cuts” (Section~\\ref{sec:put-together}); and providing an algorithm to find the needed cut (Section~\\ref{sec:sep-find}).", + "expanded_sketch": "No expanded sketch found.", + "expanded_theorem": "\\label{thm:main}\n There is a polynomial time algorithm that, given a matching covered graph $G$, finds a lattice basis for $\\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$.", + "theorem_type": [ + "Algorithmic or Constructive", + "Existence" + ], + "mcq": { + "question": "Let $G=(V,E)$ be a matching covered graph, meaning that $G$ has a perfect matching and every edge of $G$ belongs to some perfect matching. For a perfect matching $M$, let $\\mathbf{1}_M\\in\\{0,1\\}^E$ denote its incidence vector, and let the perfect matching lattice be\n\\[\n\\mathcal L(G)=\\left\\{\\sum_{M} k_M\\mathbf{1}_M : k_M\\in\\mathbb Z,\\ M\\text{ a perfect matching of }G\\right\\}.\n\\]\nA lattice basis for $\\mathcal L(G)$ means a set of vectors in $\\mathcal L(G)$ that is integer-linearly independent and whose integer span is exactly $\\mathcal L(G)$. Which existence statement holds for such graphs?", + "correct_choice": { + "label": "A", + "text": "There exists a polynomial-time algorithm that, given any matching covered graph $G$, outputs a lattice basis for $\\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$." + }, + "choices": [ + { + "label": "B", + "text": "There exists a polynomial-time algorithm that, given any graph $G$ with a perfect matching, outputs a lattice basis for $\\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$." + }, + { + "label": "C", + "text": "For every matching covered graph $G$, there exists a lattice basis for $\\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$." + }, + { + "label": "D", + "text": "There exists a single polynomial-time algorithm that, given any matching covered graph $G$, outputs a basis of the real linear span of $\\mathcal L(G)$ consisting of incidence vectors of perfect matchings of $G$." + }, + { + "label": "E", + "text": "There exists a polynomial-time algorithm that, given any matching covered graph $G$, outputs all incidence vectors of perfect matchings of $G$, and these vectors form a lattice basis for $\\mathcal L(G)$." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "matching_covered_hypothesis", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "polynomial_time_algorithmic_claim", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "lattice_basis_vs_real_basis", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "finiteness", + "tampered_component": "output_basis_not_all_perfect_matchings", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the relevant objects but does not state or strongly hint at the algorithmic existence claim in choice A. The correct answer is not leaked explicitly or implicitly." + }, + "TAS": { + "score": 1, + "justification": "The item is close to theorem recognition: the correct option appears to be essentially the target theorem statement. However, it is not a pure restatement because the alternatives vary meaningfully in graph class, algorithmic strength, and type of basis." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish the precise valid claim from nearby overstatements and weakenings, especially B, C, and D. Still, it mainly tests recognition of the exact theorem rather than substantial derivation or synthesis." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mostly plausible and target natural failure modes: overgeneralizing from matching covered to all graphs with a perfect matching (B), confusing mere existence with polynomial-time constructibility (C), confusing lattice basis with real-span basis (D), and overclaiming that all perfect matchings are output and form a basis (E)." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with no answer leakage and generally strong distractors, but it leans toward recalling the exact theorem statement rather than eliciting deep generative reasoning." + } + }, + { + "id": "2511.21583v1", + "paper_link": "http://arxiv.org/abs/2511.21583v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:main}\nFor $s >1 $ and $\\delta>0$ there exist $c_s>0$ and $\\ep_0>0$ with the following. For any $0< \\epsilon \\leq \\ep_0$ and any zero-mean initial data $\\om_{in} \\in H^s(\\mathbb{T} \\times \\mathbb{R})$ such that \n\\begin{equation}\\label{eq thm main 1}\n\\| \\om_{in}\\|_{H^{s} } \\leq \\epsilon \\quad \\text{and} \\quad \\| y \\om_{in}\\|_{L^{2} } \\leq \\epsilon ,\n\\end{equation}\ndefine the lifespan \n\\begin{equation}\\label{eq thm main 2}\nT_\\epsilon: = c_s \\epsilon^{-\\delta_s } \\quad \\text{with} \\quad \\delta_s = \n \\begin{cases}\n \\frac{1}{4-s} \\quad\\text{if $1 2 $ }. & \n\\end{cases}\n\\end{equation} \n\n Then the (unique) solution to \\eqref{eq:eu_eq} satisfies the following.\n\n\\begin{itemize}\n\n\t\\item (Regularity):\n\tThe re-normalized vorticity $W( t, X,Y): = \\omega(t,X + Yt, Y)$ satisfies\n\t\\begin{eqnarray}\\label{eq:reg intro}\n\t\t\\| W \\|_{L^\\infty_t ([0,T_\\epsilon]; H^s)} \\leq 3 \\epsilon .\n\t\\end{eqnarray}\n\n\t\\item (Invisicid damping): The velocity field $u = (u^x, u^y )$ satisfies\n\t\\begin{equation}\\label{eq:damping intro}\n\t\t\\begin{aligned}\n\t\t\t\\|P_{\\neq 0} u^x (t)\\|_{L^2} & \\lesssim \\epsilon \\langle t \\rangle^{-1} \\\\\n\t\t\t\\| u^y (t) \\|_{L^2} & \\lesssim \\epsilon \\langle t \\rangle^{- \\min \\{ 2,s\\}} \n\t\t\\end{aligned}\n\t\t\\quad \\text{for all $t \\in [0, T_\\epsilon] $ }\n\t\\end{equation}\n\twhere $P_{\\neq 0}f : = f -\\int_{\\mathbb{T} } f(x,y) \\, dx $ denotes the projection of removing the $x$-mean.\n\n\\end{itemize}", + "start_pos": 83571, + "end_pos": 85032, + "label": "thm:main" + }, + "ref_dict": { + "eq:reg intro": "\\begin{eqnarray}\\label{eq:reg intro}\n\t\t\\| W \\|_{L^\\infty_t ([0,T_\\ep]; H^s)} \\leq 3 \\ep .\n\t\\end{eqnarray}", + "eq:eu_eq": "\\begin{equation}\\label{eq:eu_eq}\n\\begin{cases}\n\t\\p_t \\om + y\\p_x \\om + u \\cdot \\nabla \\om = 0 &\\\\\n\tu = \\nabla^\\perp \\psi, \\quad \\Delta \\psi = \\om &\\\\\n\t\\om|_{t= 0 } = \\om_{in} &\n\\end{cases}\t\n\\end{equation}", + "eq:damping intro": "\\begin{equation}\\label{eq:damping intro}\n\t\t\\begin{aligned}\n\t\t\t\\|P_{\\neq 0} u^x (t)\\|_{L^2} & \\lesssim \\ep \\langle t \\rangle^{-1} \\\\\n\t\t\t\\| u^y (t) \\|_{L^2} & \\lesssim \\ep \\langle t \\rangle^{- \\min \\{ 2,s\\}} \n\t\t\\end{aligned}\n\t\t\\quad \\text{for all $t \\in [0, T_\\ep] $ }\n\t\\end{equation}" + }, + "pre_theorem_intro_text_len": 2009, + "pre_theorem_intro_text": "\\label{sec:intro}\n\nIn this note, we consider the 2D Euler equation on $\\mathbb{T} \\times \\mathbb{R}$ near the Couette flow $(y,0)$. The vorticity perturbation $\\omega : \\mathbb{T} \\times \\mathbb{R} \\to \\mathbb{R}$ satisfies \n\\begin{equation}\\label{eq:eu_eq}\n\\begin{cases}\n\t\\p_t \\omega + y\\p_x \\omega + u \\cdot \\nabla \\omega = 0 &\\\\\n\tu = \\nabla^\\perp \\psi, \\quad \\Delta \\psi = \\omega &\\\\\n\t\\omega|_{t= 0 } = \\om_{in} &\n\\end{cases}\t\n\\end{equation}\nwhere $ \\nabla^\\perp = (- \\p_y , \\p_x)$ and $\\om_{in}$ is the initial data.\n\nWe are concerned with the long-time behavior of solutions to this system, specifically the phenomenon of inviscid damping~\\cite{Kelvin1887,Rayleigh1879,Orr1907}--the decay of velocity perturbations in the absence of viscosity.\n\nFor the Couette flow, the linearized dynamics are well understood: phase mixing transfers energy to small scales in the vorticity, leading to algebraic decay of the velocity.\n\n At the nonlinear level, the situation is subtler. The breakthrough work of Bedrossian and Masmoudi \\cite{MR3415068} established nonlinear inviscid damping for small perturbations in Gevrey class regularity. Important generalizations to more complex shear flows were obtained in subsequent works \\cite{MR4740211,MR4628607}.\n\nIn the studies \\cite{MR3415068,MR4076093,MR4740211,MR4628607}, inviscid damping follows from asymptotic stability, which requires Gevrey-2 regularity. This assumption appears to be sharp-- below Gevrey-2, instability can occur~\\cite{MR4630602}.\n\n For Sobolev regularity, global inviscid damping is believed to fail. In particular, there exist steady states~\\cite{MR2796139} and traveling waves~\\cite{MR4595614} arbitrarily close to Couette flow in $H^{3/2-}$, which prohibits the global damping in such a regime.\n\n\\subsection{Main result}\n\nThe purpose of this note is to show that although global-in-time inviscid damping near Couette does not hold for Sobolev perturbations in $H^{3/2-}$, a \\emph{finite but long-time} version of damping remains valid.", + "context": "In this note, we consider the 2D Euler equation on $\\mathbb{T} \\times \\mathbb{R}$ near the Couette flow $(y,0)$. The vorticity perturbation $\\omega : \\mathbb{T} \\times \\mathbb{R} \\to \\mathbb{R}$ satisfies \n\\begin{equation}\\label{eq:eu_eq}\n\\begin{cases}\n \\p_t \\omega + y\\p_x \\omega + u \\cdot \\nabla \\omega = 0 &\\\\\n u = \\nabla^\\perp \\psi, \\quad \\Delta \\psi = \\omega &\\\\\n \\omega|_{t= 0 } = \\om_{in} &\n\\end{cases} \n\\end{equation}\nwhere $ \\nabla^\\perp = (- \\p_y , \\p_x)$ and $\\om_{in}$ is the initial data.\n\nWe are concerned with the long-time behavior of solutions to this system, specifically the phenomenon of inviscid damping~\\cite{Kelvin1887,Rayleigh1879,Orr1907}--the decay of velocity perturbations in the absence of viscosity.\n\nAt the nonlinear level, the situation is subtler. The breakthrough work of Bedrossian and Masmoudi \\cite{MR3415068} established nonlinear inviscid damping for small perturbations in Gevrey class regularity. Important generalizations to more complex shear flows were obtained in subsequent works \\cite{MR4740211,MR4628607}.\n\nFor Sobolev regularity, global inviscid damping is believed to fail. In particular, there exist steady states~\\cite{MR2796139} and traveling waves~\\cite{MR4595614} arbitrarily close to Couette flow in $H^{3/2-}$, which prohibits the global damping in such a regime.\n\n\\subsection{Main result}\n\nThe purpose of this note is to show that although global-in-time inviscid damping near Couette does not hold for Sobolev perturbations in $H^{3/2-}$, a \\emph{finite but long-time} version of damping remains valid.\n\n\\begin{equation}\\label{eq:eu_eq}\n\\begin{cases}\n\t\\p_t \\om + y\\p_x \\om + u \\cdot \\nabla \\om = 0 &\\\\\n\tu = \\nabla^\\perp \\psi, \\quad \\Delta \\psi = \\om &\\\\\n\t\\om|_{t= 0 } = \\om_{in} &\n\\end{cases}\t\n\\end{equation}", + "full_context": "In this note, we consider the 2D Euler equation on $\\mathbb{T} \\times \\mathbb{R}$ near the Couette flow $(y,0)$. The vorticity perturbation $\\omega : \\mathbb{T} \\times \\mathbb{R} \\to \\mathbb{R}$ satisfies \n\\begin{equation}\\label{eq:eu_eq}\n\\begin{cases}\n \\p_t \\omega + y\\p_x \\omega + u \\cdot \\nabla \\omega = 0 &\\\\\n u = \\nabla^\\perp \\psi, \\quad \\Delta \\psi = \\omega &\\\\\n \\omega|_{t= 0 } = \\om_{in} &\n\\end{cases} \n\\end{equation}\nwhere $ \\nabla^\\perp = (- \\p_y , \\p_x)$ and $\\om_{in}$ is the initial data.\n\nWe are concerned with the long-time behavior of solutions to this system, specifically the phenomenon of inviscid damping~\\cite{Kelvin1887,Rayleigh1879,Orr1907}--the decay of velocity perturbations in the absence of viscosity.\n\nAt the nonlinear level, the situation is subtler. The breakthrough work of Bedrossian and Masmoudi \\cite{MR3415068} established nonlinear inviscid damping for small perturbations in Gevrey class regularity. Important generalizations to more complex shear flows were obtained in subsequent works \\cite{MR4740211,MR4628607}.\n\nFor Sobolev regularity, global inviscid damping is believed to fail. In particular, there exist steady states~\\cite{MR2796139} and traveling waves~\\cite{MR4595614} arbitrarily close to Couette flow in $H^{3/2-}$, which prohibits the global damping in such a regime.\n\n\\subsection{Main result}\n\nThe purpose of this note is to show that although global-in-time inviscid damping near Couette does not hold for Sobolev perturbations in $H^{3/2-}$, a \\emph{finite but long-time} version of damping remains valid.\n\n\\begin{equation}\\label{eq:eu_eq}\n\\begin{cases}\n\t\\p_t \\om + y\\p_x \\om + u \\cdot \\nabla \\om = 0 &\\\\\n\tu = \\nabla^\\perp \\psi, \\quad \\Delta \\psi = \\om &\\\\\n\t\\om|_{t= 0 } = \\om_{in} &\n\\end{cases}\t\n\\end{equation}\n\nThe purpose of this note is to show that although global-in-time inviscid damping near Couette does not hold for Sobolev perturbations in $H^{3/2-}$, a \\emph{finite but long-time} version of damping remains valid.\n\n\\item (Invisicid damping): The velocity field $u = (u^x, u^y )$ satisfies\n \\begin{equation}\\label{eq:damping intro}\n \\begin{aligned}\n \\|P_{\\neq 0} u^x (t)\\|_{L^2} & \\lesssim \\ep \\langle t \\rangle^{-1} \\\\\n \\| u^y (t) \\|_{L^2} & \\lesssim \\ep \\langle t \\rangle^{- \\min \\{ 2,s\\}} \n \\end{aligned}\n \\quad \\text{for all $t \\in [0, T_\\ep] $ }\n \\end{equation}\n where $P_{\\neq 0}f : = f -\\int_{\\T } f(x,y) \\, dx $ denotes the projection of removing the $x$-mean.\n\n\\begin{remark}\n\\hfill\n\\begin{enumerate}\n \\item Our nonlinear estimates remain valid down to $H^{1+}$ matching the well-posedness threshold of 2D Euler. There is no contradiction with steady states\\footnote{Note that the examples in \\cite{MR2796139} are on the periodic channel $\\T \\times [-1,1]$, where the stationary structure appears near $y=0$. Nevertheless, the result in this paper is likely to hold on $\\T \\times [-1 ,1]$ when restricting away from the boundary. }~\\cite{MR2796139} or traveling waves~\\cite{MR4595614} constructions, since the initial velocities in those examples are already smaller than the decay rate predicted by \\eqref{eq:damping intro} at $t=T_\\ep$.\n\n\\begin{lemma}\\label{lemma:Hs a priori}\nLet $s>1$. The system \\eqref{eq:eu_XY} satisfies the a priori estimates:\n\\begin{equation}\n\\frac{ d \\| W(t) \\|_{H^s} }{dt} \\lesssim \\| \\nabla U \\|_{L^\\infty } \\| W \\|_{H^s} +\n\\begin{cases}\n\\| J^{ 2} U \\|_{ L^2 } \\| W \\|_{H^s} & \\quad \\text{12}.\n\\end{cases} \n\\end{equation}\n\nIf $\\int_{\\T \\times \\R} \\om_{in} = 0 $ and $ y \\om_{in} \\in L^2 $, then the system \\eqref{eq:eu_XY} admits the following a priori estimates:\n\\begin{equation}\\label{eq prop:apriori_yw}\n\\frac{ d \\| W (t) \\|_{ \\bar{H}^s } }{dt} \\lesssim \\langle t\\rangle^{\\beta_s} \\|W \\|_{\\bar{H}^s}^2 ,\n\\end{equation} \nwhere the exponent $ \\beta_s =3-s $ if $10$ if $s=2 $, and $ \\beta_s =1 $ if $s > 2 $.\n\nCombining \\eqref{eq aux apriori 1b}, \\eqref{eq aux apriori 5} and \\eqref{eq aux apriori 6} with Lemma \\ref{lemma:Hs a priori}, we have:\n\\begin{equation*}\n\\frac{ d \\| W(t) \\|_{\\bar H^s} }{dt} \n\\begin{cases}\n\\lesssim \\langle t\\rangle^{3- s} \\| W(t) \\|_{\\bar H^s}^2 \\quad & \\text{if $11$ since the extra $\\p_X$ can be absorbed in $H^s$. We have thus proven Theorem \\ref{thm:main}\n\nBy standard duality, it suffices to bound\n\\begin{equation}\n \\| P_{ \\neq 0 } \\om \\|_{\\dot H^{-1} } = \\sup_{ \n |\\varphi \\|_{\\dot H^1} \\leq 1 ; P_{=0} \\varphi = 0} \\int \\om \\varphi \\, dx dy.\n\\end{equation}\nSo we fix such a smooth test function $\\varphi \\in C^\\infty_c(\\T \\times \\R)$ with zero $x$-mean and consider\n\\begin{equation}\\label{eq:om_int}\n \\int \\om \\varphi \\, dx dy =\\int W (X, Y )\\varphi( X+ t Y, Y) \\,dX dY.\n\\end{equation}\nBecause $\\varphi$ has zero x-mean and periodic in $x$, there exists a unique zero x-mean function $ \\phi= \\p_x^{-1} \\varphi$. \nThen we use the identity\n\\begin{equation}\\label{eq:om_int 2}\n \\varphi(\\cdot) = \\frac{1}{t}\\p_Y \\left( \\phi(\\cdot) \\right) - \\frac{1}{t}\\p_y \\phi(\\cdot)\n\\end{equation}\nwhere the argument $(\\cdot) = (X+ t Y, Y )$ is omitted.\nInserting this into \\eqref{eq:om_int} gives\n\\begin{align}\n \\int \\om \\varphi \\, dx dy & =-\\frac{1}{t} \\int \\p_Y W \\phi(\\cdot) \\,dX dY \\nonumber -\\frac{1}{t} \\int W \\p_y \\phi(\\cdot) \\,dX dY \\nonumber \\\\\n & \\leq t^{-1} \\| \\p_Y W \\|_{L^2 } \\| \\phi \\|_{L^2} + t^{-1} \\| W \\|_{L^2 } \\| \\p_y \\phi \\|_{L^2 } . \\label{eq:aux10} \n\\end{align}\nNote that $\\left| \\widehat{\\phi}(k,\\xi) \\right|=\\left| \\frac{\\widehat{\\varphi}(k,\\xi)}{k} \\right|$, we have $\\| \\phi \\|_{H^1} \\lesssim \\| \\varphi \\|_{\\dot H^1} $, which gives\n\\begin{align}\n \\Big|\\int \\om \\varphi \\, dx dy \\Big| & \\lesssim \\frac{1}{t} \\| W \\|_{L^2_X H^1_Y} \\| \\varphi \\|_{\\dot H^1 } \\nonumber .\n\\end{align}\n\n\\begin{equation}\\label{eq:damping intro}\n\t\t\\begin{aligned}\n\t\t\t\\|P_{\\neq 0} u^x (t)\\|_{L^2} & \\lesssim \\ep \\langle t \\rangle^{-1} \\\\\n\t\t\t\\| u^y (t) \\|_{L^2} & \\lesssim \\ep \\langle t \\rangle^{- \\min \\{ 2,s\\}} \n\t\t\\end{aligned}\n\t\t\\quad \\text{for all $t \\in [0, T_\\ep] $ }\n\t\\end{equation}\n\n\\begin{equation}\\label{eq:eu_eq}\n\\begin{cases}\n\t\\p_t \\om + y\\p_x \\om + u \\cdot \\nabla \\om = 0 &\\\\\n\tu = \\nabla^\\perp \\psi, \\quad \\Delta \\psi = \\om &\\\\\n\t\\om|_{t= 0 } = \\om_{in} &\n\\end{cases}\t\n\\end{equation}", + "post_theorem_intro_text_len": 3268, + "post_theorem_intro_text": "\\begin{remark}\n\\hfill\n\\begin{enumerate}\n \\item Our nonlinear estimates remain valid down to $H^{1+}$ matching the well-posedness threshold of 2D Euler. There is no contradiction with steady states\\footnote{Note that the examples in \\cite{MR2796139} are on the periodic channel $\\mathbb{T} \\times [-1,1]$, where the stationary structure appears near $y=0$. Nevertheless, the result in this paper is likely to hold on $\\mathbb{T} \\times [-1 ,1]$ when restricting away from the boundary. }~\\cite{MR2796139} or traveling waves~\\cite{MR4595614} constructions, since the initial velocities in those examples are already smaller than the decay rate predicted by \\eqref{eq:damping intro} at $t=T_\\epsilon$.\n\n \\item While for $s> 2$ the decaying rates match with those of the linearized problem, it is not clear to us what would be the optimal life span $T_\\epsilon =\\epsilon^{-\\delta_s}$.\n\n \\item The estimates are most effective for initial data of the form $\\omega_{in} = \\epsilon \\omega_0$ for some fixed profile $ \\omega_0$. In the limit $\\epsilon \\to 0+$, one sees that initially $ u \\sim \\omega \\sim \\epsilon $ and $ u $ exhibits the algebraic decay over the time interval $[0,T_\\epsilon]$.\n\n \\item The damping estimate \\eqref{eq:damping intro} follows from the regularity \\eqref{eq:reg intro}. Whether nonlinear inviscid damping can persist globally without asymptotic stability remains open, see~\\cite{MR4302767} for results in the linear case. \n\n \\item The instability construction in \\cite{MR4630602} also satisfies our assumptions and estimates, with an eventual breakdown of \\eqref{eq:reg intro} occurring long after $t=T_\\epsilon$. A definite counterexample to inviscid damping for low Gevrey regularity is also open.\n\\end{enumerate}\n\n\\end{remark}\n\n\\subsection{Discussion}\nThe literature on stability/inviscid damping is vast; see, for example, the works \\cite{MR3710645,MR3772399,MR3987441,MR4400903} on general shears/vortices as well as \\cite{MR3974608,MR4121130,MR4412070,MR4848788,MR4951440,2510.16378} on related viscous problems and boundary effects.\n\nIn contrast, the study of nonlinear inviscid damping of 2D Euler equations in the Sobolev regime remains largely open. The present work demonstrates that the inviscid damping remains effective on a long, but finite, time interval, which is much shorter than the nonlinear resonance breakdown time found in recent instability constructions~\\cite{MR4630602}. \n\nOur proof is short and elementary. We work in simplified shear-back coordinates of Bedrossian-Masmoudi~\\cite{MR3415068,MR4076093,MR4740211,MR4628607} and exploit the fact that the principle of ``\\emph{decay costs regularity}'' can, in some sense, be reversed. In these coordinates, the new velocity field remains one derivative smoother than the re-normalized vorticity, at the cost of growing time factors. \n\n It is desired to find the optimal exponent $T_\\epsilon\\sim \\epsilon^{-\\delta_s}$. However, whether the lifespan obtained here can be extended, with or without additional assumptions, is unclear to us. Another important open question is whether nonlinear inviscid damping persists without the regularity of re-normalized vorticity \\eqref{eq:reg intro}. See \\cite{MR4302767} for the study of a linear case.", + "sketch": "Our proof is described as \"short and elementary.\" To prove Theorem~\\ref{thm:main}, the authors \"work in simplified shear-back coordinates of Bedrossian-Masmoudi\" and \"exploit the fact that the principle of `\\emph{decay costs regularity}' can, in some sense, be reversed.\" In these coordinates, they use that \"the new velocity field remains one derivative smoother than the re-normalized vorticity, at the cost of growing time factors,\" which yields the damping estimates (indeed, they also remark that \"The damping estimate \\eqref{eq:damping intro} follows from the regularity \\eqref{eq:reg intro}.\")", + "expanded_sketch": "Our proof is described as \"short and elementary.\" To prove the main theorem, the authors \"work in simplified shear-back coordinates of Bedrossian-Masmoudi\" and \"exploit the fact that the principle of `\\emph{decay costs regularity}' can, in some sense, be reversed.\" In these coordinates, they use that \"the new velocity field remains one derivative smoother than the re-normalized vorticity, at the cost of growing time factors,\" which yields the damping estimates\n\\begin{equation}\\label{eq:damping intro}\n\t\t\\begin{aligned}\n\t\t\t\\|P_{\\neq 0} u^x (t)\\|_{L^2} & \\lesssim \\ep \\langle t \\rangle^{-1} \\\\\n\t\t\t\\| u^y (t) \\|_{L^2} & \\lesssim \\ep \\langle t \\rangle^{- \\min \\{ 2,s\\}} \n\t\t\\end{aligned}\n\t\t\\quad \\text{for all $t \\in [0, T_\\ep] $ }\n\t\\end{equation}\n(indeed, they also remark that the damping estimate above follows from the regularity\n\\begin{eqnarray}\\label{eq:reg intro}\n\t\t\\| W \\|_{L^\\infty_t ([0,T_\\ep]; H^s)} \\leq 3 \\ep .\n\t\\end{eqnarray}\n).", + "expanded_theorem": "\\label{thm:main}\nFor $s >1 $ and $\\delta>0$ there exist $c_s>0$ and $\\ep_0>0$ with the following. For any $0< \\epsilon \\leq \\ep_0$ and any zero-mean initial data $\\om_{in} \\in H^s(\\mathbb{T} \\times \\mathbb{R})$ such that \n\\begin{equation}\\label{eq thm main 1}\n\\| \\om_{in}\\|_{H^{s} } \\leq \\epsilon \\quad \\text{and} \\quad \\| y \\om_{in}\\|_{L^{2} } \\leq \\epsilon ,\n\\end{equation}\ndefine the lifespan \n\\begin{equation}\\label{eq thm main 2}\nT_\\epsilon: = c_s \\epsilon^{-\\delta_s } \\quad \\text{with} \\quad \\delta_s = \n \\begin{cases}\n \\frac{1}{4-s} \\quad\\text{if $1 2 $ }. & \n\\end{cases}\n\\end{equation} \n\n Then the (unique) solution to\n\\begin{equation}\\label{eq:eu_eq}\n\\begin{cases}\n\t\\p_t \\om + y\\p_x \\om + u \\cdot \\nabla \\om = 0 &\\\\\n\tu = \\nabla^\\perp \\psi, \\quad \\Delta \\psi = \\om &\\\\\n\t\\om|_{t= 0 } = \\om_{in} &\n\\end{cases}\t\n\\end{equation}\nsatisfies the following.\n\n\\begin{itemize}\n\n\t\\item (Regularity):\n\tThe re-normalized vorticity $W( t, X,Y): = \\omega(t,X + Yt, Y)$ satisfies\n\t\\begin{eqnarray}\\label{eq:reg intro}\n\t\t\\| W \\|_{L^\\infty_t ([0,T_\\epsilon]; H^s)} \\leq 3 \\epsilon .\n\t\\end{eqnarray}\n\n\t\\item (Invisicid damping): The velocity field $u = (u^x, u^y )$ satisfies\n\t\\begin{equation}\\label{eq:damping intro}\n\t\t\\begin{aligned}\n\t\t\t\\|P_{\\neq 0} u^x (t)\\|_{L^2} & \\lesssim \\epsilon \\langle t \\rangle^{-1} \\\\\n\t\t\t\\| u^y (t) \\|_{L^2} & \\lesssim \\epsilon \\langle t \\rangle^{- \\min \\{ 2,s\\}} \n\t\t\\end{aligned}\n\t\t\\quad \\text{for all $t \\in [0, T_\\epsilon] $ }\n\t\\end{equation}\n\twhere $P_{\\neq 0}f : = f -\\int_{\\mathbb{T} } f(x,y) \\, dx $ denotes the projection of removing the $x$-mean.\n\n\\end{itemize}", + "theorem_type": [ + "Existential–Universal", + "Inequality or Bound" + ], + "mcq": { + "question": "Let \\(s>1\\) and \\(\\delta>0\\). Consider the 2D Euler vorticity equation on \\(\\mathbb T\\times\\mathbb R\\):\n\\[\n\\partial_t\\omega+y\\partial_x\\omega+u\\cdot\\nabla\\omega=0,\\qquad u=\\nabla^\\perp\\psi,\\qquad \\Delta\\psi=\\omega,\\qquad \\omega|_{t=0}=\\omega_{in},\n\\]\nwhere \\(\\nabla^\\perp=(-\\partial_y,\\partial_x)\\). Define the renormalized vorticity by \\(W(t,X,Y):=\\omega(t,X+Yt,Y)\\), the projection removing the \\(x\\)-mean by \\(P_{\\neq 0}f:=f-\\int_{\\mathbb T}f(x,y)\\,dx\\), and \\(\\langle t\\rangle=(1+t^2)^{1/2}\\). Which explicit uniform small-data statement is valid for this problem?", + "correct_choice": { + "label": "A", + "text": "There exist \\(c_s>0\\) and \\(\\epsilon_0>0\\) such that for every \\(0<\\epsilon\\le \\epsilon_0\\) and every zero-mean initial datum \\(\\omega_{in}\\in H^s(\\mathbb T\\times\\mathbb R)\\) satisfying\n\\[\n\\int_{\\mathbb T\\times\\mathbb R}\\omega_{in}\\,dx\\,dy=0,\\qquad \\|\\omega_{in}\\|_{H^s}\\le \\epsilon,\\qquad \\|y\\omega_{in}\\|_{L^2}\\le \\epsilon,\n\\]\nif\n\\[\nT_\\epsilon:=c_s\\epsilon^{-\\delta_s},\\qquad\n\\delta_s=\n\\begin{cases}\n\\dfrac{1}{4-s}, & \\text{if } 12,\n\\end{cases}\n\\]\nthen the unique solution of the above initial-value problem satisfies\n\\[\n\\|W\\|_{L_t^\\infty([0,T_\\epsilon];H^s)}\\le 3\\epsilon,\n\\]\nand, for all \\(t\\in[0,T_\\epsilon]\\), with \\(u=(u^x,u^y)\\),\n\\[\n\\|P_{\\neq 0}u^x(t)\\|_{L^2}\\lesssim \\epsilon\\langle t\\rangle^{-1},\n\\qquad\n\\|u^y(t)\\|_{L^2}\\lesssim \\epsilon\\langle t\\rangle^{-\\min\\{2,s\\}}.\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "There exist \\(c_s>0\\) and \\(\\epsilon_0>0\\) such that for every \\(0<\\epsilon\\le \\epsilon_0\\) and every zero-mean initial datum \\(\\omega_{in}\\in H^s(\\mathbb T\\times\\mathbb R)\\) satisfying\n\\[\n\\int_{\\mathbb T\\times\\mathbb R}\\omega_{in}\\,dx\\,dy=0,\\qquad \\|\\omega_{in}\\|_{H^s}\\le \\epsilon,\\qquad \\|y\\omega_{in}\\|_{L^2}\\le \\epsilon,\n\\]\nif\n\\[\nT_\\epsilon:=c_s\\epsilon^{-\\tilde\\delta_s},\\qquad\n\\tilde\\delta_s=\n\\begin{cases}\n\\dfrac{1}{3-s}, & \\text{if } 12,\n\\end{cases}\n\\]\nthen the unique solution of the above initial-value problem satisfies\n\\[\n\\|W\\|_{L_t^\\infty([0,T_\\epsilon];H^s)}\\le 3\\epsilon,\n\\]\nand, for all \\(t\\in[0,T_\\epsilon]\\), with \\(u=(u^x,u^y)\\),\n\\[\n\\|P_{\\neq 0}u^x(t)\\|_{L^2}\\lesssim \\epsilon\\langle t\\rangle^{-1},\n\\qquad\n\\|u^y(t)\\|_{L^2}\\lesssim \\epsilon\\langle t\\rangle^{-\\min\\{2,s\\}}.\n\\]" + }, + { + "label": "C", + "text": "There exist \\(c_s>0\\) and \\(\\epsilon_0>0\\) such that for every \\(0<\\epsilon\\le \\epsilon_0\\) and every zero-mean initial datum \\(\\omega_{in}\\in H^s(\\mathbb T\\times\\mathbb R)\\) satisfying\n\\[\n\\int_{\\mathbb T\\times\\mathbb R}\\omega_{in}\\,dx\\,dy=0,\\qquad \\|\\omega_{in}\\|_{H^s}\\le \\epsilon,\\qquad \\|y\\omega_{in}\\|_{L^2}\\le \\epsilon,\n\\]\nif\n\\[\nT_\\epsilon:=c_s\\epsilon^{-\\delta_s},\\qquad\n\\delta_s=\n\\begin{cases}\n\\dfrac{1}{4-s}, & \\text{if } 12,\n\\end{cases}\n\\]\nthen the unique solution of the above initial-value problem satisfies\n\\[\n\\|W\\|_{L_t^\\infty([0,T_\\epsilon];H^s)}\\le 3\\epsilon,\n\\]\nand, for all \\(t\\in[0,T_\\epsilon]\\), with \\(u=(u^x,u^y)\\),\n\\[\n\\|P_{\\neq 0}u^x(t)\\|_{L^2}\\lesssim \\epsilon\\langle t\\rangle^{-1}.\n\\]" + }, + { + "label": "D", + "text": "There exist \\(c_s>0\\) and \\(\\epsilon_0>0\\) such that for every \\(0<\\epsilon\\le \\epsilon_0\\) and every zero-mean initial datum \\(\\omega_{in}\\in H^s(\\mathbb T\\times\\mathbb R)\\) satisfying\n\\[\n\\int_{\\mathbb T\\times\\mathbb R}\\omega_{in}\\,dx\\,dy=0,\\qquad \\|\\omega_{in}\\|_{H^s}\\le \\epsilon,\n\\]\nif\n\\[\nT_\\epsilon:=c_s\\epsilon^{-\\delta_s},\\qquad\n\\delta_s=\n\\begin{cases}\n\\dfrac{1}{4-s}, & \\text{if } 12,\n\\end{cases}\n\\]\nthen the unique solution of the above initial-value problem satisfies\n\\[\n\\|W\\|_{L_t^\\infty([0,T_\\epsilon];H^s)}\\le 3\\epsilon,\n\\]\nand, for all \\(t\\in[0,T_\\epsilon]\\), with \\(u=(u^x,u^y)\\),\n\\[\n\\|P_{\\neq 0}u^x(t)\\|_{L^2}\\lesssim \\epsilon\\langle t\\rangle^{-1},\n\\qquad\n\\|u^y(t)\\|_{L^2}\\lesssim \\epsilon\\langle t\\rangle^{-\\min\\{2,s\\}}.\n\\]" + }, + { + "label": "E", + "text": "There exist \\(c_s>0\\) and \\(\\epsilon_0>0\\) such that for every \\(0<\\epsilon\\le \\epsilon_0\\) and every zero-mean initial datum \\(\\omega_{in}\\in H^s(\\mathbb T\\times\\mathbb R)\\) satisfying\n\\[\n\\int_{\\mathbb T\\times\\mathbb R}\\omega_{in}\\,dx\\,dy=0,\\qquad \\|\\omega_{in}\\|_{H^s}\\le \\epsilon,\\qquad \\|y\\omega_{in}\\|_{L^2}\\le \\epsilon,\n\\]\nif\n\\[\nT_\\epsilon:=c_s\\epsilon^{-\\delta_s},\\qquad\n\\delta_s=\n\\begin{cases}\n\\dfrac{1}{4-s}, & \\text{if } 12,\n\\end{cases}\n\\]\nthen the unique solution of the above initial-value problem satisfies\n\\[\n\\|W\\|_{L_t^\\infty([0,T_\\epsilon];H^{s+1})}\\le 3\\epsilon,\n\\]\nand, for all \\(t\\in[0,T_\\epsilon]\\), with \\(u=(u^x,u^y)\\),\n\\[\n\\|P_{\\neq 0}u^x(t)\\|_{L^2}\\lesssim \\epsilon\\langle t\\rangle^{-1},\n\\qquad\n\\|u^y(t)\\|_{L^2}\\lesssim \\epsilon\\langle t\\rangle^{-\\min\\{2,s+1\\}}.\n\\]" + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "B" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "lifespan exponent derived from time-growth in shear coordinates", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "dropped the \\(u^y\\) damping conclusion while keeping the same hypotheses and lifespan", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "weighted moment assumption \\(\\|y\\omega_{in}\\|_{L^2}\\le \\epsilon\\)", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "one-derivative smoothing used for velocity upgraded to an \\(H^{s+1}\\) bound for \\(W\\)", + "template_used": "stronger_trap" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives the PDE setup and notation but does not reveal the theorem statement or explicitly indicate the correct option. There is no direct answer leakage." + }, + "TAS": { + "score": 1, + "justification": "The item is very close to a theorem-recall question: the correct choice is essentially the exact small-data statement. Although the distractors introduce competing variants, the task is still largely selecting the theorem as stated rather than drawing a new conclusion." + }, + "GPS": { + "score": 1, + "justification": "There is some reasoning pressure because the options differ in subtle ways (lifespan exponent, missing hypothesis, weaker/stronger conclusions). However, solving it mainly depends on recognizing the exact theorem statement, not on substantial generative mathematical reasoning from the stem alone." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically meaningful: altered lifespan exponents, omission of the weighted assumption, a weaker true-looking conclusion, and an unjustified regularity upgrade. These reflect realistic failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid multiple-choice item with no answer leakage and strong distractors, but it functions more as precise theorem recognition than as a genuinely generative reasoning question." + } + }, + { + "id": "2511.07058v1", + "paper_link": "http://arxiv.org/abs/2511.07058v1", + "theorems_cnt": 6, + "theorem": { + "env_name": "lemma", + "content": "\\label{boundedind}\n Let $G$ be a definable group of finite dimension and $\\{H_i\\}_{i\\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\\omega$ such that there is no $J=\\{j_1,...,j_n\\}\\subseteq I$ of cardinality $n$ such that $|\\bigcap_{i=1}^kH_{j_i}:\\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\\leq n-1$.", + "start_pos": 9114, + "end_pos": 9493, + "label": "boundedind" + }, + "ref_dict": { + "boundedind": "\\begin{lemma}\\label{boundedind}\n Let $G$ be a definable group of finite dimension and $\\{H_i\\}_{i\\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\\omega$ such that there is no $J=\\{j_1,...,j_n\\}\\subseteq I$ of cardinality $n$ such that $|\\bigcap_{i=1}^kH_{j_i}:\\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\\leq n-1$.\n\\end{lemma}" + }, + "pre_theorem_intro_text_len": 6977, + "pre_theorem_intro_text": "The present work characterizes, under the model-theoretic hypothesis of finite-dimensionality,\nirreducible bi-modules, i.e., abelian groups together with two commuting subrings of endomorphisms, and which are \"irreducible\" for the bi-action.\\\\\nThis result is well-known for groups of finite Morley rank \\cite{zilber1984some}. It is also known for $o$-minimal theories (see, for example \\cite{peterzil2000simple}). The main problem is that the two proofs use techniques proper to the two families of theories involved (in the finite Morley rank case, the indecomposability theorem is necessary). A natural question arises: to which generality can we extend the linearization?\\\\\nAn interesting family of theories is finite-dimensional theories, in the sense of \\cite{wagner2020dimensional}.\n\\begin{defn}\n A theory $T$ is said to be \\emph{finite-dimensional} if there exists a function $\\operatorname{dim}$ from the class of all interpretable subset in any model $\\mathcal{M}$ of $T$ into $\\omega\\cup\\{-\\infty\\}$ such that for any $\\phi(x,y)$ formula, $X,Y$ interpretable sets in $T$ and $f$ interpretable function from $X$ to $Y$,\n\\begin{itemize}\n \\item If $a,a'$ have same type over $\\emptyset$, $\\operatorname{dim}(\\phi(x,a))=\\operatorname{dim}(\\phi(x,a'))$ \n \\item $\\operatorname{dim}(\\emptyset)=-\\infty$ and $\\operatorname{dim}(X)=0$ if and only if $X$ is finite;\n \\item $\\operatorname{dim}(X\\cup Y)=\\max\\{\\operatorname{dim}(X),\\operatorname{dim}(Y)\\}$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\geq k$ if for any $y\\in Y$, then $\\operatorname{dim}(X)\\geq \\operatorname{dim}(Y)+k$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\leq k$ for any $y\\in Y$, then $\\operatorname{dim}(X)\\leq \\operatorname{dim}(Y)+k$.\n\\end{itemize}\n\\end{defn}\nExamples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\\text{\\thorn}}$-rank.\\\\\n A first proof of Zilber's Field Theorem in this context has been given by Deloro in \\cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\\\\n In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\\phi$ of an abelian group $A$ is a subgroup of $A\\times A$ such that $\\pi_1(\\phi)=A$ and $\\{a\\in A:\\ (0,a)\\in \\phi\\}$ is finite. The latter subgroup is called the \\emph{katakernel} of $\\phi$. A \\emph{quasi-endomorphism} $\\phi$ is a subgroup of $A$ such that $\\pi_1(\\phi)$, that is called the domain of $\\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\\\\nEndogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\\Gamma\\leq \\operatorname{End}(G)$. Assume that $H$ and $H_1$ are \\emph{$\\Gamma$-almost invariant} \\hbox{i.e.} $\\gamma(H)\\cap H$ is of finite index in $\\gamma(H)$ for any $\\gamma\\in \\Gamma $. Then, $\\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\\gamma$ equal to $\\gamma(H)+H/H$) and $\\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\\{h\\in H:\\ \\gamma(h)\\in H\\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\\Gamma$-almost invariant, the connected component $H^0$ is $\\Gamma$-invariant and the action of $\\Gamma$ on $G/H^0$ is again by endomorphisms.\\\\\nThe article follows the ideas of \\cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\\\\nIn the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\\\\nThen, we proceed to prove the following result.\n\\begin{thmA}\nIn a finite-dimensional theory, let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that:\n\\begin{itemize}\n \\item $\\Gamma,\\Delta$ sharply commute;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa.\n\\end{itemize}\n There is a finite $\\Gamma$ and $\\Delta$-invariant subgroup $F$ such that $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act by endomorphisms on $A/F$.\n\\end{thmA}\nMoreover, $F$ will be given explicitly.\\\\\nIn the second section, we verify the easy case in which both $\\Gamma$ and $\\Delta$ act by endogenies with finite kernel or finite image.\\\\\nThe third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\\\\nThe main result of the fifth section is the following linearization theorem.\n\\begin{thmB}\n Let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that \n \\begin{itemize}\n \\item $C^{\\#}(\\Gamma)=\\Delta$ and $C^{\\#}(\\Delta)=\\Gamma$;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa;\n \\item $A_0=\\Kat(\\Gamma,\\Delta)$.\n \\end{itemize}\n Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\\big(\\Gamma/{\\sim}\\big)\\cap \\big(\\Delta/{\\sim}\\big)$ and $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act $K$-linearly on $A/A_0$.\n\\end{thmB}\n\\textbf{Remark}:\\\\\nThese two theorems are not direct extensions of the Theorems in \\cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \\emph{absolutely minimal} \\hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\\\\nIn the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\\Gamma,\\Delta$ is of quasi-endomorphisms and the other of endogenies.\\\\\nIn section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.\n\\subsection{Finite-dimensional groups}\nIn this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \\ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.", + "context": "mptyset)=-\\infty$ and $\\operatorname{dim}(X)=0$ if and only if $X$ is finite;\n \\item $\\operatorname{dim}(X\\cup Y)=\\max\\{\\operatorname{dim}(X),\\operatorname{dim}(Y)\\}$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\geq k$ if for any $y\\in Y$, then $\\operatorname{dim}(X)\\geq \\operatorname{dim}(Y)+k$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\leq k$ for any $y\\in Y$, then $\\operatorname{dim}(X)\\leq \\operatorname{dim}(Y)+k$.\n\\end{itemize}\n\\end{defn}\nExamples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\\text{\\thorn}}$-rank.\\\\\n A first proof of Zilber's Field Theorem in this context has been given by Deloro in \\cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\\\\n In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\\phi$ of an abelian group $A$ is a subgroup of $A\\times A$ such that $\\pi_1(\\phi)=A$ and $\\{a\\in A:\\ (0,a)\\in \\phi\\}$ is finite. The latter subgroup is called the \\emph{katakernel} of $\\phi$. A \\emph{quasi-endomorphism} $\\phi$ is a subgroup of $A$ such that $\\pi_1(\\phi)$, that is called the domain of $\\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\\\\nEndogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\\Gamma\\leq \\operatorname{End}(G)$. Assume that $H$ and $H_1$ are \\emph{$\\Gamma$-almost invariant} \\hbox{i.e.} $\\gamma(H)\\cap H$ is of finite index in $\\gamma(H)$ for any $\\gamma\\in \\Gamma $. Then, $\\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\\gamma$ equal to $\\gamma(H)+H/H$) and $\\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\\{h\\in H:\\ \\gamma(h)\\in H\\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\\Gamma$-almost invariant, the connected component $H^0$ is $\\Gamma$-invariant and the action of $\\Gamma$ on $G/H^0$ is again by endomorphisms.\\\\\nThe article follows the ideas of \\cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\\\\nIn the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\\\\nThen, we proceed to prove the following result.\n\\begin{thmA}\nIn a finite-dimensional theory, let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that:\n\\begin{itemize}\n \\item $\\Gamma,\\Delta$ sharply commute;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa.\n\\end{itemize}\n There is a finite $\\Gamma$ and $\\Delta$-invariant subgroup $F$ such that $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act by endomorphisms on $A/F$.\n\\end{thmA}\nMoreover, $F$ will be given explicitly.\\\\\nIn the second section, we verify the easy case in which both $\\Gamma$ and $\\Delta$ act by endogenies with finite kernel or finite image.\\\\\nThe third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\\\\nThe main result of the fifth section is the following linearization theorem.\n\\begin{thmB}\n Let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that \n \\begin{itemize}\n \\item $C^{\\#}(\\Gamma)=\\Delta$ and $C^{\\#}(\\Delta)=\\Gamma$;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa;\n \\item $A_0=\\Kat(\\Gamma,\\Delta)$.\n \\end{itemize}\n Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\\big(\\Gamma/{\\sim}\\big)\\cap \\big(\\Delta/{\\sim}\\big)$ and $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act $K$-linearly on $A/A_0$.\n\\end{thmB}\n\\textbf{Remark}:\\\\\nThese two theorems are not direct extensions of the Theorems in \\cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \\emph{absolutely minimal} \\hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\\\\nIn the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\\Gamma,\\Delta$ is of quasi-endomorphisms and the other of endogenies.\\\\\nIn section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.\n\\subsection{Finite-dimensional groups}\nIn this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \\ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.\n\n\\begin{lemma}\\label{boundedind}\n Let $G$ be a definable group of finite dimension and $\\{H_i\\}_{i\\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\\omega$ such that there is no $J=\\{j_1,...,j_n\\}\\subseteq I$ of cardinality $n$ such that $|\\bigcap_{i=1}^kH_{j_i}:\\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\\leq n-1$.\n\\end{lemma}", + "full_context": "mptyset)=-\\infty$ and $\\operatorname{dim}(X)=0$ if and only if $X$ is finite;\n \\item $\\operatorname{dim}(X\\cup Y)=\\max\\{\\operatorname{dim}(X),\\operatorname{dim}(Y)\\}$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\geq k$ if for any $y\\in Y$, then $\\operatorname{dim}(X)\\geq \\operatorname{dim}(Y)+k$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\leq k$ for any $y\\in Y$, then $\\operatorname{dim}(X)\\leq \\operatorname{dim}(Y)+k$.\n\\end{itemize}\n\\end{defn}\nExamples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\\text{\\thorn}}$-rank.\\\\\n A first proof of Zilber's Field Theorem in this context has been given by Deloro in \\cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\\\\n In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\\phi$ of an abelian group $A$ is a subgroup of $A\\times A$ such that $\\pi_1(\\phi)=A$ and $\\{a\\in A:\\ (0,a)\\in \\phi\\}$ is finite. The latter subgroup is called the \\emph{katakernel} of $\\phi$. A \\emph{quasi-endomorphism} $\\phi$ is a subgroup of $A$ such that $\\pi_1(\\phi)$, that is called the domain of $\\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\\\\nEndogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\\Gamma\\leq \\operatorname{End}(G)$. Assume that $H$ and $H_1$ are \\emph{$\\Gamma$-almost invariant} \\hbox{i.e.} $\\gamma(H)\\cap H$ is of finite index in $\\gamma(H)$ for any $\\gamma\\in \\Gamma $. Then, $\\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\\gamma$ equal to $\\gamma(H)+H/H$) and $\\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\\{h\\in H:\\ \\gamma(h)\\in H\\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\\Gamma$-almost invariant, the connected component $H^0$ is $\\Gamma$-invariant and the action of $\\Gamma$ on $G/H^0$ is again by endomorphisms.\\\\\nThe article follows the ideas of \\cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\\\\nIn the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\\\\nThen, we proceed to prove the following result.\n\\begin{thmA}\nIn a finite-dimensional theory, let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that:\n\\begin{itemize}\n \\item $\\Gamma,\\Delta$ sharply commute;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa.\n\\end{itemize}\n There is a finite $\\Gamma$ and $\\Delta$-invariant subgroup $F$ such that $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act by endomorphisms on $A/F$.\n\\end{thmA}\nMoreover, $F$ will be given explicitly.\\\\\nIn the second section, we verify the easy case in which both $\\Gamma$ and $\\Delta$ act by endogenies with finite kernel or finite image.\\\\\nThe third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\\\\nThe main result of the fifth section is the following linearization theorem.\n\\begin{thmB}\n Let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that \n \\begin{itemize}\n \\item $C^{\\#}(\\Gamma)=\\Delta$ and $C^{\\#}(\\Delta)=\\Gamma$;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa;\n \\item $A_0=\\Kat(\\Gamma,\\Delta)$.\n \\end{itemize}\n Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\\big(\\Gamma/{\\sim}\\big)\\cap \\big(\\Delta/{\\sim}\\big)$ and $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act $K$-linearly on $A/A_0$.\n\\end{thmB}\n\\textbf{Remark}:\\\\\nThese two theorems are not direct extensions of the Theorems in \\cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \\emph{absolutely minimal} \\hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\\\\nIn the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\\Gamma,\\Delta$ is of quasi-endomorphisms and the other of endogenies.\\\\\nIn section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.\n\\subsection{Finite-dimensional groups}\nIn this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \\ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.\n\n\\begin{lemma}\\label{boundedind}\n Let $G$ be a definable group of finite dimension and $\\{H_i\\}_{i\\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\\omega$ such that there is no $J=\\{j_1,...,j_n\\}\\subseteq I$ of cardinality $n$ such that $|\\bigcap_{i=1}^kH_{j_i}:\\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\\leq n-1$.\n\\end{lemma}\n\n\\def\\notind#1#2{#1\\setbox0=\\hbox{$#1x$}\\kern\\wd0\n\\hbox to 0pt{\\mathchardef\\nn=12854\\hss$#1\\nn$\\kern1.4\\wd0\\hss}\n\\hbox to 0pt{\\hss$#1\\mid$\\hss}\\lower.9\\ht0 \\hbox to 0pt{\\hss$#1\\smile$\\hss}\\kern\\wd0}\n\n\\end{theorem}\n\\begin{proof}\n We prove the base case. Given $\\Gamma$ by hypothesis essentialy unbounded then it not admits an ascending chain of finite $\\Gamma$-invariant subgroups. Therefore let $A_0$ this subgroup, it contains the katakernel of $\\Delta$, that therefore is finite.\\\\\n Being the katakernel finite, there exists only boundedly many elements for $\\Delta$ or $\\Delta$ is unbounded (and then the proof follows using the previous idea).\\\\\n Then we can take a representative with maximal image and we have to prove that \n\\end{proof}\n\\begin{theorem}", + "post_theorem_intro_text_len": 2223, + "post_theorem_intro_text": "\\begin{proof}\nLet $n=\\dim(G)$. We will work in a sufficiently saturated structure $\\mathfrak{M}$ in which $G$ is definable and assume that $\\phi(x,y)$ is the formula defining the family. Assume, by contrary, that the conclusion is false then for $N=n+2$ and for any $k\\in \\omega$, there exists $g_1,...,g_N$ such that $|\\bigcap_{j\\leq k-1} \\phi(\\mathfrak{M},g_j)/\\bigcap_{j\\leq k} \\phi(\\mathfrak{M},g_j)|\\geq k$. Therefore, the partial type given by the formulas \n$$\\exists a^1_1,...,a^1_k,...,a^N_1,...,a_k^N: a^i_j\\in \\bigcap_{j=1}^i \\phi(\\mathfrak{M},x_j)\\wedge a^i_j{a^i_k}^{-1}\\not\\in \\phi(\\mathfrak{M},x_j)\\wedge \\forall_{i\\leq N}\\ \\phi(\\mathfrak{M},x_i)\\leq G(\\mathfrak{M})$$\nfor all $k<\\omega$, is finitely satisfable. By compactness and saturation, there exist subgroups $\\{H_i\\}_{i\\leq N}$ such that $|\\bigcap_{i0$,\n\\begin{align*}\nn^{2- {c}{\\sqrt{\\log n}}}<\\textmd{ex}_3^L(n,C_3^3)=o(n^2),\n\\end{align*}\nwhere the lower bound is given by Behrend~\\cite{behrend46} and the upper bound is given by \nRusza and Szemer\\'{e}di~\\cite{ruz78}. \nErd\\H{o}s, Frankl and R\\\"{o}dl~\\cite{erd86}\nshowed that for every $r\\geqslant 3$, $\\textmd{ex}_r^L(n,C_3^r)=o(n^2)$ and \n$\\textmd{ex}_r^L(n,C_3^r)>n^{2-o(1)}$. \nUsing the so-called $2$-fold Sidon sets,\nLazebnik and Verstra\\\"{e}te~\\cite{laz03} constructed linear $3$-graphs with girth $5$\nand showed that \n\\begin{align*}\n\\textmd{ex}_3^L\\bigl(n,\\{B_3^3, B_4^3\\}\\bigr)= \\frac{1}{6}n^{ \\frac{3}{2}}+O(n).\n\\end{align*}\nKostochka, Mubayi and Verstra\\\"{e}te asked if for integers $r\\geqslant 3$ and $\\ell\\geqslant 4$,\n\\begin{align*}\n\\textmd{ex}_r^L\\bigl(n,C_\\ell^r\\bigr)=\\Theta\\bigl(n^{1+ \\frac{1}{\\lfloor \\ell/2\\rfloor}}\\bigr).\n\\end{align*} Later, Collier-Cartaino, Graber and Jiang~\\cite{coll14} \nproved that $\\textmd{ex}_r^L(n,C_\\ell^r)=O(n^{1+ \\frac{1}{\\lfloor \\ell/2\\rfloor}})$.\n In detail,\nthey obtained there exist positive constants $c(r, k)$ and $d(r, k)$\n such that \n \\begin{align*}\n \\textmd{ex}_r^L(n,C_{2k}^r)\\leqslant c(r,k)n^{1+ \\frac{1}{k}}\\ \\text{and }\n \\textmd{ex}_r^L(n,C_{2k+1}^r)\\leqslant d(r,k)n^{1+ \\frac{1}{k}},\n \\end{align*}\nwhere the integer $k\\geqslant 2$, the constants $c(r, k)$ and $d(r, k)$ are \nexponential in $k$ for fixed $r$.\nAs an immediate corollary of the main results in~\\cite{jiang30}, Jiang, Ma and Yepremyan \nfurther improved the coefficient $c(r, k)$ to be linear in $k$. \nErgemlidze et al.~\\cite{ergem2019} strengthened some results in $3$-graphs to be\n$\\textmd{ex}_3^L(n, B_4^3)= \\frac{1}{6}n^{ 3/2}+O(n)$, $\\textmd{ex}_3^L(n, B_5^3)= \\frac{1}{3\\sqrt{3}}n^{ 3/2}+O(n)$,\nand for $k=2,3,4$ and $6$,\n\\begin{align*}\n\\textmd{ex}_3^L\\bigl(n,\\{C_3^3,C_5^3,\\ldots,C_{2k+1}^3\\}\\bigr)= \\Theta\\bigl(n^{1+ \\frac{1}{k}}\\bigr).\n\\end{align*}\nIn general, for $k\\geqslant 2$, they also showed that \\begin{align*}\n\\textmd{ex}_3^L\\bigl(n,\\{C_3^3,C_5^3,\\ldots,C_{2k+1}^3\\}\\bigr)= \\Omega\\bigl(n^{1+ \\frac{2}{3k-4+\\epsilon}}\\bigr),\n\\end{align*}\nwhere $\\epsilon=0$ if $k$ is odd and $\\epsilon=1$ if $k$ is even.\nAs a special case of the linear hypergraph extension of K\\H{o}v\\'{a}ri–S\\'{o}s–Tur\\'{a}n's theorem in~\\cite{gao2021},\nGao and Chang showed that $\\textmd{ex}_3^L(n, C_4^3)= \\frac{1}{6}n^{ 3/2}+O(n)$, \nand Gao et al. in~\\cite{gao2023} further proved that $\\textmd{ex}_3^L(n,C_5^3)= \\frac{1}{3\\sqrt{3}}n^{ 3/2}+O(n)$.\nFor the lower bound of $\\textmd{ex}_r^L(n,C_\\ell^r)$, the best known lower bound for $r=3$ is \\begin{align*}\n\\textmd{ex}_3^L\\bigl(n,C_\\ell^3\\bigr)=\\Omega(n^{1+ \\frac{1}{\\ell-1}})\n\\end{align*}\nthat was shown in~\\cite{coll14,ergem2019} due to Verstra\\\"{e}te, by taking a random subgraph of a Steiner\ntriple system. \nUsing generalized Sidon sets~\\cite{ergem2019}, it is conceivable that one can obtain a similar (or\nbetter) constructive lower bound of $\\textmd{ex}_3^L(n,C_\\ell^3)$\n(and maybe also for all $r\\geqslant 3$), which is an area worth some exploration.\nThe lower bound on the linear Tur\\'{a}n number of linear cycles is still\nfar from what is conjectured, especially in higher uniformities $r\\geqslant 4$. \nSometimes to explore the lower bound of the corresponding\nTur\\'{a}n problem is more difficult than the upper one.\n\nFollowing the same logic with the Tur\\'{a}n problems of cycles, \nfor integers $r\\geqslant 3$ and $\\ell\\geqslant 4$, Balogh and Li~\\cite{balgoh17} conjectured that \n$ |\\textmd{Forb}_r^L(n,C_\\ell^r)|=2^{\\Theta(n^{1+1/\\lfloor \\ell/2\\rfloor})}$.\nThey confirmed the conjecture in the case of $\\ell=4$.\nFor $\\ell>4$, they provided a result on the girth version, that is,\nthere exists a constant $c=c(r,\\ell)$ such that \nthe number of linear $r$-graphs with girth larger than $\\ell$ is at most \n$2^{c\\cdot n^{1+1/\\lfloor \\ell/2\\rfloor}}$.\nDefine $\\textmd{ex}_L(n,r,\\ell)$ and $\\textmd{Forb}_L(n,r,\\ell)$\nas $\\textmd{ex}_r^L(n,\\mathcal{H})$ and $\\textmd{Forb}_r^L(n,\\mathcal{H})$ of\n$\\mathcal{H}=\\{C_i^r:\\, 3\\leqslant i\\leqslant \\ell\\}$, respectively.\nHence, it implies that $|\\textmd{Forb}_L(n,r,\\ell)|=2^{O(n^{1+1/\\lfloor \\ell/2\\rfloor})}$, while its lower bound remains open.\nThe idea of the proof in~\\cite{balgoh17} is to reduce the \n hypergraph enumeration problems to some graph enumeration problems, and \nthen the graph container method is implemented in it.\nBalogh, Narayanan and Skokan~\\cite{bal18} indeed made use of \nthe hypergraph container method and provided a balanced supersaturation theorem\nfor linear cycles to resolve a conjecture on the \nenumeration of $r$-uniform hypergraphs with a forbidden $C_\\ell^r$.\n\nIn order to solve a conjecture of Erd\\H{o}s about the existence\nof sparse Steiner triple systems,\nGlock, K\\\"{u}hn, Lo and Osthus~\\cite{glock20}, and \nBohman and Warnke~\\cite{boh19}, studied a constrained triple process along \nsimilar lines, which can be\nseen as a generalization of the \\textit{random greedy triangle removal process}~\\cite{bohman15},\nto iteratively build a sparse partial Steiner triple system,\nthat is,\nadding the sparseness constraint does not affect \nthe evolution of the process significantly.\nTo be precise, a Steiner triple system is called $k$-sparse if it does not\ncontain any $(i+2,i)$-configurations for integers $2\\leqslant i\\leqslant k$,\nwhere a $(i+2,i)$-configuration \nis a set of $i$ triples which span at most $i+2$ vertices.\nErd\\H{o}s made his conjecture\nin Steiner triple systems.\nSome further problems to \nhigher uniformities have been proposed \nand considered in~\\cite{fure13,glock20}. \n\nThe\n \\textit{random greedy $r$-clique removal process} starts with a complete graph\n on vertex set $[n]$, denoted as $\\mathbb{G}(0)$,\n and $\\mathbb{G}(i+1)$ is the remaining graph from $\\mathbb{G}(i)$ by\n selecting one $r$-clique uniformly at random out of all $r$-cliques in $\\mathbb{G}(i)$\n and deleting all its edges from the edge set ${E}(i)$ of $\\mathbb{G}(i)$.\n The process terminates once the remaining graph contains no $r$-cliques,\n and along the way produces a linear $r$-graph, where it starts with the \n empty $r$-graph on vertex set $[n]$, iteratively adding the vertex set of each \n chosen $r$-clique as a new edge in the hypergraph. \n Let $M=\\min\\{i\\,:\\mathbb{G}(i)\\ {\\mbox{is\\ } r{\\mbox {-clique free}}}\\}$,\n and ${E}(M)$ be the set of edges left unsaturated \n by the produced $r$-cliques. In particular, this problem attracted much attention when $r=3$ and it is also called the random greedy triangle removal process. \nFewer results are known on $r\\geqslant 4$.\nBennett and Bohman~\\cite{bennett15} considered the random greedy\nhypergraph matching process, which can be seen as a generalization\nof the $r$-clique removal process. \nTian et al.~\\cite{tian23} directly discussed the structure \nof the random greedy $r$-clique removal process to a natural barrier,\n using a heuristic assumption to find the trajectories of an ensemble\n of random variables when the process evolves. \nInspired by the above works, for integers $r\\geqslant 3$ and $\\ell\\geqslant 3$,\nwe consider the constrained {random greedy $r$-clique removal process} \nso that it \ndoes not just produce a linear $r$-graph but with girth larger than $\\ell$,\n and the corresponding process is called the \\textit{random greedy high girth $r$-clique removal process}\n or \\textit{high girth linear $r$-uniform hypergraph process}.\nCompared with their analysis in~\\cite{glock20,boh19} discussing sparse Steiner triple systems, \nthe challenge \narising in the analysis of random greedy high girth $r$-clique removal process is that each individual \nforbidden configuration $C_i^r$ with $3\\leqslant i\\leqslant \\ell$ should be considered.\nIt is more sensitive and stricter to establish \n the trajectories and error functions of an ensemble\n of random variables which we track in this constrained process with a higher uniformity,\nsuch that the one-step\n changes, trend hypotheses and boundedness hypotheses of these tracked variables can support the proof about the concentration of all these variables to the end. \n Finally, we show the following theorem.", + "context": "Following the same logic with the Tur\\'{a}n problems of cycles, \nfor integers $r\\geqslant 3$ and $\\ell\\geqslant 4$, Balogh and Li~\\cite{balgoh17} conjectured that \n$ |\\textmd{Forb}_r^L(n,C_\\ell^r)|=2^{\\Theta(n^{1+1/\\lfloor \\ell/2\\rfloor})}$.\nThey confirmed the conjecture in the case of $\\ell=4$.\nFor $\\ell>4$, they provided a result on the girth version, that is,\nthere exists a constant $c=c(r,\\ell)$ such that \nthe number of linear $r$-graphs with girth larger than $\\ell$ is at most \n$2^{c\\cdot n^{1+1/\\lfloor \\ell/2\\rfloor}}$.\nDefine $\\textmd{ex}_L(n,r,\\ell)$ and $\\textmd{Forb}_L(n,r,\\ell)$\nas $\\textmd{ex}_r^L(n,\\mathcal{H})$ and $\\textmd{Forb}_r^L(n,\\mathcal{H})$ of\n$\\mathcal{H}=\\{C_i^r:\\, 3\\leqslant i\\leqslant \\ell\\}$, respectively.\nHence, it implies that $|\\textmd{Forb}_L(n,r,\\ell)|=2^{O(n^{1+1/\\lfloor \\ell/2\\rfloor})}$, while its lower bound remains open.\nThe idea of the proof in~\\cite{balgoh17} is to reduce the \n hypergraph enumeration problems to some graph enumeration problems, and \nthen the graph container method is implemented in it.\nBalogh, Narayanan and Skokan~\\cite{bal18} indeed made use of \nthe hypergraph container method and provided a balanced supersaturation theorem\nfor linear cycles to resolve a conjecture on the \nenumeration of $r$-uniform hypergraphs with a forbidden $C_\\ell^r$.\n\nIn order to solve a conjecture of Erd\\H{o}s about the existence\nof sparse Steiner triple systems,\nGlock, K\\\"{u}hn, Lo and Osthus~\\cite{glock20}, and \nBohman and Warnke~\\cite{boh19}, studied a constrained triple process along \nsimilar lines, which can be\nseen as a generalization of the \\textit{random greedy triangle removal process}~\\cite{bohman15},\nto iteratively build a sparse partial Steiner triple system,\nthat is,\nadding the sparseness constraint does not affect \nthe evolution of the process significantly.\nTo be precise, a Steiner triple system is called $k$-sparse if it does not\ncontain any $(i+2,i)$-configurations for integers $2\\leqslant i\\leqslant k$,\nwhere a $(i+2,i)$-configuration \nis a set of $i$ triples which span at most $i+2$ vertices.\nErd\\H{o}s made his conjecture\nin Steiner triple systems.\nSome further problems to \nhigher uniformities have been proposed \nand considered in~\\cite{fure13,glock20}.\n\nThe\n \\textit{random greedy $r$-clique removal process} starts with a complete graph\n on vertex set $[n]$, denoted as $\\mathbb{G}(0)$,\n and $\\mathbb{G}(i+1)$ is the remaining graph from $\\mathbb{G}(i)$ by\n selecting one $r$-clique uniformly at random out of all $r$-cliques in $\\mathbb{G}(i)$\n and deleting all its edges from the edge set ${E}(i)$ of $\\mathbb{G}(i)$.\n The process terminates once the remaining graph contains no $r$-cliques,\n and along the way produces a linear $r$-graph, where it starts with the \n empty $r$-graph on vertex set $[n]$, iteratively adding the vertex set of each \n chosen $r$-clique as a new edge in the hypergraph. \n Let $M=\\min\\{i\\,:\\mathbb{G}(i)\\ {\\mbox{is\\ } r{\\mbox {-clique free}}}\\}$,\n and ${E}(M)$ be the set of edges left unsaturated \n by the produced $r$-cliques. In particular, this problem attracted much attention when $r=3$ and it is also called the random greedy triangle removal process. \nFewer results are known on $r\\geqslant 4$.\nBennett and Bohman~\\cite{bennett15} considered the random greedy\nhypergraph matching process, which can be seen as a generalization\nof the $r$-clique removal process. \nTian et al.~\\cite{tian23} directly discussed the structure \nof the random greedy $r$-clique removal process to a natural barrier,\n using a heuristic assumption to find the trajectories of an ensemble\n of random variables when the process evolves. \nInspired by the above works, for integers $r\\geqslant 3$ and $\\ell\\geqslant 3$,\nwe consider the constrained {random greedy $r$-clique removal process} \nso that it \ndoes not just produce a linear $r$-graph but with girth larger than $\\ell$,\n and the corresponding process is called the \\textit{random greedy high girth $r$-clique removal process}\n or \\textit{high girth linear $r$-uniform hypergraph process}.\nCompared with their analysis in~\\cite{glock20,boh19} discussing sparse Steiner triple systems, \nthe challenge \narising in the analysis of random greedy high girth $r$-clique removal process is that each individual \nforbidden configuration $C_i^r$ with $3\\leqslant i\\leqslant \\ell$ should be considered.\nIt is more sensitive and stricter to establish \n the trajectories and error functions of an ensemble\n of random variables which we track in this constrained process with a higher uniformity,\nsuch that the one-step\n changes, trend hypotheses and boundedness hypotheses of these tracked variables can support the proof about the concentration of all these variables to the end. \n Finally, we show the following theorem.", + "full_context": "Following the same logic with the Tur\\'{a}n problems of cycles, \nfor integers $r\\geqslant 3$ and $\\ell\\geqslant 4$, Balogh and Li~\\cite{balgoh17} conjectured that \n$ |\\textmd{Forb}_r^L(n,C_\\ell^r)|=2^{\\Theta(n^{1+1/\\lfloor \\ell/2\\rfloor})}$.\nThey confirmed the conjecture in the case of $\\ell=4$.\nFor $\\ell>4$, they provided a result on the girth version, that is,\nthere exists a constant $c=c(r,\\ell)$ such that \nthe number of linear $r$-graphs with girth larger than $\\ell$ is at most \n$2^{c\\cdot n^{1+1/\\lfloor \\ell/2\\rfloor}}$.\nDefine $\\textmd{ex}_L(n,r,\\ell)$ and $\\textmd{Forb}_L(n,r,\\ell)$\nas $\\textmd{ex}_r^L(n,\\mathcal{H})$ and $\\textmd{Forb}_r^L(n,\\mathcal{H})$ of\n$\\mathcal{H}=\\{C_i^r:\\, 3\\leqslant i\\leqslant \\ell\\}$, respectively.\nHence, it implies that $|\\textmd{Forb}_L(n,r,\\ell)|=2^{O(n^{1+1/\\lfloor \\ell/2\\rfloor})}$, while its lower bound remains open.\nThe idea of the proof in~\\cite{balgoh17} is to reduce the \n hypergraph enumeration problems to some graph enumeration problems, and \nthen the graph container method is implemented in it.\nBalogh, Narayanan and Skokan~\\cite{bal18} indeed made use of \nthe hypergraph container method and provided a balanced supersaturation theorem\nfor linear cycles to resolve a conjecture on the \nenumeration of $r$-uniform hypergraphs with a forbidden $C_\\ell^r$.\n\nIn order to solve a conjecture of Erd\\H{o}s about the existence\nof sparse Steiner triple systems,\nGlock, K\\\"{u}hn, Lo and Osthus~\\cite{glock20}, and \nBohman and Warnke~\\cite{boh19}, studied a constrained triple process along \nsimilar lines, which can be\nseen as a generalization of the \\textit{random greedy triangle removal process}~\\cite{bohman15},\nto iteratively build a sparse partial Steiner triple system,\nthat is,\nadding the sparseness constraint does not affect \nthe evolution of the process significantly.\nTo be precise, a Steiner triple system is called $k$-sparse if it does not\ncontain any $(i+2,i)$-configurations for integers $2\\leqslant i\\leqslant k$,\nwhere a $(i+2,i)$-configuration \nis a set of $i$ triples which span at most $i+2$ vertices.\nErd\\H{o}s made his conjecture\nin Steiner triple systems.\nSome further problems to \nhigher uniformities have been proposed \nand considered in~\\cite{fure13,glock20}.\n\nThe\n \\textit{random greedy $r$-clique removal process} starts with a complete graph\n on vertex set $[n]$, denoted as $\\mathbb{G}(0)$,\n and $\\mathbb{G}(i+1)$ is the remaining graph from $\\mathbb{G}(i)$ by\n selecting one $r$-clique uniformly at random out of all $r$-cliques in $\\mathbb{G}(i)$\n and deleting all its edges from the edge set ${E}(i)$ of $\\mathbb{G}(i)$.\n The process terminates once the remaining graph contains no $r$-cliques,\n and along the way produces a linear $r$-graph, where it starts with the \n empty $r$-graph on vertex set $[n]$, iteratively adding the vertex set of each \n chosen $r$-clique as a new edge in the hypergraph. \n Let $M=\\min\\{i\\,:\\mathbb{G}(i)\\ {\\mbox{is\\ } r{\\mbox {-clique free}}}\\}$,\n and ${E}(M)$ be the set of edges left unsaturated \n by the produced $r$-cliques. In particular, this problem attracted much attention when $r=3$ and it is also called the random greedy triangle removal process. \nFewer results are known on $r\\geqslant 4$.\nBennett and Bohman~\\cite{bennett15} considered the random greedy\nhypergraph matching process, which can be seen as a generalization\nof the $r$-clique removal process. \nTian et al.~\\cite{tian23} directly discussed the structure \nof the random greedy $r$-clique removal process to a natural barrier,\n using a heuristic assumption to find the trajectories of an ensemble\n of random variables when the process evolves. \nInspired by the above works, for integers $r\\geqslant 3$ and $\\ell\\geqslant 3$,\nwe consider the constrained {random greedy $r$-clique removal process} \nso that it \ndoes not just produce a linear $r$-graph but with girth larger than $\\ell$,\n and the corresponding process is called the \\textit{random greedy high girth $r$-clique removal process}\n or \\textit{high girth linear $r$-uniform hypergraph process}.\nCompared with their analysis in~\\cite{glock20,boh19} discussing sparse Steiner triple systems, \nthe challenge \narising in the analysis of random greedy high girth $r$-clique removal process is that each individual \nforbidden configuration $C_i^r$ with $3\\leqslant i\\leqslant \\ell$ should be considered.\nIt is more sensitive and stricter to establish \n the trajectories and error functions of an ensemble\n of random variables which we track in this constrained process with a higher uniformity,\nsuch that the one-step\n changes, trend hypotheses and boundedness hypotheses of these tracked variables can support the proof about the concentration of all these variables to the end. \n Finally, we show the following theorem.\n\nIn order to solve a conjecture of Erd\\H{o}s about the existence\nof sparse Steiner triple systems,\nGlock, K\\\"{u}hn, Lo and Osthus~\\cite{glock20}, and \nBohman and Warnke~\\cite{boh19}, studied a constrained triple process along \nsimilar lines, which can be\nseen as a generalization of the \\textit{random greedy triangle removal process}~\\cite{bohman15},\nto iteratively build a sparse partial Steiner triple system,\nthat is,\nadding the sparseness constraint does not affect \nthe evolution of the process significantly.\nTo be precise, a Steiner triple system is called $k$-sparse if it does not\ncontain any $(i+2,i)$-configurations for integers $2\\leqslant i\\leqslant k$,\nwhere a $(i+2,i)$-configuration \nis a set of $i$ triples which span at most $i+2$ vertices.\nErd\\H{o}s made his conjecture\nin Steiner triple systems.\nSome further problems to \nhigher uniformities have been proposed \nand considered in~\\cite{fure13,glock20}.\n\nThe remainder of the paper is structured as follows. \nNotation and auxiliary\nresults used throughout the paper are presented in Section 2.\nWe define the random greedy high girth $r$-clique\nremoval process in Section 3, introducing some key random variables of the process that we wish to track, \nestimating the corresponding expected trajectory and \n choosing error function for each tracked variable.\n We formally prove the concentration of all these variables in Section 4,\n where the required one-step changes, trend hypotheses and boundedness hypotheses for \nthese tracked variables are analyzed in higher uniformities.\n Two important claims to bound the overcounting and boundedness parameters\n are discussed in detail in the final section.\n\nThe \\textit{random greedy high girth $r$-clique removal process} starts with a complete graph\n on vertex set $[n]$, denoted by $\\mathbb{G}(0)$,\n and $\\mathbb{G}(i+1)$ is the remaining graph from $\\mathbb{G}(i)$ by\n selecting one $r$-clique uniformly at random out of all \\textit{available} $r$-cliques in $\\mathbb{G}(i)$\n and deleting all its edges from the edge set ${E}(i)$ of $\\mathbb{G}(i)$,\n where an $r$-clique is called \\textit{available} in $\\mathbb{G}(i)$ means that adding\n the vertex set of this $r$-clique as an edge\n does not produce an $r$-graph in $\\mathcal{L}$ with the edges generated by\n the vertices of previously chosen available $r$-cliques.\n Let ${M}=\\min\\{i:\\mathbb{G}(i) {\\text{ is} \\textit{ available } r{\\text{-}\\textit{clique} \\text{ free}}}\\}$.\n This process is equivalently viewed as creating\n a random linear $r$-graph with girth greater than $\\ell$, that is, the \n \\textit{random greedy high girth linear $r$-uniform hypergraph process}.\n Beginning with the empty $r$-graph $\\mathbb{H}(0)$ on the same vertex set $[n]$ we \nsequentially set $\\mathbb{H}({i+1})=\\mathbb{H}(i)+\\boldsymbol{e}_{i+1}$, where the added $r$-set $\\boldsymbol{e}_{i+1}$\nis the vertex set of the chosen available $r$-clique in $\\mathbb{G}(i)$, and \n$\\boldsymbol{e}_{i+1}$ is also called to be an available $r$-set or available edge for $\\mathbb{H}(i)$. \n The process terminates at a maximal linear $r$-graph $\\mathbb{H}({M})$ with girth larger than $\\ell$\n and no available $r$-sets or available edges. At time $i$, graphs are defined with \n respect to edge sets on the vertex set $[n]$, where\n both ${\\mathbb{G}}(i)$ and $\\mathbb{H}(i)$ depend on the outcomes of the process up\n to this point, and ${\\mathbb{G}}(i)$ is the graph given by the pairs of vertices \n that do not appear in any edge of $\\mathbb{H}(i)$. \nIt will be convenient to study the relation \n between $\\mathbb{G}(i)$ and $\\mathbb{H}(i)$ via\n\\begin{align}\n{E}(i)=\\binom{[n]}{2}\\Bigl\\backslash \\bigcup_{\\boldsymbol{f}\\in \\mathbb{H}(i)}\n\\biggl\\{\\bigl\\{x,y\\bigr\\}\\in \\binom{\\boldsymbol{f}}{2}\\biggr\\}. \n\\end{align}\n\n\\begin{theorem}\nGiven any $\\mu>0$ and fixed integers $r,\\ell\\geqslant 3$, there exist $\\lambda>0$ and $\\alpha\\in (0,1)$\nsuch that, for any ${\\boldsymbol{f}}_m\\in {K}_m(i)$ with $2\\leqslant m\\leqslant r-1$,\n${\\boldsymbol{f}}\\in {Q}(i)$, $L\\in \\mathcal{L}$ and $0\\leqslant k\\leqslant e_L-2$, with probability at least $1-n^{-\\mu}$, \n\\begin{align}\n|{Q}(i)|&={q}(t)\\pm\\epsilon_{q}(t),\\\\\n|{Y}_{{\\boldsymbol{f}}_m}(i)|&={y}_m(t)\\pm \\epsilon_{{y}_m}(t),\\\\ \n|{W}_{{\\boldsymbol{f}},L,k}(i)|&={w}_{L,k}(t)\\pm \\epsilon_{w_{L,k}}(t),\n\\end{align}\nholding when $i$ satisfying $0\\leqslant i\\leqslant {M}$ \nwith ${M}$ defined as\n\\begin{align}{M}=n^{1+ \\frac{1}{\\ell-1}-\\lambda \\frac{\\log\\log n}{\\log n}}.\n\\end{align}\nHere, for $t\\in [0,t_M]$,\n\\begin{align}{q}(t)&= \\frac{n^r}{r!}p_i^{\\binom{r}{2}}\\xi_i,\\\\\n{y}_m(t)&= \\frac{n^{r-m}}{(r-m)!}p_i^{ \\binom{r}{2}- \\binom{m}{2}}\\xi_i,\\\\\n{w}_{L,k}(t)&= \\frac{r!e_L}{|\\texttt{Aut}(L)|}\\binom{e_L-1}{k}(r!t)^{k}(p_i^{\\binom{r}{2}}\\xi_i)^{e_L-1-k}n^{(r-1)(e_L-k)+k-r},\n\\end{align}\nand \\begin{align}\n\\epsilon_{q}(t)&=\\sigma_i n^{\\alpha+r-1},\\\\\\epsilon_{y_m}(t)&=\\sigma_i n^{\\alpha+r-m-1},\\\\\\epsilon_{w_{L,k}}(t)&=\\beta_k\\sigma_i^{k+2} n^{\\alpha+(r-1)(e_L-k)+k-r-1}t_{\\textmd{M}}^k,\n\\end{align}in which \\begin{align}\n\\sigma_i=\\sigma(t)=\\log(n^\\alpha+n^2t),\n\\end{align} $\\beta_k$ is a constant depending on $k$ and\n\\begin{align}\nt_{{M}}= \\frac{{M}}{n^2}=n^{-1+ \\frac{1}{\\ell-1}-\\lambda \\frac{\\log\\log n}{\\log n}}.\n\\end{align}\n\\end{theorem}\n\nTheorem~3.4 is proved in Section 4 and we \nalways tacitly assume $0\\leqslant i\\leqslant {M}$ with ${M}$ defined in~(3.18).\n It implies that the process produces\na linear $r$-graph of size at least ${M}$ with its girth greater than $\\ell$.\nTheorem~3.4 verifies Theorem~1.2 \nwith room to spare in the power of the logarithmic factor.\nWe make no attempt to optimize the constants $\\lambda$, $\\alpha$ and \nthe coefficients $\\beta_k$ for any $L\\in \\mathcal{L}$\n and $0\\leqslant k\\leqslant e_L-2$. \nThere are many choices of\nthem that can be balanced to satisfy certain inequalities. \nFor example, for given positive integers $r$, $\\ell$ and $k$, we\nchoose these constants to satisfy the equations \\begin{align}\n\\lambda> \\frac{\\ell}{\\ell-1},\\quad\\alpha_0<\\alpha<1 \\quad\\text{and}\\quad\\beta_k= \\Bigl(\\frac{3\\ell r!}{p_{{M}}^{ \\binom{r}{2}}\\xi_{{M}}}\\Bigr)^{k},\n\\end{align}\nwhere $\\alpha_0= \\frac{\\ell-2}{\\ell-1}$, $p_{{M}}$ and $\\xi_{{M}}$ are constants \nwhen $i=M$ defined in~(3.5) and~(3.9). We will see these choices are sufficient for our proof of Theorem 3.4 in next section.\n We do not replace them with their actual values, which is for the interest of\nunderstanding the role of these constants played in the calculations.", + "post_theorem_intro_text_len": 1600, + "post_theorem_intro_text": "\\noindent In the following proof of Theorem~1.1, we will show the term $O({\\log\\log n}/{\\log n})$ \n is necessary in the exponent of\n the lower bound of\n $M$. Furthermore, it is impossible to apply this \n approach to obtain a better lower bound than $\\Omega(n^{1+ \\frac{1}{\\ell-1}})$. \n As a corollary of Theorem 1.1, it is natural to obtain lower bounds \nof $\\textmd{ex}_L(n,r,\\ell)$ and $|\\textmd{Forb}_L(n,r,\\ell)|$\nwhen $\\ell\\geqslant 3$.\n\\begin{corollary} For every pair of fixed integers $r,\\ell\\geqslant 3$, \nthere exists $\\lambda=\\lambda(\\ell)>0$ such that as $n\\rightarrow\\infty$, \\begin{align*}\n\\textmd{ex}_L(n,r,\\ell)> n^{1+ \\frac{1}{\\ell-1}- \\frac{\\lambda\\log\\log n}{\\log n}}\\quad \\text{and}\\quad \n|\\textmd{Forb}_L(n,r,\\ell)|> 2^{n^{1+ \\frac{1}{\\ell-1}- \\frac{\\lambda\\log\\log n}{\\log n}}}.\n\\end{align*}\n\\end{corollary}\n\\noindent \n\nThe remainder of the paper is structured as follows. \nNotation and auxiliary\nresults used throughout the paper are presented in Section 2.\nWe define the random greedy high girth $r$-clique\nremoval process in Section 3, introducing some key random variables of the process that we wish to track, \nestimating the corresponding expected trajectory and \n choosing error function for each tracked variable.\n We formally prove the concentration of all these variables in Section 4,\n where the required one-step changes, trend hypotheses and boundedness hypotheses for \nthese tracked variables are analyzed in higher uniformities.\n Two important claims to bound the overcounting and boundedness parameters\n are discussed in detail in the final section.", + "sketch": "We define the random greedy high girth $r$-clique removal process (Section 3) and introduce “some key random variables of the process that we wish to track,” then estimate “the corresponding expected trajectory” and “choos[e] error function for each tracked variable.” Next, we “formally prove the concentration of all these variables” (Section 4) by analyzing “the required one-step changes, trend hypotheses and boundedness hypotheses for these tracked variables … in higher uniformities.” Finally, “two important claims to bound the overcounting and boundedness parameters are discussed in detail in the final section.” The proof also notes that the exponent’s term $O({\\log\\log n}/{\\log n})$ is “necessary,” and that “it is impossible to apply this approach to obtain a better lower bound than $\\Omega(n^{1+ \\frac{1}{\\ell-1}})$.”", + "expanded_sketch": "We define the random greedy high girth $r$-clique removal process (Section 3) and introduce “some key random variables of the process that we wish to track,” then estimate “the corresponding expected trajectory” and “choos[e] error function for each tracked variable.” Next, we “formally prove the concentration of all these variables” (Section 4) by analyzing “the required one-step changes, trend hypotheses and boundedness hypotheses for these tracked variables … in higher uniformities.” Finally, “two important claims to bound the overcounting and boundedness parameters are discussed in detail in the final section.” The proof also notes that the exponent’s term $O({\\log\\log n}/{\\log n})$ is “necessary,” and that “it is impossible to apply this approach to obtain a better lower bound than $\\Omega(n^{1+ \\frac{1}{\\ell-1}})$.”", + "expanded_theorem": "For every pair of fixed integers $r,\\ell\\geqslant 3$, consider the random greedy high girth $r$-clique\nremoval process on vertex set $[n]$. Let $M$ be the number of edges in the\ngenerated linear $r$-graph with girth larger than $\\ell$ when the process terminates. \nWith high probability, there exists some positive constant $\\lambda=\\lambda(\\ell)$ such that\n\\begin{align*}\nM\\geqslant n^{1+ \\frac{1}{\\ell-1}-\\lambda \\frac{\\log\\log n}{\\log n}}.\n\\end{align*}", + "theorem_type": [ + "Existential–Universal", + "Inequality or Bound" + ], + "mcq": { + "question": "For fixed integers $r,\\ell\\ge 3$, consider the random greedy high girth $r$-clique removal process on vertex set $[n]$: starting from the complete graph on $[n]$, one repeatedly chooses an $r$-clique uniformly at random among the currently available choices subject to the condition that the $r$-sets chosen so far form a linear $r$-uniform hypergraph of girth greater than $\\ell$, adds the chosen $r$-set as a hyperedge, and deletes all graph edges of that clique; the process stops when no further such $r$-clique can be chosen. Let $M$ denote the number of hyperedges in the final generated linear $r$-graph. Which statement is guaranteed with high probability (that is, with probability tending to $1$ as $n\\to\\infty$)?", + "correct_choice": { + "label": "A", + "text": "There exists a positive constant $\\lambda=\\lambda(\\ell)$ such that $M\\ge n^{1+\\frac{1}{\\ell-1}-\\lambda\\frac{\\log\\log n}{\\log n}}$." + }, + "choices": [ + { + "label": "B", + "text": "There exists a positive constant $\\lambda=\\lambda(\\ell)$ such that $M\\ge n^{1+\\frac{1}{\\ell-1}+\\lambda\\frac{\\log\\log n}{\\log n}}$." + }, + { + "label": "C", + "text": "There exists a positive constant $\\lambda=\\lambda(\\ell)$ such that $M\\ge n^{1+\\frac{1}{\\ell-1}-\\lambda}$." + }, + { + "label": "D", + "text": "There exists a positive constant $\\lambda=\\lambda(r,\\ell)$ such that $M\\ge n^{1+\\frac{1}{\\ell-1}-\\lambda\\frac{\\log\\log n}{\\log n}}$." + }, + { + "label": "E", + "text": "With high probability, for every positive constant $\\lambda=\\lambda(\\ell)$ one has $M\\ge n^{1+\\frac{1}{\\ell-1}-\\lambda\\frac{\\log\\log n}{\\log n}}$." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "necessary negative logarithmic correction", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "replaced the sharper $\\frac{\\log\\log n}{\\log n}$ loss by a fixed constant loss in the exponent", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "dependence of the constant on parameters; theorem states $\\lambda=\\lambda(\\ell)$, not on $r$", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "existential choice of $\\lambda$ replaced by universal quantification over all positive constants", + "template_used": "quantifier_dependence" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not reveal the correct option explicitly or through obvious wording cues; the answer must be identified from the fine details of the asymptotic statement." + }, + "TAS": { + "score": 1, + "justification": "The item is very close to theorem recall: the correct choice is essentially the precise theorem statement, though the alternatives introduce nearby variants in exponent loss, parameter dependence, and quantification." + }, + "GPS": { + "score": 1, + "justification": "Selecting A requires some comparison of subtle asymptotic and quantifier differences, especially against the weaker true statement C and the near-miss options B, D, and E, but it mainly tests precise recall rather than substantial derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic failure modes: wrong sign in the logarithmic correction, weakening the exponent, incorrect parameter dependence, and existential-versus-universal quantifier confusion." + }, + "total_score": 6, + "overall_assessment": "A solid but theorem-recall-heavy MCQ: no answer leakage and strong distractors, but only moderate resistance to tautology and limited generative reasoning." + } + }, + { + "id": "2511.04027v1", + "paper_link": "http://arxiv.org/abs/2511.04027v1", + "theorems_cnt": 4, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm1}\nThere exists a constant $C>1$ such that\n\\begin{equation}\\label{equnthm1}\nC^{-1}\\lambda^{d_S/2}\\leq{N}(u)\\leq C\\lambda^{d_S/2}\n\\end{equation}\nholds for any global Dirichlet or Neumann eigenfunction $u$ on $\\mathcal {SG}$ with eigenvalue $\\lambda$, except for the first non-constant Neumann eigenfunction where $N(u)=0$.", + "start_pos": 79332, + "end_pos": 79694, + "label": "thm1" + }, + "ref_dict": { + "prothm2": "\\begin{proposition}\\label{prothm2}\nIf there exists a pre-localized eigenfunction, then\n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{\\log N(x)}{\\log x}\\geq\\kappa\n\\end{equation*}\nfor some $0<\\kappa\\leq d_S/2$.\nIn particular, for the lattice case (see the exact meaning in Section \\ref{sec2}), \n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\end{equation*}\n\\end{proposition}", + "thm4": "\\begin{theorem}\\label{thm4}\nFor $0<\\lambda<\\lambda_1^D$, we have\n\\begin{equation}\\label{eq3}\n\\begin{aligned}\n\\mathcal{A}_\\lambda=\\ & \\mathcal{C}_\\lambda=\\mathcal{D}_{\\psi^{-1}(\\lambda)},\\quad \\mathcal{B}_\\lambda^1=\\big(\\bigcup_{ij\\in S_1}\\mathcal{L}_{\\psi^{-1}(\\lambda),ij}\\big)\\cup\\{\\boldsymbol{\\theta}\\},\\quad\\text{and}\\\\\n&(\\mathcal{T}_\\lambda^i)^{-1}(\\mathcal{A}_{5^{-1}\\lambda})=\n\\mathcal{G}_{\\psi^{-1}(\\lambda),i}\\quad\\text{for }i\\in S.\n\\end{aligned}\n\\end{equation}\nIn particular, condition (A) holds.\n\\end{theorem}", + "equnthm1": "\\begin{equation}\\label{equnthm1}\nC^{-1}\\lambda^{d_S/2}\\leq{N}(u)\\leq C\\lambda^{d_S/2}\n\\end{equation}", + "figure1": "\\begin{center}\n\\includegraphics[width=5.5cm]{SG2.pdf}\n\\caption{The Sierpinski gasket $\\mathcal{SG}$.}\n\\label{figure1}\n\\end{center}", + "thm5": "\\begin{theorem}\\label{thm5}\nFor $\\eps\\in\\{-1,1\\}^n$ of length $n\\geq 1$ with $\\eps_n=1$, we have \n\\begin{equation}\\label{equn}\nc^{-1}(\\lambda^{\\eps})^{d_S/2}\\leq N(u^{\\eps})\\leq c(\\lambda^{\\eps})^{d_S/2}\n\\end{equation}\nwith some constant $c>1$ independent of $\\lambda_0$, $\\mathbf a$ and $\\eps$.\n\\end{theorem}", + "thm1": "\\begin{theorem}\\label{thm1}\nThere exists a constant $C>1$ such that\n\\begin{equation}\\label{equnthm1}\nC^{-1}\\lambda^{d_S/2}\\leq{N}(u)\\leq C\\lambda^{d_S/2}\n\\end{equation}\nholds for any global Dirichlet or Neumann eigenfunction $u$ on $\\mathcal {SG}$ with eigenvalue $\\lambda$, except for the first non-constant Neumann eigenfunction where $N(u)=0$.\n\\end{theorem}", + "prothm1": "\\begin{proposition}\\label{prothm1}\nIf condition (A) holds, then\n\\begin{equation*}\n\\limsup_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}< \\infty.\n\\end{equation*}\n\\end{proposition}", + "def1": "\\begin{definition}\\label{def1}\n\n(a). \nLet $u\\in\\mathscr{D}_\\mu$, $\\emptyset\\neq A\\subset K$ and $c\\in\\mathbb{R}$, we say that $u$ has a \\textit{local maximum ({\\rm resp.} minimum)} at $A$ with value $c$ if\n\n(a-1). $A$ is a connected component of $u^{-1}(c)$;\n\n(a-2). $A\\cap V_0=\\emptyset$;\n\n(a-3). there exists $\\delta>0$ such that $u(p)\\leq c$ (resp. $u(p)\\geq c$) for any $ p\\in A_\\delta$, where\n$A_\\delta$ is the $\\delta$-neighborhood of $A$.\n\nCall such a set $A$ a \\textit{(local) extreme set} of $u$. Further, if $A$ is a singleton, call $A$ an {\\it extreme point} of $u$.\n\n(b). For $u\\in\\mathscr{D}_\\mu$, denote\n\\[{N}(u)=\\#\\{A\\subset K: A \\text{ is an extreme set of } u\\}.\\]\nFor $x>0$, define\n\\[N(x)=\\sup\\{{N}(u): u \\text { is a $\\lambda$-eigenfunction with}\\ 0\\leq \\lambda\\leq x\\},\\]\nand call it \\textit{the extremum counting function} of $-\\Delta_\\mu$.\n\\end{definition}" + }, + "pre_theorem_intro_text_len": 4258, + "pre_theorem_intro_text": "\\label{sec1}\nThe study of Laplacians on fractals and their spectral properties constitutes a cornerstone of analysis on fractals, a field largely pioneered by Kigami \\cite{K2,K4}. Unlike their Euclidean counterparts, Laplacians on fractals exhibit a wealth of novel and unexpected phenomena, leading to a rich and distinct spectral theory \\cite{FS, Ka1,Ka2,K3,KL, Sh, Sh2, T}. Among these phenomena, the intricate oscillatory behavior of eigenfunctions — such as the distribution and growth of their local extrema — remains a topic of profound interest.\n\nIn the classical setting of a smooth, compact Riemannian manifold (or a bounded Euclidean domain with suitable boundary conditions), the celebrated Courant nodal domain theorem \\cite{CH} provides a fundamental upper bound on the number of nodal domains of an eigenfunction, which in turn implies an upper bound for its local extrema. Moreover, the Hörmander-type estimates \\cite{Ho} show that the $L^\\infty$-norm of an eigenfunction grows at most polynomially with the eigenvalue. These results paint a picture of eigenfunctions whose complexity increases in a controlled and predictable manner as the energy (eigenvalue) increases.\n\nThe landscape on fractals is quite different. The existence of pre-localized eigenfunctions — eigenfunctions that vanish identically on the boundary along with their normal derivatives — is a hallmark of fractals \\cite{K3,K4}. Such eigenfunctions can be highly localized and give rise to a cascade of new eigenfunctions through a localization process. This structure fundamentally alters the asymptotic distribution of eigenvalues and the qualitative behavior of eigenfunctions. Consequently, classical tools and intuition from elliptic PDEs often fail, necessitating new frameworks for understanding the fine properties of eigenfunctions on fractals.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=5.5cm]{SG2.pdf}\n\\caption{The Sierpinski gasket $\\mathcal{SG}$.}\n\\label{figure1}\n\\end{center}\n\\end{figure}\n\nThis paper is devoted to a study of the oscillatory behavior of eigenfunctions on post-critically finite (p.c.f.) self-similar sets, among which the Sierpinski gasket ($\\mathcal{SG}$, see Figure \\ref{figure1}) serves as a typical example. We introduce and investigate the extremum counting function \n$N(x)$ for the Laplacian $-\\Delta_\\mu$ where $\\mu$ is the reference measure. For an eigenfunction $u$ with eigenvalue $\\lambda$, the count \n$N(u)$ enumerates its distinct local maximum and minimum sets (formally defined in Definition \\ref{def1}). The function $N(x)$ then captures the maximum of this number over all eigenfunctions with eigenvalues up to $x$:\n\n$$N(x)=\\sup\\{N(u): u \\text{ is a }\\lambda \\text{-eigenfunction with } 0\\leq \\lambda\\leq x\\}.$$\n\nOur primary goal is to establish the asymptotic growth rate of $N(x)$ as $x\\to \\infty$, linking it directly to the spectral exponent $d_S$ of the fractal, which characterizes a sharp phase transition in the behavior of $N(x)$.\n\nFor the upper bound, we prove in Proposition \\ref{prothm1} that under a natural condition (A) — namely, that eigenfunctions with sufficiently small eigenvalues possess at most one extreme set — the growth of $N(x)$ is at most of the order $x^{d_S/2}$:\n$$\\limsup_{x\\to\\infty} \\frac{N(x)}{x^{d_S/2}}<\\infty.$$\nCondition (A) can be interpreted as a form of ``low-energy simplicity''. It is not universal for all fractals, as evidenced by the modified Koch curve \\cite{M,Sh2}, where even low-energy eigenfunctions can have infinitely many extrema. Verifying its validity is therefore a key step in the analysis for a given fractal.\n\nConversely, for the lower bound, Proposition \\ref{prothm2} shows that the existence of a pre-localized eigenfunction forces \\( N(x) \\) to grow at least polynomially. In the lattice case, this lower bound is sharp, matching the upper bound:\n\\[\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\]\nThe proof constructs sums of copies of a pre-localized eigenfunction over appropriately chosen cells — an inherently fractal method with no direct analogue in the smooth setting.\n\nOur main result is the following theorem, which gives a precise, uniform, two-sided estimate of $N(u)$ for eigenfunctions on the Sierpinski gasket $\\mathcal{SG}$.", + "context": "In the classical setting of a smooth, compact Riemannian manifold (or a bounded Euclidean domain with suitable boundary conditions), the celebrated Courant nodal domain theorem \\cite{CH} provides a fundamental upper bound on the number of nodal domains of an eigenfunction, which in turn implies an upper bound for its local extrema. Moreover, the Hörmander-type estimates \\cite{Ho} show that the $L^\\infty$-norm of an eigenfunction grows at most polynomially with the eigenvalue. These results paint a picture of eigenfunctions whose complexity increases in a controlled and predictable manner as the energy (eigenvalue) increases.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=5.5cm]{SG2.pdf}\n\\caption{The Sierpinski gasket $\\mathcal{SG}$.}\n\\label{figure1}\n\\end{center}\n\\end{figure}\n\nThis paper is devoted to a study of the oscillatory behavior of eigenfunctions on post-critically finite (p.c.f.) self-similar sets, among which the Sierpinski gasket ($\\mathcal{SG}$, see Figure \\ref{figure1}) serves as a typical example. We introduce and investigate the extremum counting function \n$N(x)$ for the Laplacian $-\\Delta_\\mu$ where $\\mu$ is the reference measure. For an eigenfunction $u$ with eigenvalue $\\lambda$, the count \n$N(u)$ enumerates its distinct local maximum and minimum sets (formally defined in Definition \\ref{def1}). The function $N(x)$ then captures the maximum of this number over all eigenfunctions with eigenvalues up to $x$:\n\n$$N(x)=\\sup\\{N(u): u \\text{ is a }\\lambda \\text{-eigenfunction with } 0\\leq \\lambda\\leq x\\}.$$\n\nConversely, for the lower bound, Proposition \\ref{prothm2} shows that the existence of a pre-localized eigenfunction forces \\( N(x) \\) to grow at least polynomially. In the lattice case, this lower bound is sharp, matching the upper bound:\n\\[\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\]\nThe proof constructs sums of copies of a pre-localized eigenfunction over appropriately chosen cells — an inherently fractal method with no direct analogue in the smooth setting.\n\nOur main result is the following theorem, which gives a precise, uniform, two-sided estimate of $N(u)$ for eigenfunctions on the Sierpinski gasket $\\mathcal{SG}$.\n\n\\begin{definition}\\label{def1}\n\n(a). \nLet $u\\in\\mathscr{D}_\\mu$, $\\emptyset\\neq A\\subset K$ and $c\\in\\mathbb{R}$, we say that $u$ has a \\textit{local maximum ({\\rm resp.} minimum)} at $A$ with value $c$ if\n\n(a-1). $A$ is a connected component of $u^{-1}(c)$;\n\n(a-2). $A\\cap V_0=\\emptyset$;\n\n(a-3). there exists $\\delta>0$ such that $u(p)\\leq c$ (resp. $u(p)\\geq c$) for any $ p\\in A_\\delta$, where\n$A_\\delta$ is the $\\delta$-neighborhood of $A$.\n\nCall such a set $A$ a \\textit{(local) extreme set} of $u$. Further, if $A$ is a singleton, call $A$ an {\\it extreme point} of $u$.\n\n(b). For $u\\in\\mathscr{D}_\\mu$, denote\n\\[{N}(u)=\\#\\{A\\subset K: A \\text{ is an extreme set of } u\\}.\\]\nFor $x>0$, define\n\\[N(x)=\\sup\\{{N}(u): u \\text { is a $\\lambda$-eigenfunction with}\\ 0\\leq \\lambda\\leq x\\},\\]\nand call it \\textit{the extremum counting function} of $-\\Delta_\\mu$.\n\\end{definition}\n\n\\begin{proposition}\\label{prothm2}\nIf there exists a pre-localized eigenfunction, then\n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{\\log N(x)}{\\log x}\\geq\\kappa\n\\end{equation*}\nfor some $0<\\kappa\\leq d_S/2$.\nIn particular, for the lattice case (see the exact meaning in Section \\ref{sec2}), \n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\end{equation*}\n\\end{proposition}", + "full_context": "In the classical setting of a smooth, compact Riemannian manifold (or a bounded Euclidean domain with suitable boundary conditions), the celebrated Courant nodal domain theorem \\cite{CH} provides a fundamental upper bound on the number of nodal domains of an eigenfunction, which in turn implies an upper bound for its local extrema. Moreover, the Hörmander-type estimates \\cite{Ho} show that the $L^\\infty$-norm of an eigenfunction grows at most polynomially with the eigenvalue. These results paint a picture of eigenfunctions whose complexity increases in a controlled and predictable manner as the energy (eigenvalue) increases.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=5.5cm]{SG2.pdf}\n\\caption{The Sierpinski gasket $\\mathcal{SG}$.}\n\\label{figure1}\n\\end{center}\n\\end{figure}\n\nThis paper is devoted to a study of the oscillatory behavior of eigenfunctions on post-critically finite (p.c.f.) self-similar sets, among which the Sierpinski gasket ($\\mathcal{SG}$, see Figure \\ref{figure1}) serves as a typical example. We introduce and investigate the extremum counting function \n$N(x)$ for the Laplacian $-\\Delta_\\mu$ where $\\mu$ is the reference measure. For an eigenfunction $u$ with eigenvalue $\\lambda$, the count \n$N(u)$ enumerates its distinct local maximum and minimum sets (formally defined in Definition \\ref{def1}). The function $N(x)$ then captures the maximum of this number over all eigenfunctions with eigenvalues up to $x$:\n\n$$N(x)=\\sup\\{N(u): u \\text{ is a }\\lambda \\text{-eigenfunction with } 0\\leq \\lambda\\leq x\\}.$$\n\nConversely, for the lower bound, Proposition \\ref{prothm2} shows that the existence of a pre-localized eigenfunction forces \\( N(x) \\) to grow at least polynomially. In the lattice case, this lower bound is sharp, matching the upper bound:\n\\[\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\]\nThe proof constructs sums of copies of a pre-localized eigenfunction over appropriately chosen cells — an inherently fractal method with no direct analogue in the smooth setting.\n\nOur main result is the following theorem, which gives a precise, uniform, two-sided estimate of $N(u)$ for eigenfunctions on the Sierpinski gasket $\\mathcal{SG}$.\n\n\\begin{definition}\\label{def1}\n\n(a). \nLet $u\\in\\mathscr{D}_\\mu$, $\\emptyset\\neq A\\subset K$ and $c\\in\\mathbb{R}$, we say that $u$ has a \\textit{local maximum ({\\rm resp.} minimum)} at $A$ with value $c$ if\n\n(a-1). $A$ is a connected component of $u^{-1}(c)$;\n\n(a-2). $A\\cap V_0=\\emptyset$;\n\n(a-3). there exists $\\delta>0$ such that $u(p)\\leq c$ (resp. $u(p)\\geq c$) for any $ p\\in A_\\delta$, where\n$A_\\delta$ is the $\\delta$-neighborhood of $A$.\n\nCall such a set $A$ a \\textit{(local) extreme set} of $u$. Further, if $A$ is a singleton, call $A$ an {\\it extreme point} of $u$.\n\n(b). For $u\\in\\mathscr{D}_\\mu$, denote\n\\[{N}(u)=\\#\\{A\\subset K: A \\text{ is an extreme set of } u\\}.\\]\nFor $x>0$, define\n\\[N(x)=\\sup\\{{N}(u): u \\text { is a $\\lambda$-eigenfunction with}\\ 0\\leq \\lambda\\leq x\\},\\]\nand call it \\textit{the extremum counting function} of $-\\Delta_\\mu$.\n\\end{definition}\n\n\\begin{proposition}\\label{prothm2}\nIf there exists a pre-localized eigenfunction, then\n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{\\log N(x)}{\\log x}\\geq\\kappa\n\\end{equation*}\nfor some $0<\\kappa\\leq d_S/2$.\nIn particular, for the lattice case (see the exact meaning in Section \\ref{sec2}), \n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\end{equation*}\n\\end{proposition}\n\n\\noindent{\\bf Remark.}\nFor a $\\lambda$-eigenfunction with $\\lambda>0$, the values of its local maxima (resp. minima) are positive (resp. negative). \nIndeed, suppose that $u$ has a local maximum at $A$ with value $c\\leq 0$. Since $u$ is a $\\lambda$-eigenfunction, there exists $\\delta>0$ such that $u\\leq c\\leq 0$ and $\\Delta_\\mu u=-\\lambda u\\geq 0$ on $A_\\delta$. Moreover, $u$ is non-constant on $A_\\delta$. Let $h$ be the harmonic function satisfying $h|_{\\partial A_\\delta}=u|_{\\partial A_\\delta}$, and denote by $g(\\cdot,\\cdot)$ the Green function for $-\\Delta_\\mu$ on $A_\\delta$. It is known that $g(p,q)>0$ for all $p,q\\in A_\\delta$. Then, for $p\\in A$, we have\n\\begin{equation*}\nu(p)=\\int_{A_\\delta}g(p,q)\\lambda u(q)\\mathrm{d}\\mu(q)+h(p)0\\text{ for all }i, \\text{ or }a_i<0\\text{ for all }i\\}$ and $\\mathcal{C}_\\lambda=\\pi\\circ\\tau_\\lambda^D\\circ(\\tau_\\lambda^N)^{-1}(\\mathbf{C})$. By the symmetry of $\\mathcal{SG}$, we have $\\boldsymbol{\\theta}\\in\\mathcal{C}_\\lambda$. So the above calculation gives\n$\\mathcal{C}_{\\lambda}=\\mathcal{D}_{\\lambda_0}$.\n\n(b). For $5^{-m}\\lambda<\\lambda_1^D$ with $m\\geq 1$, it is easy to see that \n$\\lambda_{m}=\\psi^{-1}(5^{-m}\\lambda)\\in(0,6)$ by Remark \\ref{re33}. Considering the function $v=u\\circ F_2F_3^{m-1}$, by (\\ref{gamma-delta01pre}), (\\ref{gamma-delta02pre}) and $p_{23}=F_2F_3^{m-1}p_3\\in A$, we see \n\\begin{equation}\\label{gamma-delta0102r}\n\\begin{aligned}\n&2v(p_1)+2v(p_2)-(4-\\lambda_m)v(p_3)=0,\\\\\n&\\big(v(p_{3})-v(p_2)\\big)\\big(v(p_{3})-v(p_1)\\big)\\geq 0.\n\\end{aligned}\n\\end{equation}\nUsing (\\ref{pixxi}) we find that\n\\begin{equation*}\n\\begin{aligned}\n&\\tau(v)\\in\\pi\\circ\\tau_\\lambda^D\\Big(\\big\\{v\\in E(5^{-m}\\lambda)\\setminus\\{0\\}: v\\text{ satisfies (\\ref{gamma-delta0102r})}\\big\\}\\Big)\\\\\n=&\\ \\big\\{(\\xi^{(1)},\\xi^{(2)})\\in\\mathbb{R}^2:\\xi^{(2)}=\\sqrt{3}\\xi^{(1)}-\\frac{\\lambda_m}{6-\\lambda_m},\\ 0\\leq\\xi^{(1)}\\leq\\frac{\\sqrt{3}\\lambda_m}{2(6-\\lambda_m)}\\big\\}\\\\\n=&\\ \\mathcal{I}_{\\lambda_m,3}\\cup\\{\\boldsymbol{\\zeta}_{\\lambda_m,23}\\}\\cup\\{\\boldsymbol{\\zeta}_{\\lambda_m,31}\\},\n\\end{aligned}\n\\end{equation*}\nsince\n\\begin{equation*}\n\\begin{aligned}\n&(2,2,-(4-\\lambda_m))(\\mathbf{1},Q)^{-\\mathrm{t}}=\\frac{1}{3}\\big(\\lambda_m,-\\sqrt{3}(6-\\lambda_m),6-\\lambda_m\\big),\\\\\n&(0,-1,1)(\\mathbf{1},Q)^{-\\mathrm{t}}=\\frac{1}{3}\\big(0,2\\sqrt{3},0\\big),\\\\\n&(-1,0,1)(\\mathbf{1},Q)^{-\\mathrm{t}}=\\frac{1}{3}\\big(0,\\sqrt{3},-3\\big),\n\\end{aligned}\n\\end{equation*}\nand $\\pi\\circ \\tau_\\lambda^D(v')=\\frac{\\lambda_m}{4(6-\\lambda_m)}(\\sqrt{3},-1)^\\mathrm{t}\\in\\mathcal{I}_{\\lambda_m,3}$ for a function $v'\\in E(5^{-m}\\lambda)$ satisfying (\\ref{gamma-delta0102r}) with $v'(p_1)=v'(p_2)=4-\\lambda_m$ and $v'(p_3)=4$.\n\n(b). for $u$ in the (D5) or (N5) case, there exists a constant $c_5>1$ such that\n\\begin{equation}\\label{esti5}\nc_5^{-1}\\lambda^{d_S/2}\\leq N(u)\\leq c_5\\lambda^{d_S/2};\n\\end{equation}\n\n\\begin{lemma}\\label{lm62}\nLet $u\\in E_D(\\lambda)\\setminus\\{0\\}$ or $E_N(\\lambda)\\setminus\\{0\\}$ with $\\mathrm{supp}\\,u=\\mathcal{SG}$. For $u$ in the (D6) or (N6) case and $\\eps=\\varnothing$, we have\n\\begin{equation}\\label{esti6}\nc_6^{-1}\\lambda^{d_S/2}\\leq N(u)\\leq c_6\\lambda^{d_S/2}\n\\end{equation}\nfor some constant $c_6>1$.\n\\end{lemma}\n\\begin{proof}\nFor $u$ in the (D6) or (N6) case, we have $\\lambda_m=3$, $\\lambda_{m-1}=6$ and $\\lambda=5^m\\psi(3)$. Now we consider $u\\circ F_{w'}$ for each $w'\\in W_{m-2}$. Since $-\\Delta_{m-1}u|_{V_{m-1}}=\\lambda_{m-1}u|_{V_{m-1}}$, for distinct $i,j,k\\in S$ we have\n\\begin{equation*}\n\\begin{aligned}\n-2u(p_{ij}^{w'})=u(p_{i}^{w'})+u(p_{j}^{w'})+u(p_{ki}^{w'})+u(p_{jk}^{w'}),\n\\end{aligned}\n\\end{equation*}\nand solving the above equations gives $u(p_i^{w'})=-u(p_{ki}^{w'})-u(p_{ij}^{w'})$. Without loss of generality, we assume $u(p_{23}^{w'})\\leq u(p_{31}^{w'})\\leq u(p_{12}^{w'})$, then", + "post_theorem_intro_text_len": 5772, + "post_theorem_intro_text": "The result demonstrates the highly regular behavior of $\\mathcal{SG}$, where eigenfunction complexity — quantified by the count of local extrema — grows precisely as the power law $d_S/2$. The proof relies on a detail analysis on the eigenfunction decimation, originated developed by Rammal and Toulouse \\cite{RT} and later rigorously established by Shima and Fukushima \\cite{Sh, FS}. \n\nIn the classical setting, the number of critical points of an eigenfunction on an $n$-dimensional manifold is generally expected to grow linearly with $\\lambda^{(n-1)/2}$, a rate related to the wave propagation and the Bohr-Sommerfeld quantization. However, in highly symmetric setting such as $n$-dimensional rectangles or balls — where eigenfunctions can oscillate at high frequencies independently along different directions — this order improves to $\\lambda^{n/2}$. \n\nOn fractals, as shown in Theorem \\ref{thm1} and Propositions \\ref{prothm1} and \\ref{prothm2}, the growth is governed by the spectral exponent $d_S$, which takes the role played by the geometric Hausdorff dimension $n$ in the Euclidean setting. The observed order $d_S/2$ for $\\mathcal{SG}$ suggests that its high symmetry is a key factor in realizing this maximal growth rate. We therefore conjecture that for a broader class of p.c.f. self-similar sets, the growth order of local extrema is at most $d_S/2$, with this upper bound being attained only in highly symmetric cases — such as nested fractals, or even beyond the p.c.f. setting, as exemplified by the Sierpinski carpet — and strictly smaller in the presence of lower symmetry. \n\n\\subsection{Notation and Propositions \\ref{prothm1} and \\ref{prothm2}}Before ending this section, let us introduce the exact definition of $N(u)$ and $N(x)$ on a p.c.f. self-similar set $K$, which is always assumed to be connected.\n\nLet $V_0$ denote the boundary of $K$, and $\\mathscr{D}_\\mu$ denote the domain of $\\Delta_\\mu$. For a function $u\\in \\mathscr{D}_\\mu$ and $p\\in V_0$, denote by $(du)_p$ the normal derivative of $u$ at $p$. \nFor $\\lambda\\geq 0$, call a non-trivial function $u\\in\\mathscr{D}_\\mu$ satisfying $-\\Delta_\\mu u=\\lambda u$ on $K\\setminus V_0$ a {\\it $\\lambda$-eigenfunction} of $-\\Delta_\\mu$. Say an eigenfunction $u$ a \\textit{Dirichlet (Neumann) eigenfunction} if $u|_{V_0}=0$ ($du|_{V_0}=0$). In particular, say $u$ a \\textit{pre-localized eigenfunction} if both $u|_{V_0}=0$ and $du|_{V_0}=0$ hold; and a \\textit{global eigenfunction} if $\\mathrm{supp}\\,u=K$.\n\n\\begin{definition}\\label{def1}\n\n(a). \nLet $u\\in\\mathscr{D}_\\mu$, $\\emptyset\\neq A\\subset K$ and $c\\in\\mathbb{R}$, we say that $u$ has a \\textit{local maximum ({\\rm resp.} minimum)} at $A$ with value $c$ if\n\n(a-1). $A$ is a connected component of $u^{-1}(c)$;\n\n(a-2). $A\\cap V_0=\\emptyset$;\n\n(a-3). there exists $\\delta>0$ such that $u(p)\\leq c$ (resp. $u(p)\\geq c$) for any $ p\\in A_\\delta$, where\n$A_\\delta$ is the $\\delta$-neighborhood of $A$.\n\nCall such a set $A$ a \\textit{(local) extreme set} of $u$. Further, if $A$ is a singleton, call $A$ an {\\it extreme point} of $u$.\n\n(b). For $u\\in\\mathscr{D}_\\mu$, denote\n\\[{N}(u)=\\#\\{A\\subset K: A \\text{ is an extreme set of } u\\}.\\]\nFor $x>0$, define\n\\[N(x)=\\sup\\{{N}(u): u \\text { is a $\\lambda$-eigenfunction with}\\ 0\\leq \\lambda\\leq x\\},\\]\nand call it \\textit{the extremum counting function} of $-\\Delta_\\mu$.\n\\end{definition}\n\n\\noindent{\\bf Remark.}\nFor a $\\lambda$-eigenfunction with $\\lambda>0$, the values of its local maxima (resp. minima) are positive (resp. negative). \nIndeed, suppose that $u$ has a local maximum at $A$ with value $c\\leq 0$. Since $u$ is a $\\lambda$-eigenfunction, there exists $\\delta>0$ such that $u\\leq c\\leq 0$ and $\\Delta_\\mu u=-\\lambda u\\geq 0$ on $A_\\delta$. Moreover, $u$ is non-constant on $A_\\delta$. Let $h$ be the harmonic function satisfying $h|_{\\partial A_\\delta}=u|_{\\partial A_\\delta}$, and denote by $g(\\cdot,\\cdot)$ the Green function for $-\\Delta_\\mu$ on $A_\\delta$. It is known that $g(p,q)>0$ for all $p,q\\in A_\\delta$. Then, for $p\\in A$, we have\n\\begin{equation*}\nu(p)=\\int_{A_\\delta}g(p,q)\\lambda u(q)\\mathrm{d}\\mu(q)+h(p)0$ such that ${N}(u)\\leq 1$ for any $\\lambda$-eigenfunction $u$ with $0\\leq\\lambda<\\lambda_0$.\n\n\\begin{proposition}\\label{prothm1}\nIf condition (A) holds, then\n\\begin{equation*}\n\\limsup_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}< \\infty.\n\\end{equation*}\n\\end{proposition}\n\n\\begin{proposition}\\label{prothm2}\nIf there exists a pre-localized eigenfunction, then\n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{\\log N(x)}{\\log x}\\geq\\kappa\n\\end{equation*}\nfor some $0<\\kappa\\leq d_S/2$.\nIn particular, for the lattice case (see the exact meaning in Section \\ref{sec2}), \n\\begin{equation*}\n\\liminf_{x\\to\\infty}\\frac{N(x)}{x^{d_S/2}}>0.\n\\end{equation*}\n\\end{proposition}\n\nWe structure the paper as follows. \n\nIn Section \\ref{sec2}, we present the proofs of Propositions \\ref{prothm1} and \\ref{prothm2}, and provide an equivalent characterization of condition (A). \n\nBeginning in Section \\ref{sec3}, we focus on the canonic Laplacian on the Sierpinski gasket $\\mathcal{SG}$. There, we recall the spectral decimation method and state two key preparatory theorems — Theorems \\ref{thm4} and \\ref{thm5}. The former confirms the validity of condition (A), while the later establishes two-sided estimate of $N(u)$ for a special class of eigenfunctions. \n\nThe proof of Theorem \\ref{thm4} is given in Section \\ref{sec4}, followed by the proof of Theorem \\ref{thm5} in Section \\ref{sec5}. We conclude in Section \\ref{sec6} with the proof of Theorem \\ref{equnthm1}.", + "sketch": "The post-theorem introduction says that the proof of Theorem~\\ref{thm1} “relies on a detail analysis on the eigenfunction decimation,” i.e. the spectral (eigenfunction) decimation method “originated developed by Rammal and Toulouse \\cite{RT} and later rigorously established by Shima and Fukushima \\cite{Sh, FS}.” It further indicates the overall organization supporting Theorem~\\ref{thm1}: in Section~\\ref{sec3} the paper “recall[s] the spectral decimation method” and states “two key preparatory theorems — Theorems~\\ref{thm4} and \\ref{thm5},” where “\\ref{thm4} confirms the validity of condition (A), while the later establishes two-sided estimate of $N(u)$ for a special class of eigenfunctions,” and then “conclude[s] in Section~\\ref{sec6} with the proof” of the main estimate \\eqref{equnthm1} (i.e. Theorem~\\ref{thm1}).", + "expanded_sketch": "The post-theorem introduction says that the proof of the main theorem “relies on a detail analysis on the eigenfunction decimation,” i.e. the spectral (eigenfunction) decimation method “originated developed by Rammal and Toulouse \\cite{RT} and later rigorously established by Shima and Fukushima \\cite{Sh, FS}.” It further indicates the overall organization supporting the main theorem: next the paper “recall[s] the spectral decimation method” and states “two key preparatory theorems.” We first record the following theorem.\n\n\\begin{theorem}\\label{thm4}\nFor $0<\\lambda<\\lambda_1^D$, we have\n\\begin{equation}\\label{eq3}\n\\begin{aligned}\n\\mathcal{A}_\\lambda=\\ & \\mathcal{C}_\\lambda=\\mathcal{D}_{\\psi^{-1}(\\lambda)},\\quad \\mathcal{B}_\\lambda^1=\\big(\\bigcup_{ij\\in S_1}\\mathcal{L}_{\\psi^{-1}(\\lambda),ij}\\big)\\cup\\{\\boldsymbol{\\theta}\\},\\quad\\text{and}\\\\\n&(\\mathcal{T}_\\lambda^i)^{-1}(\\mathcal{A}_{5^{-1}\\lambda})=\n\\mathcal{G}_{\\psi^{-1}(\\lambda),i}\\quad\\text{for }i\\in S.\n\\end{aligned}\n\\end{equation}\nIn particular, condition (A) holds.\n\\end{theorem}\n\nThe introduction then points to the later preparatory result\n\\begin{theorem}\\label{thm5}\nFor $\\eps\\in\\{-1,1\\}^n$ of length $n\\geq 1$ with $\\eps_n=1$, we have \n\\begin{equation}\\label{equn}\nc^{-1}(\\lambda^{\\eps})^{d_S/2}\\leq N(u^{\\eps})\\leq c(\\lambda^{\\eps})^{d_S/2}\n\\end{equation}\nwith some constant $c>1$ independent of $\\lambda_0$, $\\mathbf a$ and $\\eps$.\n\\end{theorem}\n\nand finally says it “conclude[s] later with the proof” of the main estimate\n\\begin{equation}\\label{equnthm1}\nC^{-1}\\lambda^{d_S/2}\\leq{N}(u)\\leq C\\lambda^{d_S/2}\n\\end{equation}\n(i.e. the main theorem).", + "expanded_theorem": "\\label{thm1}\nThere exists a constant $C>1$ such that\n\\begin{equation}\\label{equnthm1}\nC^{-1}\\lambda^{d_S/2}\\leq{N}(u)\\leq C\\lambda^{d_S/2}\n\\end{equation}\nholds for any global Dirichlet or Neumann eigenfunction $u$ on $\\mathcal {SG}$ with eigenvalue $\\lambda$, except for the first non-constant Neumann eigenfunction where $N(u)=0$.", + "theorem_type": [ + "Existential–Universal", + "Inequality or Bound" + ], + "mcq": { + "question": "On the Sierpiński gasket \\(\\mathcal{SG}\\), let \\(N(u)\\) denote the number of extreme sets of a function \\(u\\), where an extreme set is a connected component \\(A\\) of a level set \\(u^{-1}(c)\\), disjoint from the boundary vertex set \\(V_0\\), such that for some \\(\\delta>0\\) the function is everywhere \\(\\le c\\) on the \\(\\delta\\)-neighborhood of \\(A\\) (local maximum case) or everywhere \\(\\ge c\\) there (local minimum case). Let \\(d_S\\) be the spectral dimension of \\(\\mathcal{SG}\\). A global Dirichlet or Neumann eigenfunction means an eigenfunction of the Laplacian on all of \\(\\mathcal{SG}\\) satisfying Dirichlet or Neumann boundary conditions on \\(V_0\\). Which uniform estimate holds for such eigenfunctions?", + "correct_choice": { + "label": "A", + "text": "There exists a constant \\(C>1\\) such that for every global Dirichlet or Neumann eigenfunction \\(u\\) on \\(\\mathcal{SG}\\) with eigenvalue \\(\\lambda\\), one has \\(C^{-1}\\lambda^{d_S/2}\\le N(u)\\le C\\lambda^{d_S/2}\\), except for the first non-constant Neumann eigenfunction, for which \\(N(u)=0\\)." + }, + "choices": [ + { + "label": "B", + "text": "There exists a constant \\(C>1\\) such that for every global Dirichlet or Neumann eigenfunction \\(u\\) on \\(\\mathcal{SG}\\) with eigenvalue \\(\\lambda\\), one has \\(C^{-1}\\lambda^{d_S/2}\\le N(u)\\le C\\lambda^{d_S/2}\\), with no exceptional Neumann eigenfunction." + }, + { + "label": "C", + "text": "There exists a constant \\(C>1\\) such that for every global Dirichlet or Neumann eigenfunction \\(u\\) on \\(\\mathcal{SG}\\) with eigenvalue \\(\\lambda\\), one has \\(N(u)\\le C\\lambda^{d_S/2}\\)." + }, + { + "label": "D", + "text": "For every global Dirichlet or Neumann eigenfunction \\(u\\) on \\(\\mathcal{SG}\\) with eigenvalue \\(\\lambda\\), there exists a constant \\(C=C(u)>1\\) such that \\(C^{-1}\\lambda^{d_S/2}\\le N(u)\\le C\\lambda^{d_S/2}\\), except for the first non-constant Neumann eigenfunction where \\(N(u)=0\\)." + }, + { + "label": "E", + "text": "There exists a constant \\(C>1\\) such that for every global Dirichlet or Neumann eigenfunction \\(u\\) on \\(\\mathcal{SG}\\) with eigenvalue \\(\\lambda\\), one has \\(C^{-1}\\lambda^{d_S/2}\\le N(u)\\le C\\lambda\\), except for the first non-constant Neumann eigenfunction, for which \\(N(u)=0\\)." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "B" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "characteristic", + "tampered_component": "first_nonconstant_Neumann_exception", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "finiteness", + "tampered_component": "dropped_lower_bound_and_exception_detail", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "uniformity_effectivity", + "tampered_component": "constant_uniformity_independent_of_eigenfunction", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "characteristic", + "tampered_component": "correct_growth_exponent_d_S_over_2_on_upper_bound", + "template_used": "boundary_range" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives definitions and setup but does not state or strongly hint at the exact estimate, exception, or exponent. The correct answer is not leaked directly." + }, + "TAS": { + "score": 1, + "justification": "The item is largely theorem-recognition: it asks which estimate holds, and the correct option is essentially the theorem statement. However, it is not a pure tautology because the options vary in meaningful ways (exception, quantifier uniformity, exponent, one-sided vs two-sided bound)." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish the strongest correct uniform estimate from nearby variants, especially against the weaker true statement and the quantifier/exponent perturbations. Still, the task is closer to precise recall/recognition than to genuine derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically targeted: omission of the exceptional Neumann case, weakening to a one-sided bound, loss of uniformity via dependence on u, and substitution of an incorrect growth exponent. These reflect realistic failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it tests precise recall more than fully generative mathematical reasoning." + } + }, + { + "id": "2511.22531v1", + "paper_link": "http://arxiv.org/abs/2511.22531v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "theorem", + "content": "Let $\\Delta$ be a spherical building.\nThere are order-preserving maps\n\\[ \\Gamma: \\mathrm{sd} \\PD(\\Delta) \\to {\\rm CB}(\\Delta),\\]\n\\[ \\phi :\\OPD(\\Delta) \\to \\Delta * \\Delta,\\]\nsuch that for every group $H$ acting on $\\Delta$ by simplicial automorphisms, $\\Gamma$ and $\\phi$ are $H$-equivariant and induce homotopy equivalences between the fixed point subposets:\n\\[ \\Gamma_H : \\PD(\\Delta)^H \\to {\\rm CB}(\\Delta)^H,\\]\n\\[ \\phi_H :\\OPD(\\Delta)^H \\to \\Delta^H * \\Delta^H.\\]\nIn particular, $\\Gamma$ and $\\phi$ induce homotopy equivalences and equivariant isomorphisms in (co)homology.", + "start_pos": 14834, + "end_pos": 15422, + "label": null + }, + "ref_dict": { + "def:ODandOPDVectorSpaces": "\\begin{definition}\n\\label{def:ODandOPDVectorSpaces}\nLet $V$ be a finite-dimensional vector space defined over a field $k$.\nThe poset of ordered partial decompositions of $V$, denoted by $\\OPD(V)$ consists of tuples of distinct subspaces $(S_1,\\ldots,S_r)$ such that $\\{S_1,\\ldots,S_r\\}\\in \\PD(V)$.\nThe ordering in $\\OPD(V)$ is given by refinement that preserves the order of the elements of the tuples.\nThat is, if $d_1 = (S_1,\\ldots,S_r)$ and $d_2 = (W_1,\\ldots,W_t)$, then $d_1\\leq d_2$ if for all $1\\leq i\\leq j\\leq r$ there are $1\\leq k\\leq l\\leq t$ such that $S_i\\leq W_k$ and $S_j\\leq W_l$.\n\nThe poset $\\OD(V)$ of ordered full decompositions of $V$ is the subposet of\n$\\OPD(V)$ on the set of $(S_1,\\ldots,S_r)$ such that $\\{S_1,\\ldots,S_r\\}\\in \\D(V)$.\n\\end{definition}", + "sub:convexity": "\\begin{proof}\nThis is part of Proposition 2.1 in \\cite{vH}.\n\\end{proof}\n\nHere, $\\proj_{\\sigma}(\\tau)$ denotes the projection of a simplex $\\tau$ to $\\sigma$, in the sense of Tits \\cite[2.30]{Tits1} (see next subsection for the definition).\n\n\\subsubsection{Convexity}\n\\label{sub:convexity}\nNow we look at convex subcomplexes of $\\Delta$, in the sense we defined above.\nOur notion of convexity coincides with Tits' definition \\cite[1.5]{Tits1}, and it is stronger than the notion given in \\cite[Definition 4.120]{AB}.\nSee also \\cite[Remark 4.122]{AB}.\nMoreover, we will mostly work with convex subcomplexes of $\\Delta$ that are contained in some apartment (from the complete system of apartments).\n\nSuppose first that $\\Sigma$ is a Coxeter complex.\nA root of $\\Sigma$ is the image of a folding \\cite[1.8]{Tits1}.\nIf $\\alpha$ is a root of $\\Sigma$, we write $-\\alpha$ for its opposite root, and $\\partial \\alpha =\\alpha \\cap (-\\alpha)$ is (by definition) a wall (see discussion at the top of page 11 in \\cite{Tits1}).\nNote that $-\\alpha = \\op_{\\Sigma}(\\alpha)$.\nWrite $\\roots(\\Sigma)$ for the set of roots of $\\Sigma$.\nRecall that a panel is a codimension-one simplex.\nGiven a chamber $\\sigma\\in \\Sigma$ and a panel $\\tau\\subseteq \\sigma$, there exists a unique root $\\alpha\\in \\roots(\\Sigma)$ such that $\\tau\\in \\partial \\alpha$ and $\\sigma\\in \\alpha$.\nEvery root (and hence every wall) is a convex subcomplex of $\\Sigma$.\nIndeed, by \\cite[2.19]{Tits1}, convex subcomplexes of $\\Sigma$ are exactly those obtained as intersections of roots.\nIf $K$ is a subcomplex of $\\Sigma$, we write $\\roots_{\\Sigma}(K)$ for the set of roots $\\alpha\\in \\roots(\\Sigma)$ that contain $K$.\nThus, $K$ is convex if and only if\n\\[ K = \\bigcap_{\\alpha\\in \\roots_{\\Sigma}(K)} \\alpha.\\]\n\nThe convex hull of two opposite simplices $\\sigma,\\sigma'\\in \\Sigma$ is termed a Levi sphere of the Coxeter complex $\\Sigma$.\nIndeed, Levi spheres are exactly the subcomplexes of $\\Sigma$ obtained as intersections of walls.\nThis terminology is\nborrowed from the work in \\cite{Serre} of Serre, who defines a Levi sphere as an intersection of $|\\Sigma|$, viewed as the classical Euclidean sphere, with sets of reflecting hyperplanes of the underlying Coxeter group.\nThis concept generalizes the notion of Levi subgroups in algebraic groups to the context of buildings, as we will explain later in Lemma \\ref{lm:LeviSubgroupsAndLeviSpheres}.\n\nNow we go back to $\\Delta$.\nLet $S$ be a set of simplices of $\\Delta$.\nIf $S \\subseteq \\Sigma$ for some apartment $\\Sigma$, then the convex hull of $S$ is the same whether it is taken in $\\Sigma$ or in $\\Delta$, that is, $\\Conv_{\\Sigma}(S) = \\Conv_{\\Delta}(S)$.\nIf $\\sigma,\\tau\\in \\Delta$ are two simplices, then they lie in some apartment $\\Sigma$.\nThe projection of $\\tau$ onto $\\sigma$, denoted by $\\proj_{\\sigma}(\\tau)$ is the unique maximal simplex containing $\\sigma$ in the convex hull $\\Conv_{\\Delta}(\\sigma,\\tau)=\\Conv_{\\Sigma}(\\sigma,\\tau)$ (see \\cite[2.30, 3.19]{Tits1}).\n\nA Levi sphere of $\\Delta$ is the convex hull of two opposite simplices, which is then a Levi sphere of any apartment containing these simplices.\nIf $\\sigma,\\sigma'$ are opposite chambers, then $\\Conv_{\\Delta}(\\sigma,\\sigma')$ is the unique apartment that contains them and hence a $\\dim \\Delta$-sphere.\nIf $\\sigma,\\sigma'$ are opposite vertices, then $\\Conv_{\\Delta}(\\sigma,\\sigma') = \\{\\sigma,\\sigma'\\}$ is a $0$-sphere.\n\nMore generally, Levi spheres are always spheres (see also Remarques 2 \non the bottom of page 200 in \\cite{Serre}).\n\n\\begin{lemma}\n\\label{lm:convexIsleviSphere}\nLet $\\Sigma$ be a finite Coxeter complex, and let $K\\subseteq \\Sigma$ be a convex subcomplex.\nThen $K$ is a Levi sphere if and only if $K$ contains a pair of opposite simplices $\\sigma_1,\\sigma_2$ with dimension $\\dim \\sigma_i = \\dim K$.\nIn such a case, $K$ is the convex hull of $\\sigma_1,\\sigma_2$ and $|K|$ is a sphere of dimension $\\dim K$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $K$ contains a pair of opposite simplices $\\sigma_1,\\sigma_2$ of dimension $\\dim K$.\nThen, in the terminology of \\cite[Théorème 2.1]{Serre}, $K$ must be completely reducible. This means that every point of $K$ has an opposite.\nBut every point of $K$ has at most one opposite as $K$ lies in the sphere $\\Sigma$, so $K$ must be exactly the Levi sphere spanned by $\\sigma_1$ and $\\sigma_2$.\nIn particular, $K$ is the convex hull of $\\sigma_1,\\sigma_2$, and it triangulates a sphere of dimension $\\dim K$.\n\nThe converse of this statement is clear (see \\cite{Serre}).\n\\end{proof}", + "def:decompBuilding": "\\begin{definition}\n\\label{def:decompBuilding}\nLet $\\Delta$ be a spherical building.\nWe denote by $\\D(\\Delta)$ the poset of all non-empty Levi spheres, with order induced by reverse inclusion.\n\\end{definition}", + "prop:lowLeviIntervalPD": "\\begin{proposition}\n\\label{prop:lowLeviIntervalPD}\nLet $L\\in \\D(\\GG,k)$.\nThen $\\PD(\\GG,k)_{\\prec L} \\simeq \\Delta(\\GG,k)^L * \\PD(L,k)$. \nMoreover, in homology, we have an isomorphism of $L(k)$-modules\n\\begin{equation}\n\\label{eq:isoHomologyIntervalsPDLevi}\n\\widetilde{H}_*\\big(\\PD(\\GG,k)_{\\prec L},R \\big) \\cong \\widetilde{H}_*\\big(\\, \\Delta(\\GG,k)^L *\\PD(L,k), R \\, \\big) \\cong \\widetilde{H}_*\\big(\\,\\PD(L,k), R \\, \\big),\n\\end{equation}\nprovided that $RL(k)$ is semisimple.\n\\end{proposition}", + "sub:opposition": "\\begin{theorem}\n\\label{thm:homologyWedge}\nSuppose that $G$ is a finite group, and $f:X\\to Y$ is a $G$-equivariant order-preserving map between finite-dimensional $G$-posets.\nLet $R$ be a ring such that $RG$ is semisimple.\nThen, under the conditions of Theorem \\ref{thm:wedgeDecomposition}, for all $m\\geq 0$ we have an isomorphism of $RG$-modules:\n\\[ \\widetilde{H}_{m}(X, R) \\cong_G \\widetilde{Y}_m(Y, R) \\oplus \\bigoplus_{\\overline{y} \\in Y/G} \\bigoplus_{i+j=m-1} \\Ind_{\\Stab_G(y)}^G \\big( \\widetilde{H}_i(f^{-1}(Y_{\\leq y})) \\otimes \\widetilde{H}_j(Y_{> y}) \\big). \\]\n\\end{theorem}\n\nHere we are denoting by $\\overline{y}$ the image of an element $y\\in Y$ in the orbit poset $Y/G$.\n\n\\subsection{Buildings}\nWe work with spherical buildings in the sense of \\cite{AB}, so we do not assume that our buildings are thick.\nFrom now on, $\\Delta$ will denote a (spherical) building.\nBy the Solomon-Tits theorem, $\\Delta$ is spherical in the sense that it has the homotopy type of a wedge of spheres of dimension $\\dim \\Delta$ \\cite[Theorem 4.73]{AB}.\nOn the other hand, for every simplex $\\sigma\\in \\Delta$, its link $\\Lk_{\\Delta}(\\sigma)$ is a building of dimension $\\codim_{\\Delta}(\\sigma)$ \\cite[Proposition 4.9]{AB}.\nIn particular, buildings are Cohen-Macaulay.\n\nWe write $\\A(\\Delta)$ for the complete system of apartments of $\\Delta$.\nFrom now on, when we speak of an apartment of $\\Delta$ we mean an\nelement of $\\A(\\Delta)$. \nIf $\\sigma_1,\\ldots,\\sigma_r\\in \\Delta$ are simplices, we denote by $\\A(\\Delta,\\sigma_1,\\ldots,\\sigma_r)$ the set of apartments $\\Sigma\\in \\A(\\Delta)$ such that $\\sigma_i\\in \\Sigma$ for all $i$.\nRecall that, since we work with spherical buildings, apartments are finite Coxeter complexes.\n\n\\subsubsection{Opposition}\n\\label{sub:opposition}\nLet $\\Sigma$ be a (finite) Coxeter complex.\nTwo chambers of $\\Sigma$ are called opposite if their distance coincides with the diameter of $\\Sigma$.\nIt is well known that every chamber of $\\Sigma$ has a unique opposite, giving rise to an involutory bijection $\\cham \\Sigma\\to \\cham \\Sigma$ that extends uniquely to an involutory automorphism $\\op_{\\Sigma}:\\Sigma\\to\\Sigma$ \\cite[2.39]{Tits1}.\nThus, two simplices $\\sigma,\\sigma' \\in \\Sigma$ are opposite if $\\op_{\\Sigma}(\\sigma) = \\sigma'$.\nBy convention, the opposite of the empty simplex is the empty simplex.\n\nTwo simplices of $\\Delta$ are called opposite if they are opposite in some apartment (and hence in every apartment that contains both of them).\nIt follows that two opposite chambers lie in a unique apartment (see \\cite[Lemma 4.69]{AB} and \\cite[3.25]{Tits1}).\nNote that a simplex may have multiple opposites in $\\Delta$.\n\nThere is a bijection between apartments containing two given opposite simplices and apartments in the link of one of these simplices:\n\n\\begin{lemma}\n\\label{lm:bijectionApartmentsLinkOpposite}\nLet $\\Delta$ be a spherical building, and let $\\sigma,\\sigma'$ be two opposite simplices.\nThen we have a bijection:\n\\[ \\Sigma\\in \\A(\\Delta, \\sigma,\\sigma') \\longmapsto \\Lk_{\\Sigma}(\\sigma)\\in \\A(\\Lk_{\\Delta}(\\sigma)).\\]\nThe inverse of this map is given as follows.\nIf $\\widetilde{\\Sigma}\\in \\A(\\Lk_{\\Delta}(\\sigma))$ and $c,c'\\in \\widetilde{\\Sigma}$ are two opposite chambers there, then $\\widetilde{\\Sigma} = \\Lk_{\\Sigma}(\\sigma)$ where $\\Sigma$ is the convex hull in $\\Delta$ of the opposite chambers $c$ and $\\proj_{\\sigma'}(c')$. Thus we map $\\widetilde{\\Sigma}$ to $\\Sigma$.\n\\end{lemma}\n\n\\begin{proof}\nThis is part of Proposition 2.1 in \\cite{vH}.\n\\end{proof}\n\nHere, $\\proj_{\\sigma}(\\tau)$ denotes the projection of a simplex $\\tau$ to $\\sigma$, in the sense of Tits \\cite[2.30]{Tits1} (see next subsection for the definition).\n\n\\subsubsection{Convexity}\n\\label{sub:convexity}\nNow we look at convex subcomplexes of $\\Delta$, in the sense we defined above.\nOur notion of convexity coincides with Tits' definition \\cite[1.5]{Tits1}, and it is stronger than the notion given in \\cite[Definition 4.120]{AB}.\nSee also \\cite[Remark 4.122]{AB}.\nMoreover, we will mostly work with convex subcomplexes of $\\Delta$ that are contained in some apartment (from the complete system of apartments).\n\nSuppose first that $\\Sigma$ is a Coxeter complex.\nA root of $\\Sigma$ is the image of a folding \\cite[1.8]{Tits1}.\nIf $\\alpha$ is a root of $\\Sigma$, we write $-\\alpha$ for its opposite root, and $\\partial \\alpha =\\alpha \\cap (-\\alpha)$ is (by definition) a wall (see discussion at the top of page 11 in \\cite{Tits1}).\nNote that $-\\alpha = \\op_{\\Sigma}(\\alpha)$.\nWrite $\\roots(\\Sigma)$ for the set of roots of $\\Sigma$.\nRecall that a panel is a codimension-one simplex.\nGiven a chamber $\\sigma\\in \\Sigma$ and a panel $\\tau\\subseteq \\sigma$, there exists a unique root $\\alpha\\in \\roots(\\Sigma)$ such that $\\tau\\in \\partial \\alpha$ and $\\sigma\\in \\alpha$.\nEvery root (and hence every wall) is a convex subcomplex of $\\Sigma$.\nIndeed, by \\cite[2.19]{Tits1}, convex subcomplexes of $\\Sigma$ are exactly those obtained as intersections of roots.\nIf $K$ is a subcomplex of $\\Sigma$, we write $\\roots_{\\Sigma}(K)$ for the set of roots $\\alpha\\in \\roots(\\Sigma)$ that contain $K$.\nThus, $K$ is convex if and only if\n\\[ K = \\bigcap_{\\alpha\\in \\roots_{\\Sigma}(K)} \\alpha.\\]\n\nThe convex hull of two opposite simplices $\\sigma,\\sigma'\\in \\Sigma$ is termed a Levi sphere of the Coxeter complex $\\Sigma$.\nIndeed, Levi spheres are exactly the subcomplexes of $\\Sigma$ obtained as intersections of walls.\nThis terminology is\nborrowed from the work in \\cite{Serre} of Serre, who defines a Levi sphere as an intersection of $|\\Sigma|$, viewed as the classical Euclidean sphere, with sets of reflecting hyperplanes of the underlying Coxeter group.\nThis concept generalizes the notion of Levi subgroups in algebraic groups to the context of buildings, as we will explain later in Lemma \\ref{lm:LeviSubgroupsAndLeviSpheres}.\n\nNow we go back to $\\Delta$.\nLet $S$ be a set of simplices of $\\Delta$.\nIf $S \\subseteq \\Sigma$ for some apartment $\\Sigma$, then the convex hull of $S$ is the same whether it is taken in $\\Sigma$ or in $\\Delta$, that is, $\\Conv_{\\Sigma}(S) = \\Conv_{\\Delta}(S)$.\nIf $\\sigma,\\tau\\in \\Delta$ are two simplices, then they lie in some apartment $\\Sigma$.\nThe projection of $\\tau$ onto $\\sigma$, denoted by $\\proj_{\\sigma}(\\tau)$ is the unique maximal simplex containing $\\sigma$ in the convex hull $\\Conv_{\\Delta}(\\sigma,\\tau)=\\Conv_{\\Sigma}(\\sigma,\\tau)$ (see \\cite[2.30, 3.19]{Tits1}).\n\nA Levi sphere of $\\Delta$ is the convex hull of two opposite simplices, which is then a Levi sphere of any apartment containing these simplices.\nIf $\\sigma,\\sigma'$ are opposite chambers, then $\\Conv_{\\Delta}(\\sigma,\\sigma')$ is the unique apartment that contains them and hence a $\\dim \\Delta$-sphere.\nIf $\\sigma,\\sigma'$ are opposite vertices, then $\\Conv_{\\Delta}(\\sigma,\\sigma') = \\{\\sigma,\\sigma'\\}$ is a $0$-sphere.\n\nMore generally, Levi spheres are always spheres (see also Remarques 2 \non the bottom of page 200 in \\cite{Serre}).\n\n\\begin{lemma}\n\\label{lm:convexIsleviSphere}\nLet $\\Sigma$ be a finite Coxeter complex, and let $K\\subseteq \\Sigma$ be a convex subcomplex.\nThen $K$ is a Levi sphere if and only if $K$ contains a pair of opposite simplices $\\sigma_1,\\sigma_2$ with dimension $\\dim \\sigma_i = \\dim K$.\nIn such a case, $K$ is the convex hull of $\\sigma_1,\\sigma_2$ and $|K|$ is a sphere of dimension $\\dim K$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $K$ contains a pair of opposite simplices $\\sigma_1,\\sigma_2$ of dimension $\\dim K$.\nThen, in the terminology of \\cite[Théorème 2.1]{Serre}, $K$ must be completely reducible. This means that every point of $K$ has an opposite.\nBut every point of $K$ has at most one opposite as $K$ lies in the sphere $\\Sigma$, so $K$ must be exactly the Levi sphere spanned by $\\sigma_1$ and $\\sigma_2$.\nIn particular, $K$ is the convex hull of $\\sigma_1,\\sigma_2$, and it triangulates a sphere of dimension $\\dim K$.\n\nThe converse of this statement is clear (see \\cite{Serre}).\n\\end{proof}\n\nA root (resp. a wall) of $\\Delta$ is a root (resp. a wall) of some apartment.\nWe write $\\roots(\\Delta)$ for the set of roots of $\\Delta$, so\n\\[ \\roots(\\Delta) = \\bigcup_{\\Sigma \\in \\A(\\Delta)} \\roots(\\Sigma).\\]\nSimilarly, if $K\\subseteq \\Delta$, $\\roots_{\\Delta}(K)$ denotes the set of roots $\\alpha\\in \\roots(\\Delta)$ that contain $K$.\nIf $K$ is a convex subcomplex of $\\Delta$ that is contained in some apartment, then $K$ is a convex subcomplex of any apartment containing it.\nIn particular, if $K\\subseteq \\Sigma\\in \\A(\\Delta)$, then \n$K = \\bigcap_{\\alpha \\in \\roots_{\\Sigma}(K)} \\alpha$, and more generally,\n\\[ K =\\bigcap_{\\alpha\\in \\roots_{\\Delta}(K)} \\alpha.\\]\n\n\\subsubsection{The CAT(1) metric}\n\\label{subsub:cat1metric}\nThe geometric realization of a spherical building $\\Delta$ admits a canonical metric $\\dcat$ that makes it a complete CAT(1) space.\nSee \\cite[II.10 Theorem 10A.4]{BH} and \\cite[Example 12.39]{AB}.\nWith this metric, an apartment becomes isometric to the unit sphere $\\SS^d$ of the vector space on which the underlying reflection group acts, where $d = \\dim \\Delta$.\nIn particular, the opposition map $\\op_{\\Sigma}:\\Sigma\\to\\Sigma$ of an apartment $\\Sigma$ gives rise to the involution $-\\id_{\\SS^d}:\\SS^d\\to \\SS^d$ of the unit sphere.\nHence, two simplices $\\sigma,\\sigma'\\in \\Sigma$ are opposite if and only if their barycenters are opposite points when regarded in $\\SS^d$ via this identification.\n\nThe diameter of $|\\Delta|$, which is $\\pi$ with this metric, is also the diameter of any apartment, which equals the distance between two opposite points.\nIn particular, for two points $x,y\\in |\\Delta|$ at distance $\\dcat(x,y) < \\pi$, there exists a unique geodesic from $x$ to $y$.\nRecall that a subspace $X$ of a CAT(1) space is convex if for every two points $x,y\\in X$ at distance $<\\pi$, the unique geodesic segment joining $x,y$ is completely contained in $X$.\nIt follows that a convex subcomplex $K$ of $\\Delta$ gives rise to a convex subspace of $|\\Delta|$.\nNotice that not every convex subspace of $|\\Delta|$ arises in this way.\nNow, if $\\sigma,\\sigma'$ are two non-opposite simplices in $\\Delta$, then there is a unique geodesic (in the geometric realization) that joins their barycenters.\n\n\\subsubsection{Automorphism group}\n\\label{subsub:autDelta}\nWe denote by $\\Aut(\\Delta)$ the group of simplicial automorphisms of $\\Delta$.\nThese automorphisms might not be type-preserving.\nBy a group acting on $\\Delta$, we mean a group inducing simplicial automorphisms on $\\Delta$.\n\nAny simplicial automorphism on $\\Delta$ gives rise to an isometry of $|\\Delta|$ with the metric $\\dcat$.\nTherefore, if $H$ is a group acting simplicially on $\\Delta$, and $x,y\\in |\\Delta|^H$ are two points at distance $<\\pi$, then $H$ must fix the unique geodesic joining $x$ and $y$.\nIn particular, if $\\sigma,\\sigma'$ are non-opposite simplices of $\\Delta$ that are invariant under the action of $H$, then $H$ fixes the unique geodesic in $|\\Delta|$ that joins their barycenters.\n\nFrom these observations, we get the following lemma.\n\n\\begin{lemma}\n\\label{lm:contractibleIntersectionApartments}\nLet $\\Delta$ be a spherical building, and let $H$ be a group acting on $\\Delta$ by simplicial automorphisms.\nLet $\\tau\\in \\Delta^H$ be a non-empty simplex fixed by $H$, and let $\\S \\subseteq \\A(\\Delta,\\tau)$ be a set of apartments containing $\\tau$.\n\nIf $\\bigcap_{\\Sigma\\in \\S} (\\Sigma^H)$ does not contain an opposite of $\\tau$, then it is contractible.\n\\end{lemma}\n\n\\begin{proof}\nLet $X := \\bigcap_{\\Sigma\\in \\S} \\Sigma^H$, and note that $|X| = \\bigcap_{\\Sigma\\in \\S} |\\Sigma'|^H$.\nAssume that there is no opposite of $\\tau$ in $X$.\nThen every point $x\\in X$ is at distance $ < \\pi$ from the barycenter of $\\tau$ (say, $x_0$).\nHence, by the discussion in Subsection \\ref{subsub:autDelta}, $H$ fixes the unique geodesic joining $x$ with $x_0$.\nThus, we can contract $X$ to $x_0$ using these geodesics.\n\\end{proof}\n\n\\section{Posets and simplicial complexes in the case of vector spaces}\n\\label{sec:vectorspaces}\n\nIn this section we take a closer look at the poset of partial decompositions, the poset of ordered partial decompositions\nand the common bases complex for vector spaces. \nWe demonstrate how building-related constructions show up.\nThe results then motivate and guide the more general definitions for buildings in Sections \\ref{sec:CBBuildings}, \\ref{sec:leviSpheres} and\n\\ref{sec:opd}.\n\nLet $V$ be a finite-dimensional vector space defined over a field $k$.\nWrite $T(V)$ for the poset of proper non-zero subspaces of $V$, ordered by inclusion.\nWe denote by $\\Delta(V)$ the order complex of $T(V)$, which is also the building associated with the group $\\GL(V)$.\n\n\\begin{definition}\n\\label{def:DandPDVectorSpaces}\nLet $V$ be a finite-dimensional vector space defined over a field $k$.\nA partial decomposition of $V$ is a subset $\\{S_1,\\ldots,S_r\\} \\subseteq T(V)$ such that\n\\[ \\gen{S_1,\\ldots,S_r} \\cong S_1\\oplus \\cdots \\oplus S_r.\\]\n\nWe denote by $\\PD(V)$ the poset of partial decompositions of $V$ other than $\\emptyset$ and $\\{V\\}$, with order given by refinement; that is\nfor $d_1,d_2\\in \\PD(V)$\n\\[ d_1\\leq d_2 \\text{ if for all } S\\in d_1 \\text{ there is } T\\in d_2 \\text{ such that } S\\leq T.\\]\n\nA full decomposition $V$ is a partial\ndecomposition $\\{S_1,\\ldots, S_r\\}$ such that $V \\cong S_1 \\oplus\\cdots \\oplus S_r$. The poset of full decompositions of $V$ is the subposet\n$\\D(V)$ of $\\PD(V)$ on the set of full decompositions of $V$.\n\\end{definition}\n\nNote that we are not including $\\{V\\}$ in the poset $\\D(V)$\nand that \n\\[ d \\text{ is a partial decomposition } \\Leftrightarrow\n\\ \\dim \\gen{S\\tq S\\in d} = \\sum_{S\\in d} \\dim S.\\]\n\nWe will also work with the ordered versions of $\\PD(V)$ and $\\D(V)$:\n\n\\begin{definition}\n\\label{def:ODandOPDVectorSpaces}\nLet $V$ be a finite-dimensional vector space defined over a field $k$.\nThe poset of ordered partial decompositions of $V$, denoted by $\\OPD(V)$ consists of tuples of distinct subspaces $(S_1,\\ldots,S_r)$ such that $\\{S_1,\\ldots,S_r\\}\\in \\PD(V)$.\nThe ordering in $\\OPD(V)$ is given by refinement that preserves the order of the elements of the tuples.\nThat is, if $d_1 = (S_1,\\ldots,S_r)$ and $d_2 = (W_1,\\ldots,W_t)$, then $d_1\\leq d_2$ if for all $1\\leq i\\leq j\\leq r$ there are $1\\leq k\\leq l\\leq t$ such that $S_i\\leq W_k$ and $S_j\\leq W_l$.\n\nThe poset $\\OD(V)$ of ordered full decompositions of $V$ is the subposet of\n$\\OPD(V)$ on the set of $(S_1,\\ldots,S_r)$ such that $\\{S_1,\\ldots,S_r\\}\\in \\D(V)$.\n\\end{definition}\n\nAs observed first in \\cite{LR}, the poset $\\OD(V)$ is naturally isomorphic to the poset of opposite pairs of the building $\\Delta(V)$.\n\n\\begin{remark}\n[{Opposite simplices}]\n\\label{rk:ODlinearCase}\nAn ordered full decomposition $d = (S_1,\\ldots,S_r)\\in \\OD(V)$ determines the pair $\\big(\\,P(d),Q(d)\\,\\big)$ of flags given by\n\\[ P(d) = (\\,S_1 < S_1 \\oplus S_2 < \\cdots < S_1\\oplus\\cdots \\oplus S_{r-1}\\,),\\]\nand\n\\[ Q(d) = (\\, S_r < S_r\\oplus S_{r-1} < \\cdots < S_2\\oplus\\cdots \\oplus S_r\\,).\\]\nIn the language of buildings, this means that $P(d),Q(d)$ are opposite simplices of $\\Delta(V)$.\nFor $\\Delta(V)$ the notion of opposition\nof simplices from Subsection \\ref{sub:opposition} translates into\nthe following. Two simplices $\\sigma$ and $\\tau$ from $\\Delta(V)$ are opposite if they have the same dimension and for all $S\\in \\sigma$ there exists a unique $T\\in \\tau$ such that $V = \\gen{S,T} \\cong S\\oplus T$.\n\nTherefore, the poset $\\OD(V)$ can be alternatively described as the poset of pairs of opposite simplices of the building $\\Delta(V)$, \nwhere the ordering is given by coordinate-wise reverse inclusion.\nThe isomorphism is given by $d\\mapsto \\big(\\,P(d), Q(d)\\,\\big)$.\n\nAnother description of $\\OD(V)$ is in terms of the Charney poset.\nRecall that the Charney poset $\\Ch(V)$ consists of pairs $(S,T)$ of proper non-zero subspaces of $V$ such that $S\\oplus T = V$.\nThe ordering in $\\Ch(V)$ is given by zig-zag containment:\n\\[ (S_1,T_1) \\leq (S_2,T_2) \\ \\Leftrightarrow \\ S_1\\leq S_2 \\text{ and } T_1\\geq T_2.\\]\nThen it is not hard to see that\n\\[ \\OD(V) = \\X\\big(\\,\\K(\\,\\Ch(V)\\,)\\,\\big)^{\\op}.\\]\nThe Charney poset was introduced by R. Charney \\cite{Charney} in the context of free modules over Dedekind domains $R$, where it was used to establish homological stability results for the linear groups $\\GL_n(R)$.\n\\end{remark}\n\nWe have seen that $\\OD(V)$ has an intrinsic description in terms of building properties.\nOur next theorem describes $\\D(V)$ in terms of Levi subgroups, hinting at a possible definition of the full decomposition poset for arbitrary buildings that arise from the BN-pair of the $k$-points of a connected reductive algebraic group.\n\nIn what follows, suppose that $\\overline{k}$ is the algebraic closure of a field $k$.\nWe say that $L$ is a $k$-Levi subgroup of $\\GL_n(k)$ if $L$ is an algebraic group defined over $k$ and it is the Levi complement in a parabolic subgroup of $\\GL_n$ that is also defined over $k$.\n\n\\begin{theorem}\n\\label{thm:decompAndLeviVectorSpaces}\nLet $\\L(\\GL_n,k)$ denote the poset of (proper) $k$-Levi subgroups of $\\GL_n$ ordered by inclusion.\nThen we have a $\\GL_n(k)$-equivariant poset isomorphism $\\L(\\GL_n,k)\\groupiso_{\\GL_n(k)} \\D(k^n)$.\n\\end{theorem}", + "def:ODandOPDbuildings": "\\begin{definition}\n\\label{def:ODandOPDbuildings}\nLet $\\Delta$ be a spherical building.\nWe define the posets:\n\\begin{itemize}\n\\item $\\OD(\\Delta) := \\Opp(\\Delta) = \\{ (\\sigma,\\sigma')\\tq \\sigma,\\sigma'$ are opposite simplices$\\}$, with order relation \n$(\\sigma,\\sigma') \\leq (\\tau,\\tau')$ if $\\tau\\subseteq \\sigma$ and $\\tau'\\subseteq \\sigma'$.\n\\item $\\OPD(\\Delta) = \\X(\\Delta) \\cup \\Opp(\\Delta)$ with order relation $\\preceq$ defined as follows. Among elements of $\\X(\\Delta)$ and \namong elements of $\\Opp(\\Delta)$ the order $\\preceq$ is inherited from the order on the respective posets. If $\\sigma \\in \\X(\\Delta)$ and\n$(\\tau,\\tau') \\in \\Opp(\\Delta)$ then we set \n$\\sigma \\prec (\\tau,\\tau')$ if there is an apartment containing $\\sigma,\\tau$, and $\\tau'$.\n\\end{itemize}\nWe call $\\OD(\\Delta)$ the ordered decomposition poset associated with $\\Delta$, and $\\OPD(\\Delta)$ the ordered partial decomposition poset.\n\\end{definition}", + "def:PDbuildings": "\\begin{definition}\n\\label{def:PDbuildings}\nLet $\\Delta$ be a spherical building.\nWe define $\\PD(\\Delta)$ as the poset on the disjoint union of \n$\\X(\\Delta)$ and ${\\D(\\Delta)}$ with order relation $\\preceq$\ndefined as follows. \nAmong elements of $\\X(\\Delta)$ and among elements of $\\D(\\Delta)$ the\norder $\\preceq$ is the one inherited from the order on the \nrespective posets.\nIf $\\sigma \\in \\Delta$ and $S\\in {\\D(\\Delta)}$, then we set $\\sigma \\prec S$ if and only if there exists an apartment $\\Sigma$ of $\\Delta$ such that $\\sigma\\in \\Sigma$ and $S$ is a Levi sphere of $\\Sigma$.\n\\end{definition}", + "coro:DDeltaCM": "\\begin{corollary}\n\\label{coro:DDeltaCM}\nLet $\\Delta$ be a spherical building.\nThen the poset of Levi spheres $\\D(\\Delta)$ is Cohen-Macaulay of dimension $\\dim \\Delta$.\n\\end{corollary}", + "thm:PDandPDBuildingVectorSpace": "\\begin{theorem}\n\\label{thm:PDandPDBuildingVectorSpace}\nLet $V$ be an $n$-dimensional vector space over a field $k$.\nLet $\\sigma \\in \\OPD(V)'$, and write $\\sigma = \\sigma_0\\cup \\sigma_1$ where $\\sigma_0$ is the set of ordered partial decompositions in $\\sigma$ that span a proper subspace of $V$, and $\\sigma_1$ is the set of full ordered decompositions in $\\sigma$.\nThen, for $\\sigma \\in \\OPD(V)'$, we set\n\\[ \\phi(\\sigma) = \\begin{cases}\n\\{ \\gen{s} \\tq s\\in \\sigma_0\\} & \\sigma_1 = \\emptyset,\\\\\n\\max \\sigma_1 & \\sigma_1\\neq\\emptyset.\n\\end{cases}\\]\nThen $\\phi$ is a $\\GL(V)$-equivariant poset map $\\phi: \\OPD(V)' \\to \\OKtwo{V}$ such that for all $H\\leq \\GL(V)$, the induced map $\\phi_H: (\\OPD(V)')^H \\to \\OKtwo{V}^H$ is a homotopy equivalence.\n\nWe get analogous conclusions for the corresponding map $\\PD(V)'\\to \\Ktwo{V}$.\n\\end{theorem}", + "lm:convexIsleviSphere": "\\begin{lemma}\n\\label{lm:convexIsleviSphere}\nLet $\\Sigma$ be a finite Coxeter complex, and let $K\\subseteq \\Sigma$ be a convex subcomplex.\nThen $K$ is a Levi sphere if and only if $K$ contains a pair of opposite simplices $\\sigma_1,\\sigma_2$ with dimension $\\dim \\sigma_i = \\dim K$.\nIn such a case, $K$ is the convex hull of $\\sigma_1,\\sigma_2$ and $|K|$ is a sphere of dimension $\\dim K$.\n\\end{lemma}", + "def:commonBasis": "\\begin{definition}\n\\label{def:commonBasis}\nLet $\\Delta$ be a spherical building.\nWe write $\\CB(\\Delta)$ for the simplicial complex on the same\nvertex set as $\\Delta$ whose maximal simplices are \n$\\displaystyle{\\bigcup_{\\sigma \\in \\Sigma}} \\sigma$ for \n$\\Sigma \\in \\A(\\Delta)$. \n\n\\end{definition}" + }, + "pre_theorem_intro_text_len": 9650, + "pre_theorem_intro_text": "\\label{sec:intro}\n\nThe order complex of the lattice of non-trivial subspaces of a finite-dimensional vector space is well known to be the spherical building of type A (see e.g., \\cite{Tits1}).\nRecently, simplicial complexes defined \non the set of non-trivial subspaces \nvia constraints on common bases (see e.g., \\cite{Rognes,MPW}) or the relative position of the subspaces were studied (see e.g., \\cite{LR,BPW24}). Such complexes are often order complexes of partially ordered sets.\nThe goals of this paper are to provide definitions of such\nsimplicial complexes and posets that are independent of Lie type, and to provide results on the associated homology groups\nand homotopy types. In some cases, the complexes we construct in type A are the same as the previously studied ones. In other cases, our type A complexes are different than those previously studied but have the same equivariant homotopy type. We start by defining the motivating posets and simplicial complexes and reviewing the relevant literature. \n\nLet $V$ be an $n$-dimensional vector space over a field $k$.\nA partial decomposition of $V$ is a set $\\{V_1,\\ldots,V_r\\}$ of non-zero and proper subspaces of $V$ such that $\\langle V_1,\\ldots, V_r \\rangle \\cong V_1\\oplus\\cdots\\oplus V_r$;\nthat is, the $V_1,\\ldots, V_r$ are in internal direct sum.\nThe set $\\PD(V)$ of all non-empty partial decompositions of $V$ is a poset with order given by refinement; i.e., $\\{V_1,\\ldots, V_r\\}\\leq \\{W_1,\\ldots, W_s\\}$ if for all $1\\leq i \\leq r$ there is $1\\leq j\\leq s$ such that $V_i\\leq W_j$.\n\nIn \\cite{HHS}, P. Hanlon, P. Hersh and J. Shareshian proved that $\\PD(V)$ is Cohen-Macaulay of dimension $2n-3$ if $k$ is a finite field.\nIn particular, $\\PD(V)$ is spherical (see Section \\ref{sec:preliminaries} for definitions).\nMore recently, in \\cite{BPW24}, it was proved that for any field $k$,\\,$\\PD(V)$ has the homotopy type of the common bases complex ${\\rm CB}(V)$ of $V$, as defined by J. Rognes \\cite{Rognes}.\nConcretely, ${\\rm CB}(V)$ is the simplicial complex whose simplices are sets $\\sigma$ of non-zero proper subspaces of $V$ for which there exists a basis $B$ of $V$ that each subspace in $\\sigma$ is generated by some subset of $B$.\nIn this case, we say that the elements of $\\sigma$ have a common basis.\nRognes showed that, although ${\\rm CB}(V)$ has dimension $2^n-3$, its homology is concentrated in degrees $\\leq 2n-3$, and he conjectured that ${\\rm CB}(V)$ must indeed be $(2n-4)$-connected.\nIn \\cite{MPW}, J. Miller, P. Patzt and J. Wilson established Rognes' conjecture by showing that there is a continuous map from a $(2n-4)$-connected simplicial complex to ${\\rm CB}(V)$ that induces an isomorphism on homotopy groups up to degree $2n-4$.\nIn particular, via \\cite{BPW24} this implies that $\\PD(V)$ is spherical for any field.\n\nConsidering the ordered version $\\OPD(V)$ of $\\PD(V)$ \nappears to be equally natural.\nBy definition, $\\OPD(V)$ is the poset whose elements are tuples \n$(V_1,\\ldots,V_r)$, $r\\geq 1$, of distinct subspaces of $V$ such that $\\{V_1,\\ldots,V_r\\} \\in \\PD(V)$.\nThe ordering in $\\OPD(V)$ is given by refinement consistent with the ordering of the tuples (see Definition \\ref{def:ODandOPDVectorSpaces}).\nWe call the elements of $\\OPD(V)$ ordered partial decompositions.\nIt follows from \\cite[Theorem 6.8]{PW25} that $\\OPD(V)$ is homotopy equivalent to a wedge of spheres of dimension $2n-3$.\nIf $V$ is a finite vector space then \\cite[Theorem 6.10]{PW25} implies \nthat $\\OPD(V)$ has the homotopy type of the two-fold join $\\Delta * \\Delta$ of the Tits building $\\Delta$ of $V$.\nUsing GAP we also verified in some small examples that the top homology group of\n$\\OPD(V)$, as a $\\GL(V)$-module, is the tensor square of the Steinberg module.\n\nLater, we found an alternative approach to determining the \nhomotopy type of $\\OPD(V)$ using closely related constructions, which inspired the work in this paper.\nLet $\\OD(V)$ denote the subposet of $\\OPD(V)$ consisting of full ordered decompositions $(V_1,\\ldots, V_r)$; i.e., those decompositions satisfying\n$V \\cong V_1\\oplus \\cdots \\oplus V_r$.\nLet $X = \\Delta \\cup \\OD(V)$ be the poset on the disjoint union of \n$\\Delta$ (regarded as a poset without the empty simplex) and $\\OD(V)$ with the following\nordering.\nWe keep the inclusion ordering among elements of $\\Delta$, and the refinement ordering in $\\OD(V)$. No element of $\\OD(V)$ lies below any element of $\\Delta$. If $\\sigma\\in \\Delta$ and $d\\in \\OD(V)$, we set $\\sigma \\prec d$ if and only if there is a basis of $V$ containing a basis for $d$ and a basis for every subspace appearing in $\\sigma$.\nIn Theorem \\ref{thm:PDandPDBuildingVectorSpace} we show that\n$\\OPD(V) \\simeq X$.\nThe poset $X$ in turn is homotopy equivalent to the complex $T^{1,1}(V)$ used in \\cite{MPW}, which is shown therein to have\nthe homotopy type of $\\Delta * \\Delta$.\nThis again allows us to conclude that $\\OPD(V) \\simeq \\Delta * \\Delta$ for any finite-dimensional vector space $V$.\nSince all homotopy equivalences can be seen to be $\\GL(V)$-equivariant, it now follows that the top homology group of\n$\\OPD(V)$ carries the tensor square of the Steinberg representation. \n\nThe poset $\\OD(V)$ already appears in work {\\cite{LR} of G.I. Lehrer and L.J. Rylands, who observe that \n$\\OD(V)$ can be identified with poset of pairs of opposite parabolics of $\\GL(V)$.\n\\footnote{This observation is motivated by previous work by R. Charney \\cite{Charney}, and $\\OD(V)$ is commonly known as the Charney complex. See also \\cite[Proposition 4.15]{PW25}.}\nRecall that two parabolic subgroups are called opposite if their intersection is a Levi complement in both of them.\nIndeed, two parabolic subgroups are opposite if and only if they are opposite as simplices of the building $\\Delta$ of $V$, which is a notion intrinsically defined for every building (see Subsection \\ref{sub:opposition}).\nThus, $\\OD(V)$ can be described in terms of simplices of $\\Delta$, and we define\n\\[\\, \\OD(\\Delta) = \\big\\{\\,(\\sigma,\\sigma') \\tq \\sigma,\\sigma'\\in \\Delta \\text{ are (non-empty) opposite simplices}\\,\\big\\},\\]\nwhere the ordering of this poset is inclusion-reversing in each coordinate (so a pair of opposite maximal simplices is an ordered frame of $V$, which is a minimal element of the poset).\nUsing this identification the ordering between crossed terms in $X = \\Delta \\cup \\OD(\\Delta)$ becomes $\\sigma\\prec (\\sigma_1,\\sigma_2)$ if there is a basis of $V$ spanning the subspaces from $\\sigma\\cup \\sigma_1\\cup \\sigma_2$.\nThis condition is equivalent to saying that $\\sigma,\\sigma_1,\\sigma_2$ lie in a common apartment of $\\Delta$. This shows\nthat\nthe poset $\\Delta\\cup \\OD(\\Delta)$ can be defined in terms of intrinsic combinatorial properties of the building.\n\nThese observations suggest that there may be a more general phenomenon in the context of algebraic groups or even spherical buildings.\nIn this paper, we study the following questions and answer all of them positively.\nLet $\\Delta$ be a (spherical) building.\n\\begin{enumerate}\n \\item Can we define a poset $\\OPD(\\Delta)$ that, up to homotopy, coincides with $\\OPD(V)$ when $\\Delta$ is the building of the vector space $V$?\n \\item Can we similarly define $\\PD(\\Delta)$? Is it highly connected?\n \\item Do we have a common basis complex ${\\rm CB}(\\Delta)$ for $\\Delta$ that coincides with ${\\rm CB}(V)$ when $\\Delta$ is the building of the vector space $V$?\n \\item If questions (2) and (3) have a positive answer, do we have ${\\rm CB}(\\Delta) \\simeq \\PD(\\Delta)$?\n \\item Is $\\OPD(\\Delta)\\simeq \\Delta * \\Delta$?\n\\end{enumerate}\n\nWhile the definitions of $\\OPD(\\Delta)$ and ${\\rm CB}(\\Delta)$ arise naturally, a suitable definition of $\\PD(\\Delta)$ is less obvious.\nIndeed, as in the case of $\\OPD$, Theorem \\ref{thm:PDandPDBuildingVectorSpace} shows that $\\PD(V)\\simeq \\Delta \\cup \\D(V)$, where $\\D(V)$ is the poset of full decompositions of $V$, and the ordering between crossed terms in $\\Delta \\cup \\D(V)$ is defined as in the ordered case: if $\\sigma\\in \\Delta$ and $d\\in \\D(V)$, then we set $\\sigma\\prec d$ if $\\sigma\\cup d\\in {\\rm CB}(V)$ ($\\sigma$ and $d$ have a common basis).\nOur next observation is, that $\\D(V)$ can be identified with the poset of (split) Levi subgroups of $\\GL(V)$.\nIn the language of buildings, split Levi subgroups correspond to Levi spheres, as introduced by J.P. Serre in \\cite{Serre} (see Subsection \\ref{sub:convexity} and Section \\ref{sec:leviSpheres}).\nIf we identify a full decomposition $d\\in \\D(V)$ with a Levi sphere, then the condition $\\sigma \\prec d$ becomes ``$\\sigma$ and the Levi sphere $d$ lie in a common apartment\".\nTherefore, for a spherical building $\\Delta$, we propose:\n\n\\begin{itemize}\n \\item ${\\rm CB}(\\Delta) = \\{\\sigma \\tq \\sigma$ is a subset of the vertex set of some apartment of $\\Delta\\}$ (Definition \\ref{def:commonBasis}) \n \\item $\\D(\\Delta) = $ poset of non-empty Levi spheres ordered by reverse inclusion (Definition \\ref{def:decompBuilding}).\n \\item $\\PD(\\Delta) = \\Delta\\cup \\D(\\Delta)$, with crossed-term ordering $\\sigma\\prec S$ if there is an apartment containing the Levi sphere $S$ and the simplex $\\sigma$ (Definition \\ref{def:PDbuildings}).\n \\item $\\OD(\\Delta) = \\{(\\sigma_1,\\sigma_2)\\tq \\sigma_1,\\sigma_2$ are non-empty opposite simplices of $\\Delta\\}$, with coordinate-wise reverse-inclusion ordering (Definition \\ref{def:ODandOPDbuildings}).\n \\item $\\OPD(\\Delta) = \\Delta \\cup \\OD(\\Delta)$, with crossed-term ordering $\\sigma\\prec (\\sigma_1,\\sigma_2)$ if there is an apartment containing $\\sigma,\\sigma_1,\\sigma_2$ (Definition \\ref{def:ODandOPDbuildings}).\n\\end{itemize}\n\nThe main theorems of the paper are the following:", + "context": "In \\cite{HHS}, P. Hanlon, P. Hersh and J. Shareshian proved that $\\PD(V)$ is Cohen-Macaulay of dimension $2n-3$ if $k$ is a finite field.\nIn particular, $\\PD(V)$ is spherical (see Section \\ref{sec:preliminaries} for definitions).\nMore recently, in \\cite{BPW24}, it was proved that for any field $k$,\\,$\\PD(V)$ has the homotopy type of the common bases complex ${\\rm CB}(V)$ of $V$, as defined by J. Rognes \\cite{Rognes}.\nConcretely, ${\\rm CB}(V)$ is the simplicial complex whose simplices are sets $\\sigma$ of non-zero proper subspaces of $V$ for which there exists a basis $B$ of $V$ that each subspace in $\\sigma$ is generated by some subset of $B$.\nIn this case, we say that the elements of $\\sigma$ have a common basis.\nRognes showed that, although ${\\rm CB}(V)$ has dimension $2^n-3$, its homology is concentrated in degrees $\\leq 2n-3$, and he conjectured that ${\\rm CB}(V)$ must indeed be $(2n-4)$-connected.\nIn \\cite{MPW}, J. Miller, P. Patzt and J. Wilson established Rognes' conjecture by showing that there is a continuous map from a $(2n-4)$-connected simplicial complex to ${\\rm CB}(V)$ that induces an isomorphism on homotopy groups up to degree $2n-4$.\nIn particular, via \\cite{BPW24} this implies that $\\PD(V)$ is spherical for any field.\n\nConsidering the ordered version $\\OPD(V)$ of $\\PD(V)$ \nappears to be equally natural.\nBy definition, $\\OPD(V)$ is the poset whose elements are tuples \n$(V_1,\\ldots,V_r)$, $r\\geq 1$, of distinct subspaces of $V$ such that $\\{V_1,\\ldots,V_r\\} \\in \\PD(V)$.\nThe ordering in $\\OPD(V)$ is given by refinement consistent with the ordering of the tuples (see Definition \\ref{def:ODandOPDVectorSpaces}).\nWe call the elements of $\\OPD(V)$ ordered partial decompositions.\nIt follows from \\cite[Theorem 6.8]{PW25} that $\\OPD(V)$ is homotopy equivalent to a wedge of spheres of dimension $2n-3$.\nIf $V$ is a finite vector space then \\cite[Theorem 6.10]{PW25} implies \nthat $\\OPD(V)$ has the homotopy type of the two-fold join $\\Delta * \\Delta$ of the Tits building $\\Delta$ of $V$.\nUsing GAP we also verified in some small examples that the top homology group of\n$\\OPD(V)$, as a $\\GL(V)$-module, is the tensor square of the Steinberg module.\n\nLater, we found an alternative approach to determining the \nhomotopy type of $\\OPD(V)$ using closely related constructions, which inspired the work in this paper.\nLet $\\OD(V)$ denote the subposet of $\\OPD(V)$ consisting of full ordered decompositions $(V_1,\\ldots, V_r)$; i.e., those decompositions satisfying\n$V \\cong V_1\\oplus \\cdots \\oplus V_r$.\nLet $X = \\Delta \\cup \\OD(V)$ be the poset on the disjoint union of \n$\\Delta$ (regarded as a poset without the empty simplex) and $\\OD(V)$ with the following\nordering.\nWe keep the inclusion ordering among elements of $\\Delta$, and the refinement ordering in $\\OD(V)$. No element of $\\OD(V)$ lies below any element of $\\Delta$. If $\\sigma\\in \\Delta$ and $d\\in \\OD(V)$, we set $\\sigma \\prec d$ if and only if there is a basis of $V$ containing a basis for $d$ and a basis for every subspace appearing in $\\sigma$.\nIn Theorem \\ref{thm:PDandPDBuildingVectorSpace} we show that\n$\\OPD(V) \\simeq X$.\nThe poset $X$ in turn is homotopy equivalent to the complex $T^{1,1}(V)$ used in \\cite{MPW}, which is shown therein to have\nthe homotopy type of $\\Delta * \\Delta$.\nThis again allows us to conclude that $\\OPD(V) \\simeq \\Delta * \\Delta$ for any finite-dimensional vector space $V$.\nSince all homotopy equivalences can be seen to be $\\GL(V)$-equivariant, it now follows that the top homology group of\n$\\OPD(V)$ carries the tensor square of the Steinberg representation.\n\nThese observations suggest that there may be a more general phenomenon in the context of algebraic groups or even spherical buildings.\nIn this paper, we study the following questions and answer all of them positively.\nLet $\\Delta$ be a (spherical) building.\n\\begin{enumerate}\n \\item Can we define a poset $\\OPD(\\Delta)$ that, up to homotopy, coincides with $\\OPD(V)$ when $\\Delta$ is the building of the vector space $V$?\n \\item Can we similarly define $\\PD(\\Delta)$? Is it highly connected?\n \\item Do we have a common basis complex ${\\rm CB}(\\Delta)$ for $\\Delta$ that coincides with ${\\rm CB}(V)$ when $\\Delta$ is the building of the vector space $V$?\n \\item If questions (2) and (3) have a positive answer, do we have ${\\rm CB}(\\Delta) \\simeq \\PD(\\Delta)$?\n \\item Is $\\OPD(\\Delta)\\simeq \\Delta * \\Delta$?\n\\end{enumerate}\n\n\\begin{itemize}\n \\item ${\\rm CB}(\\Delta) = \\{\\sigma \\tq \\sigma$ is a subset of the vertex set of some apartment of $\\Delta\\}$ (Definition \\ref{def:commonBasis}) \n \\item $\\D(\\Delta) = $ poset of non-empty Levi spheres ordered by reverse inclusion (Definition \\ref{def:decompBuilding}).\n \\item $\\PD(\\Delta) = \\Delta\\cup \\D(\\Delta)$, with crossed-term ordering $\\sigma\\prec S$ if there is an apartment containing the Levi sphere $S$ and the simplex $\\sigma$ (Definition \\ref{def:PDbuildings}).\n \\item $\\OD(\\Delta) = \\{(\\sigma_1,\\sigma_2)\\tq \\sigma_1,\\sigma_2$ are non-empty opposite simplices of $\\Delta\\}$, with coordinate-wise reverse-inclusion ordering (Definition \\ref{def:ODandOPDbuildings}).\n \\item $\\OPD(\\Delta) = \\Delta \\cup \\OD(\\Delta)$, with crossed-term ordering $\\sigma\\prec (\\sigma_1,\\sigma_2)$ if there is an apartment containing $\\sigma,\\sigma_1,\\sigma_2$ (Definition \\ref{def:ODandOPDbuildings}).\n\\end{itemize}\n\nThe main theorems of the paper are the following:", + "full_context": "In \\cite{HHS}, P. Hanlon, P. Hersh and J. Shareshian proved that $\\PD(V)$ is Cohen-Macaulay of dimension $2n-3$ if $k$ is a finite field.\nIn particular, $\\PD(V)$ is spherical (see Section \\ref{sec:preliminaries} for definitions).\nMore recently, in \\cite{BPW24}, it was proved that for any field $k$,\\,$\\PD(V)$ has the homotopy type of the common bases complex ${\\rm CB}(V)$ of $V$, as defined by J. Rognes \\cite{Rognes}.\nConcretely, ${\\rm CB}(V)$ is the simplicial complex whose simplices are sets $\\sigma$ of non-zero proper subspaces of $V$ for which there exists a basis $B$ of $V$ that each subspace in $\\sigma$ is generated by some subset of $B$.\nIn this case, we say that the elements of $\\sigma$ have a common basis.\nRognes showed that, although ${\\rm CB}(V)$ has dimension $2^n-3$, its homology is concentrated in degrees $\\leq 2n-3$, and he conjectured that ${\\rm CB}(V)$ must indeed be $(2n-4)$-connected.\nIn \\cite{MPW}, J. Miller, P. Patzt and J. Wilson established Rognes' conjecture by showing that there is a continuous map from a $(2n-4)$-connected simplicial complex to ${\\rm CB}(V)$ that induces an isomorphism on homotopy groups up to degree $2n-4$.\nIn particular, via \\cite{BPW24} this implies that $\\PD(V)$ is spherical for any field.\n\nConsidering the ordered version $\\OPD(V)$ of $\\PD(V)$ \nappears to be equally natural.\nBy definition, $\\OPD(V)$ is the poset whose elements are tuples \n$(V_1,\\ldots,V_r)$, $r\\geq 1$, of distinct subspaces of $V$ such that $\\{V_1,\\ldots,V_r\\} \\in \\PD(V)$.\nThe ordering in $\\OPD(V)$ is given by refinement consistent with the ordering of the tuples (see Definition \\ref{def:ODandOPDVectorSpaces}).\nWe call the elements of $\\OPD(V)$ ordered partial decompositions.\nIt follows from \\cite[Theorem 6.8]{PW25} that $\\OPD(V)$ is homotopy equivalent to a wedge of spheres of dimension $2n-3$.\nIf $V$ is a finite vector space then \\cite[Theorem 6.10]{PW25} implies \nthat $\\OPD(V)$ has the homotopy type of the two-fold join $\\Delta * \\Delta$ of the Tits building $\\Delta$ of $V$.\nUsing GAP we also verified in some small examples that the top homology group of\n$\\OPD(V)$, as a $\\GL(V)$-module, is the tensor square of the Steinberg module.\n\nLater, we found an alternative approach to determining the \nhomotopy type of $\\OPD(V)$ using closely related constructions, which inspired the work in this paper.\nLet $\\OD(V)$ denote the subposet of $\\OPD(V)$ consisting of full ordered decompositions $(V_1,\\ldots, V_r)$; i.e., those decompositions satisfying\n$V \\cong V_1\\oplus \\cdots \\oplus V_r$.\nLet $X = \\Delta \\cup \\OD(V)$ be the poset on the disjoint union of \n$\\Delta$ (regarded as a poset without the empty simplex) and $\\OD(V)$ with the following\nordering.\nWe keep the inclusion ordering among elements of $\\Delta$, and the refinement ordering in $\\OD(V)$. No element of $\\OD(V)$ lies below any element of $\\Delta$. If $\\sigma\\in \\Delta$ and $d\\in \\OD(V)$, we set $\\sigma \\prec d$ if and only if there is a basis of $V$ containing a basis for $d$ and a basis for every subspace appearing in $\\sigma$.\nIn Theorem \\ref{thm:PDandPDBuildingVectorSpace} we show that\n$\\OPD(V) \\simeq X$.\nThe poset $X$ in turn is homotopy equivalent to the complex $T^{1,1}(V)$ used in \\cite{MPW}, which is shown therein to have\nthe homotopy type of $\\Delta * \\Delta$.\nThis again allows us to conclude that $\\OPD(V) \\simeq \\Delta * \\Delta$ for any finite-dimensional vector space $V$.\nSince all homotopy equivalences can be seen to be $\\GL(V)$-equivariant, it now follows that the top homology group of\n$\\OPD(V)$ carries the tensor square of the Steinberg representation.\n\nThese observations suggest that there may be a more general phenomenon in the context of algebraic groups or even spherical buildings.\nIn this paper, we study the following questions and answer all of them positively.\nLet $\\Delta$ be a (spherical) building.\n\\begin{enumerate}\n \\item Can we define a poset $\\OPD(\\Delta)$ that, up to homotopy, coincides with $\\OPD(V)$ when $\\Delta$ is the building of the vector space $V$?\n \\item Can we similarly define $\\PD(\\Delta)$? Is it highly connected?\n \\item Do we have a common basis complex ${\\rm CB}(\\Delta)$ for $\\Delta$ that coincides with ${\\rm CB}(V)$ when $\\Delta$ is the building of the vector space $V$?\n \\item If questions (2) and (3) have a positive answer, do we have ${\\rm CB}(\\Delta) \\simeq \\PD(\\Delta)$?\n \\item Is $\\OPD(\\Delta)\\simeq \\Delta * \\Delta$?\n\\end{enumerate}\n\n\\begin{itemize}\n \\item ${\\rm CB}(\\Delta) = \\{\\sigma \\tq \\sigma$ is a subset of the vertex set of some apartment of $\\Delta\\}$ (Definition \\ref{def:commonBasis}) \n \\item $\\D(\\Delta) = $ poset of non-empty Levi spheres ordered by reverse inclusion (Definition \\ref{def:decompBuilding}).\n \\item $\\PD(\\Delta) = \\Delta\\cup \\D(\\Delta)$, with crossed-term ordering $\\sigma\\prec S$ if there is an apartment containing the Levi sphere $S$ and the simplex $\\sigma$ (Definition \\ref{def:PDbuildings}).\n \\item $\\OD(\\Delta) = \\{(\\sigma_1,\\sigma_2)\\tq \\sigma_1,\\sigma_2$ are non-empty opposite simplices of $\\Delta\\}$, with coordinate-wise reverse-inclusion ordering (Definition \\ref{def:ODandOPDbuildings}).\n \\item $\\OPD(\\Delta) = \\Delta \\cup \\OD(\\Delta)$, with crossed-term ordering $\\sigma\\prec (\\sigma_1,\\sigma_2)$ if there is an apartment containing $\\sigma,\\sigma_1,\\sigma_2$ (Definition \\ref{def:ODandOPDbuildings}).\n\\end{itemize}\n\nThe main theorems of the paper are the following:\n\nLater, we found an alternative approach to determining the \nhomotopy type of $\\OPD(V)$ using closely related constructions, which inspired the work in this paper.\nLet $\\OD(V)$ denote the subposet of $\\OPD(V)$ consisting of full ordered decompositions $(V_1,\\ldots, V_r)$; i.e., those decompositions satisfying\n$V \\cong V_1\\oplus \\cdots \\oplus V_r$.\nLet $X = \\Delta \\cup \\OD(V)$ be the poset on the disjoint union of \n$\\Delta$ (regarded as a poset without the empty simplex) and $\\OD(V)$ with the following\nordering.\nWe keep the inclusion ordering among elements of $\\Delta$, and the refinement ordering in $\\OD(V)$. No element of $\\OD(V)$ lies below any element of $\\Delta$. If $\\sigma\\in \\Delta$ and $d\\in \\OD(V)$, we set $\\sigma \\prec d$ if and only if there is a basis of $V$ containing a basis for $d$ and a basis for every subspace appearing in $\\sigma$.\nIn Theorem \\ref{thm:PDandPDBuildingVectorSpace} we show that\n$\\OPD(V) \\simeq X$.\nThe poset $X$ in turn is homotopy equivalent to the complex $T^{1,1}(V)$ used in \\cite{MPW}, which is shown therein to have\nthe homotopy type of $\\Delta * \\Delta$.\nThis again allows us to conclude that $\\OPD(V) \\simeq \\Delta * \\Delta$ for any finite-dimensional vector space $V$.\nSince all homotopy equivalences can be seen to be $\\GL(V)$-equivariant, it now follows that the top homology group of\n$\\OPD(V)$ carries the tensor square of the Steinberg representation.\n\nThe main theorems of the paper are the following:\n\nHere, $\\CB(\\Delta)$ and $\\Delta$ are regarded as posets via their face posets, and the superscript $H$ means that we are taking the $H$-fixed point subposet.\n\nIf $\\Delta$ is the building of a finite-dimensional vector space $V$, then $\\CB(V) = \\CB(\\Delta)$, $\\PD(\\Delta)\\simeq \\PD(V)$ and $\\OPD(\\Delta)\\simeq \\OPD(V)$.\nIn particular, our results recover the connectivity result --- i.e., Rognes's conjecture --- for the common bases complex of a vector space, proved first in \\cite{MPW}.\nOur proof is independent of that of \\cite{MPW}, and it is based on geometric properties of buildings.\nTo prove our homotopy equivalences, we use the notion of convexity and complete reducibility introduced by Serre.\nKey ingredients in our proofs are statements of the form ``a certain (convex) subcomplex $K$ of the building is contractible.\" For that, we show that $K$ is not completely reducible, that is, there is a simplex in $K$ without an opposite in $K$.\nThis implies that $K$ is contractible by \\cite{Serre}.\nWe extend this idea to prove that certain fixed-point subposets are contractible.\n\n\\begin{proof}\nFirst, we show that $\\Gamma$ is well-defined.\nNote that if $c\\in \\PD(\\Delta)'$, then there is an apartment $\\Sigma$ containing all the vertices involved in $c$ by definition of $\\PD(\\Delta)$.\nIn particular, the convex hull of $c$ is a convex subcomplex of $\\Sigma$, so it lies in $Y(\\Delta)$.\nIf in addition the maximal element of $c$, say $x$, satisfies $x\\prec S$, we can take $\\Sigma$ containing $S$ by definition of the ordering in $\\PD(\\Delta)$, and hence $\\Gamma(c) = \\Conv_{\\Delta}(c,S)\\in Y(\\Delta)_{\\supseteq S}$.\nThus $\\Gamma$ is an order-preserving map between posets.\nAlso, as pointed out at the end of Subsection \\ref{subsub:chambercomplexes}, $\\Gamma$ is $H$-equivariant.\n\nNext, we prove that $\\OPD(\\Delta)$ has the homotopy type of the simplicial join $\\Delta*\\Delta$, and that it carries the tensor-square Steinberg representation in homology.\nFor a poset $X$, recall that $X^{(i)}$ denotes the subposet of elements of height at most $i$.\nIf $X = \\X(K)$ where $K$ is a Cohen-Macaulay simplicial complex, then $X$ is Cohen-Macaulay as a poset.\nAlso, every rank selection of a Cohen-Macaulay poset is spherical (see Theorem 6.2 of \\cite{Baclawski}).\nIn particular, rank selections of the face poset of a (spherical) building are Cohen-Macaulay.\n\n\\begin{theorem}\n\\label{thm:OPDandDeltaDelta}\nLet $\\Delta$ be a spherical building of dimension $m$ and let $H$ be a group acting on $\\Delta$ by simplicial automorphisms.\nWrite $D_1 = \\X(\\Delta)$ and $D_2 = \\X(\\Delta)^{\\op}$.\nNote that $\\OPD(\\Delta) = D_1 \\cup \\OD(\\Delta)$.\n\nLet $\\phi:\\OPD(\\Delta)\\to D_1 \\ojoin D_2$ be the following map:\n\\begin{align*}\n \\phi(\\sigma,\\sigma') & = \\sigma \\in D_2 \\quad \\text{ if } (\\sigma,\\sigma')\\in \\OD(\\Delta) \\subset \\OPD(\\Delta),\\\\\n \\phi(\\sigma) & = \\sigma \\in D_1 \\quad \\text{ if } \\sigma\\in D_1 \\subset \\OPD(\\Delta).\n\\end{align*}\nThen $\\phi$ is an $H$-equivariant order-preserving map.\nMoreover, the following hold:\n\\begin{enumerate}\n\\item The restriction\n\\[ \\phi_H: (\\OPD(\\Delta)^{(i+m+1)})^H \\to (D_1 \\ojoin (D_2^{(i)}))^H\\]\nis a homotopy equivalence for all $0 \\leq i \\leq m$.\nThus, $\\OPD(\\Delta)^{(i+m+1)} \\simeq D_1 \\ojoin D_2^{(i)}$ is spherical of dimension $m + i + 1$.\n\\item In particular, $\\phi$ gives rise to a homotopy equivalence\n\\[\\OPD(\\Delta) \\simeq \\Delta * \\Delta,\\]\nwhich is $H$-equivariant if $H$ is a compact Lie group, and to an $H$-equivariant isomorphism\n\\[ \\widetilde{H}_{*}(\\OPD(\\Delta)) \\cong_H \\widetilde{H}_*(\\Delta*\\Delta).\\]\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{theorem}\n\\label{thm:PDandPDBuildingVectorSpace}\nLet $V$ be an $n$-dimensional vector space over a field $k$.\nLet $\\sigma \\in \\OPD(V)'$, and write $\\sigma = \\sigma_0\\cup \\sigma_1$ where $\\sigma_0$ is the set of ordered partial decompositions in $\\sigma$ that span a proper subspace of $V$, and $\\sigma_1$ is the set of full ordered decompositions in $\\sigma$.\nThen, for $\\sigma \\in \\OPD(V)'$, we set\n\\[ \\phi(\\sigma) = \\begin{cases}\n\\{ \\gen{s} \\tq s\\in \\sigma_0\\} & \\sigma_1 = \\emptyset,\\\\\n\\max \\sigma_1 & \\sigma_1\\neq\\emptyset.\n\\end{cases}\\]\nThen $\\phi$ is a $\\GL(V)$-equivariant poset map $\\phi: \\OPD(V)' \\to \\OKtwo{V}$ such that for all $H\\leq \\GL(V)$, the induced map $\\phi_H: (\\OPD(V)')^H \\to \\OKtwo{V}^H$ is a homotopy equivalence.\n\nWe get analogous conclusions for the corresponding map $\\PD(V)'\\to \\Ktwo{V}$.\n\\end{theorem}", + "post_theorem_intro_text_len": 5706, + "post_theorem_intro_text": "Here, ${\\rm CB}(\\Delta)$ and $\\Delta$ are regarded as posets via their face posets, and the superscript $H$ means that we are taking the $H$-fixed point subposet.\n\nThere is a canonical map $F:\\OD(\\Delta)\\to \\D(\\Delta)$ that maps a pair of opposite simplices $(\\sigma_1,\\sigma_2)\\in\\OD(\\Delta)$ to the Levi sphere spanned by $\\sigma_1,\\sigma_2$ (which is actually their convex hull, see Subsection \\ref{sub:convexity}).\n\nIn \\cite{vH}, A. von Heydebreck showed that $\\OD(\\Delta)$ is Cohen-Macaulay of dimension $\\dim\\Delta$.\nUsing the map $F$, we conclude:\n\n\\begin{corollary}\n\\label{coro:DDeltaCM}\nLet $\\Delta$ be a spherical building.\nThen the poset of Levi spheres $\\D(\\Delta)$ is Cohen-Macaulay of dimension $\\dim \\Delta$.\n\\end{corollary}\n\nWe mention that Corollary \\ref{coro:DDeltaCM} can potentially be applied to compute the homotopy type of Quillen's $p$-subgroup posets \\cite{Qui78} of connected reductive algebraic groups $G$ over finite fields $\\GF{q}$, where $p$ is a ``good\" prime for which $q$ has order $1$ modulo $p$.\nThis is based on the work by D. Rossi \\cite{Rossi}, where it is proved that for such a good prime $p$, the $p$-subgroup poset of the finite group $G(\\GF{q})$ has the homotopy type of the poset of $\\GF{q}$-Levi subgroups, which coincides with the poset $\\D(\\Delta(G,\\GF{q}))$.\nHere $\\Delta(G,\\GF{q})$ is the building of $G(\\GF{q})$.\n\nWe will write $|S|$ to denote the geometric realization of the Levi sphere $S$, which is a sphere of dimension $\\dim S$ (see Lemma \\ref{lm:convexIsleviSphere}).\n\nThe map $F$ naturally extends to an order-preserving map $F:\\OPD(\\Delta)\\to \\PD(\\Delta)$ as the identity on the $\\Delta$-part.\nStudying fibers of this map, we prove:\n\n\\begin{theorem}\nLet $\\Delta$ be a spherical building.\nThen we have a homotopy equivalence\n\\begin{equation}\n\\OPD(\\Delta) \\simeq \\PD(\\Delta) \\bigvee_{ T \\in {\\D(\\Delta)}} |T| * |T|*\\PD(\\Lk_{\\Delta}(\\sigma_T)),\n\\end{equation}\nwhere $\\sigma_T\\in T$ is some maximal simplex.\nIn particular, $\\PD(\\Delta)$ is spherical of dimension $2\\dim\\Delta + 1$, and ${\\rm CB}(\\Delta)$ has the homotopy type of a wedge of spheres of dimension $2\\dim\\Delta + 1$.\n\nMoreover, for a Levi sphere $S$ and a pair of opposite maximal simplices $\\sigma_1,\\sigma_2\\in S$, we have\n\\begin{align*}\n\\PD(\\Delta)_{\\prec S} & \\simeq |S| * \\PD(\\Lk_{\\Delta}(\\sigma_1))\\\\\n\\OPD(\\Delta)_{\\prec (\\sigma_1,\\sigma_2)} & \\simeq |S| * \\OPD(\\Lk_{\\Delta}(\\sigma_1)). \n\\end{align*}\n\\end{theorem}\n\nHere we are using the bar notation for Levi spheres specifically to distinguish between $S$ as a sphere (via $|S|$) and $S$ as an element of the poset $\\D(\\Delta)$.\n\nIf $\\Delta$ is the building of a finite-dimensional vector space $V$, then ${\\rm CB}(V) = {\\rm CB}(\\Delta)$, $\\PD(\\Delta)\\simeq \\PD(V)$ and $\\OPD(\\Delta)\\simeq \\OPD(V)$.\nIn particular, our results recover the connectivity result --- i.e., Rognes's conjecture --- for the common bases complex of a vector space, proved first in \\cite{MPW}.\nOur proof is independent of that of \\cite{MPW}, and it is based on geometric properties of buildings.\nTo prove our homotopy equivalences, we use the notion of convexity and complete reducibility introduced by Serre.\nKey ingredients in our proofs are statements of the form ``a certain (convex) subcomplex $K$ of the building is contractible.\" For that, we show that $K$ is not completely reducible, that is, there is a simplex in $K$ without an opposite in $K$.\nThis implies that $K$ is contractible by \\cite{Serre}.\nWe extend this idea to prove that certain fixed-point subposets are contractible.\n\n\\subsection*{Organization of the article}\n\nThe paper is organized as follows.\nIn Section \\ref{sec:preliminaries} we introduce the notation from algebra, combinatorics, geometry, and topology that we will use throughout the paper. We also recall results that will play an important role in the rest of the paper. In particular, we outline the results of the theory of Tits buildings, mostly following \\cite{AB} and sometimes \\cite{Tits1}.\nIn Section \\ref{sec:vectorspaces}, we study the motivating type A situation.\nMore precisely, we give exact definitions of the ordered and unordered (partial) decomposition posets and the common basis complex for vector spaces. We also provide the proof that the ordered partial vector space decompositions are equivariantly homotopy equivalent to the poset of ordered partial decompositions of the corresponding building.\n\nIn Sections \\ref{sec:CBBuildings}, \\ref{sec:leviSpheres} and \\ref{sec:opd}, we define common basis complexes, ordered and unordered decomposition, and partial decomposition posets for arbitrary buildings.\nWe provide results about homotopy types and homotopy equivalences, including equivariant versions and fixed points. Some of them extend\nfacts from the linear case discussed in\nSection \\ref{sec:vectorspaces}. \n\nFinally, in Section \\ref{sec:algebraicgroups} we specialize these results to the case of rational points of connected reductive algebraic groups, obtaining explicit descriptions of the posets and complexes in terms of rational parabolics and Levi subgroups.\nIn particular, as an application of the equivariance of our maps and the connectivity results, in Proposition \\ref{prop:lowLeviIntervalPD} we provide a long exact sequence in terms of Steinberg-square modules of rational Levi subgroups for the rational points of a connected reductive algebraic group, ending in the Steinberg module of such a group.\n\nComputer calculations were performed with GAP \\cite{GAP4}\nand software package \\cite{posets}.\n\n\\subsection*{Acknowledgments}\nWe thank Bernhard Mühlherr and Richard Weiss for helpful and motivating conversations.\nThe first author was supported by the FWO grant 12K1223N.", + "sketch": "To prove the stated homotopy equivalences, the authors say they “use the notion of convexity and complete reducibility introduced by Serre.” The “key ingredients” are statements that “a certain (convex) subcomplex $K$ of the building is contractible.” Their method is: “we show that $K$ is not completely reducible, that is, there is a simplex in $K$ without an opposite in $K$,” and then “this implies that $K$ is contractible by \\cite{Serre}.” They also note that they “extend this idea to prove that certain fixed-point subposets are contractible.”\n\nThey further indicate a structural approach via the canonical map $F:\\OD(\\Delta)\\to \\D(\\Delta)$ (sending an opposite pair $(\\sigma_1,\\sigma_2)$ to the Levi sphere they span), which “naturally extends to an order-preserving map $F:\\OPD(\\Delta)\\to \\PD(\\Delta)$ as the identity on the $\\Delta$-part,” and that “studying fibers of this map” yields the wedge-type homotopy decomposition stated in the later theorem.", + "expanded_sketch": "To prove the stated homotopy equivalences, the authors say they “use the notion of convexity and complete reducibility introduced by Serre.” The “key ingredients” are statements that “a certain (convex) subcomplex $K$ of the building is contractible.” Their method is: “we show that $K$ is not completely reducible, that is, there is a simplex in $K$ without an opposite in $K$,” and then “this implies that $K$ is contractible by \\cite{Serre}.” They also note that they “extend this idea to prove that certain fixed-point subposets are contractible.”\n\nThey further indicate a structural approach via the canonical map $F:\\OD(\\Delta)\\to \\D(\\Delta)$ (sending an opposite pair $(\\sigma_1,\\sigma_2)$ to the Levi sphere they span), which “naturally extends to an order-preserving map $F:\\OPD(\\Delta)\\to \\PD(\\Delta)$ as the identity on the $\\Delta$-part,” and that “studying fibers of this map” yields the wedge-type homotopy decomposition stated in the later theorem.", + "expanded_theorem": "Let $\\Delta$ be a spherical building.\nThere are order-preserving maps\n\\[ \\Gamma: \\mathrm{sd} \\PD(\\Delta) \\to {\\rm CB}(\\Delta),\\]\n\\[ \\phi :\\OPD(\\Delta) \\to \\Delta * \\Delta,\\]\nsuch that for every group $H$ acting on $\\Delta$ by simplicial automorphisms, $\\Gamma$ and $\\phi$ are $H$-equivariant and induce homotopy equivalences between the fixed point subposets:\n\\[ \\Gamma_H : \\PD(\\Delta)^H \\to {\\rm CB}(\\Delta)^H,\\]\n\\[ \\phi_H :\\OPD(\\Delta)^H \\to \\Delta^H * \\Delta^H.\\]\nIn particular, $\\Gamma$ and $\\phi$ induce homotopy equivalences and equivariant isomorphisms in (co)homology.", + "theorem_type": "unknown", + "mcq": { + "question": "Let \\(\\Delta\\) be a spherical building. Define the common-basis complex \\({\\rm CB}(\\Delta)\\) to be the simplicial complex whose simplices are subsets of the vertex set of some apartment of \\(\\Delta\\). Let \\(\\mathcal D(\\Delta)\\) be the poset of nonempty Levi spheres ordered by reverse inclusion, and let \\(\\mathrm{PD}(\\Delta)=\\Delta\\cup \\mathcal D(\\Delta)\\), with cross relation \\(\\sigma\\prec S\\) when there is an apartment containing both the simplex \\(\\sigma\\) and the Levi sphere \\(S\\). Let \\(\\mathrm{OPD}(\\Delta)\\) be the associated ordered partial-decomposition poset, let \\(\\mathrm{sd}\\,\\mathrm{PD}(\\Delta)\\) denote the barycentric subdivision, and let \\(\\Delta*\\Delta\\) be the simplicial join. For a group \\(H\\) acting on \\(\\Delta\\) by simplicial automorphisms, write \\(X^H\\) for the fixed-point subposet or subcomplex. Which statement is valid for these objects?", + "correct_choice": { + "label": "A", + "text": "There exist order-preserving maps \\(\\Gamma:\\mathrm{sd}\\,\\mathrm{PD}(\\Delta)\\to {\\rm CB}(\\Delta)\\) and \\(\\phi:\\mathrm{OPD}(\\Delta)\\to \\Delta*\\Delta\\) such that, for every group \\(H\\) acting on \\(\\Delta\\) by simplicial automorphisms, both \\(\\Gamma\\) and \\(\\phi\\) are \\(H\\)-equivariant and induce homotopy equivalences on fixed points, namely \\(\\Gamma_H:\\mathrm{PD}(\\Delta)^H\\to {\\rm CB}(\\Delta)^H\\) and \\(\\phi_H:\\mathrm{OPD}(\\Delta)^H\\to \\Delta^H*\\Delta^H\\). In particular, \\(\\Gamma\\) and \\(\\phi\\) themselves are homotopy equivalences and induce equivariant isomorphisms in homology and cohomology." + }, + "choices": [ + { + "label": "B", + "text": "There exist order-preserving maps \\(\\Gamma:\\mathrm{PD}(\\Delta)\\to {\\rm CB}(\\Delta)\\) and \\(\\phi:\\mathrm{OPD}(\\Delta)\\to \\Delta*\\Delta\\) such that, for every group \\(H\\) acting on \\(\\Delta\\) by simplicial automorphisms, both \\(\\Gamma\\) and \\(\\phi\\) are \\(H\\)-equivariant and induce homotopy equivalences on fixed points, namely \\(\\Gamma_H:\\mathrm{PD}(\\Delta)^H\\to {\\rm CB}(\\Delta)^H\\) and \\(\\phi_H:\\mathrm{OPD}(\\Delta)^H\\to \\Delta^H*\\Delta^H\\). In particular, \\(\\Gamma\\) and \\(\\phi\\) themselves are homotopy equivalences and induce equivariant isomorphisms in homology and cohomology." + }, + { + "label": "C", + "text": "There exist order-preserving maps \\(\\Gamma:\\mathrm{sd}\\,\\mathrm{PD}(\\Delta)\\to {\\rm CB}(\\Delta)\\) and \\(\\phi:\\mathrm{OPD}(\\Delta)\\to \\Delta*\\Delta\\) such that \\(\\Gamma\\) and \\(\\phi\\) are homotopy equivalences. In particular, they induce isomorphisms in homology and cohomology." + }, + { + "label": "D", + "text": "There exist order-preserving maps \\(\\Gamma:\\mathrm{sd}\\,\\mathrm{PD}(\\Delta)\\to {\\rm CB}(\\Delta)\\) and \\(\\phi:\\mathrm{OPD}(\\Delta)\\to \\Delta*\\Delta\\) such that, for every group \\(H\\) acting on \\(\\Delta\\) by simplicial automorphisms, both \\(\\Gamma\\) and \\(\\phi\\) are \\(H\\)-equivariant and induce homotopy equivalences on fixed points, namely \\(\\Gamma_H:\\mathrm{PD}(\\Delta)^H\\to {\\rm CB}(\\Delta)^H\\) and \\(\\phi_H:\\mathrm{OPD}(\\Delta)^H\\to (\\Delta*\\Delta)^H\\). In particular, \\(\\Gamma\\) and \\(\\phi\\) themselves are homotopy equivalences and induce equivariant isomorphisms in homology and cohomology." + }, + { + "label": "E", + "text": "There exist order-preserving maps \\(\\Gamma:\\mathrm{sd}\\,\\mathrm{PD}(\\Delta)\\to {\\rm CB}(\\Delta)\\) and \\(\\phi:\\mathrm{OPD}(\\Delta)\\to \\Delta*\\Delta\\) such that, for every group \\(H\\) acting on \\(\\Delta\\) by simplicial automorphisms, both \\(\\Gamma\\) and \\(\\phi\\) are \\(H\\)-equivariant and induce homotopy equivalences between the underlying spaces. Moreover, for every such \\(H\\), the fixed-point subposets \\(\\mathrm{PD}(\\Delta)^H\\) and \\(\\mathrm{OPD}(\\Delta)^H\\) are contractible." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "barycentric-subdivision requirement for the PD-to-CB map", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "fixed-point equivariance and fixed-point-level homotopy equivalences for every H", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "characteristic", + "tampered_component": "identification of the fixed-point target as \\(\\Delta^H*\\Delta^H\\), not \\((\\Delta*\\Delta)^H\\)", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "contractibility applies only to certain convex fibers/fixed-point subposets used in the proof, not to all fixed-point subposets themselves", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives only definitions and notation; it does not state the theorem’s conclusion or uniquely signal choice A. The correct answer is not leaked explicitly or by an obvious cue." + }, + "TAS": { + "score": 2, + "justification": "This is not a direct restatement of information already given in the stem. The respondent must distinguish between several closely related technical claims, so the item is non-tautological." + }, + "GPS": { + "score": 1, + "justification": "The item requires some reasoning or precise theorem-level recognition to separate subtle variants (e.g., barycentric subdivision, fixed points of joins, strength of equivariant claims). However, it mainly tests recall/discrimination of a known result rather than substantial generative mathematical construction." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong: they are plausible, mathematically nearby, and reflect realistic failure modes such as omitting subdivision, weakening the theorem, confusing \\((\\Delta*\\Delta)^H\\) with \\(\\Delta^H*\\Delta^H\\), or overclaiming contractibility." + }, + "total_score": 7, + "overall_assessment": "A high-quality MCQ with no meaningful answer leakage and very good distractors. Its main limitation is that it tests precise theorem recognition more than deep generative reasoning." + } + }, + { + "id": "2511.21628v1", + "paper_link": "http://arxiv.org/abs/2511.21628v1", + "theorems_cnt": 7, + "theorem": { + "env_name": "theorem", + "content": "\\label{t.main}\n\tLet $n,s,\\ell,c$ be positive integers such that $n=2s+c=3s-\\ell$, and $c,\\ell\\in [s-1]$. Then $$e(n, s) = \\max\\big\\{|\\mathcal{P}(s, \\ell)|, |\\mathcal{P}'(\\ell)|, |\\mathcal{Q}(s, \\ell)|, |\\mathcal{W}(s, \\ell)|\\big\\}.$$ Moreover, if $s \\geq 3$, ${\\mathcal F}\\subset 2^{[n]}$ is shifted, has no $s$-matching and $|{\\mathcal F}|=e(n,s)$, then ${\\mathcal F}$ must coincide with one of the families above.", + "start_pos": 76309, + "end_pos": 76727, + "label": "t.main" + }, + "ref_dict": { + "eq008": "\\begin{equation}\\label{eq008} e_k(n,s) = \\max\\bigl\\{|\\aaa_1^{(k)}(n,s-1)|,|\\aaa_k^{(k)}(n,s-1)|\\bigr\\}.\n\\end{equation}", + "thmfk": "\\begin{thm}[\\cite{FK9,FK8}]\\label{thmfk} \\ $e(sm+s-\\ell,s) = |\\pp(s,m,\\ell)|$ holds for\n\\begin{align*} &\\mathrm{(i)}\\ \\ \\ \\ \\ell = 2, \\\\\n &\\mathrm{(ii)}\\ \\ \\ m=1,\\\\\n &\\mathrm{(iii)} \\ \\ s\\ge \\ell m+3\\ell+3.\n\\end{align*}\n\\end{thm}", + "fig2": "\\begin{figure}\n\n\\centering\n\n\\includegraphics[width=0.6\\linewidth]{extremal_families_2.png}\n\n\\caption{Extremal families for small $c,s$. Note that for $c=1, s=5$ there are $3$ different extremal families. For $\\ell = 1$, that is, $s = c + 1$, the family $\\mathcal{P}'$ coincides with $\\mathcal{P}$ (both are equal to ${n \\choose \\geq (m + 1)}$) and thus is formally extremal. When $\\mathcal{P}' \\neq \\mathcal{P}$, for $c \\leq 9$ $\\mathcal{P}'$ cannot be extremal.}\n\n\\label{fig2}\n\n\\end{figure}", + "eq002": "\\begin{align}\\label{eq001} e(sm-1,s) &= \\sum_{t=m}^{sm-1}{sm-1\\choose t},\\\\\n\\label{eq002} e(sm,s) &= {sm-1\\choose m}+\\sum_{t=m+1}^{sm}{sm\\choose t}.\n\\end{align}", + "conj1": "\\begin{gypo}[\\cite{FK9}]\\label{conj1}\nSuppose that $s\\ge 2, m\\ge 1$, and $n = sm+s-\\ell$ for some integer $0<\\ell\\le \\lceil \\frac s2\\rceil$. Then\n\\begin{equation}\\label{eq007} e(sm+s-\\ell,s) = |\\pp(s,m,\\ell)|.\n\\end{equation}\n\\end{gypo}", + "fig1": "\\begin{figure}\n\n\\centering\n\n\\includegraphics[width=0.8\\linewidth]{extremal_families.png}\n\n\\caption{Extremal families. $\\mathcal{W}$ is one of the extremal families for $c = 1, s \\geq 5$.}\n\n\\label{fig1}\n\n\\end{figure}" + }, + "pre_theorem_intro_text_len": 9223, + "pre_theorem_intro_text": "Let $[n] := \\{1,2,\\ldots, n\\}$ and, more generally, $[a,b]=\\{a,a+1,\\ldots, b\\}$. For a set $X$ and an integer $k$, let $2^{X}$, ${X\\choose k}$ and ${X\\choose \\geq k}$ stand for the power set of $X$, the set of its $k$-element subsets and the set of its subsets with size at least $k$, respectively. Any collection of sets is called a {\\it family.} A {\\it matching} is a collection of pairwise disjoint sets. An {\\it $s$-matching} is a matching of size $s$. Given a family ${\\mathcal F},$ its {\\it matching number}\n$\\nu(\\mathcal F)$ is the size of the largest matching in ${\\mathcal F}$.\n\nOne of the classical topics in extremal set theory is the study of {\\it intersecting} families, that is, families with matching number $1$. Erd\\H os, Ko and Rado~\\cite{EKR} showed that the largest intersecting family ${\\mathcal F}\\subset 2^{[n]}$ has size at most $2^{n-1}$, and that for $n\\ge 2k$ the largest intersecting family ${\\mathcal F}\\subset {[n]\\choose k}$ has size ${n-1\\choose k-1}.$ In the several years that followed, Erd\\H os asked for the size of the largest family avoiding an $s$-matching. Let us introduce the following two quantities.\n\\begin{align*}\n e(n,s)&=\\max\\big\\{|{\\mathcal F}|: {\\mathcal F}\\subset 2^{[n]}, \\nu({\\mathcal F})0$). We refer the reader to \\cite{aletal, FK21} for the connections of the Erd\\H os Matching Conjecture and other questions, such as Dirac thresholds and small deviations in probability theory. In \\cite{HLS}, \\cite{K49}, the multi-family variant of the EMC was addressed. In \\cite{FK6}, a Hilton--Milner type stability result for the EMC is obtained.\n\n\\subsection{The non-uniform case}\nThe study of $e(n,s)$ was also initiated by Erd\\H os at around the same time. The behavior of $e(n,s)$ heavily depends on $n\\ ({\\rm mod\\ } s)$. Answering a question of Erd\\H os, Kleitman proved the following theorem. \n\\begin{thm}[Kleitman \\cite{Kl}]\\label{thmkl}\n\\begin{align}\\label{eq001} e(sm-1,s) &= \\sum_{t=m}^{sm-1}{sm-1\\choose t},\\\\\n\\label{eq002} e(sm,s) &= {sm-1\\choose m}+\\sum_{t=m+1}^{sm}{sm\\choose t}.\n\\end{align}\n\\end{thm}\nThe matching example for the first case is the family ${[n]\\choose \\ge m}$ of all subsets of $[n]$ of size at least $m$. It is also not difficult to see that $e(sm,s) = 2e(sm-1,s)$. In general, $e(n+1,s)\\ge 2e(n,s)$ because of the {\\it doubling} construction. Given a family ${\\mathcal F}\\subset 2^{[n]}$ with $\\nu({\\mathcal F})\\min\\big\\{|\\overline{\\mathcal{P'}(s, \\ell)}|, |\\overline{\\mathcal{P}(s, \\ell)}|, |\\overline{\\mathcal{Q}(s, \\ell)}|, |\\overline{\\mathcal{W}(s, \\ell)}|\\big\\}.\n \\end{multline*}\n Moreover,\n \\begin{multline*}\n \\min\\big\\{|\\overline{\\mathcal{P'}(s-1, \\ell-2)}^{(\\leq 3)}|, |\\overline{\\mathcal{P}(s-1, \\ell-2)}^{(\\leq 3)}|, |\\overline{\\mathcal{Q}(s-1, \\ell-2)}^{(\\leq 3)}|, \\\\|\\overline{\\mathcal{W}(s-1, \\ell-2)}^{(\\leq 3)}|\\big\\} >\\min\\big\\{|\\overline{\\mathcal{P'}(s, \\ell)}^{(\\leq 3)}|, |\\overline{\\mathcal{P}(s, \\ell)}^{(\\leq 3)}|, |\\overline{\\mathcal{Q}(s, \\ell)}^{(\\leq 3)}|, |\\overline{\\mathcal{W}(s, \\ell)}^{(\\leq 3)}|\\big\\}.\n \\end{multline*}\n\\label{c.no_siggletons}\n\\end{restatable}\n\nThe case of odd $d$ requires a more careful analysis. In this case, we use the inequality \\eqref{eqd2c} which states that $d(\\mathcal{F}) \\leq 2c$. \n\\begin{lemma} \\label{l.odd_d_to_y2}\n Let $d$ be a positive odd integer, $d \\leq 2c$. If $\\mathcal{F} \\subset 2^{[n]}$ is a shifted family with $d(\\mathcal{F}) = d$, then {\\small \\begin{equation}\\label{eqy22}y(2) \\geq \\min\\Big\\{\\frac{(4\\ell+3c+d-2)(3c-d+1)}{2}, \\frac{(\\ell+3c-\\frac{d-1}{2})(\\ell+3c-\\frac{d+1}2)}{2}\\Big\\}.\\end{equation}} Moreover, equality is achieved only if $\\mathcal{F}^{(2)} = {[2\\ell+d-1] \\choose 2}$ or $\\mathcal{F}^{(2)} = \\{F \\in {n \\choose 2 }: F\\cap[\\ell+\\frac{d-1}{2}] \\neq \\emptyset\\}$.\n\\end{lemma}\n\n\\subsection{$c \\in \\{3, 4\\}$}\n\\begin{lemma} \\label{l.c_eq_3}\n Let $n = 2s+3$ and $\\mathcal{F} \\subset 2^{[n]}$ is a shifted up-set with $\\nu(\\mathcal{F}) < s$ and $\\mathcal{F} \\cap {[n] \\choose 1} = \\emptyset$. Then $|\\mathcal{F}| \\leq \\max(|\\mathcal{P}(s, \\ell)|, |\\mathcal{P}'(s, \\ell)|, |\\mathcal{Q}(s, \\ell)|)$. Moreover, equality is achieved only if $\\mathcal{F}$ is one of the families $\\mathcal{P}(s, \\ell), \\mathcal{P}'(s, \\ell), \\mathcal{Q}(s, \\ell)$.\n\\end{lemma}", + "post_theorem_intro_text_len": 2648, + "post_theorem_intro_text": "\\begin{figure}\n\n\\centering\n\n\\includegraphics[width=0.8\\linewidth]{extremal_families.png}\n\n\\caption{Extremal families. $\\mathcal{W}$ is one of the extremal families for $c = 1, s \\geq 5$.}\n\n\\label{fig1}\n\n\\end{figure}\n\n\\begin{figure}\n\n\\centering\n\n\\includegraphics[width=0.6\\linewidth]{extremal_families_2.png}\n\n\\caption{Extremal families for small $c,s$. Note that for $c=1, s=5$ there are $3$ different extremal families. For $\\ell = 1$, that is, $s = c + 1$, the family $\\mathcal{P}'$ coincides with $\\mathcal{P}$ (both are equal to ${n \\choose \\geq (m + 1)}$) and thus is formally extremal. When $\\mathcal{P}' \\neq \\mathcal{P}$, for $c \\leq 9$ $\\mathcal{P}'$ cannot be extremal.}\n\n\\label{fig2}\n\n\\end{figure}\nOn Figures~\\ref{fig1} and~\\ref{fig2} we show, which families are extremal for different regimes of the parameters $s,c$. For some values we get that three different families are extremal at the same time. \n\nWe define shifted families in the next section. We should note that actually there are rather natural examples of families with no $s$-matching interpolating between $\\mathcal{P}'(s, l)$ and $\\mathcal{Q}(s, l)$ in a somewhat similar way as $\\aaa_i$ interpolate between $\\aaa_0$ and $\\aaa_k$, but, as in the case of the EMC, there is a certain convexity that leads to the fact that it is the endpoints that must be extremal. \n\nIn the proof we will work only with sets of size $3$ or less. Therefore, any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family. We thus get the following theorem about the truncated boolean lattice, confirming a conjecture of Frankl and the first author \\cite{FK9} in our regime of the parameters. \n\n\\begin{theorem} \\label{t.truncated_lattice}\n Let $n,s,\\ell,c$ be positive integers such that $n=2s+c=3s-\\ell$, and $c,\\ell\\in [s-1]$. If $\\mathcal{F} \\subset {[n] \\choose \\leq 3}$ has no $s$-matching, then\n $$|\\mathcal{F}| \\leq \\max\\big\\{|\\mathcal{P}(s, \\ell)^{(\\leq 3)}|, |\\mathcal{P}'(s, \\ell)^{(\\leq 3)}|, |\\mathcal{Q}(s, \\ell)^{(\\leq 3)}|, |\\mathcal{W}(s, \\ell)^{(\\leq 3)}|\\big\\}.$$\n\\end{theorem}\n\nNote that a similar statement about $2$ first layers of boolean lattice is obviously false. Indeed, one of the families $\\aaa_1^{(2)}(n,s-1), \\aaa_2^{(2)}(n,s-1)$ has a larger cardinality than families $\\mathcal{P}(s, \\ell)^{(\\leq 2)}, \\mathcal{P}'(s, \\ell)^{(\\leq 2)}, \\mathcal{Q}(s, \\ell)^{(\\leq 2)}, \\mathcal{W}(s, \\ell)^{(\\leq 2)}$.\n\nIn Section~\\ref{sec2}, we prove several easy facts and make the necessary preparations for the proof of the main theorem. In Section~\\ref{sec3} we describe the strategy of the proof of the main theorem.", + "sketch": "To prove Theorem~\\ref{t.main}, the authors note that they will \"work only with sets of size $3$ or less.\" Hence, \"any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family.\" This reduction leads to an auxiliary result on the truncated Boolean lattice: Theorem~\\ref{t.truncated_lattice}, which bounds $|\\mathcal F\\subset {[n]\\choose \\le 3}|$ with no $s$-matching by the maximum of the truncated versions of the candidate extremal families $\\mathcal P,\\mathcal P',\\mathcal Q,\\mathcal W$. They also mention that while there are \"natural examples\" interpolating between $\\mathcal P'$ and $\\mathcal Q$, \"there is a certain convexity\" implying \"it is the endpoints that must be extremal.\" Finally, they indicate the paper structure: in Section~\\ref{sec2} they \"prove several easy facts and make the necessary preparations\" and in Section~\\ref{sec3} they \"describe the strategy of the proof of the main theorem.\"", + "expanded_sketch": "To prove the main theorem, the authors note that they will \"work only with sets of size $3$ or less.\" Hence, \"any family, avoiding $s$-matching, must miss at least as many sets of size $3$ or less, as the extremal family.\" This reduction leads to an auxiliary result on the truncated Boolean lattice: Theorem~\\ref{t.truncated_lattice}, which bounds $|\\mathcal F\\subset {[n]\\choose \\le 3}|$ with no $s$-matching by the maximum of the truncated versions of the candidate extremal families $\\mathcal P,\\mathcal P',\\mathcal Q,\\mathcal W$. They also mention that while there are \"natural examples\" interpolating between $\\mathcal P'$ and $\\mathcal Q$, \"there is a certain convexity\" implying \"it is the endpoints that must be extremal.\" Finally, they indicate the paper structure: next they \"prove several easy facts and make the necessary preparations\" and later they \"describe the strategy of the proof of the main theorem.\"", + "expanded_theorem": "\\label{t.main}\n\tLet $n,s,\\ell,c$ be positive integers such that $n=2s+c=3s-\\ell$, and $c,\\ell\\in [s-1]$. Then $$e(n, s) = \\max\\big\\{|\\mathcal{P}(s, \\ell)|, |\\mathcal{P}'(\\ell)|, |\\mathcal{Q}(s, \\ell)|, |\\mathcal{W}(s, \\ell)|\\big\\}.$$ Moreover, if $s \\geq 3$, ${\\mathcal F}\\subset 2^{[n]}$ is shifted, has no $s$-matching and $|{\\mathcal F}|=e(n,s)$, then ${\\mathcal F}$ must coincide with one of the families above.", + "theorem_type": [ + "Classification or Bijection", + "Equality or Bound" + ], + "mcq": { + "question": "Let [n] = {1,2,...,n}, and define e(n,s) = max{|F| : F ⊂ 2^[n] and ν(F) < s}, where ν(F) is the maximum size of a pairwise disjoint subfamily of F. A family F ⊂ 2^[n] is shifted if whenever A ∈ F, j ∈ A, i ∉ A, and 1 ≤ i < j ≤ n, then (A \\ {j}) ∪ {i} ∈ F. Suppose n, s, ℓ, c are positive integers satisfying n = 2s + c = 3s - ℓ with c, ℓ ∈ [s - 1]. Let P(s,ℓ) = {P ⊂ [n] : |P| + |P ∩ [ℓ - 1]| ≥ 3}, and let P'(s,ℓ), Q(s,ℓ), and W(s,ℓ) be the other three explicit candidate extremal families on 2^[n] from the same setup. Which statement is valid for e(n,s) and for shifted extremal families with no s-matching?", + "correct_choice": { + "label": "A", + "text": "e(n,s) = max{|P(s,ℓ)|, |P'(s,ℓ)|, |Q(s,ℓ)|, |W(s,ℓ)|}. Moreover, if s ≥ 3 and F ⊂ 2^[n] is shifted, satisfies ν(F) < s, and has |F| = e(n,s), then F must be exactly one of the four families P(s,ℓ), P'(s,ℓ), Q(s,ℓ), or W(s,ℓ)." + }, + "choices": [ + { + "label": "B", + "text": "e(n,s) = \\max\\{|P(s,\\ell)|, |P'(s,\\ell)|, |Q(s,\\ell)|, |W(s,\\ell)|\\}. Moreover, if s \\ge 3 and F \\subset 2^{[n]} is shifted, satisfies \\nu(F) < s, and has |F| = e(n,s), then F must be exactly one of the four families P(s,\\ell), P'(s,\\ell), Q(s,\\ell), or W(s,\\ell) up to changing its members of size at least 4." + }, + { + "label": "C", + "text": "e(n,s) \\ge \\max\\{|P(s,\\ell)|, |P'(s,\\ell)|, |Q(s,\\ell)|, |W(s,\\ell)|\\}. Moreover, if s \\ge 3, each of the four families P(s,\\ell), P'(s,\\ell), Q(s,\\ell), and W(s,\\ell) satisfies \\nu(F) < s." + }, + { + "label": "D", + "text": "e(n,s) = \\max\\{|P(s,\\ell)|, |P'(s,\\ell)|, |Q(s,\\ell)|, |W(s,\\ell)|\\}. Moreover, if s \\ge 3 and F \\subset 2^{[n]} satisfies \\nu(F) < s and has |F| = e(n,s), then F must be exactly one of the four families P(s,\\ell), P'(s,\\ell), Q(s,\\ell), or W(s,\\ell)." + }, + { + "label": "E", + "text": "e(n,s) = \\max\\{|P(s,\\ell)|, |P'(s,\\ell)|, |Q(s,\\ell)|, |W(s,\\ell)|\\}. Moreover, if s \\ge 3 and F \\subset 2^{[n]} is shifted, satisfies \\nu(F) < s, and has |F| = e(n,s), then F must be exactly one of the two endpoint families P'(s,\\ell) or Q(s,\\ell)." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "B" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "exact_global_uniqueness_from_truncated_reduction", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "finiteness", + "tampered_component": "replace_exact_maximum_and_classification_by_lower_bound_and_feasibility", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "shifted_hypothesis_in_uniqueness_clause", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "convexity_endpoints_only_as_global_extremals", + "template_used": "stronger_trap" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly reveal the correct option. It provides the setup and names the candidate families, but the exact maximality and classification statement still has to be selected from several close variants." + }, + "TAS": { + "score": 1, + "justification": "The item is fairly close to asking for the precise theorem statement from the setup, so it is partly a recognition/recollection task rather than a substantially reformulated problem. Still, the options introduce meaningful variations in strength and hypotheses." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish exact equality from a lower bound, shifted versus arbitrary families, and full classification versus endpoint-only or weakened uniqueness. However, it mainly tests precise theorem recall/verification rather than deeper generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and target realistic failure modes: dropping the shifted hypothesis, weakening equality to a lower bound, adding an unjustified 'up to large sets' caveat, and overrestricting extremals to endpoint families." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no direct answer leakage, but it leans more toward exact statement recognition than genuinely generative reasoning." + } + }, + { + "id": "2511.19410v1", + "paper_link": "http://arxiv.org/abs/2511.19410v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "theorem", + "content": "\\label{theorem:special}\n Let $g=2^d$ with $d \\geq 4$. Then there exists an abelian variety $A$ of dimension $g$ over $\\overline{\\mathbb{Q}(t)}$ with no power $A^n$ isogenous to a Jacobian.", + "start_pos": 20222, + "end_pos": 20444, + "label": "theorem:special" + }, + "ref_dict": { + "definition:nonsplit": "\\begin{definition}\\label{definition:nonsplit}\n Let $r \\geq 3$ be an integer. A \\emph{Shimura curve of Mumford type} in $\\Agr$ (see \\cite{viehwegzuo-2004}) is a proper Shimura curve $C \\subset \\Agr$ of the following form. Consider a polarized abelian scheme $B_1 \\to C_1$ as defined above. \nPick an étale cover $C_2 \\to C_1$ with pull-back $B_2 \\to C_2$, so that there is a principally polarized abelian scheme $A_2 \\to C_2$ isogenous to the polarized abelian scheme $B_2 \\to C_2$, such that $A_2 \\to C_2$ carries a symplectic level $r$ structure. This gives a morphism $C_2 \\to \\ca A_{g,[r]}$, and we let $C \\subset \\ca A_{g,[r]}$ denote its image. \n\nLet us say that the Shimura curve of Mumford type $C$ is \\emph{split} (resp.\\ \\emph{non-split}) if we are in case \\eqref{case:i} (resp.\\ \\eqref\n{case:ii}) of Lemma \\ref{lemma:cases}. \n\\end{definition}", + "theorem:special": "\\begin{theorem} \\label{theorem:special}\n Let $g=2^d$ with $d \\geq 4$. Then there exists an abelian variety $A$ of dimension $g$ over $\\overline{\\mathbb{Q}(t)}$ with no power $A^n$ isogenous to a Jacobian. \n\\end{theorem}", + "theorem:generic": "\\begin{theorem} \\label{theorem:generic}\nLet $g = 2^d \\geq 16$ and $N \\geq 1$. Then there exists a Hodge generic point $x = [A] \\in \\Ag(\\Qtbar)$ such that the corresponding abelian variety $A$ over $\\Qtbar$ has no power $A^n$ with $n \\leq N$ which is isogenous to a Jacobian.\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 1380, + "pre_theorem_intro_text": "Every abelian variety $A$ over an algebraically closed field is dominated by a Jacobian, but not always isogenous to one: over $\\mathbb{C}$ this follows from a countability argument, over $\\overline{\\Q}$ this was proven by Chai--Oort and Tsimerman \\cite{chaioort, tsimerman-jacobian}, and over $\\overline{\\F_p(t)}$ this was proven \nby the second named author and Tsimerman \n\\cite{shankartsimerman-Fptbar}.\\footnote{The case of finite fields is expected to be different as suggested by heuristics, independently offered by \\cite{shankar-tsimerman-heuristics} and Poonen.}\n\nThis leaves open the question whether some power of $A$ is isogenous to a Jacobian. \nIn \\cite{dGFSchreieder-2025}, the first named author and Schreieder \nshowed that the answer is again no in general: for any $g \\geq 4$, \nthere exists a \ncomplex abelian variety of dimension $g$ with no power isogenous to a Jacobian, and for $g=5$ one can find intermediate Jacobians of smooth cubic threefolds among such examples. The authors ask whether one can find \nabelian varieties \nover $\\overline{\\Q}$ with no power isogenous to a Jacobian, see \\cite[Remark 1.6]{dGFSchreieder-2025}. \n\nThe first main result of this paper says that if $g \\geq 16$ is a power of $2$, then such examples exist at least over $\\smash{\\overline{\\Q(t)}}$. By a Jacobian, we mean the Jacobian of a projective stable curve of compact type.", + "context": "Every abelian variety $A$ over an algebraically closed field is dominated by a Jacobian, but not always isogenous to one: over $\\mathbb{C}$ this follows from a countability argument, over $\\overline{\\Q}$ this was proven by Chai--Oort and Tsimerman \\cite{chaioort, tsimerman-jacobian}, and over $\\overline{\\F_p(t)}$ this was proven \nby the second named author and Tsimerman \n\\cite{shankartsimerman-Fptbar}.\\footnote{The case of finite fields is expected to be different as suggested by heuristics, independently offered by \\cite{shankar-tsimerman-heuristics} and Poonen.}\n\nThis leaves open the question whether some power of $A$ is isogenous to a Jacobian. \nIn \\cite{dGFSchreieder-2025}, the first named author and Schreieder \nshowed that the answer is again no in general: for any $g \\geq 4$, \nthere exists a \ncomplex abelian variety of dimension $g$ with no power isogenous to a Jacobian, and for $g=5$ one can find intermediate Jacobians of smooth cubic threefolds among such examples. The authors ask whether one can find \nabelian varieties \nover $\\overline{\\Q}$ with no power isogenous to a Jacobian, see \\cite[Remark 1.6]{dGFSchreieder-2025}.\n\nThe first main result of this paper says that if $g \\geq 16$ is a power of $2$, then such examples exist at least over $\\smash{\\overline{\\Q(t)}}$. By a Jacobian, we mean the Jacobian of a projective stable curve of compact type.", + "full_context": "Every abelian variety $A$ over an algebraically closed field is dominated by a Jacobian, but not always isogenous to one: over $\\mathbb{C}$ this follows from a countability argument, over $\\overline{\\Q}$ this was proven by Chai--Oort and Tsimerman \\cite{chaioort, tsimerman-jacobian}, and over $\\overline{\\F_p(t)}$ this was proven \nby the second named author and Tsimerman \n\\cite{shankartsimerman-Fptbar}.\\footnote{The case of finite fields is expected to be different as suggested by heuristics, independently offered by \\cite{shankar-tsimerman-heuristics} and Poonen.}\n\nThis leaves open the question whether some power of $A$ is isogenous to a Jacobian. \nIn \\cite{dGFSchreieder-2025}, the first named author and Schreieder \nshowed that the answer is again no in general: for any $g \\geq 4$, \nthere exists a \ncomplex abelian variety of dimension $g$ with no power isogenous to a Jacobian, and for $g=5$ one can find intermediate Jacobians of smooth cubic threefolds among such examples. The authors ask whether one can find \nabelian varieties \nover $\\overline{\\Q}$ with no power isogenous to a Jacobian, see \\cite[Remark 1.6]{dGFSchreieder-2025}.\n\nThe first main result of this paper says that if $g \\geq 16$ is a power of $2$, then such examples exist at least over $\\smash{\\overline{\\Q(t)}}$. By a Jacobian, we mean the Jacobian of a projective stable curve of compact type.\n\nThe first main result of this paper says that if $g \\geq 16$ is a power of $2$, then such examples exist at least over $\\smash{\\Qtbar}$. By a Jacobian, we mean the Jacobian of a projective stable curve of compact type.\n\nLet $\\Ag$ be the moduli space of principally polarized abelian varieties of dimension $g$. A point $x \\in \\Ag$ is called \\emph{Hodge generic} if $x$ is not contained in a special subvariety of $\\Ag$ of positive codimension---equivalently, the associated Mumford--Tate group is $\\GSp_{2g}$. \nIn \\cite{tsimerman-jacobian}, Tsimerman showed that one can find CM abelian varieties of dimension $g \\geq 4$ which are not isogenous to a Jacobian; in \\cite{masser-zannier} and \\cite{shankartsimerman-Fptbar}, the authors (using different methods) showed that there are also Hodge generic $\\Qbar$ points of $\\Ag$ with that property.\n\n\\begin{theorem} \\label{theorem:generic}\nLet $g = 2^d \\geq 16$ and $N \\geq 1$. Then there exists a Hodge generic point $x = [A] \\in \\Ag(\\Qtbar)$ such that the corresponding abelian variety $A$ over $\\Qtbar$ has no power $A^n$ with $n \\leq N$ which is isogenous to a Jacobian.\n\\end{theorem}\n\nThe situation may be depicted in the following diagram, in which the three curves $C', \\tilde C$ and $C$ are smooth, \nthe morphism $\\tilde C \\to C'$ is finite, the morphism $\\tilde C \\to C$ is finite étale, \nand the two squares are cartesian:\n\\[\n\\xymatrix{\n&&\\tilde Q\\ar[dl] \\ar[dr]&&\\tilde A^n \\ar[dr]\\ar[ll] \\ar[dl]& &\\\\\n&Q'\\ar[dr]&& \\tilde C\\ar[dl] \\ar[dr] && A^n \\ar[dl] &\n\\\\\n \\ca A_{ng,[s]}&& C' \\ar@{_{(}->}[ll] & & C \\ar@{^{(}->}[rr]& &\\ca A_{g,[r]}.\n}\n\\]\nConsider the abelian schemes $f \\colon A \\to C, g \\colon A^n \\to C$, $\\tilde h \\colon \\tilde Q \\to \\tilde C$, and $h' \\colon Q' \\to C'$, and define $L_f, L_g, L_{\\tilde h}$ and $L_{h'}$ as the respective Hodge line bundles of these families. \nAs $C \\subset \\Agr$ is a smooth Shimura curve of Mumford type, we have by Corollary \\ref{corollary:equality} that $\\deg(L_f) = (g/2) \\cdot \\deg(\\Omega_C)$. \nBy Lemmas \\ref{lemma:inequality} and \\ref{lemma:higgs-power}, we therefore obtain:\n\\begin{align} \\label{first}\n\\deg(L_{\\tilde h}) = \\frac{ng}{2} \\cdot \\deg(\\Omega_{\\tilde C}). \n\\end{align}\nThe geometric generic fibers of $\\tilde Q \\to \\tilde C$ and $Q' \\to C'$ are canonically isomorphic as principally polarized abelian varieties, and by construction isomorphic to the Jacobian $JX$ of the smooth projective connected curve $X$ over $\\smash{\\overline{\\mathbb{Q}(t)}}$. Therefore, the smooth curve $C' \\subset \\ca A_{ng,[s]}$ is contained in the closed Torelli locus, and intersects the open Torelli locus non-trivially. As $g = 2^d>11$, this implies by \\cite[Theorem 1.4]{luzuo-2019} that\n\\begin{align}\\label{align:inequality-proof}\n\\deg(L_{h'}) < \\frac{ng}{2} \\cdot \\deg(\\Omega_{C'}). \n\\end{align}\nThe morphism \n$\n\\tilde C \\to C'\n$\nis a finite morphism of smooth curves. Therefore, by Lemma \\ref{lemma:inequality}, the strict inequality \\eqref{align:inequality-proof} implies that we have a strict inequality\n$\n\\deg(L_{\\tilde h}) < (ng/2) \\cdot \\deg(\\Omega_{\\tilde C})$, which contradicts \\eqref{first}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{theorem:special}]\nWe must provide an abelian variety of dimension $g = 2^d \\geq 16$ over $\\smash{\\overline{\\mathbb{Q}(t)}}$ with no power isogenous to a Jacobian. Let $r \\geq 3$ be an integer and $C \\subset \\Agr$ a non-split Shimura curve of Mumford type (cf.\\ Definition \\ref{definition:nonsplit}); such a pair $(r,C)$ exists by Remark \\ref{remark}. Since $C \\subset \\Agr$ is a Shimura curve, it is defined over $\\Qbar$. Let $A \\to C$ be the natural abelian scheme. The geometric generic fiber $A_{\\bar \\eta}$ is a $g$-dimensional abelian variety over $\\smash{\\overline{\\mathbb{Q}(t)}}$ which, by Theorem \\ref{theorem:shimuracurve}, has no power isogenous to a Jacobian. \n\\end{proof}\n\nLet $\\eta \\in C$ be the generic point of $C$ and let $A \\to C$ be the induced abelian scheme. Assume for a contradiction that for some integer $n$ with $1 \\leq n \\leq N$, we have an isogeny \\begin{align} \\label{align:isogeny:JX}A_{\\bar \\eta}^n \\to JX, \\end{align} where $X$ is a compact type curve over $\\Qtbarsmash$. Let $\\lambda_{A, \\bar \\eta}$ (resp.\\ $\\lambda_{JX}$) be the principal polarization of $A_{\\bar \\eta}$ (resp.\\ $JX$), and let $\\mu$ be the polarization on $A^n_{\\bar \\eta}$ obtained by pulling back $\\lambda_{JX}$ to $A^n_{\\bar \\eta}$ along \\eqref{align:isogeny:JX}. As $C$ has maximal monodromy, we have $\\End(A_{\\bar \\eta}) = \\Z$, see Lemma \\ref{lemma:endo-mono}. By \\cite[Lemma 5.2]{dGFSchreieder-2025}, this implies that there exists a symmetric positive definite matrix $\\alpha \\in \\rm{M}_n(\\Z)$ such that $\\mu = \\lambda_\\alpha$, see Notation \\ref{notation:alpha}. Consider the morphism of moduli spaces\n$$\n\\Phi \\colon \\ca A_{g,[r]} \\to \\ca A_{ng, \\delta,[r]}, \\quad \\quad (A, \\lambda_A, \\bar \\phi) \\mapsto (A^n, \\lambda_\\alpha, \\bar \\phi_\\alpha),\n$$\nwhere\n$\n\\delta = (\\delta_1, \\dotsc, \\delta_{ng})$ is the type of the polarization $\\lambda_\\alpha$. \nConsider the generic point $\\eta \\in C \\subset \\Agr$ of the curve $C$. \nThere exists an element $a \\in \\GSp_{2ng}(\\QQ)_+$ \nand a $\\Qtbar$ point $\\xi \\in \\ca A_{ng,[r]}$, with $(JX, \\lambda_{JX})$ as underlying principally polarized abelian variety over $\\Qtbar$, such that with respect to the correspondence $\\ca A_{ng,[r]} \\xleftarrow{p_1} {\\ca A}_a \\xrightarrow{p_2} \\ca A_{ng,\\delta,[r]}$ of Definition \\ref{def:a}, we have\n\\begin{align*}\n\\xi \\in \\tau_a(\\Phi(\\eta)) = p_1(p_2^{-1}(\\Phi(\\eta))).\n\\end{align*}\nWe define $S' = \\Phi(S)$ and $C' = \\Phi(C)$, so that \n$\nC' \\subset S' \\subset \\ca A_{ng,\\delta, [r]}. \n$\nLet $C^\\# \\subset \\tau_a(C')$ be the irreducible component of $\\tau_a(C')$ with $\\xi \\in C^\\#$. By Corollary \\ref{singlecurve-upgraded}, there is no irreducible curve $D \\neq C$ with $D\\subset S$ such that $C^\\# \\subset \\tau_a(\\Phi(D))$. \nTherefore, by Theorem \\ref{theorem:degree}, there exists an irreducible component $S^\\# \\subset \\tau_a(S')$ with $C^\\# \\subset S^\\#$, such that \n \\begin{align}\\label{align:ratio:proof}\n \\frac{\\deg(C^\\#)}{\\deg(S^\\#)} = \\frac{\\deg(C)}{\\deg(S)}. \n \\end{align}\nAs $\\xi \\in \\ca T_{ng,[r]}$, we have $C^\\# \\subset \\ca T_{ng,[r]}$, hence \\eqref{intersection} implies that\n$\nC^\\# \\subset H_{n, i}$ for each $ i \\in I_n$. \nBy construction, $S \\subset \\ca A_{g,[r]}$ contains a Shimura curve of Mumford type, hence by Theorem \\ref{theorem:shimuracurve}, we have $\nS^\\# \\not \\subset \\ca T_{ng,[r]} = \\bigcap_{n, i \\in I_n} H_{n,i}$. Thus, $S^\\# \\not \\subset H_{n,i}$ for some $i \\in I_n$. Since the irreducible components of $S^\\# \\cap H_{n,i}$ have dimension at least one, they have dimension exactly one. In particular, the intersection $S^\\# \\cap H_{n,i} = S^\\# \\cap \\overline H_{n,i}$ is transverse, so that $$\\deg(S^\\# \\cap \\overline H_{n,i} \\subset \\PP^{N_n}) = \\deg(S^\\# \\subset \\PP^{N_n}) \\cdot \\deg(\\overline H_{n,i} \\subset \\PP^{N_n}).$$ \nAs $C^\\# \\subset \\overline H_{n,i}$, we have \n$\nC^\\# \\subset S^\\# \\cap \\overline H_{n, i} \\subsetneq S^\\#$, hence $C^\\#$ is an irreducible component of the purely one-dimensional scheme $S^\\# \\cap \\overline H_{n, i}$. Consequently, \n\\begin{align*}\n\\begin{split}\n\\deg(C^\\# \\subset \\PP^{N_n}) &\\leq \\deg( S^\\# \\cap \\overline H_{n,i} \\subset \\PP^{N_n}) \\\\\n& = \n\\deg(S^\\# \\subset \\PP^{N_n}) \\cdot \\deg(\\overline H_{n,i} \\subset \\PP^{N_n}),\\end{split}\n\\end{align*}\nand hence, by \\eqref{align:deg:CS} and \\eqref{align:def:d}, we get\n\\begin{align} \\label{inequality:intersection}\n\\deg(C^\\#) \\leq k_n \\cdot \\deg(S^\\#) \\cdot \\deg(\\overline H_{n,i} \\subset \\PP^{N_n}) \\leq d \\cdot \\deg(S^\\#). \n\\end{align}\nFrom \\eqref{align:ratio:proof} and \\eqref{inequality:intersection}, we deduce $\n\\deg(C) \\leq d\\cdot \\deg(S)$, which contradicts \\eqref{degree-strict}. Thus, \nthe $n$-th power $A_{\\bar \\eta}^n$ of \nthe geometric generic fiber $A_{\\bar \\eta}$ of the abelian scheme $A \\to C$ is not isogenous to a Jacobian, for every $n \\leq N$. \n\\end{proof}", + "post_theorem_intro_text_len": 3773, + "post_theorem_intro_text": "Let $\\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties of dimension $g$. A point $x \\in \\mathcal{A}_g$ is called \\emph{Hodge generic} if $x$ is not contained in a special subvariety of $\\mathcal{A}_g$ of positive codimension---equivalently, the associated Mumford--Tate group is $\\GSp_{2g}$. \nIn \\cite{tsimerman-jacobian}, Tsimerman showed that one can find CM abelian varieties of dimension $g \\geq 4$ which are not isogenous to a Jacobian; in \\cite{masser-zannier} and \\cite{shankartsimerman-Fptbar}, the authors (using different methods) showed that there are also Hodge generic $\\overline{\\Q}$ points of $\\mathcal{A}_g$ with that property. \n\nAs the proof of Theorem \\ref{theorem:special} shows, for $g = \\smash{ 2^d} \\geq 16$, there are $x\\in \\smash{\\mathcal{A}_g(\\overline{\\Q(t)})}$, the geometric generic point of a special curve $C \\subset \\mathcal{A}_g$, such that $A_x$ has no power isogenous to a Jacobian. It is natural to ask whether there exist Hodge generic points $x \\in \\smash{\\mathcal{A}_g(\\overline{\\Q(t)})}$ with that same property. Our second main result says that at least for bounded powers, the answer is yes.\n\n\\begin{theorem} \\label{theorem:generic}\nLet $g = 2^d \\geq 16$ and $N \\geq 1$. Then there exists a Hodge generic point $x = [A] \\in \\mathcal{A}_g(\\overline{\\Q(t)})$ such that the corresponding abelian variety $A$ over $\\overline{\\Q(t)}$ has no power $A^n$ with $n \\leq N$ which is isogenous to a Jacobian.\n\\end{theorem}\n\n\\subsection{Outline of proof}\nWe now briefly describe our methods. Let $\\mathcal A_{g,[r]}$ be the moduli space of principally polarized abelian varieties of dimension $g$ with symplectic level $r\\geq 3$ structure. The proof of Theorem \\ref{theorem:special} uses an Arakelov inequality due to Lu and Zuo \\cite[Theorem 1.4]{luzuo-2019}, which says in particular that given a smooth proper curve $C \\subset \\mathcal A_{g,[r]}$ with induced abelian scheme $\\pi \\colon A \\to C$, one has $\\deg(\\pi_\\ast \\Omega_{A/C}) < (g/2) \\cdot \\deg(\\Omega_C)$ whenever $g \\geq 12$ and $C$ is generically contained in the Torelli locus. On the other hand, this becomes an equality if $C$ is a Shimura curve of Mumford type (cf.\\ Definition \\ref{definition:nonsplit}), from which Lu and Zuo deduce that such Shimura curves cannot be generically contained in the Torelli locus if $g \\geq 12$. We apply their Arakelov inequality to show that, more generally, the $n$-th power $A_{\\bar \\eta}^n$ cannot be isogenous to a Jacobian for every $n$, for $\\eta \\in C$ the generic point. \n\nWe prove the above for Shimura curves of Mumford type $C \\subset \\mathcal A_{g,[r]}$ which are not necessarily smooth. Such curves exist in $\\mathcal A_{g,[r]}$ for each $r \\geq 3$. For technical reasons, we proceed in the second part of the paper to assume that $r$ is prime. We pick a suitable surface $S\\subset\\mathcal A_{g,[r]}$ that contains $C$ as well as Hodge generic points. Theorem \\ref{theorem:special} implies that the induced abelian scheme $B\\rightarrow S$ has the property that the $n$-th power $B_{\\bar \\eta}^n$ is not isogenous to a Jacobian for every $n$, where now $\\eta \\in S$ denotes the generic point of $S$. Bounding $n$, we then use an intersection-theoretic argument to find a curve $C_{\\textrm{gen}} \\subset S$ over $\\overline{\\Q}$ that satisfies the conclusion of Theorem \\ref{theorem:generic}. \n\n\\subsection{Acknowledgements}\n\nWe would like to thank Emiliano Ambrosi, Philip Engel, Stefan Schreieder and Jacob Tsimerman for stimulating discussions. \n\nO.d.G.F.~has received funding from the ERC Consolidator Grant FourSurf \\textnumero 101087365. A.S.~was partially supported by the NSF grant DMS-2338942, the Institute for advanced studies (via the NSF grant DMS-2424441), and a Sloan research fellowship.", + "sketch": "The post-theorem discussion explains that the proof of Theorem~\\ref{theorem:special} proceeds by applying an Arakelov inequality of Lu--Zuo \\cite[Theorem 1.4]{luzuo-2019} to a (possibly singular) Shimura curve of Mumford type $C\\subset \\mathcal A_{g,[r]}$ with induced abelian scheme $\\pi:A\\to C$. The inequality gives, for $g\\ge 12$ and $C$ generically contained in the Torelli locus, a strict bound\n\\[\n\\deg(\\pi_*\\Omega_{A/C}) < (g/2)\\cdot \\deg(\\Omega_C),\n\\]\nwhile for a Shimura curve of Mumford type this “becomes an equality”. Since Lu--Zuo deduce that such Shimura curves cannot be generically contained in the Torelli locus for $g\\ge 12$, the authors “apply their Arakelov inequality to show that, more generally, the $n$-th power $A_{\\bar\\eta}^n$ cannot be isogenous to a Jacobian for every $n$, for $\\eta\\in C$ the generic point,” yielding the conclusion for $A$ over $\\overline{\\mathbb Q(t)}$ when $g=2^d\\ge 16$.", + "expanded_sketch": "The post-theorem discussion explains that the proof of the main theorem proceeds by applying an Arakelov inequality of Lu--Zuo \\cite[Theorem 1.4]{luzuo-2019} to a (possibly singular) Shimura curve of Mumford type $C\\subset \\mathcal A_{g,[r]}$ with induced abelian scheme $\\pi:A\\to C$. The inequality gives, for $g\\ge 12$ and $C$ generically contained in the Torelli locus, a strict bound\n\\[\n\\deg(\\pi_*\\Omega_{A/C}) < (g/2)\\cdot \\deg(\\Omega_C),\n\\]\nwhile for a Shimura curve of Mumford type this “becomes an equality”. Since Lu--Zuo deduce that such Shimura curves cannot be generically contained in the Torelli locus for $g\\ge 12$, the authors “apply their Arakelov inequality to show that, more generally, the $n$-th power $A_{\\bar\\eta}^n$ cannot be isogenous to a Jacobian for every $n$, for $\\eta\\in C$ the generic point,” yielding the conclusion for $A$ over $\\overline{\\mathbb Q(t)}$ when $g=2^d\\ge 16$.", + "expanded_theorem": "\\label{theorem:special}\n Let $g=2^d$ with $d \\geq 4$. Then there exists an abelian variety $A$ of dimension $g$ over $\\overline{\\mathbb{Q}(t)}$ with no power $A^n$ isogenous to a Jacobian.", + "theorem_type": [ + "Existence", + "Nonexistence" + ], + "mcq": { + "question": "Let \\(g=2^d\\) with \\(d\\ge 4\\), so \\(g\\ge 16\\). Here a Jacobian means the Jacobian of a projective stable curve of compact type. Which statement holds about abelian varieties of dimension \\(g\\) over \\(\\overline{\\mathbb{Q}(t)}\\)?", + "correct_choice": { + "label": "A", + "text": "There exists an abelian variety \\(A\\) of dimension \\(g\\) over \\(\\overline{\\mathbb{Q}(t)}\\) such that for every integer \\(n\\ge 1\\), the power \\(A^n\\) is not isogenous to a Jacobian." + }, + "choices": [ + { + "label": "B", + "text": "There exists an abelian variety \\(A\\) of dimension \\(g\\) over \\(\\overline{\\mathbb{Q}(t)}\\) such that for every integer \\(n\\ge 1\\), the power \\(A^n\\) is not isomorphic to a Jacobian." + }, + { + "label": "C", + "text": "There exists an abelian variety \\(A\\) of dimension \\(g\\) over \\(\\overline{\\mathbb{Q}(t)}\\) such that \\(A\\) itself is not isogenous to a Jacobian." + }, + { + "label": "D", + "text": "For every abelian variety \\(A\\) of dimension \\(g\\) over \\(\\overline{\\mathbb{Q}(t)}\\), and for every integer \\(n\\ge 1\\), the power \\(A^n\\) is not isogenous to a Jacobian." + }, + { + "label": "E", + "text": "There exists an abelian variety \\(A\\) of dimension \\(g\\) over \\(\\overline{\\mathbb{Q}(t)}\\) such that for every integer \\(1\\le n\\le g\\), the power \\(A^n\\) is not isogenous to a Jacobian." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "B" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "property_confusion", + "tampered_component": "isogenous_vs_isomorphic", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped_all_higher_powers", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "geometric_construction", + "tampered_component": "existential_conclusion_for_special_construction_replaced_by_universal_statement", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "quantifier_dependence", + "tampered_component": "for_all_n_replaced_by_bounded_range_1_to_g", + "template_used": "boundary_range" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not reveal the correct statement or give away the key quantifier pattern; it only sets the dimension hypothesis and asks which claim is true." + }, + "TAS": { + "score": 2, + "justification": "This is not a direct restatement in the stem. The item requires distinguishing among nearby existential, universal, and weakened variants rather than simply echoing a theorem verbatim." + }, + "GPS": { + "score": 1, + "justification": "The item requires some reasoning about strength of conclusions, quantifiers, and isogeny versus isomorphism, but it mainly tests precise theorem recognition rather than substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and target realistic failure modes: confusing isomorphic with isogenous, accepting a weaker true statement, overgeneralizing to a universal claim, or weakening the quantifier range." + }, + "total_score": 7, + "overall_assessment": "A strong MCQ with no answer leakage and very good distractors; its main limitation is that it tests theorem discrimination/recall more than deeper constructive reasoning." + } + }, + { + "id": "2511.16095v1", + "paper_link": "http://arxiv.org/abs/2511.16095v1", + "theorems_cnt": 5, + "theorem": { + "env_name": "thm", + "content": "\\label{main-lemma} \\cite[Theorem 1.3]{KR1}\nLet $X$ and $F$ be as above, and let $y \\in E(F,\\infty)$; then\n\\begin{equation}\\label{dim1}\n \\dim\\big(\\{ \\alpha \\in \\R : x_\\alpha \\in A(F_+,y) \\cap E(F_+,\\infty)\\}\\big) = 1.\n\\end{equation}", + "start_pos": 14425, + "end_pos": 14682, + "label": "main-lemma" + }, + "ref_dict": { + "dim1": "\\begin{equation}\\label{dim1}\n \\dim\\big(\\{ \\alpha \\in \\R : x_\\alpha \\in A(F_+,y) \\cap E(F_+,\\infty)\\}\\big) = 1.\n\\end{equation}", + "thm: main theorem": "\\begin{thm}\\label{thm: main theorem}\n Let $X = G/\\Gamma$ and $F = \\{g_t : t\\in\\R\\}\\subset G$, where \n \\begin{itemize}\n \\item $G$ is a Lie group with discrete center,\n \\item $\\Gamma$ is a non-uniform \n lattice in $G$,\n\\item $g_t$ is as in \\eqref{gt}, where $a_0 \\in \\fg\\nz$ is $\\ad$-diagonalizable over $\\Co$;\n \\item the action of $F$ on $(X,m_X)$ is mixing.\n\\end{itemize}\n Let $y \\in E(F,\\infty)$. Then for any non-empty open $U\\subset X$ we have \n \\begin{equation}\\label{eq: dim X - dim Z + 1}\n \\dim \\big(U\\cap A(F_+,y) \\cap E(F_+,\\infty)\\big) \\geq \\dim X - \\dim Z + 1,\n \\end{equation}\n where $Z \\subset G$ is the {neutral subgroup of $G$ with respect to $F$} . \n\\end{thm}", + "eq: x to y_0": "\\begin{equation}\\label{eq: x to y_0}\n \\lim_{k \\to \\infty} g_{s_k} x = y.\n\\end{equation}", + "main-lemma": "\\begin{thm}\\label{main-lemma} \\cite[Theorem 1.3]{KR1}\nLet $X$ and $F$ be as above, and let $y \\in E(F,\\infty)$; then\n\\begin{equation}\\label{dim1}\n \\dim\\big(\\{ \\alpha \\in \\R : x_\\alpha \\in A(F_+,y) \\cap E(F_+,\\infty)\\}\\big) = 1.\n\\end{equation}\n\\end{thm}", + "eq: tx notin W": "\\begin{equation}\\label{eq: tx notin W}\n g_tx \\notin W \\text{ for all }t> t_0.\n\\end{equation}", + "thm: dense-orbits": "\\begin{thm}\\label{thm: dense-orbits} \nLet $G$, $\\Gamma$ and $F$ be as in Theorem \\ref{thm: main theorem}, and assume in addition that \\linebreak {$\\dim Z = \n1$}.\nThen for any {compact} \nF-invariant\nsubset $B$ of $E(F,\\infty)$ there exists a thick set of $x \\in E(F_+,\\infty)$ such that the closure of the (bounded) trajectory $F_+x$ contains $B$.\n \\end{thm}", + "eq: y_0 to y_1": "\\begin{equation}\\label{eq: y_0 to y_1}\n \\lim_{k \\to \\infty} g_{t_k} y= z\n\\end{equation}", + "gt": "\\begin{equation}\\label{gt}\n \\big\\{ g_t := \\exp(ta_0) \\in G: t \\in \\R \\big\\},\n \\end{equation}", + "defn: expanding, contracting and neutral groups": "\\begin{defn}\\label{horosph}[Expanding, contracting and neutral subgroups]\\label{defn: expanding, contracting and neutral groups}\n We let $H$, $H^-$ and $Z$ be the connected Lie subgroups of $G$ associated to the algebras $\\fh$, $\\fh^-$ and $\\fn$ respectively. We fix $m_H$, a Haar measure on $H$. We further consider the connected subgroup $\\tilde{H}$ associated to the Lie algebra $\\fn \\oplus \\fh^-$ and a left Haar measure $m_{\\tilde{H}}$ on $\\tilde{H}$, scaled so that $m_G$ is a direct product of $m_H$ and $m_{\\tilde{H}}$, see \\cite[Ch.~VII, \\S 9, Proposition 13]{B} or \\cite[Theorem 8.32]{Kn} for justification.\n\\end{defn}" + }, + "pre_theorem_intro_text_len": 7753, + "pre_theorem_intro_text": "\\label{intro}\n\n Let $X$ be a metric space, and let $F$ be {either of the (semi) groups $\\R$ or $\\Z$ or $\\R_{\\ge 0}$ or $\\Z_{\\ge 0}$ represented as a family of} self-maps of $X$. We will write $F = \\{g_t\\}$. For $g\\in F$ we will denote its action on $X$ by \n$(g,x) \\mapsto g\nx$.\n Now fix a subset $Y\\subset X$ and define the set of points of $X$ {\\sl approaching} $Y$ by $F$ as\n \\begin{equation*}\n A(F,Y) := \\big\\{ x \\in X : \\exists \\text{ an unbounded sequence }(g_{t_k})_{k \\in \\N} \\subset F \\text{ with }\n {\\lim_{k \\to \\infty} g_{t_k} x \\in Y}\n \\big\\}.\n \\end{equation*}\n Its complement, the set of points of $X$ {\\sl escaping} {$Y$} by $F$ will be denoted by $E(F,{Y})$, namely\n \\begin{equation*}\n E(F,{Y}) := \\bigcap_{y \\in Y}\\big\\{ x\\in X : y\n \\notin\\overline{(F\\smallsetminus F_0) x} \n \\text{ for some bounded }F_0\\subset F\\big\\}.\n \\end{equation*}\n If $X$ is not bounded, we make the convention that $x\\in A(F,\\infty)$ when the orbit $F\n x$ is unbounded.\n\n {Take $y\\in Y$ and consider $A(F,y):= A(F,\\{y\\})$ and $E(F,y):= E(F,\\{y\\})$.} A natural question one could ask is how large the sets $A(F,y)$ and $E(F,y)$ can be. If the action admits an ergodic invariant probability measure, or has some hyperbolic behavior, some instances of this question can be answered. Namely, under the assumption of existence of an ergodic measure $\\mu$ of full support the set $A(F,y)$ has full measure for any $y\\in X\\cup \\{\\infty\\}$. Additionally, for many hyperbolic systems one can prove that the $\\mu$-null sets $E(F,y)$ are \\textsl{thick}, that is, their intersection with any nonempty open set has full Hausdorff dimension. For results of this kind see e.g.\\ \\cite{Da3} for toral endomorphisms, \\cite{U, Do} for Anosov flows, and {\\cite{Da1, KM, Kl2, AGK}} for partially hyperbolic flows on \\hs.\nFurthermore, in many cases these sets can be proved to be winning in the sense of Schmidt games (see e.g. \\cite{AGK, BFK,Da1,Da3,Ts1,Ts2,wu1,wu2}).\nConsequently in these cases it follows that both $A(F,y)$ and $E(F,y)$ are thick.\nMoreover, both full-measure and winning conditions are stable with respect to countable intersections; thus\nfor any choice of countably many semigroups $F_i$ and points $y_i \\in X\\cup \\{\\infty\\}$ for which the above conclusions can be established, it holds that both $\\bigcap_i A(F_i,y_i)$ and $\\bigcap_i E(F_i,y_i)$ are thick. \n\n\\smallskip\nThis paper addresses the following question: what if one considers \na mixed case, that is, investigate the intersection \\eq{mixed}{A(F,y) \\cap E(F,z),} where $y,z\\in X\\cup \\{\\infty\\}$? Problems of this type are amenable neither to the full-measure argument, nor to the technique based on Schmidt games. Also it is clear that some conditions on $y,z$ should be imposed; for example if $y=z$ it is clear that the intersection \\eqref{eq:mixed} is empty. \n\n\\smallskip\nIn what follows we restrict our attention to $F\\cong \\R$, also denoting $F_+ := \\{g_t: t\\ge 0\\}$.\nAssume that the $F$-action on $X$ is continuous. Then let us record {a natural} \nobstruction to \nthe set \\eq{mixedplus}{A(F_+,y) \\cap E(F_+,z)} being non-empty: \n\n\\begin{lem}\\label{lem: obstruction}\n If {$y \\in X$ and $z \\in X \\cup \\{\\infty\\}$} are such that $y \\in A(F,z)$ (that is, $z$ can be approached by the trajectory of $y$ either in the positive or in the negative time direction), then \n \\begin{equation*}\n A(F_+,y) \\subset A(F_+,z).\n \\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nLet $x \\in A(F_+,y)$, and let $(s_\\ell)_{\\ell \\in \\N}$ be an unbounded sequence in $\\R_{\\geq 0}$ with\n\\begin{equation*}\n \\lim_{\\ell \\to \\infty} g_{s_\\ell}x = y.\n\\end{equation*}\nLet $W \\subset X$ be an open neighborhood of $z$. In the case, when $z = \\infty$, we take $W$ to be an open set with bounded complement. \nBy the hypothesis, there exists a time $t \\in \\R$ for which $g_t y \\in W$. By the continuity of the action we have\n\\begin{equation*}\n \\lim_{\\ell \\to \\infty} g_{s_\\ell}g_tx = g_t y \\in W.\n\\end{equation*}\nNote that the sequence $(s_\\ell + t)_{\\ell \\in \\N}$ is unbounded and eventually positive. Since $W$ was arbitrary, we conclude that $x \\in A(F_+, z)$.\n\\end{proof}\n\\ignore{{Suppose} we have a point $x \\in A(F_+,y) \\cap E(F_+,z)$. By {the} hypothesis, take an unbounded sequence $(t_k)_{k \\in \\N} \\subset \\R$ with\n\\begin{equation}\\label{eq: y_0 to y_1}\n \\lim_{k \\to \\infty} g_{t_k} y= z\n\\end{equation}\n(in the case $z = \\infty$ this means that $g_{t_k} y$ leaves every compact subset of $X$).\nSince $x$ is an element of $E(F_+,z)$, we can take an open neighborhood $W$ containing $z$ (if $z = \\infty$ we can let $W$ be the complement of a compact subset of $X$) and $t_0>0$ such that\n\\begin{equation}\\label{eq: tx notin W}\n g_tx \\notin W \\text{ for all }t> t_0.\n\\end{equation}\nUsing the fact that $x \\in A(F_+,y)$, we take an unbounded sequence $(s_k)_{k \\in \\N} \\subset \\R_{\\geq 0}$ with\n\\begin{equation}\\label{eq: x to y_0}\n \\lim_{k \\to \\infty} g_{s_k} x = y.\n\\end{equation}\nThe convergence in \\eqref{eq: y_0 to y_1} guarantees us some $p \\in \\N$ with \n\\begin{equation*}\n {g_{t_{p}} y \\in W}.\n\\end{equation*}\nThe continuity of the action and the convergence in \\eqref{eq: x to y_0} guarantees us that \n\\begin{equation*}\n \\lim_{k \\to \\infty} g_{t_p + s_k} x = g_{t_p} y \\in W. \n\\end{equation*}\nHowever, for $k$ large enough, $t_p + s_k$ will be larger than $t_0$, contradicting \\eqref{eq: tx notin W}.}\n\n{In view of the above lemma it makes sense to look for conditions guaranteeing that the set \\eqref{eq:mixedplus} is reasonably large \nunder the assumption that $y \\in E(F,z)$. \nIn the present paper we will do it in the following set-up: we let $X = G/\\Gamma$, where $G$ is a real Lie group \nand $\\Gamma \\subset G$ is a lattice,\nand consider the $\\R$-action on $X$ via the left multiplication by a one-parameter subgroup of $G$:\n \\begin{equation}\\label{gt}\n \\big\\{ g_t := \\exp(ta_0) \\in G: t \\in \\R \\big\\},\n \\end{equation}\n where ${a_0}$ is an element of {$\\fg:=\\operatorname{Lie}(G)$, the Lie algebra of $G$}. \n We shall fix once and for all an inner product on $\\fg$ to induce a right-invariant Riemannian metric $\\dist_G$ on $G$.\n Note that the invariance of the metric implies that it is geodesically complete, and so the Hopf--Rinow theorem implies that it is proper (closed and bounded sets are compact). Then one obtains a metric on $X = G/\\Gamma$ by setting\n\\begin{equation*}\n \\dist_X(g_1\\Gamma, g_2\\Gamma) := \\inf\\big\\{\\dist_G(g_1\\gamma, g_2): \\gamma \\in \\Gamma \\big\\}.\n\\end{equation*}\nThis is the set-up in which we will attempt to study sets of the form \\eqref{eq:mixedplus}. We will denote by $m_X$ the $G$-invariant Haar probability measure on $X$, and by $m_G$ the Haar measure on $G$ scaled so that the measure of any fundamental domain for $\\Gamma$ is equal to $1$. Further,\n {in this paper} we will assume that $X$ is noncompact and $z = \\infty$; that is, we will seek to construct bounded $F^+$-trajectories with prescribed limit points. \n\n Recently the second- and third-named authors proved the following result. Take \\linebreak $G = \\SL_2(\\R)$, $\\Gamma = \\SL_2(\\Z)$ and $X = G/\\Gamma$. For $t,\\alpha\\in\\R$ consider \n\\begin{equation*}\\label{eq: gtsl2}\n g_t := \\left[ {\\begin{array}{cc} e^t & 0 \\\\ \n 0 & e^{-t} \\end{array}}\\right]\\in G,\\ F = \\{g_t: t\\in\\R\\},\n \\end{equation*}\n and $${x_\\alpha := \\left[ {\\begin{array}{cc} 1 & \\alpha \\\\ \n 0 & 1 \\end{array}}\\right]\\Gamma\\in X.} \n \\ignore{the latter being identified with the lattice in $\\R^2$ generated by $\\left[ {\\begin{array}{cc} 1 \\\\ \n 0 \\end{array}}\\right]$ and $ \\left[ {\\begin{array}{cc} \\alpha \\\\ \n 1 \\end{array}}\\right]$.}$$", + "context": "{Take $y\\in Y$ and consider $A(F,y):= A(F,\\{y\\})$ and $E(F,y):= E(F,\\{y\\})$.} A natural question one could ask is how large the sets $A(F,y)$ and $E(F,y)$ can be. If the action admits an ergodic invariant probability measure, or has some hyperbolic behavior, some instances of this question can be answered. Namely, under the assumption of existence of an ergodic measure $\\mu$ of full support the set $A(F,y)$ has full measure for any $y\\in X\\cup \\{\\infty\\}$. Additionally, for many hyperbolic systems one can prove that the $\\mu$-null sets $E(F,y)$ are \\textsl{thick}, that is, their intersection with any nonempty open set has full Hausdorff dimension. For results of this kind see e.g.\\ \\cite{Da3} for toral endomorphisms, \\cite{U, Do} for Anosov flows, and {\\cite{Da1, KM, Kl2, AGK}} for partially hyperbolic flows on \\hs.\nFurthermore, in many cases these sets can be proved to be winning in the sense of Schmidt games (see e.g. \\cite{AGK, BFK,Da1,Da3,Ts1,Ts2,wu1,wu2}).\nConsequently in these cases it follows that both $A(F,y)$ and $E(F,y)$ are thick.\nMoreover, both full-measure and winning conditions are stable with respect to countable intersections; thus\nfor any choice of countably many semigroups $F_i$ and points $y_i \\in X\\cup \\{\\infty\\}$ for which the above conclusions can be established, it holds that both $\\bigcap_i A(F_i,y_i)$ and $\\bigcap_i E(F_i,y_i)$ are thick.\n\n\\begin{lem}\\label{lem: obstruction}\n If {$y \\in X$ and $z \\in X \\cup \\{\\infty\\}$} are such that $y \\in A(F,z)$ (that is, $z$ can be approached by the trajectory of $y$ either in the positive or in the negative time direction), then \n \\begin{equation*}\n A(F_+,y) \\subset A(F_+,z).\n \\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nLet $x \\in A(F_+,y)$, and let $(s_\\ell)_{\\ell \\in \\N}$ be an unbounded sequence in $\\R_{\\geq 0}$ with\n\\begin{equation*}\n \\lim_{\\ell \\to \\infty} g_{s_\\ell}x = y.\n\\end{equation*}\nLet $W \\subset X$ be an open neighborhood of $z$. In the case, when $z = \\infty$, we take $W$ to be an open set with bounded complement. \nBy the hypothesis, there exists a time $t \\in \\R$ for which $g_t y \\in W$. By the continuity of the action we have\n\\begin{equation*}\n \\lim_{\\ell \\to \\infty} g_{s_\\ell}g_tx = g_t y \\in W.\n\\end{equation*}\nNote that the sequence $(s_\\ell + t)_{\\ell \\in \\N}$ is unbounded and eventually positive. Since $W$ was arbitrary, we conclude that $x \\in A(F_+, z)$.\n\\end{proof}\n\\ignore{{Suppose} we have a point $x \\in A(F_+,y) \\cap E(F_+,z)$. By {the} hypothesis, take an unbounded sequence $(t_k)_{k \\in \\N} \\subset \\R$ with\n\\begin{equation}\\label{eq: y_0 to y_1}\n \\lim_{k \\to \\infty} g_{t_k} y= z\n\\end{equation}\n(in the case $z = \\infty$ this means that $g_{t_k} y$ leaves every compact subset of $X$).\nSince $x$ is an element of $E(F_+,z)$, we can take an open neighborhood $W$ containing $z$ (if $z = \\infty$ we can let $W$ be the complement of a compact subset of $X$) and $t_0>0$ such that\n\\begin{equation}\\label{eq: tx notin W}\n g_tx \\notin W \\text{ for all }t> t_0.\n\\end{equation}\nUsing the fact that $x \\in A(F_+,y)$, we take an unbounded sequence $(s_k)_{k \\in \\N} \\subset \\R_{\\geq 0}$ with\n\\begin{equation}\\label{eq: x to y_0}\n \\lim_{k \\to \\infty} g_{s_k} x = y.\n\\end{equation}\nThe convergence in \\eqref{eq: y_0 to y_1} guarantees us some $p \\in \\N$ with \n\\begin{equation*}\n {g_{t_{p}} y \\in W}.\n\\end{equation*}\nThe continuity of the action and the convergence in \\eqref{eq: x to y_0} guarantees us that \n\\begin{equation*}\n \\lim_{k \\to \\infty} g_{t_p + s_k} x = g_{t_p} y \\in W. \n\\end{equation*}\nHowever, for $k$ large enough, $t_p + s_k$ will be larger than $t_0$, contradicting \\eqref{eq: tx notin W}.}\n\n{In view of the above lemma it makes sense to look for conditions guaranteeing that the set \\eqref{eq:mixedplus} is reasonably large \nunder the assumption that $y \\in E(F,z)$. \nIn the present paper we will do it in the following set-up: we let $X = G/\\Gamma$, where $G$ is a real Lie group \nand $\\Gamma \\subset G$ is a lattice,\nand consider the $\\R$-action on $X$ via the left multiplication by a one-parameter subgroup of $G$:\n \\begin{equation}\\label{gt}\n \\big\\{ g_t := \\exp(ta_0) \\in G: t \\in \\R \\big\\},\n \\end{equation}\n where ${a_0}$ is an element of {$\\fg:=\\operatorname{Lie}(G)$, the Lie algebra of $G$}. \n We shall fix once and for all an inner product on $\\fg$ to induce a right-invariant Riemannian metric $\\dist_G$ on $G$.\n Note that the invariance of the metric implies that it is geodesically complete, and so the Hopf--Rinow theorem implies that it is proper (closed and bounded sets are compact). Then one obtains a metric on $X = G/\\Gamma$ by setting\n\\begin{equation*}\n \\dist_X(g_1\\Gamma, g_2\\Gamma) := \\inf\\big\\{\\dist_G(g_1\\gamma, g_2): \\gamma \\in \\Gamma \\big\\}.\n\\end{equation*}\nThis is the set-up in which we will attempt to study sets of the form \\eqref{eq:mixedplus}. We will denote by $m_X$ the $G$-invariant Haar probability measure on $X$, and by $m_G$ the Haar measure on $G$ scaled so that the measure of any fundamental domain for $\\Gamma$ is equal to $1$. Further,\n {in this paper} we will assume that $X$ is noncompact and $z = \\infty$; that is, we will seek to construct bounded $F^+$-trajectories with prescribed limit points.\n\nRecently the second- and third-named authors proved the following result. Take \\linebreak $G = \\SL_2(\\R)$, $\\Gamma = \\SL_2(\\Z)$ and $X = G/\\Gamma$. For $t,\\alpha\\in\\R$ consider \n\\begin{equation*}\\label{eq: gtsl2}\n g_t := \\left[ {\\begin{array}{cc} e^t & 0 \\\\ \n 0 & e^{-t} \\end{array}}\\right]\\in G,\\ F = \\{g_t: t\\in\\R\\},\n \\end{equation*}\n and $${x_\\alpha := \\left[ {\\begin{array}{cc} 1 & \\alpha \\\\ \n 0 & 1 \\end{array}}\\right]\\Gamma\\in X.} \n \\ignore{the latter being identified with the lattice in $\\R^2$ generated by $\\left[ {\\begin{array}{cc} 1 \\\\ \n 0 \\end{array}}\\right]$ and $ \\left[ {\\begin{array}{cc} \\alpha \\\\ \n 1 \\end{array}}\\right]$.}$$", + "full_context": "{Take $y\\in Y$ and consider $A(F,y):= A(F,\\{y\\})$ and $E(F,y):= E(F,\\{y\\})$.} A natural question one could ask is how large the sets $A(F,y)$ and $E(F,y)$ can be. If the action admits an ergodic invariant probability measure, or has some hyperbolic behavior, some instances of this question can be answered. Namely, under the assumption of existence of an ergodic measure $\\mu$ of full support the set $A(F,y)$ has full measure for any $y\\in X\\cup \\{\\infty\\}$. Additionally, for many hyperbolic systems one can prove that the $\\mu$-null sets $E(F,y)$ are \\textsl{thick}, that is, their intersection with any nonempty open set has full Hausdorff dimension. For results of this kind see e.g.\\ \\cite{Da3} for toral endomorphisms, \\cite{U, Do} for Anosov flows, and {\\cite{Da1, KM, Kl2, AGK}} for partially hyperbolic flows on \\hs.\nFurthermore, in many cases these sets can be proved to be winning in the sense of Schmidt games (see e.g. \\cite{AGK, BFK,Da1,Da3,Ts1,Ts2,wu1,wu2}).\nConsequently in these cases it follows that both $A(F,y)$ and $E(F,y)$ are thick.\nMoreover, both full-measure and winning conditions are stable with respect to countable intersections; thus\nfor any choice of countably many semigroups $F_i$ and points $y_i \\in X\\cup \\{\\infty\\}$ for which the above conclusions can be established, it holds that both $\\bigcap_i A(F_i,y_i)$ and $\\bigcap_i E(F_i,y_i)$ are thick.\n\n\\begin{lem}\\label{lem: obstruction}\n If {$y \\in X$ and $z \\in X \\cup \\{\\infty\\}$} are such that $y \\in A(F,z)$ (that is, $z$ can be approached by the trajectory of $y$ either in the positive or in the negative time direction), then \n \\begin{equation*}\n A(F_+,y) \\subset A(F_+,z).\n \\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nLet $x \\in A(F_+,y)$, and let $(s_\\ell)_{\\ell \\in \\N}$ be an unbounded sequence in $\\R_{\\geq 0}$ with\n\\begin{equation*}\n \\lim_{\\ell \\to \\infty} g_{s_\\ell}x = y.\n\\end{equation*}\nLet $W \\subset X$ be an open neighborhood of $z$. In the case, when $z = \\infty$, we take $W$ to be an open set with bounded complement. \nBy the hypothesis, there exists a time $t \\in \\R$ for which $g_t y \\in W$. By the continuity of the action we have\n\\begin{equation*}\n \\lim_{\\ell \\to \\infty} g_{s_\\ell}g_tx = g_t y \\in W.\n\\end{equation*}\nNote that the sequence $(s_\\ell + t)_{\\ell \\in \\N}$ is unbounded and eventually positive. Since $W$ was arbitrary, we conclude that $x \\in A(F_+, z)$.\n\\end{proof}\n\\ignore{{Suppose} we have a point $x \\in A(F_+,y) \\cap E(F_+,z)$. By {the} hypothesis, take an unbounded sequence $(t_k)_{k \\in \\N} \\subset \\R$ with\n\\begin{equation}\\label{eq: y_0 to y_1}\n \\lim_{k \\to \\infty} g_{t_k} y= z\n\\end{equation}\n(in the case $z = \\infty$ this means that $g_{t_k} y$ leaves every compact subset of $X$).\nSince $x$ is an element of $E(F_+,z)$, we can take an open neighborhood $W$ containing $z$ (if $z = \\infty$ we can let $W$ be the complement of a compact subset of $X$) and $t_0>0$ such that\n\\begin{equation}\\label{eq: tx notin W}\n g_tx \\notin W \\text{ for all }t> t_0.\n\\end{equation}\nUsing the fact that $x \\in A(F_+,y)$, we take an unbounded sequence $(s_k)_{k \\in \\N} \\subset \\R_{\\geq 0}$ with\n\\begin{equation}\\label{eq: x to y_0}\n \\lim_{k \\to \\infty} g_{s_k} x = y.\n\\end{equation}\nThe convergence in \\eqref{eq: y_0 to y_1} guarantees us some $p \\in \\N$ with \n\\begin{equation*}\n {g_{t_{p}} y \\in W}.\n\\end{equation*}\nThe continuity of the action and the convergence in \\eqref{eq: x to y_0} guarantees us that \n\\begin{equation*}\n \\lim_{k \\to \\infty} g_{t_p + s_k} x = g_{t_p} y \\in W. \n\\end{equation*}\nHowever, for $k$ large enough, $t_p + s_k$ will be larger than $t_0$, contradicting \\eqref{eq: tx notin W}.}\n\n{In view of the above lemma it makes sense to look for conditions guaranteeing that the set \\eqref{eq:mixedplus} is reasonably large \nunder the assumption that $y \\in E(F,z)$. \nIn the present paper we will do it in the following set-up: we let $X = G/\\Gamma$, where $G$ is a real Lie group \nand $\\Gamma \\subset G$ is a lattice,\nand consider the $\\R$-action on $X$ via the left multiplication by a one-parameter subgroup of $G$:\n \\begin{equation}\\label{gt}\n \\big\\{ g_t := \\exp(ta_0) \\in G: t \\in \\R \\big\\},\n \\end{equation}\n where ${a_0}$ is an element of {$\\fg:=\\operatorname{Lie}(G)$, the Lie algebra of $G$}. \n We shall fix once and for all an inner product on $\\fg$ to induce a right-invariant Riemannian metric $\\dist_G$ on $G$.\n Note that the invariance of the metric implies that it is geodesically complete, and so the Hopf--Rinow theorem implies that it is proper (closed and bounded sets are compact). Then one obtains a metric on $X = G/\\Gamma$ by setting\n\\begin{equation*}\n \\dist_X(g_1\\Gamma, g_2\\Gamma) := \\inf\\big\\{\\dist_G(g_1\\gamma, g_2): \\gamma \\in \\Gamma \\big\\}.\n\\end{equation*}\nThis is the set-up in which we will attempt to study sets of the form \\eqref{eq:mixedplus}. We will denote by $m_X$ the $G$-invariant Haar probability measure on $X$, and by $m_G$ the Haar measure on $G$ scaled so that the measure of any fundamental domain for $\\Gamma$ is equal to $1$. Further,\n {in this paper} we will assume that $X$ is noncompact and $z = \\infty$; that is, we will seek to construct bounded $F^+$-trajectories with prescribed limit points.\n\nRecently the second- and third-named authors proved the following result. Take \\linebreak $G = \\SL_2(\\R)$, $\\Gamma = \\SL_2(\\Z)$ and $X = G/\\Gamma$. For $t,\\alpha\\in\\R$ consider \n\\begin{equation*}\\label{eq: gtsl2}\n g_t := \\left[ {\\begin{array}{cc} e^t & 0 \\\\ \n 0 & e^{-t} \\end{array}}\\right]\\in G,\\ F = \\{g_t: t\\in\\R\\},\n \\end{equation*}\n and $${x_\\alpha := \\left[ {\\begin{array}{cc} 1 & \\alpha \\\\ \n 0 & 1 \\end{array}}\\right]\\Gamma\\in X.} \n \\ignore{the latter being identified with the lattice in $\\R^2$ generated by $\\left[ {\\begin{array}{cc} 1 \\\\ \n 0 \\end{array}}\\right]$ and $ \\left[ {\\begin{array}{cc} \\alpha \\\\ \n 1 \\end{array}}\\right]$.}$$\n\n\\begin{prop}\\label{prop: equidistirbution for small balls}\n Let $V \\subset H$ be bounded measurable with $m_H(\\partial V) =0$. Let $K \\subset X$ be compact and let $0<\\sigma<\\rho(K)$. \n Let $L \\subset X$ be another compact set, and let $\\eta >0$. Then there exists $t_0 := t_0(V, K, \\sigma, L, \\eta) >0$ such that, for all $t > t_0$, base point $x \\in L$ and center $y \\in K$,\n \\begin{equation*}\n m_H\\left\\lbrace h \\in V: g_thx \\in B_X(y,\\sigma) \\right\\rbrace > m_H(V) m_G\\big(B_G(\\sigma)\\big) - \\eta.\n \\end{equation*}\n\\end{prop}\n\\begin{proof}\n The proof is a modification of that of \\cite[Proposition 2.4]{Kl2}. \n By partitioning $V$ into finitely many smaller subsets with boundary of measure zero and noting the invariance of $m_H$, one can without loss of generality assume that $e_H\\in V$, $m_H(V)<1$ and $\\textup{diam}(V) < \\rho(L)$. The latter condition implies that there is an open ball $U \\subset G$ centered at $e_G$ and containing $V$ such that for all $x \\in L$ the map\n $\\pi_x$ is injective on $U$, and hence\n \\begin{equation*}\n m_X(Ux) = m_G(U).\n \\end{equation*}\nSimilarly we have, for any $0 < \\alpha < 1$ and $y \\in K$,\n \\begin{equation*}\n m_X\\big(B_X(y,\\alpha \\sigma)\\big) = m_G\\big(B_G(\\alpha \\sigma)\\big).\n \\end{equation*}\n Fix $\\alpha <1$ such that\n \\begin{equation}\\label{eq: alpha ball close}\n m_G\\big(B_G(\\alpha \\sigma)\\big) > m_G\\big(B_G(\\sigma)\\big) - \\frac{\\eta}{2}. \n \\end{equation}\n Now let $\\tilde{H}$ and $m_{\\tilde{H}}$ \n be as in Definition~\\ref{horosph}.\nChoose a nonempty open ball $\\tilde V \\subset \\tilde{H}$ such that $\\tilde VV \\subset U$ and such that for all $\\tilde v \\in \\tilde V$ and $t>0$ we have\n \\begin{equation*}\n \\dist_G\\big(e_G,\\Phi_t(\\tilde v)\\big) < (1-\\alpha)\\sigma. \\end{equation*}\n Now for $y \\in K$, $x \\in L$ and $t>0$ denote\n \\begin{equation*}\n W := \\left\\lbrace h \\in V : g_t hx \\in B_X(y,\\sigma)\\right\\rbrace\\ \\text{ and }\\ U' := \\tilde V V.\n \\end{equation*}\n \\begin{claim}\\label{claim: 99 containment}\n The set $U'x \\cap g_t^{-1} B_X(y, \\alpha \\sigma)$ is contained in $\\tilde V W x$.\n \\end{claim}\n \\begin{proof} Let $\\tilde v \\in \\tilde V$, $h \\in V$ be such that $g_t\\tilde vhx \\in B_X(y, \\alpha \\sigma)$. We have\n \\begin{equation*}\n \\begin{split}\n \\dist_X(g_t h x, y) &\\leq \\dist_X(g_t hx, \\Phi_t(\\tilde v)g_thx) + \\dist_X(g_t\\tilde vhx, y) \\\\\n &< (1-\\alpha)\\sigma + \\alpha \\sigma = \\sigma.\n \\end{split}\n \\end{equation*}\n Thus $g_thx \\in B_X(y,\\sigma)$.\n \\end{proof}\n We now use Proposition \\ref{thm: KM mixing} with \\begin{equation*}\n \\Xi := \\left\\lbrace \\mathds{1}_{U'x} : x \\in L\\right\\rbrace,\\ \n\\Psi := \\{\\mathds{1}_{B_X(y,\\alpha \\sigma}) : y \\in K\\},\n \\end{equation*} \n and with $\\eta$ replaced by $\\eta m_{\\tilde{H}}(\\tilde V)/2$.\n This gives $t_0 >0$ such that, for all $t>t_0$, $x \\in L$ and $y \\in K$, we have\n \\begin{equation*}\n \\left|m_X\\big(U'x \\cap g_t^{-1}B_X(y,\\alpha \\sigma)\\big) - m_G(U')m_G\\big(B_G(\\alpha \\sigma)\\big) \\right| < \\eta m_{\\tilde{H}}(\\tilde{V})/2.\n \\end{equation*}\n In particular, using the choice of $m_{\\tilde{H}}$ and Claim \\ref{claim: 99 containment}, we compute\n \\begin{equation*}\n \\begin{split}\n m_{\\tilde{H}}(\\tilde{V})m_H(W) &=m_X(\\tilde VWx) \\\\\n &\\geq m_X\\big(U'x \\cap g_t^{-1}B_X(y,\\alpha \\sigma)\\big) \\\\\n &\\geq m_G(U') m_G\\big(B_G(\\alpha \\sigma)\\big) - \\eta m_{\\tilde{H}}(\\tilde V)/2 \\\\\n &= m_{\\tilde{H}}(\\tilde V)m_H(V) m_G\\big(B_G(\\alpha \\sigma)\\big) - \\eta m_{\\tilde{H}}(\\tilde V)/2.\n \\end{split}\n \\end{equation*}\n Canceling $m_{\\tilde{H}}(\\tilde V)$, using \\eqref{eq: alpha ball close} and the fact that $m_H(V) < 1$, we get\n \\begin{equation*}\n \\begin{split}\n m_{H}(W) &\\geq m_H(V) m_G\\big(B_G(\\alpha \\sigma)\\big) - \\eta/2 \\\\\n &\\geq m_H(V) m_G\\big(B_G( \\sigma)\\big) - m_H(V)\\eta/2 - \\eta/2 \\\\\n &\\geq m_H(V)m_G\\big(B_G( \\sigma)\\big) - \\eta\n \\end{split}\n \\end{equation*}\n which gives the desired result.\n\\end{proof}\n\n\\begin{defn}[Strongly tree-like collections and Cantor sets]\\label{defn: tree like collection}\\label{defn: tree-like collection}\n We say a collection of subsets $\\ca{E} \\subset 2^H$ is \\textsl{strongly tree-like} if:\n \\begin{itemize}\n \\item Each $E \\in \\ca{E}$ is a compact set with nonempty interior.\n \\item We have a partition\n \\begin{equation*}\n \\ca{E} = \\bigcup_{k \\in \\Z_{\\geq 0}} \\ca{E}_k\n \\end{equation*}\n with each $\\ca{E}_k$ being finite and $\\ca{E}_0$ being a singleton.\n \\item If $E' \\in \\ca{E}_k$ with $k \\in \\N$, there is a unique $E \\in \\ca{E}_{k-1}$ with $E' \\subset E$.\n \\item For each $E \\in \\ca{E}_k$ with $k \\in \\Z_{\\geq 0}$, there exists $E' \\in \\ca{E}_{k+1}$ with $E' \\subset E$.\n \\item If $E_1, E_2 \\in \\ca{E}_k$ are distinct, then $m_H(E_1 \\cap E_2)=0$. \n \\item If we define \n \\begin{equation}\\label{eq: diam} \n d_k := \\sup\\{\\on{diam}_H(E) : E \\in \\ca{E}_k\\}, \\text{ then } \\lim_{k \\to \\infty} d_k = 0.\n \\end{equation}\n \\end{itemize}\n We write\n \\begin{equation*}\n \\cup \\ca{E}_k := \\bigcup_{E \\in \\ca{E}_k} E\n \\end{equation*}\n and define the {\\sl limit set} of the collection to be $E_\\infty := \\bigcap_{k \\in \\N} \\cup \\ca{E}_k$.\n\\end{defn}\nAssociated to a tree-like collection $\\ca{E}$ we also define, for $E \\in \\ca{E}_k$,\n\\begin{equation*}\n \\on{density}(\\ca{E}_{k+1},E) := \\frac{m_H((\\cup \\ca{E}_{k+1}) \\cap E)}{m_H(E)}.\n\\end{equation*}\nFurther, for each $k \\in \\Z_{\\geq 0}$, we set \n\\begin{equation}\\label{eq: density}\n \\Delta_k := \\inf\\left\\lbrace \\operatorname{density}(\\mathcal{E}_{{k+1}}, E) : E \\in \\mathcal{E}_k \\right\\rbrace\n\\end{equation}\n\\begin{thm}[Lemma 2.1 in \\cite{U}]\\label{thm: urbanski}\n For a tree-like collection $\\ca{E}$ and the resulting limit set $E_\\infty$, we have the dimension estimate\n \\begin{equation*}\n \\dim H - \\dim E_\\infty \\leq \\limsup_{k \\to \\infty} \\frac{\\sum_{j=0}^{k} \\log \\Delta_j}{\\log d_k}.\n \\end{equation*}\n\\end{thm}\n\nConversely, let us assume that $x_\\alpha \\in E(F_+, K_i)$. Then, by the compactness of $K_i$ and by Mahler's compactness criterion, there exists an $r$ with $0< r < \\sqrt{r_0}$ and a time $s_0 >0$ such that for all $s > s_0$,\n \\begin{equation*}\n g_s x_\\alpha \\notin K_i(r).\n \\end{equation*}\n In particular, \n \\begin{equation*}\n x_\\alpha \\cap \\left\\lbrace (z,w)\\in \\Co^2: |z| \\leq e^{-s}r, |w| \\leq e^{s}r \\right\\rbrace \\neq \\{(0,0)\\}.\n \\end{equation*}\n In particular, $\\alpha \\in \\DI_i$ with constant $\\frac{r^2}{r_0}$.\n\\end{proof}\nThis dynamical perspective on $\\DI_i$ allows one to effortlessly establish certain properties of $\\DI_i$ using existing techniques in ergodic theory and dynamics: the question of measure and dimension. (The interested reader may consult \\cite[Theorem 1.5]{KW2} and \\cite[Theorem 2.8]{AGK} where very general dynamical analogs of these questions are considered.)\nIn particular, we may use the above dynamical results to conclude that $\\DI_i$ has measure zero but full Hausdorff dimension.\nWith respect to this current article, we are primarily concerned with the relation between Dirichlet improvability and bad approximability (cf. \\cite[Theorems 1 and 2]{DS}).\nConsider the set of badly approximable complex numbers with respect to $\\Z[i]$; those $\\alpha \\in \\Co$ for which there is a constant $c >0$ such that for all nonzero $q \\in \\Z[i]$,\n\\begin{equation*}\n |q| \\min_{p \\in \\Z[i]} |q\\alpha - p| >c.\n\\end{equation*}\nWe write this set as $\\BA_i$.\n\\begin{ques}\\label{ques: BA DI i}\n What is the relationship between Dirichlet improvability and bad approximabilty in the case of the Gaussian integers? Is it true that $\\BA_i$ is a subset of $\\DI_i$?\n\\end{ques}\nWe are able to provide a partial answer to this question as follows. However, it seems that a complete answer might require a further study of the set $K_i$ in Proposition \\ref{prop: DIi dynamical}.", + "post_theorem_intro_text_len": 4162, + "post_theorem_intro_text": "Here and hereafter \"dim\" stands for \\hd.\n In fact, even though it was not mentioned in \\cite{KR1}, the proof actually demonstrates the thickness of the set in the left-hand side of \\eqref{dim1}. Also, by the standard \"slicing\" argument (see \\S \\ref{sec: reduction to H}), it can be easily derived from Theorem \\ref{main-lemma} \nthat for any $y \\in E(F,\\infty)$ the set $A(F_+,y) \\cap E(F_+,\\infty)$ is thick.\nThe proof of the above theorem used the technique of continued fractions, and in fact the result can be interpreted within the framework of Diophantine approximation. Namely it constructs numbers that are badly approximable but not $\\nu$-Dirichlet improvable, where $\\nu$ is an irreducible norm on $\\R^2$. See \\cite{KR, KR1, KRS, AD} for more detail, and \\cite{MS} for some new results in that direction. \n\n\\smallskip\nOne of the main results of this paper is a generalization of Theorem \\ref{main-lemma} to a wider class of flows on \\hs s. \n\n\\begin{thm}\\label{thm: main theorem}\n Let $X = G/\\Gamma$ and $F = \\{g_t : t\\in\\R\\}\\subset G$, where \n \\begin{itemize}\n \\item $G$ is a Lie group with discrete center,\n \\item $\\Gamma$ is a non-uniform \n lattice in $G$,\n\\item $g_t$ is as in \\eqref{gt}, where $a_0 \\in \\fg\\smallsetminus\\{0\\}$ is $\\ad$-diagonalizable over $\\Co$;\n \\item the action of $F$ on $(X,m_X)$ is mixing.\n\\end{itemize}\n Let $y \\in E(F,\\infty)$. Then for any non-empty open $U\\subset X$ we have \n \\begin{equation}\\label{eq: dim X - dim Z + 1}\n \\dim \\big(U\\cap A(F_+,y) \\cap E(F_+,\\infty)\\big) \\geq \\dim X - \\dim Z + 1,\n \\end{equation}\n where $Z \\subset G$ is the {neutral subgroup of $G$ with respect to $F$} . \n\\end{thm}\nSee Definition \\ref{defn: expanding, contracting and neutral groups} below for a precise definition of $Z$.\nIn view of Moore's Theorem \\cite{Mo}, the mixing assumption is satisfied when $G$ is a connected semi-simple \n Lie group\n with finite center and with no compact factors, and $\\Gamma\\subset G$ is irreducible. However mixing is not restricted to the semi-simple case, see e.g.\\ \\cite{BM, Kl}. \nNote that when $G = \\SL_2(\\R)$ (that is, under the assumptions of Theorem \\ref{main-lemma}) we know that {$\\dim Z=\\dim F = 1$}, and thus the set $A(F_+,y) \\cap E(F_+,\\infty)$ is thick. Moreover, in the latter case one can generalize Theorem~\\ref{main-lemma} as follows.\n\n \\begin{thm}\\label{thm: dense-orbits} \nLet $G$, $\\Gamma$ and $F$ be as in Theorem \\ref{thm: main theorem}, and assume in addition that \\linebreak {$\\dim Z = \n1$}.\nThen for any {compact} \nF-invariant\nsubset $B$ of $E(F,\\infty)$ there exists a thick set of $x \\in E(F_+,\\infty)$ such that the closure of the (bounded) trajectory $F_+x$ contains $B$.\n \\end{thm}\n\nIn addition to $G = \\SL_2(\\R)$, the assumptions of the above theorem hold when \\linebreak $G = \\SL_2(\\R)\\ltimes \\R^2$; the choice $\\Gamma = \\SL_2(\\Z)\\ltimes \\Z^2$ yields the space of two-dimensional unimodular grids, cf.\\ \\cite{Kl}.\n \\begin{cor}\\label{cor: dim-drop} \nLet $G$, $\\Gamma$, $X$ and $F$ be as in Theorem \\ref{thm: dense-orbits}; then for any $\\varepsilon > 0$ there exists a thick set of $x\\in X$ such that the orbit closure $\\overline{F^+x}$ is compact and has Hausdorff dimension greater than $\\dim X - \\varepsilon$.\n \\end{cor}\n \\begin{proof}\n When $\nX$ and $F$ are as in Theorem \\ref{thm: main theorem}, it follows from \n \\cite[Theorem 1.1]{KM} that, given $\\varepsilon>0$, one can find a large enough compact set $K \\subset X$ such that the compact $F$-invariant set\n \\begin{equation*}\n B = \\{x \\in X: Fx \\subset K\\}\n \\end{equation*}\n has Hausdorff dimension greater than $\\dim X - \\varepsilon$.\nUnder the additional hypothesis of $\\dim Z = 1$, an application of Theorem \\ref{thm: dense-orbits} to $B$ then gives the result.\n \\end{proof}\n\n \\noindent{\\bf Acknowledgements.}\n The second-named author is grateful to ETH (Z\\\"urich) for its hospitality during a sabbatical stay in 2022, and to Barak Weiss for helpful discussions. The third-named author thanks N.\\ Chandgotia, T.\\ Mesikepp and Y.\\ Peres for holding public-access office hours at TIFR-CAM, Peking University and Tsinghua University respectively.", + "sketch": "The post-theorem discussion says that (although not stated in \\cite{KR1}) “the proof actually demonstrates the thickness of the set in the left-hand side of \\eqref{dim1}.” It also notes that “by the standard ‘slicing’ argument (see \\S \\ref{sec: reduction to H}), it can be easily derived from Theorem \\ref{main-lemma} that for any $y\\in E(F,\\infty)$ the set $A(F_+,y)\\cap E(F_+,\\infty)$ is thick.” Finally, it describes the method/interpretation: “The proof of the above theorem used the technique of continued fractions,” and “can be interpreted within the framework of Diophantine approximation,” namely constructing “numbers that are badly approximable but not $\\nu$-Dirichlet improvable, where $\\nu$ is an irreducible norm on $\\R^2$.”", + "expanded_sketch": "The post-theorem discussion says that (although not stated in \n\nKenyon, R. and Ralston, J., \\emph{[Theorem 1.3]} (KR1)\n\n) “the proof actually demonstrates the thickness of the set in the left-hand side of\n\\begin{equation}\\label{dim1}\n \\dim\\big(\\{ \\alpha \\in \\R : x_\\alpha \\in A(F_+,y) \\cap E(F_+,\\infty)\\}\\big) = 1.\n\\end{equation}\n.” It also notes that “by the standard ‘slicing’ argument (see \\S \\ref{sec: reduction to H}), it can be easily derived from the main theorem that for any $y\\in E(F,\\infty)$ the set $A(F_+,y)\\cap E(F_+,\\infty)$ is thick.” Finally, it describes the method/interpretation: “The proof of the above theorem used the technique of continued fractions,” and “can be interpreted within the framework of Diophantine approximation,” namely constructing “numbers that are badly approximable but not $\\nu$-Dirichlet improvable, where $\\nu$ is an irreducible norm on $\\R^2$.”", + "expanded_theorem": "\\label{main-lemma} \\cite[Theorem 1.3]{KR1}\nLet $X$ and $F$ be as above, and let $y \\in E(F,\\infty)$; then\n\\begin{equation}\\label{dim1}\n \\dim\\big(\\{ \\alpha \\in \\R : x_\\alpha \\in A(F_+,y) \\cap E(F_+,\\infty)\\}\\big) = 1.\n\\end{equation},", + "theorem_type": "unknown", + "mcq": { + "question": "Let \n\\(X=\\mathrm{SL}_2(\\mathbb R)/\\mathrm{SL}_2(\\mathbb Z)\\), let \n\\(g_t=\\begin{pmatrix}e^t&0\\\\0&e^{-t}\\end{pmatrix}\\), and let \n\\(F=\\{g_t:t\\in\\mathbb R\\}\\) with \\(F_+=\\{g_t:t\\ge 0\\}\\). For \\(\\alpha\\in\\mathbb R\\), write \n\\(x_\\alpha=u_\\alpha\\,\\mathrm{SL}_2(\\mathbb Z)\\) where \n\\(u_\\alpha=\\begin{pmatrix}1&\\alpha\\\\0&1\\end{pmatrix}\\). For \\(y\\in X\\), define\n\\(A(F_+,y)=\\{x\\in X: g_{t_n}x\\to y\\text{ for some unbounded sequence }t_n\\ge 0\\}\\), and let \\(E(F_+,\\infty)\\) denote the set of points whose positive \\(F_+\\)-orbit does not approach the point at infinity in the one-point compactification of \\(X\\) (equivalently, whose positive orbit is bounded in \\(X\\)). If \\(y\\in E(F,\\infty)\\), i.e. the full \\(F\\)-orbit of \\(y\\) is bounded, which statement holds about the Hausdorff dimension of\n\\[\n\\{\\alpha\\in\\mathbb R: x_\\alpha\\in A(F_+,y)\\cap E(F_+,\\infty)\\}?\n\\]", + "correct_choice": { + "label": "A", + "text": "Its Hausdorff dimension is exactly \\(1\\):\n\\[\n\\dim\\big(\\{\\alpha\\in\\mathbb R: x_\\alpha\\in A(F_+,y)\\cap E(F_+,\\infty)\\}\\big)=1.\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "Its Hausdorff dimension is exactly \\(0\\); in fact,\n\\[\n\\dim\\big(\\{\\alpha\\in\\mathbb R: x_\\alpha\\in A(F_+,y)\\cap E(F_+,\\infty)\\}\\big)=0.\n\\]" + }, + { + "label": "C", + "text": "Its Hausdorff dimension is at least \\(0\\) and at most \\(1\\); in particular, the set is a subset of \\(\\mathbb R\\) and may be nonempty, but no sharper dimension conclusion is asserted here." + }, + { + "label": "D", + "text": "Its Hausdorff dimension is exactly \\(1\\) provided \\(y\\in E(F_+,\\infty)\\); that is,\n\\[\n\\dim\\big(\\{\\alpha\\in\\mathbb R: x_\\alpha\\in A(F_+,y)\\cap E(F_+,\\infty)\\}\\big)=1\n\\]\nfor every point whose positive \\(F_+\\)-orbit is bounded." + }, + { + "label": "E", + "text": "The set has full Lebesgue measure in \\(\\mathbb R\\), and hence in particular Hausdorff dimension \\(1\\):\n\\[\nm\\big(\\{\\alpha\\in\\mathbb R: x_\\alpha\\in A(F_+,y)\\cap E(F_+,\\infty)\\}\\big)=1.\n\\]" + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "B" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "thickness/full-dimension conclusion replaced by zero-dimension exceptional-set heuristic", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped the sharp equality \\(=1\\), keeping only the trivial ambient-dimension bound", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "hypothesis weakened from full-orbit boundedness \\(y\\in E(F,\\infty)\\) to only forward boundedness \\(y\\in E(F_+,\\infty)\\)", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "dimension/thickness conclusion strengthened to full Lebesgue measure", + "template_used": "stronger_trap" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not state the conclusion or embed the correct dimension claim. It gives the full setup and hypotheses, but the correct answer is not leaked beyond the natural theorem-style framing." + }, + "TAS": { + "score": 1, + "justification": "This is close to a theorem-recall item: the stem presents the hypotheses and asks for the conclusion. However, it is not a pure restatement because the options introduce meaningful nearby variants (weaker, stronger, and hypothesis-altered claims)." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to reject plausible alternatives such as the weaker trivial bound, the stronger full-measure claim, and the subtly weakened hypothesis in choice D. Still, the item mainly tests recognition of the sharp theorem statement rather than substantial derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically targeted: zero dimension reflects an exceptional-set heuristic, choice C is a weaker true-but-uninformative bound, choice D weakens the hypothesis in a subtle way, and choice E improperly strengthens dimension to full measure." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and little answer leakage, though it leans more toward precise recall/recognition than deep generative reasoning." + } + }, + { + "id": "2511.13976v1", + "paper_link": "http://arxiv.org/abs/2511.13976v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:main1}\nFor each $n \\ge 10$, the Torelli group of $X_n = 2\\mathbb{CP}^2 \\# n \\overline{\\mathbb{CP}^2}$ admits a surjective homomorphism $\\Phi : T(X_n) \\to \\mathbb{Z}^\\infty$.", + "start_pos": 6220, + "end_pos": 6433, + "label": "thm:main1" + }, + "ref_dict": { + "thm:main1": "\\begin{theorem}\\label{thm:main1}\nFor each $n \\ge 10$, the Torelli group of $X_n = 2\\mathbb{CP}^2 \\# n \\overline{\\mathbb{CP}^2}$ admits a surjective homomorphism $\\Phi : T(X_n) \\to \\mathbb{Z}^\\infty$.\n\\end{theorem}", + "thm:main2": "\\begin{theorem}\\label{thm:main2}\nThe mapping class group of $X = 2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ is not finitely generated. More precisely, there is an index $2$-subgroup $M_+(X) \\subseteq M(X)$ and a surjective homomorphism $\\Phi : M_+(X) \\to \\mathbb{Z}^\\infty$.\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 2543, + "pre_theorem_intro_text": "Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\\em mapping class group} $M(X)$ of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\\em Torelli group} $T(X) \\subseteq M(X)$ is the subgroup of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. Non-trivial elements of $T(X)$ represent exotic diffeomorphisms in the sense that they are continuously isotopic to the identity but not smoothly isotopic. Thus, the Torelli group $T(X)$ can be seen as a measure of the difference between continuous and smooth isotopy.\n\nIt was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \\cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \\ge 2$ and $m \\ge 10n+1$, the Torelli group of $2n \\mathbb{CP}^2 \\# m \\overline{\\mathbb{CP}^2}$, surjects to $\\mathbb{Z}^\\infty$ (where $\\mathbb{Z}^\\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \\cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \\mathbb{CP}^2 \\# 10n \\overline{\\mathbb{CP}^2}$ for each odd $n \\ge 3$ and also for for $n(S^2 \\times S^2) \\# nK3$ for each $n \\ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$. \n\nThe main goal of this paper is to extend the methods of \\cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \\cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:", + "context": "Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\\em mapping class group} $M(X)$ of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\\em Torelli group} $T(X) \\subseteq M(X)$ is the subgroup of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. Non-trivial elements of $T(X)$ represent exotic diffeomorphisms in the sense that they are continuously isotopic to the identity but not smoothly isotopic. Thus, the Torelli group $T(X)$ can be seen as a measure of the difference between continuous and smooth isotopy.\n\nIt was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \\cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \\ge 2$ and $m \\ge 10n+1$, the Torelli group of $2n \\mathbb{CP}^2 \\# m \\overline{\\mathbb{CP}^2}$, surjects to $\\mathbb{Z}^\\infty$ (where $\\mathbb{Z}^\\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \\cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \\mathbb{CP}^2 \\# 10n \\overline{\\mathbb{CP}^2}$ for each odd $n \\ge 3$ and also for for $n(S^2 \\times S^2) \\# nK3$ for each $n \\ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$.\n\nThe main goal of this paper is to extend the methods of \\cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \\cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:", + "full_context": "Let $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\\em mapping class group} $M(X)$ of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\\em Torelli group} $T(X) \\subseteq M(X)$ is the subgroup of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. Non-trivial elements of $T(X)$ represent exotic diffeomorphisms in the sense that they are continuously isotopic to the identity but not smoothly isotopic. Thus, the Torelli group $T(X)$ can be seen as a measure of the difference between continuous and smooth isotopy.\n\nIt was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \\cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \\ge 2$ and $m \\ge 10n+1$, the Torelli group of $2n \\mathbb{CP}^2 \\# m \\overline{\\mathbb{CP}^2}$, surjects to $\\mathbb{Z}^\\infty$ (where $\\mathbb{Z}^\\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \\cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \\mathbb{CP}^2 \\# 10n \\overline{\\mathbb{CP}^2}$ for each odd $n \\ge 3$ and also for for $n(S^2 \\times S^2) \\# nK3$ for each $n \\ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$.\n\nThe main goal of this paper is to extend the methods of \\cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \\cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:\n\n\\begin{abstract}\nLet $X$ be a compact, oriented, smooth, simply-connected $4$-manifold. The {\\em mapping class group} of $X$ is defined as the group of smooth isotopy classes of diffeomorphisms of $X$. The {\\em Torelli group} of $X$ is the subgroup of the mapping class group consisting of smooth isotopy classes of diffeomorphisms which are continuously isotopic to the identity. We prove that for each $n \\ge 10$, the Torelli group of $2\\mathbb{CP}^2 \\# n \\overline{\\mathbb{CP}^2}$ surjects to $\\mathbb{Z}^\\infty$. We also prove that the mapping class group of $2 \\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ is not finitely generated. Our proofs of these results makes use of Seiberg--Witten invariants for $1$-parameter familes of $4$-manifolds and in particular a gluing formula for connected sum families. Since the manifolds we consider have $b_+ = 2$, the chamber structure of the $1$-parameter Seiberg--Witten invariants plays an important role.\n\\end{abstract}\n\nIt was first established by Ruberman that the Torelli group of a compact, simply-connected $4$-manifold can be non-trivial \\cite{rub1,rub2}. Ruberman constructed versions of the Donaldson and Seiberg--Witten invariants for $1$-parameter families of $4$-manifolds and used these invariants to detect non-triviality of elements of the Torelli group. More specifically, his invariant can be used to show that for every $n \\ge 2$ and $m \\ge 10n+1$, the Torelli group of $2n \\mathbb{CP}^2 \\# m \\overline{\\mathbb{CP}^2}$, surjects to $\\mathbb{Z}^\\infty$ (where $\\mathbb{Z}^\\infty$ denotes a free abelian group of countably infinite rank). Baraglia--Konno proved a more general gluing formula for the families Seiberg--Witten invariants \\cite{bk1}, which for example can be used to show a similar result for the Torelli groups of $2n \\mathbb{CP}^2 \\# 10n \\overline{\\mathbb{CP}^2}$ for each odd $n \\ge 3$ and also for for $n(S^2 \\times S^2) \\# nK3$ for each $n \\ge 1$. In all of these cases the parity of $b_+$ must be even due to a dimension constraint and there is a further constraint that $b_+ > 2$ so as to avoid the presence of any chamber structure for the families Seiberg--Witten invariants. Thus the smallest possible value of $b_+$ allowed by these methods is $b_+ = 4$.\n\nThe main goal of this paper is to extend the methods of \\cite{rub1,rub2,bk1} to the case $b_+ = 2$, using $1$-parameter Seiberg--Witten invariants in the presence of chambers. In a series of papers \\cite{tom1,tom2}, the second author has proven a general gluing formula for the families Seiberg--Witten invariants which can be applied even for low values of $b_+$ where chambers are present. The results in this paper are obtained by applying this more general gluing formula to the case of $1$-parameter families. Our first main result is that the Torelli groups of certain simply-connected $4$-manifolds with $b_+ = 2$ can be infinitely generated:\n\nPrior to this result it had been shown by Konno--Mallick--Taniguchi \\cite{kmt} that $2\\mathbb{CP}^2 \\# m \\overline{\\mathbb{CP}^2}$ has an exotic diffeomorphism for each $m \\ge 11$. Qiu proves that $2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ admits an exotic diffeomorphism with infinite order in the Torelli group \\cite{qiu}. Qiu's method is similar to the one used in this paper, although the conclusion is not as strong.\n\n\\begin{theorem}\\label{thm:main2}\nThe mapping class group of $X = 2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ is not finitely generated. More precisely, there is an index $2$-subgroup $M_+(X) \\subseteq M(X)$ and a surjective homomorphism $\\Phi : M_+(X) \\to \\mathbb{Z}^\\infty$.\n\\end{theorem}\n\n\\begin{theorem}\nFor each $n \\ge 10$, the Torelli group of $X_n = 2\\mathbb{CP}^2 \\# n \\overline{\\mathbb{CP}^2}$ admits a surjective homomorphism $\\Phi : T(X_n) \\to \\mathbb{Z}^\\infty$ to a free abelian group of countably infinite rank.\n\\end{theorem}\n\\begin{proof}\nLet $\\mathbb{Z}[\\mathcal{S}(X_n)] = \\bigoplus_{\\mathfrak{s} \\in \\mathcal{S}(X_n)} \\mathbb{Z}$. Elements of $\\mathbb{Z}[\\mathcal{S}(X_n)]$ can be regarded as functions $\\mathcal{S}(X_n) \\to \\mathbb{Z}$ with finite support. Define a homomorphism $\\Phi' : T(X_n) \\to \\bigoplus_{\\mathfrak{s} \\in \\mathcal{S}(X_n) } \\mathbb{Z}$ by taking $(\\Phi'(f))(\\mathfrak{s}) = SW^c_{X_n , \\mathfrak{s} , \\mathbb{Z}}(f)$. The function $\\mathfrak{s} \\mapsto SW^c_{X_n , \\mathfrak{s} , \\mathbb{Z}}(f)$ has finite support because of the compactness properties of the Seiberg--Witten equations.\n\nThe image $im(\\Phi')$ of $\\Phi'$ is a subgroup of the free abelian group $\\mathbb{Z}[\\mathcal{S}(X_n)]$ and hence is free abelian. Let $\\Phi : T(X_n) \\to im(\\Phi')$ be the homomorphism obtained by factoring $\\Phi'$ through its image. The theorem will follow if we can show that $im(\\Phi')$ has infinite rank (since $\\mathbb{Z}[\\mathcal{S}(X_n)]$ has countably infinite rank, the rank of $im(\\Phi')$ can be at most countably infinite). For $n=10$, Lemma \\ref{lem:torelli} immediately implies that $im(\\Phi')$ has countably infinite rank. For $n > 10$ we use induction on $n$ and the blow-up formula. Write $X_n = X_{n-1} \\# \\overline{\\mathbb{CP}^2}$. Let $f \\in T(X_{n-1})$. We can isotopy $f$ so that it is the identity in a neighbourhood of a point. Then we can form the connected sum diffeomorphism $f \\# id_{\\overline{\\mathbb{CP}^2}} \\in T(X_n)$. Let $\\kappa$ be a spin$^c$-structure on $\\overline{\\mathbb{CP}^2}$ with $c(\\kappa)^2 = -1$. Let $\\mathfrak{s} \\in \\mathcal{S}(X_{n-1})$. Then $\\mathfrak{s} \\# \\kappa \\in \\mathcal{S}(X_n)$. The blowup formula gives\n\\[\nSW^c_{X_n , \\mathfrak{s} \\# \\kappa , \\mathbb{Z}}(f \\# id_{\\overline{\\mathbb{CP}^2}}) = SW^c_{X_{n-1} , \\mathfrak{s} , \\mathbb{Z}}(f).\n\\]\nHence there are infinitely many spin$^c$-structures on $X_n$ for which $SW^c_{X_n , \\mathfrak{s} , \\mathbb{Z}} : T(X_n) \\to \\mathbb{Z}$ is non-zero and hence $\\Phi'( T(X_n) )$ has infinite rank.\n\\end{proof}\n\n\\begin{theorem}\nThe mapping class group of $X = 2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ is not finitely generated. More precisely there is a surjective homomorphism $\\Phi : M_+(X) \\to \\mathbb{Z}^\\infty$.\n\\end{theorem}\n\\begin{proof}\nDefine $\\Phi' : M_+(X) \\to \\mathbb{Z}^\\infty$ to be given by\n\\[\n\\Phi' = \\bigoplus_{n=1}^{\\infty} SW_{X , \\mathcal{O}_{2n-1}} : M_+(X) \\to \\bigoplus_{n=1}^{\\infty} \\mathbb{Z}.\n\\]\nTo prove the result, it suffices to show that the image of $\\Phi'$ is not finitely generated. Then we let $\\Phi$ be the homomorphism obtained by replacing the codomain of $\\Phi'$ with the image of $\\Phi'$. The argument is almost identical to the proof of \\cite[Theorem 3.1]{bar}, except that now we use $E(1)_{2,2n+1}$ in place of $E(n)_q$.\n\\end{proof}", + "post_theorem_intro_text_len": 4933, + "post_theorem_intro_text": "Prior to this result it had been shown by Konno--Mallick--Taniguchi \\cite{kmt} that $2\\mathbb{CP}^2 \\# m \\overline{\\mathbb{CP}^2}$ has an exotic diffeomorphism for each $m \\ge 11$. Qiu proves that $2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ admits an exotic diffeomorphism with infinite order in the Torelli group \\cite{qiu}. Qiu's method is similar to the one used in this paper, although the conclusion is not as strong.\n\nOur second main result concerns infinite generation of the mapping class group $M(X)$ (by infinite generation, we mean to say that $M(X)$ is not finitely generated). Note that infinite generation of $M(X)$ does not follow from infinite generation of $T(X)$, because a finitely generated group can have infinitely generated subgroups.\n\n\\begin{theorem}\\label{thm:main2}\nThe mapping class group of $X = 2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$ is not finitely generated. More precisely, there is an index $2$-subgroup $M_+(X) \\subseteq M(X)$ and a surjective homomorphism $\\Phi : M_+(X) \\to \\mathbb{Z}^\\infty$.\n\\end{theorem}\n\nNote that by Schreier's lemma \\cite{ser} every finite index subgroup of a finitely generated group is finitely generated. So the infinite generation of $M_+(X)$ implies the infinite generation of $M(X)$.\n\nThe first examples of compact, simply-connected $4$-manifolds whose mapping class groups are not finitely generated were given by Baraglia \\cite{bar} and Konno \\cite{kon}, namely $M(X)$ is not finitely generated for $X = 2n \\mathbb{CP}^2 \\# 10n \\overline{\\mathbb{CP}^2}$ where $n \\ge 3$ is odd and also for $X = n (S^2 \\times S^2) \\# nK3$, $n \\ge 1$. Note that these are precisely the manifolds of the form $E(m) \\# (S^2 \\times S^2)$, $m \\ge 2$. Theorem \\ref{thm:main2} says the same result is true for the case $m=1$, that is, for $X = E(1) \\# (S^2 \\times S^2) = 2\\mathbb{CP}^2 \\# 10 \\overline{\\mathbb{CP}^2}$.\n\n\\subsection{Outline of the proofs of the main results}\nTo each diffeomorphism $f \\in M(X)$, one can form the mapping cylinder $E(f)$. This is the $1$-parameter family of $4$-manifolds obtained from $[0,1] \\times X$ by identifying the ends via $f$. If $\\mathfrak{s}$ is a spin$^c$-structure which is preserved by $f$ and for which the expected dimension of the families Seiberg--Witten moduli space for $E(f)$ is zero, then one obtains a numerical invariant by counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$. When $b_+ = 2$, this invariant depends on the choice of chamber and so is not strictly an invariant of $f$ alone. However under certain circumstances we find that a distinguished choice of chamber exists, and so we obtain invariants. More specifically, there are two cases that we consider:\n\\begin{itemize}\n\\item[(1)]{{\\bf The constant chamber:} assume that $f \\in T(X)$. Then $f$ acts trivially on $H^2(X ; \\mathbb{R})$ and so the local system over $S^1$ whose fibres are $H^2$ of the fibres of $E(f)$ has trivial monodromy. This leads to a trivialisation (unique up to homotopy) of the bundle $\\mathcal{H}^+ \\to S^1$ whose fibres are $H^+$ of the fibres of $E(f)$. The constant chamber is the chamber which corresponds to the homotopy class of a constant section of $\\mathcal{H}^+$ under the above trivialisation.}\n\\item[(2)]{{\\bf The zero chamber:} assume that $c(\\mathfrak{s})^2 \\ge 0$ and $c(\\mathfrak{s})$ is not torsion. Then there is a well-defined chamber corresponding to taking the self-dual $2$-form perturbation of the Seiberg--Witten equations to be zero.}\n\\end{itemize}\n\nThese two chambers are shown to coincide when they are both defined. Corresponding to the constant and zero chambers are families Seiberg--Witten invariants $SW^c_{X , \\mathfrak{s}}(f)$ and $SW^0_{X , \\mathfrak{s}}(f)$ depending only on $(X , \\mathfrak{s})$ and the isotopy class of $f$. In particular, the constant chamber invariants define homomorphisms $SW^c_{X , \\mathfrak{s}} : T(X) \\to \\mathbb{Z}$. Compactness properties of the Seiberg--Witten equations implies that for any given $f \\in T(X)$, the invariants $SW^c_{X , \\mathfrak{s}}(f)$ are non-zero for only finitely many spin$^c$-structures. Thus we obtain a homomorphism\n\\[\n\\Phi : T(X) \\to \\bigoplus_{\\mathfrak{s}} \\mathbb{Z}\n\\]\nwhere the sum is over spin$^c$-structures for which the corresponding families Seiberg--Witten moduli space is zero dimensional. The proof of Theorem \\ref{thm:main1} follows by showing that the image of $\\Phi$ has infinite rank. For this we need to construct an infinite sequence of spin$^c$-structures $\\{ \\mathfrak{s}_n\\}$ and diffeomorphisms $\\{ t_n \\}$ for which $SW^c_{X , \\mathfrak{s}_n}( t_n ) \\neq 0$. The diffeomorphisms $t_n$ are constructed in a similar fashion to \\cite{rub1}, \\cite{bk1}, making use of diffeomorphisms $E(1)_{p,q} \\# (S^2 \\times S^2) \\cong E(1) \\# (S^2 \\times S^2)$.\n\n\\noindent{\\bf Acknowledgments.} D. Baraglia was financially supported by an Australian Research Council Future Fellowship, FT230100092.", + "sketch": "To prove Theorem~\\ref{thm:main1}, the paper associates to each diffeomorphism $f\\in M(X)$ its mapping cylinder $E(f)$ (a $1$-parameter family over $S^1$), and—when $f$ preserves a spin$^c$ structure $\\mathfrak{s}$ with zero expected families Seiberg--Witten dimension—defines a numerical invariant by “counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$.” Since $b_+=2$ this count depends on a chamber, but the authors identify circumstances giving a distinguished chamber:\n\n(1) **Constant chamber:** if $f\\in T(X)$, then $f$ acts trivially on $H^2(X;\\mathbb R)$ so the local system over $S^1$ has trivial monodromy, yielding a (homotopically unique) trivialisation of $\\mathcal H^+\\to S^1$; the constant chamber corresponds to a constant section.\n\n(2) **Zero chamber:** if $c(\\mathfrak{s})^2\\ge 0$ and $c(\\mathfrak{s})$ is non-torsion, there is a chamber given by taking the self-dual $2$-form perturbation to be zero.\n\nThey show these chambers “coincide when they are both defined,” producing invariants $SW^c_{X,\\mathfrak{s}}(f)$ and $SW^0_{X,\\mathfrak{s}}(f)$ depending only on $(X,\\mathfrak{s})$ and the isotopy class of $f$. In particular, the constant-chamber invariants give homomorphisms $SW^c_{X,\\mathfrak{s}}:T(X)\\to\\mathbb Z$. By compactness, for fixed $f\\in T(X)$ only finitely many $\\mathfrak{s}$ contribute nontrivially, so one gets a combined homomorphism\n\\[\n\\Phi:T(X)\\to\\bigoplus_{\\mathfrak{s}}\\mathbb Z,\n\\]\n(summing over zero-dimensional families moduli spaces). The proof of Theorem~\\ref{thm:main1} then “follows by showing that the image of $\\Phi$ has infinite rank,” by constructing an infinite sequence of spin$^c$-structures $\\{\\mathfrak{s}_n\\}$ and diffeomorphisms $\\{t_n\\}$ with $SW^c_{X,\\mathfrak{s}_n}(t_n)\\ne 0$. The $t_n$ are built “in a similar fashion to \\cite{rub1}, \\cite{bk1},” using diffeomorphisms $E(1)_{p,q}\\#(S^2\\times S^2)\\cong E(1)\\#(S^2\\times S^2)$.", + "expanded_sketch": "To prove the main theorem, the paper associates to each diffeomorphism $f\\in M(X)$ its mapping cylinder $E(f)$ (a $1$-parameter family over $S^1$), and—when $f$ preserves a spin$^c$ structure $\\mathfrak{s}$ with zero expected families Seiberg--Witten dimension—defines a numerical invariant by “counting the number of solutions of the Seiberg--Witten equations for the family $E(f)$.” Since $b_+=2$ this count depends on a chamber, but the authors identify circumstances giving a distinguished chamber:\n\n(1) **Constant chamber:** if $f\\in T(X)$, then $f$ acts trivially on $H^2(X;\\mathbb R)$ so the local system over $S^1$ has trivial monodromy, yielding a (homotopically unique) trivialisation of $\\mathcal H^+\\to S^1$; the constant chamber corresponds to a constant section.\n\n(2) **Zero chamber:** if $c(\\mathfrak{s})^2\\ge 0$ and $c(\\mathfrak{s})$ is non-torsion, there is a chamber given by taking the self-dual $2$-form perturbation to be zero.\n\nThey show these chambers “coincide when they are both defined,” producing invariants $SW^c_{X,\\mathfrak{s}}(f)$ and $SW^0_{X,\\mathfrak{s}}(f)$ depending only on $(X,\\mathfrak{s})$ and the isotopy class of $f$. In particular, the constant-chamber invariants give homomorphisms $SW^c_{X,\\mathfrak{s}}:T(X)\\to\\mathbb Z$. By compactness, for fixed $f\\in T(X)$ only finitely many $\\mathfrak{s}$ contribute nontrivially, so one gets a combined homomorphism\n\\[\n\\Phi:T(X)\\to\\bigoplus_{\\mathfrak{s}}\\mathbb Z,\n\\]\n(summing over zero-dimensional families moduli spaces). The proof of the main theorem then “follows by showing that the image of $\\Phi$ has infinite rank,” by constructing an infinite sequence of spin$^c$-structures $\\{\\mathfrak{s}_n\\}$ and diffeomorphisms $\\{t_n\\}$ with $SW^c_{X,\\mathfrak{s}_n}(t_n)\\ne 0$. The $t_n$ are built “in a similar fashion to \\cite{rub1}, \\cite{bk1},” using diffeomorphisms $E(1)_{p,q}\\#(S^2\\times S^2)\\cong E(1)\\#(S^2\\times S^2)$.,", + "expanded_theorem": "\\label{thm:main1}\nFor each $n \\ge 10$, the Torelli group of $X_n = 2\\mathbb{CP}^2 \\# n \\overline{\\mathbb{CP}^2}$ admits a surjective homomorphism $\\Phi : T(X_n) \\to \\mathbb{Z}^\\infty$.", + "theorem_type": [ + "Universal–Existential", + "Existence" + ], + "mcq": { + "question": "For each integer $n\\ge 10$, let $X_n=2\\mathbb{CP}^2\\# n\\overline{\\mathbb{CP}^2}$. Here the mapping class group $M(X_n)$ is the group of smooth isotopy classes of diffeomorphisms of $X_n$, and the Torelli group $T(X_n)\\subseteq M(X_n)$ consists of those smooth isotopy classes represented by diffeomorphisms that are continuously isotopic to the identity. Which statement holds for $T(X_n)$?", + "correct_choice": { + "label": "A", + "text": "For every $n\\ge 10$, there exists a surjective homomorphism $\\Phi:T(X_n)\\to \\mathbb{Z}^\\infty$, where $\\mathbb{Z}^\\infty$ denotes a free abelian group of countably infinite rank." + }, + "choices": [ + { + "label": "B", + "text": "For every $n\\ge 10$, there exists a homomorphism $\\Phi:T(X_n)\\to \\mathbb{Z}^\\infty$ whose image has finite index in $\\mathbb{Z}^\\infty$." + }, + { + "label": "C", + "text": "For every $n\\ge 10$, there exists a homomorphism $\\Phi:T(X_n)\\to \\mathbb{Z}^\\infty$ with infinite image." + }, + { + "label": "D", + "text": "There exists a single surjective homomorphism $\\Phi$ such that for every $n\\ge 10$, one has $\\Phi:T(X_n)\\to \\mathbb{Z}^\\infty$." + }, + { + "label": "E", + "text": "For every $n\\ge 9$, there exists a surjective homomorphism $\\Phi:T(X_n)\\to \\mathbb{Z}^\\infty$, where $\\mathbb{Z}^\\infty$ denotes a free abelian group of countably infinite rank." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "B" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "finiteness", + "tampered_component": "surjectivity-vs-large-image via infinite-rank image inside direct sum", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "finiteness", + "tampered_component": "dropped surjectivity, retaining only existence of an infinite-image homomorphism", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "dependence of the homomorphism on the manifold parameter $n$", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "characteristic", + "tampered_component": "sharp lower bound $n\\ge 10$ coming from the base case and induction", + "template_used": "boundary_range" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the objects involved but does not reveal the conclusion. The correct statement is not encoded in the wording beyond standard setup." + }, + "TAS": { + "score": 1, + "justification": "This is close to theorem recognition: the correct option appears to restate a known result almost verbatim. However, the alternatives vary quantifiers, range bounds, and strength of conclusion, so it is not a pure tautology." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish surjectivity from weaker infinite-image claims, dependence on n, and the sharp threshold n>=10. But the item mainly tests recall of the exact theorem statement rather than deeper derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic failure modes: weakening surjectivity, confusing large image with surjective image, mishandling quantifier dependence, and shifting the lower bound from 10 to 9." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it leans more toward precise recall than genuinely generative reasoning." + } + }, + { + "id": "2511.13447v1", + "paper_link": "http://arxiv.org/abs/2511.13447v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "thmA", + "content": "\\label{introthm:transserial_char}\n The following are equivalent for an exponential o-minimal theory $T$:\n \\begin{enumerate}[ref=(\\arabic*)]\n \\item\\label{introthmenum:transserial} $T$ is transserial;\n \\item\\label{introthmenum:exp-bdd_gne_at_special} $T$ is exponentially bounded and for all $(\\mathbb E, \\mathcal{O}) \\models T_\\mathrm{convex}$, all $\\mathcal{O} < b <\\mathbb E^{>\\mathcal{O}}$, for every $y \\in \\mathbb E \\langle b \\rangle$ there is a $\\mathbb E$-definable gne $g$, such that $y \\equiv_\\mathbb E g(b)$;\n \\item\\label{introthmenum:gne_wim_1} all 1-$\\dcl_T$-dimensional wim-constructible extension of models of $T_\\mathrm{convex}$ have the gne-property;\n \\item\\label{introthmenum:gne_res_1} all 1-$\\dcl_T$-dimensional res-constructible extensions of models of $T_\\mathrm{convex}$ have the gne property;\n \\item\\label{introthmenum:gne_wim} all wim-constructible extensions of models of $T_\\mathrm{convex}$ have the gne property;\n \\item\\label{introthmenum:gne_res} all res-constructible extensions of models of $T_\\mathrm{convex}$ have the gne property.\n \\end{enumerate}", + "start_pos": 13291, + "end_pos": 14350, + "label": "introthm:transserial_char" + }, + "ref_dict": { + "ssec:first_application": "\\begin{proof}\n This is essentially already in Example~\\ref{main:examples}(2). Let $\\cO_x^+\\coloneqq \\cO_*^+\\cap \\bE\\langle x \\rangle$ and Consider the valued differential field $(\\bE \\langle x \\rangle, \\cO_x^+, \\partial_x)$. \n By Proposition~\\ref{prop:der_order-convexity} and Corollary~\\ref{cor:order-convex_implies_val-convex}(1), $\\partial_x$ is weakly-$\\cO_x$-Liouville-convex. But since it contains a $\\cO$-wim element over $\\bE$, by Corollary~\\ref{cor:der-types_and_absorbed-elements} $\\partial_x$ has type (W) and $\\res_{\\cO_*^+}(\\bE)=\\res_{\\cO_*^+}(\\bE\\langle x \\rangle)$.\n \\end{proof}\n\\end{corollary}\n\n\\subsection{Tressl signature-alternative in simply exponential o-minimal fields}\\label{ssec:first_application}\nIn this subsection we show how Proposition~\\ref{prop:height_over_absorbed}, can be used to partially answer a problem posed in \\cite{tressl2005model}. Our setting will be the one of an elementary extension $\\bE \\prec \\bE_*\\models T$ of models of an \\emph{exponential} o-minimal theory $T$.\n\nIn \\cite[Def.~3.16]{tressl2005model}, Tressl considers the following condition on a unary type $p=\\tp(x/\\bE)$ over $\\bE$: $p$ satisfies the \\emph{signature alternative} if it is either non-symmetric, or\n\\begin{enumerate}\n \\item $\\Br(x/\\bE)(\\bE)$ is cofinal in $\\Br(x/\\bE)(\\bE\\langle x\\rangle)$, that is, $\\bE\\langle x \\rangle$ does not realize the cut above $\\Br(x/\\bE)(\\bE)$;\n \\item if $y$ is the realization of such cut, i.e.\\ if $y \\in \\Br(y/\\bE)(\\bE_*)$ and $y>\\Br(x/\\bE)(\\bE)$, then $\\Br(\\log y/\\bE)$ is cofinal in $\\Br(\\log y/\\bE)(\\bE \\langle y \\rangle)$.\n\\end{enumerate}\n\nNote that if (1) above is satisfied by all types over $\\bE$, then all types over $\\bE$ satisfy (2) as well. We point out the following immediate Corollary of Proposition~\\ref{prop:breadth-ortho} and \n\n\\begin{corollary}\\label{cor:finite_hight_and_br_ortho}\n If $x \\in \\bE_* \\setminus \\bE$ is symmetric, and all $y \\in \\bE \\langle x\\rangle$ have finite height in $(\\bE\\langle x \\rangle, \\cO_*^-, \\partial_x)$ with $\\cO_*^-=\\CH(\\bZ)$, then $\\Br(x/\\bE)(\\bE)$ is cofinal in $\\Br(x/\\bE)(\\bE\\langle x\\rangle)$.\n \\begin{proof}\n By Proposition~\\ref{prop:breadth-ortho}(1) we have for all $y \\in \\bE \\langle x \\rangle$ that if $y \\in \\Br(x/\\bE)(\\bE\\langle x\\rangle)\\setminus \\CH(\\Br(x/\\bE)(\\bE))$, then\n \\[ \\lder_x^k(y)\\in \\Br(x/\\bE)(\\bE\\langle x\\rangle)\\setminus \\CH(\\Br(x/\\bE)(\\bE)) \\quad \\text{for all}\\; k \\in \\bN.\\]\n But then this would imply that $\\val_{\\cO_*^-}(\\lder^k y) \\notin \\val_{\\cO_*^-}(\\bE)$ and $y$ would have infinite height.\n \\end{proof}", + "introthm:transserial_char": "\\begin{thmA}\\label{introthm:transserial_char}\n The following are equivalent for an exponential o-minimal theory $T$:\n \\begin{enumerate}[ref=(\\arabic*)]\n \\item\\label{introthmenum:transserial} $T$ is transserial;\n \\item\\label{introthmenum:exp-bdd_gne_at_special} $T$ is exponentially bounded and for all $(\\bE, \\cO) \\models T_\\convex$, all $\\cO < b <\\bE^{>\\cO}$, for every $y \\in \\bE \\langle b \\rangle$ there is a $\\bE$-definable gne $g$, such that $y \\equiv_\\bE g(b)$;\n \\item\\label{introthmenum:gne_wim_1} all 1-$\\dcl_T$-dimensional wim-constructible extension of models of $T_\\convex$ have the gne-property;\n \\item\\label{introthmenum:gne_res_1} all 1-$\\dcl_T$-dimensional res-constructible extensions of models of $T_\\convex$ have the gne property;\n \\item\\label{introthmenum:gne_wim} all wim-constructible extensions of models of $T_\\convex$ have the gne property;\n \\item\\label{introthmenum:gne_res} all res-constructible extensions of models of $T_\\convex$ have the gne property.\n \\end{enumerate}\n\\end{thmA}", + "def:absorbed-type-der": "\\begin{definition}\\label{def:absorbed-type-der}\n Let $(\\bE_*, \\cO_*)$ be a valued field, $\\partial$ be an almost $\\cO_*$-Lioville-convex derivation on $\\bE_*$. Let $\\bE\\coloneqq\\Kr(\\partial)$.\n We will say that $\\partial$ is \\emph{weakly $\\cO_*$-absorbed} if $\\partial(\\br_{\\cO_*}(\\partial))\\prec 1$. We will say that it is \\emph{$\\cO_*$-absorbed} if furthermore there is $x\\in \\bE_* \\setminus \\bE$ such that $\\val(x-\\bE)\\subseteq \\val(\\bE)$. \n\\end{definition}", + "def:rosenlicht_levels": "\\begin{definition}\\label{def:rosenlicht_levels}\n Let $T$ be an exponential o-minimal theory and $(\\bE, \\cO)\\models T_\\convex$. Recall that $x, y \\in \\bE^{> \\cO}$ have the same \\emph{Rosenlicht level} if there is a nautral number $m$ such that $\\val(\\log_m x)=\\val(\\log_m y)$ We write $x\\asymp_Ly$ when $x$ and $y$ have the same Rosenlicht level. $\\asymp_L$ is an equivalence relation with convex classes, so the set of Rosenlicht levels is naturally ordered.\n\\end{definition}", + "ssec:few-constants": "\\begin{proof}\n Note that it is valuation convex for $\\CH^{\\le}(\\bZ)$ by Remark~\\ref{rmk:order-convex_to_val-convex}, then apply Lemma~\\ref{lem:convex_der_val_relaxation} to conclude that $\\partial$ is in fact $\\cO_*$-valuation-convex.\n\n (2) Now suppose that $\\partial$ is furthermore logarithmically order-convex. Note that for every order-convex subgroup $M<1$, we have $\\CH^{\\le}(\\lder(1+M))=\\CH(\\partial M)$ as for all $m \\in M$, $|\\partial m/2| \\le |\\lder(1+m)| \\le |2\\partial m|$.\n So to conclude that it is weakly-$\\cO_*$-Liouville convex, by Lemma~\\ref{lem:Lv-convex_implies_log-convex} it suffices to show that $1+\\co_*$ is $\\lder$-convex for $\\preceq_{\\cO_*}$. Since by logarithmic order-convexity we have $\\lder(1+\\co_*)=\\CH^{\\le}(\\lder(1+\\co_*)) \\cap \\lder(\\bE_*^{\\neq0})$, it suffices to show that $\\CH^{\\le}(\\lder(1+\\co_*))=\\CH^{\\le}(\\partial \\co_*)$ is a $\\cO_*$-module, but this follows from the first part of the statement.\n\n (3) By (1) and (2), we only need to show that $\\partial$ satisfies (\\ref{axiom:new-type-H-relative}).\n Let $\\cO_\\circ=\\CH^{\\le}_{\\bE_*}(\\bZ)$.\n Suppose that there are no constants $c$ with $x \\preceq_{\\cO_*} c \\preceq_{\\cO_*} y$, then a fortiori there are no constants $c$ with $x \\preceq_{\\cO_\\circ} c \\preceq_{\\cO_\\circ} y$, so by Lemma~\\ref{lem:der_local_monotone}, we can conclude that if $\\lder(x) \\prec \\lder(y)$, we have that for all $x\\prec_{\\cO_\\circ} z \\prec_{\\cO_\\circ} y$, and thus a fortiori for all $x \\prec_{\\cO_*} \\prec z \\prec_{\\cO_*} y$, we have $\\lder(z)\\preceq_{\\cO_\\circ} \\lder(y)$ and a fortiori $\\lder(z) \\preceq_{\\cO_*}\\lder(y)$.\n \\end{proof}\n\\end{corollary}\n\n\\subsection{The case of few constants}\\label{ssec:few-constants}\nWhen the constants are included in the valuation ring, several notions of convexity collapse to notions already present in the literature and thoroughly studied in \\cite{aschenbrenner2019asymptotic}.\n\n\\begin{remark}[Asymptotic and pre-d-valued fields]\n Recall the following notions from \\cite[Sec.~9.1, 10.1]{aschenbrenner2019asymptotic}, that a valued differential field $(\\bE_*, \\cO_*, \\partial)$ is\n \\begin{enumerate}\n \\item \\emph{asymptotic} if for all $x,y\\prec 1$ (or equivalently, by \\cite[Prop.~9.1.3]{aschenbrenner2019asymptotic}, all $x,y \\notin \\cO_* \\setminus \\co_*)$, $x\\prec y \\Leftrightarrow \\partial x \\prec \\partial y$;\n \\item \\emph{pre-d-valued} if for all $y\\preceq 1$ and all $x \\prec 1$ (or equivalently all $x \\notin \\cO_* \\setminus \\co_*$), $\\lder(x) \\succ \\partial y$;\n \\item \\emph{asymptotic of type H}, if it is asymptotic and furthermore for all $x\\prec y\\prec 1$, $\\lder(x) \\succeq \\lder(y)$. \n \\end{enumerate}\n Also recall that every pre-d-valued field is asymptotic, \\cite[Lem.~10.1.1]{aschenbrenner2019asymptotic}.\n The next Lemma shows that if the constants of $\\partial$ are included in $\\cO_*$ (i.e.\\ there are \\emph{few constants}, \\cite[p.192]{aschenbrenner2019asymptotic}), then the notion of $\\cO_*$-Liouville-convex specializes to the notion of a pre-d-valued field of type H and the notion of almost $\\cO_*$-Liouville-convex specializes to the notion of pre-d-valued field.\n\\end{remark}\n\n\\begin{lemma}\\label{lem:pre-d-valued}\n If $\\partial$ is a derivation on $(\\bE_*, \\cO_*)$ with constants $\\bE=\\Kr(\\partial) \\subseteq \\cO_*$, and $\\val_{\\cO_*}(\\bE)$ divisible, then\n \\begin{enumerate}\n \\item if $\\partial$ is $\\cO_*$-valuation-convex, then $(\\bE_*, \\cO_*, \\partial)$ is asymptotic;\n \\item if $(\\bE_*, \\cO_*, \\partial)$ is pre-d-valued if and only if $\\cO_* \\setminus \\co_*$ is $\\lder$-convex;\n \\item if $(\\bE_*, \\cO_*, \\partial)$ is pre-d-valued, then $\\partial$ is $\\cO_*$-valuation-convex, so $(\\bE_*, \\cO_*, \\partial)$ is pre-d-valued if and only if it is almost Liouville-$\\cO_*$-convex;\n \\item if $(\\bE_*, \\cO_*, \\partial)$ is pre-d-valued, then it is of type H if and only if it is logarithmically convex.\n \\end{enumerate}\n \\begin{proof}\n (1). Suppose that $\\partial$ is $\\cO_*$-valuation convex and $x,y\\prec 1$. If $\\partial x \\in \\cO_*\\partial y \\subseteq \\cO_* \\partial (y \\cO_*)$, then by convexity $\\partial x \\in \\partial (y \\cO_*)$, so $x \\in \\bE+ y \\cO_*$, but since $x, y \\prec \\bE^{\\neq 0}$ by construction, this implies $x \\in \\cO_* y$. Similarly if $\\partial x \\in \\co_* \\partial y\\subseteq \\cO_* \\partial (y \\co_*)$ (here we are using Lemma~\\ref{lem:module-derivatives}(4)),\n then $\\partial x \\in \\partial (y \\co_*)$, so by the same argument $x \\prec y$.\n\n (2). $\\cO_* \\setminus \\co_*$ is $\\lder$-convex if and only if $\\val(x) \\notin \\val(\\bE)\\Rightarrow \\lder(x) \\succ \\partial \\cO_*$. But since $\\bE \\subseteq \\cO_*$, we have $\\val(x) \\notin \\val(\\bE) \\Leftrightarrow x \\not \\asymp 1$ and thus this is equivalent to $\\lder(x) \\succ \\partial \\cO_*$ for all $x\\not \\asymp 1$ which is being pre-d-valued.\n\n (3) We need to show to show that for all $y$, $\\partial(y \\cO_*)$ is $\\preceq$-convex in $\\partial \\bE_*$ for all $y$, that is: $x \\notin \\bE+y\\cO_* \\Rightarrow \\partial x \\succ \\partial (y \\cO_*)$.\n\n Note that since $\\bE \\subseteq \\cO_*$, we have $x \\notin \\bE+y \\cO_* \\Leftrightarrow x\\succ y$.\n So if $y \\not\\asymp 1$, we get $\\partial x \\succ \\partial y$. Furthermore we have $\\lder(y)\\succ \\partial \\cO_*$, so by Lemma~\\ref{lem:module-derivatives}(1) $\\cO_*\\partial (y \\cO_*)=\\cO_*\\partial y + y \\cO_*\\partial \\cO_*=\\cO_* \\partial y$ and we are done.\n\n For the case $y \\in \\cO_* \\setminus \\co_*$, note that by the hypothesis that $\\val(\\cO_*)$ is divisible we have $x\\succ z$ for some $1\\prec z \\prec x$, so we can reduce to the previous case and deduce $x \\succ \\partial(z\\cO_*)\\supseteq \\partial (\\cO_*)$.\n\n (4) Note that since $\\bE^{\\neq0} \\subseteq \\cO_*\\setminus \\co_*$, we have that for $1\\prec x\\prec y$, $\\not\\exists c \\in \\bE,\\, 1 \\prec c \\preceq y$, so (\\ref{axiom:new-type-H}) is equivalent to $1 \\prec x\\prec y \\Rightarrow \\lder(x) \\preceq \\lder(y)$. On the other hand it is easy to see that if $1 \\prec x\\prec y \\Rightarrow \\lder(x) \\preceq \\lder(y)$, then (\\ref{axiom:new-type-H-relative}) holds.\n \\end{proof}", + "prop:height_over_absorbed": "\\begin{proposition}\\label{prop:height_over_absorbed}\n If $\\partial\\br(\\partial) \\prec 1$, then\n \\[\\height(x,\\partial)=\\inf\\{k\\in \\omega: \\val(\\lder^k(\\partial x))\\in \\val(\\bE)\\}\\in \\omega+1.\\]\n \\begin{proof}\n Set $a=\\partial x$. Clearly the height of $x$ is $0$ if and only if $\\val(a)\\in \\val(\\bE)$.\n\n It suffices to show that for all $m\\ge 0$, there is an essential approximate logarithmic scale $(y_k)_{k < m}$ with $a \\sim \\partial y_0$ (in the case $m>0$) if and only if $\\val(\\lder^k(a))\\notin \\val(\\bE)$ for all $kr \\rightarrow G_m^f(\\overline{x},t) < 1 \\big).\\]\n\\end{enumerate}\n\nIt turns out that for transserial theories, a similar property to the one outlined above for $T$-$\\lambda$-spherical completions, is shared by other kind of extensions. So, to state the main result in a compact way, let us say that an elementary extension $(\\mathbb E, \\mathcal{O}) \\prec (\\bE_*, \\cO_*)$ of models of $T_\\mathrm{convex}$,\n\\begin{enumerate}[label=(E\\arabic*)]\n \\item\\label{df:gne-ppty} has the \\emph{gne-property} if for all elements $y\\in \\bE_*$, there is a $\\mathbb E$-definable gne such that $\\mathbf{v}(g^{-1}(y)-\\mathbb E)\\subseteq \\mathbf{v}(\\mathbb E)$.\n\\end{enumerate}\nand recall that an elementary extension $(\\mathbb E, \\mathcal{O}) \\preceq (\\bE_*, \\cO_*)\\models T_\\mathrm{convex}$ is said to be\n\\begin{enumerate}[label=(E\\arabic*), resume]\n \\item\\label{df:res-constructible} \\emph{residually constructible} (or \\emph{res-constructible}, (cf \\cite[Def.~14]{freni2025residually}), if there is a $\\dcl_T$-basis $\\overline{r}$ of $\\bE_*$ over $\\mathbb E$, such that $\\mathbf{r}(\\overline{r})$ is $\\dcl_T$-independent over $\\mathbf{r}(\\mathbb E)$;\n \\item\\label{df:wim-constructible} \\emph{($\\lambda$-bounded) wim-constructible} (cf \\cite[Def.~3.15]{freni2024t}) if there is an ordinal indexed $\\dcl_T$-basis $(x_i: i<\\alpha)$ of $\\bE_*$ over $\\mathbb E$, such that for all $j<\\alpha$, $x_j$ is a pseudolimit of a p.c.\\ sequence in $\\mathbb E \\langle x_i: ir \\rightarrow G_m^f(\\overline{x},t) < 1 \\big).\\]\n\\end{enumerate}\n\nIt turns out that for transserial theories, a similar property to the one outlined above for $T$-$\\lambda$-spherical completions, is shared by other kind of extensions. So, to state the main result in a compact way, let us say that an elementary extension $(\\mathbb E, \\mathcal{O}) \\prec (\\bE_*, \\cO_*)$ of models of $T_\\mathrm{convex}$,\n\\begin{enumerate}[label=(E\\arabic*)]\n \\item\\label{df:gne-ppty} has the \\emph{gne-property} if for all elements $y\\in \\bE_*$, there is a $\\mathbb E$-definable gne such that $\\mathbf{v}(g^{-1}(y)-\\mathbb E)\\subseteq \\mathbf{v}(\\mathbb E)$.\n\\end{enumerate}\nand recall that an elementary extension $(\\mathbb E, \\mathcal{O}) \\preceq (\\bE_*, \\cO_*)\\models T_\\mathrm{convex}$ is said to be\n\\begin{enumerate}[label=(E\\arabic*), resume]\n \\item\\label{df:res-constructible} \\emph{residually constructible} (or \\emph{res-constructible}, (cf \\cite[Def.~14]{freni2025residually}), if there is a $\\dcl_T$-basis $\\overline{r}$ of $\\bE_*$ over $\\mathbb E$, such that $\\mathbf{r}(\\overline{r})$ is $\\dcl_T$-independent over $\\mathbf{r}(\\mathbb E)$;\n \\item\\label{df:wim-constructible} \\emph{($\\lambda$-bounded) wim-constructible} (cf \\cite[Def.~3.15]{freni2024t}) if there is an ordinal indexed $\\dcl_T$-basis $(x_i: i<\\alpha)$ of $\\bE_*$ over $\\mathbb E$, such that for all $j<\\alpha$, $x_j$ is a pseudolimit of a p.c.\\ sequence in $\\mathbb E \\langle x_i: ir \\rightarrow G_m^f(\\overline{x},t) < 1 \\big).\\]\n\\end{enumerate}\n\nIt turns out that for transserial theories, a similar property to the one outlined above for $T$-$\\lambda$-spherical completions, is shared by other kind of extensions. So, to state the main result in a compact way, let us say that an elementary extension $(\\mathbb E, \\mathcal{O}) \\prec (\\bE_*, \\cO_*)$ of models of $T_\\mathrm{convex}$,\n\\begin{enumerate}[label=(E\\arabic*)]\n \\item\\label{df:gne-ppty} has the \\emph{gne-property} if for all elements $y\\in \\bE_*$, there is a $\\mathbb E$-definable gne such that $\\mathbf{v}(g^{-1}(y)-\\mathbb E)\\subseteq \\mathbf{v}(\\mathbb E)$.\n\\end{enumerate}\nand recall that an elementary extension $(\\mathbb E, \\mathcal{O}) \\preceq (\\bE_*, \\cO_*)\\models T_\\mathrm{convex}$ is said to be\n\\begin{enumerate}[label=(E\\arabic*), resume]\n \\item\\label{df:res-constructible} \\emph{residually constructible} (or \\emph{res-constructible}, (cf \\cite[Def.~14]{freni2025residually}), if there is a $\\dcl_T$-basis $\\overline{r}$ of $\\bE_*$ over $\\mathbb E$, such that $\\mathbf{r}(\\overline{r})$ is $\\dcl_T$-independent over $\\mathbf{r}(\\mathbb E)$;\n \\item\\label{df:wim-constructible} \\emph{($\\lambda$-bounded) wim-constructible} (cf \\cite[Def.~3.15]{freni2024t}) if there is an ordinal indexed $\\dcl_T$-basis $(x_i: i<\\alpha)$ of $\\bE_*$ over $\\mathbb E$, such that for all $j<\\alpha$, $x_j$ is a pseudolimit of a p.c.\\ sequence in $\\mathbb E \\langle x_i: ir \\rightarrow G_m^f(\\overline{x},t) < 1 \\big).\\]\n\\end{enumerate}\n\nThe name \\emph{transserial}, comes from the field of logarithmic exponential transseries, where certain configurations such as formal asymptotic solutions of the equation $\\log(f(x)-x)=f(\\log_2(x))$ are prohibited.\n\nThe way results are obtained is an analysis of the ordered differential valued fields $(\\bE\\langle x\\rangle, \\cO_x, \\partial_x)$ where\n\\begin{enumerate}[label=(C\\arabic*)]\n \\item $\\bE\\langle x\\rangle\\coloneqq\\dcl_T(\\bE, x)$ is a 1-$\\dcl_T$-dimensional elementary extension of $\\bE\\models T$ and $\\cO_x$ is a $T$-convex valuation subring;\n \\item $\\partial_x$ is the unique non-trivial $\\bE$-linear $T$-derivation with $\\partial_x (x)=1$ (cf \\cite{fornasiero2021generic}, \\cite{kaplan2023t});\n \\item $\\val_{\\cO_x}(x-\\bE)\\subseteq \\val_{\\cO_x}(\\bE)$.\n\\end{enumerate}\n\n\\begin{corollary}\\label{cor:sOabs_to_sOabs_der}\n Let $x \\in \\bE_*\\setminus \\bE$ and $f,g,h: \\bE_* \\to \\bE_*$ be $\\bE$-definable functions. Then the following hold:\n \\begin{enumerate}\n \\item if $x \\in \\bE_* \\setminus \\bE$ is strongly $\\cO$-absorbed and $h(x) \\in (B_x^+:\\cO)$, then $h'(x) \\in \\co_*^-$, furthermore if $\\cO$ is $T$-convex, then $h'(x) \\in \\co_*^+$;\n \\item if $x \\in \\bE_* \\setminus \\bE$ is $\\cO$-wim and $f(x)\\equiv_\\bE g(x)$ are strongly $\\cO$-absorbed, then $f(x), g(x)$ are $\\cO$-wim and moreover $f'(x) \\sim_{\\cO_*^-} g'(x)$.\n \\end{enumerate}\n \\begin{proof}\n (1) is a direct application of Lemma~\\ref{lem:sOabs_to_sOabs_der} with $f(x)=x$ and $g(x)=x+h(x)$. The ``furthermore'' follows from the fact that if $\\cO$ is $T$-convex, and $x$ is strongly $\\cO$-absorbed, then by \\cite[Thm.~A]{freni2024t} $\\co_*^-\\cap \\bE \\langle x \\rangle = \\co_*^+\\cap \\bE \\langle x \\rangle$.\n (2) if $x$ is $\\cO$-wim, then $B_x^+$ is a $\\cO$-module. Furthermore since $f(x), g(x)$ are strongly $\\cO$-absorbed, by Lemma~\\ref{lem:wim-cofres_orto}, we must have that $f(x)$ and $g(x)$ are $\\cO$-wim as well so $B_{f(x)}$ is a $\\cO$-module as well, but then Lemma~\\ref{lem:sOabs_to_sOabs_der}, implies $|f'(x)-g'(x)|r \\rightarrow G_m^f(t) \\in \\cO\\big).\n \\end{equation}\n \\item $T$ is transserial if and only if for all $(\\bE, \\cO) \\models T_\\convex$ and every unary $\\bE$-definable function $f$, there is a natural number $m$ such that\n \\begin{equation}\\tag{$\\mathrm{TS}^{f}_{m,1}$}\n (\\bE, \\cO) \\models \\exists r \\in \\cO, \\forall t\\in \\cO, \\big(t>r \\rightarrow G_m^f(t) \\le 1\\big).\n \\end{equation}\n \\end{enumerate}\n \\begin{proof}\n We prove (2), the proof of (1) is similar.\n $\\Rightarrow$ is obvious.\n The converse is a compactness argument. Suppose that for some $T$-definable $n$-ary $f: \\bE^{n+1} \\to \\bE$ we have for all $m \\in \\bN$\n \\[T_\\convex \\models \\exists \\overline{x},\\, \\forall r\\in \\cO, \\exists t\\in \\cO, (t > r \\land G_m^f(\\overline{x}, t)>1)\\]\n which by weak o-minimality of $T_\\convex$ is equivalent to \\begin{equation}\\label{eqn:strong_negation}\n T_\\convex \\models \\exists \\overline{x},\\, \\exists r\\in \\cO, \\forall t\\in \\cO,\\, (t > r \\rightarrow G_m^f(\\overline{x}, t)>1).\n \\end{equation}\n Notice that since \n \\[G_{m+1}^f(\\overline{x}, t)=\\min\\{G_{m}^f(\\overline{x}, t), \\lder_n (G_m^f)(\\overline{x},t)\\} \\le G_{m}^f(\\overline{x}, t),\\]\n said \n \\[\\varphi_m(\\overline{x}):= \\big(\\exists r\\in \\cO, \\forall t\\in \\cO,\\, (t > r \\rightarrow G_m^f(\\overline{x}, t)>1)\\big),\\]\n we have $\\varphi_{m+1}(\\overline{x}) \\rightarrow \\varphi_{m}(\\overline{x})$ hence the validity of \\ref{eqn:strong_negation} for all $m$ is equivalent to the consistency of the type $p(\\overline{x})=\\{\\varphi_m(\\overline{x}): m \\in \\bN\\}$. If $(\\bE, \\cO)$ is a model of $T_\\convex$ realizing such type and $\\overline{c} \\in \\bE^n$ is a realization of $p$, we get that $g(t)=f(\\overline{c}, t)$ is such that for all $m$ $(\\bE, \\cO)\\models \\exists r \\in \\cO,\\, \\forall t \\in \\cO,\\, (t>r \\rightarrow G_m^g(t)>1)$.\n \\end{proof}\n\\end{lemma}", + "post_theorem_intro_text_len": 5416, + "post_theorem_intro_text": "The name \\emph{transserial}, comes from the field of logarithmic exponential transseries, where certain configurations such as formal asymptotic solutions of the equation $\\log(f(x)-x)=f(\\log_2(x))$ are prohibited.\n\nI also show that if a transserial theory admits an Archimedean prime model, then Tressl's signature alternative (cf \\cite[Def.~3.16]{tressl2005model} or see Subsection~\\ref{ssec:first_application}) holds true for every cut over every model. Moreover all Hardy fields of a transserial theory have the the same Rosenlicht levels (as defined in \\cite[Def.~5.2]{berarducci2015surreal} in the context of surreal numbers with the natural valuation, see Definition~\\ref{def:rosenlicht_levels}).\n\n\\begin{thmA}\\label{introthm:transserial_consequences}\n Let $T$ be a transserial theory, then:\n \\begin{enumerate}\n \\item wim-constructible and residually constructible extensions do not add new Rosenlicht levels, in particular for all $\\mathbb K\\prec \\mathbb K'\\models T$, the Hardy fields of $\\mathbb K$ and $\\mathbb K'$ have the same Rosenlicht levels;\n \\item if $T$ has an Archimedean prime model, then for every $\\mathbb E\\models T$, every unary type $p$ over $\\mathbb E$ satisfies Tressl's signature alternative.\n \\end{enumerate}\n\\end{thmA}\n\nNote that the condition (1) is a weakening of the notion of ($\\mathbb Z$-)\\emph{levelled} o-minimal theory of \\cite{marker1997levelled}. \n\n\\smallskip \n\nThe way results are obtained is an analysis of the ordered differential valued fields $(\\mathbb E\\langle x\\rangle, \\cO_x, \\partial_x)$ where\n\\begin{enumerate}[label=(C\\arabic*)]\n \\item $\\mathbb E\\langle x\\rangle\\coloneqq\\dcl_T(\\mathbb E, x)$ is a 1-$\\dcl_T$-dimensional elementary extension of $\\mathbb E\\models T$ and $\\cO_x$ is a $T$-convex valuation subring;\n \\item $\\partial_x$ is the unique non-trivial $\\mathbb E$-linear $T$-derivation with $\\partial_x (x)=1$ (cf \\cite{fornasiero2021generic}, \\cite{kaplan2023t});\n \\item $\\val_{\\cO_x}(x-\\mathbb E)\\subseteq \\val_{\\cO_x}(\\mathbb E)$.\n\\end{enumerate}\n\nNote that this situation comprises the cases in which $(\\mathbb E \\langle x\\rangle, \\cO_x)\\succ (\\mathbb E, \\mathcal{O})$ is wim-constructible or res-constructible.\n\nA key ingredient in Theroem~\\ref{introthm:transserial_char}, is that in the setting above, if $T$ is exponential, an element $y \\in \\mathbb E \\langle x \\rangle$ is of the form $g(z)$ for a $\\mathbb E$-definable gne and a $z$ such that $\\mathbf{v}(z-\\mathbb E)\\subseteq \\mathbf{v}(\\mathbb E)$, if and only if there is a natural number $m$ such that $\\mathbf{v}(\\lder_x^m \\partial_x(y))\\in \\mathbf{v}(\\mathbb E)$ (Proposition~\\ref{prop:height_over_absorbed}).\n\n\\smallskip\n\nThe treatment points out some general properties of ordered differential valued fields the form $(\\mathbb E \\langle x\\rangle, \\cO_x, \\partial_x)$ satisfying (C1) and (C2) above. When $x>\\mathbb E$, and $\\cO_x$ is the convex hull of $\\mathbb E$, $(\\mathbb E \\langle x\\rangle, \\cO_x, \\partial_x)$ is the Hardy field of the o-minimal structure $\\mathbb E$ which is an example of an $H$-field in the sense of \\cite{aschenbrenner2002fields}.\n\nIn a similar vein, even $\\mathrm{tp}(x/\\mathbb E)$ is not a definable type, many important properties are captured by the notions of an \\emph{order-convex} (resp.\\ \\emph{logarithmically ---}) derivation and a \\emph{valuation-convex} (resp.\\ \\emph{logarithmically ---}) derivation (Definitions~\\ref{def:order-convex_der} and \\ref{def:v-convex_derivation}).\n\nI call a derivation $\\partial$ on an ordered field $\\bE_*$, \\emph{order-convex} if it maps order-convex sets onto order-convex subsets of $\\partial\\bE_*$, similarly it is \\emph{logarithmically order-convex} if the corresponding logarithmic derivative $\\lder(t)\\coloneqq (\\partial t)/t$ sends order-convex sets to order-convex subsets of $\\lder(\\mathbb E^{\\neq0}_*)$. These can be regarded as traces of the mean-value theorem for functions definable in o-minimal ordered fields.\nThe notion of valuation-convex and logarithmically valuation-convex arise naturally as traces of the corresponding notions of order-convexity with respect to the minimal order-convex valuation, specifically logarithmic convexity has several weakening that are sufficient for most purposes.\n\nAs expected, when $\\cO_*$ contains the field of constants, this notions specialize to several notions already studied in the literature (\\cite{rosenlicht1980differential},\\cite{rosenlicht1983rank}, \\cite{aschenbrenner2002fields}, and \\cite{aschenbrenner2019asymptotic}) such as pre-H-field, asymptotic field, pre-d-valued field and pre-d-valued field of H-type (see Subsection~\\ref{ssec:few-constants}).\n\nGiven suitable convexity assumptions on the derivative on a valued field $(\\bE_*, \\cO_*)$ with constants $\\mathbb E\\coloneqq\\Kr(\\partial)$, the condition $\\mathbf{v}(x-\\mathbb E)\\subseteq \\mathbf{v}(\\mathbb E)$ is equivalent to the fact that the derivation ``at $x$'' $\\partial_x=\\partial/\\partial x$ has the property that $y \\succ \\partial_x\\cO_x\\Rightarrow \\lder_x(y)\\prec y$. We will say that such derivations are \\emph{absorbed} (Definition~\\ref{def:absorbed-type-der}), a property that will be at the center of several arguments.\n\n\\subsection{Acknowledgments} I thank Vincenzo Mantova for several conversations on the topic and for encouraging me to try to convert some previous proofs of the main theorems into general arguments about valued differential fields.", + "sketch": "The post-theorem introduction does not give a full step-by-step proof of Theorem~\\ref{introthm:transserial_char}, but it does describe the main ingredients and framework used.\n\n- The results (including Theorem~\\ref{introthm:transserial_char}) are obtained by “an analysis of the ordered differential valued fields $(\\mathbb E\\langle x\\rangle, \\cO_x, \\partial_x)$” satisfying:\n (C1) $\\mathbb E\\langle x\\rangle:=\\dcl_T(\\mathbb E,x)$ is a 1-$\\dcl_T$-dimensional elementary extension of $\\mathbb E\\models T$ and $\\cO_x$ is a $T$-convex valuation subring;\n (C2) $\\partial_x$ is the “unique non-trivial $\\mathbb E$-linear $T$-derivation with $\\partial_x(x)=1$”; and\n (C3) $\\val_{\\cO_x}(x-\\mathbb E)\\subseteq \\val_{\\cO_x}(\\mathbb E)$.\n This “comprises the cases in which $(\\mathbb E\\langle x\\rangle,\\cO_x)\\succ (\\mathbb E,\\mathcal O)$ is wim-constructible or res-constructible.”\n\n- A “key ingredient in Th[e]orem~\\ref{introthm:transserial_char}” is the following criterion (stated as Proposition~\\ref{prop:height_over_absorbed}): in the above setting, when $T$ is exponential, for $y\\in\\mathbb E\\langle x\\rangle$,\n \\[\n y\\text{ is of the form }g(z)\\text{ for a }\\mathbb E\\text{-definable gne and }z\\text{ with }\\mathbf v(z-\\mathbb E)\\subseteq \\mathbf v(\\mathbb E)\n \\]\n “if and only if there is a natural number $m$ such that $\\mathbf v(\\lder_x^m\\partial_x(y))\\in \\mathbf v(\\mathbb E)$.”\n\n- The treatment isolates “general properties” of such $(\\mathbb E\\langle x\\rangle,\\cO_x,\\partial_x)$ and introduces convexity conditions on derivations (order-convex / logarithmically order-convex; valuation-convex / logarithmically valuation-convex) as “traces of the mean-value theorem,” used to control behavior even when “$\\mathrm{tp}(x/\\mathbb E)$ is not a definable type.”\n\n- Under “suitable convexity assumptions,” the condition $\\mathbf v(x-\\mathbb E)\\subseteq \\mathbf v(\\mathbb E)$ is reformulated via an ‘absorbed’ condition on $\\partial_x$ (Definition~\\ref{def:absorbed-type-der}): “$y\\succ \\partial_x\\cO_x\\Rightarrow \\lder_x(y)\\prec y$,” and such absorbed derivations are said to be “at the center of several arguments.”", + "expanded_sketch": "The post-theorem introduction does not give a full step-by-step proof of the main theorem, but it does describe the main ingredients and framework used.\n\n- The results (including the main theorem) are obtained by “an analysis of the ordered differential valued fields $(\\mathbb E\\langle x\\rangle, \\cO_x, \\partial_x)$” satisfying:\n (C1) $\\mathbb E\\langle x\\rangle:=\\dcl_T(\\mathbb E,x)$ is a 1-$\\dcl_T$-dimensional elementary extension of $\\mathbb E\\models T$ and $\\cO_x$ is a $T$-convex valuation subring;\n (C2) $\\partial_x$ is the “unique non-trivial $\\mathbb E$-linear $T$-derivation with $\\partial_x(x)=1$”; and\n (C3) $\\val_{\\cO_x}(x-\\mathbb E)\\subseteq \\val_{\\cO_x}(\\mathbb E)$.\n This “comprises the cases in which $(\\mathbb E\\langle x\\rangle,\\cO_x)\\succ (\\mathbb E,\\mathcal O)$ is wim-constructible or res-constructible.”\n\n- A “key ingredient in the main theorem” is the following criterion.\n\n\\begin{proposition}\\label{prop:height_over_absorbed}\n If $\\partial\\br(\\partial) \\prec 1$, then\n \\[\\height(x,\\partial)=\\inf\\{k\\in \\omega: \\val(\\lder^k(\\partial x))\\in \\val(\\bE)\\}\\in \\omega+1.\\]\n \\begin{proof}\n Set $a=\\partial x$. Clearly the height of $x$ is $0$ if and only if $\\val(a)\\in \\val(\\bE)$.\n\n It suffices to show that for all $m\\ge 0$, there is an essential approximate logarithmic scale $(y_k)_{k < m}$ with $a \\sim \\partial y_0$ (in the case $m>0$) if and only if $\\val(\\lder^k(a))\\notin \\val(\\bE)$ for all $k\\mathcal{O}}$, for every $y \\in \\mathbb E \\langle b \\rangle$ there is a $\\mathbb E$-definable gne $g$, such that $y \\equiv_\\mathbb E g(b)$;\n \\item\\label{introthmenum:gne_wim_1} all 1-$\\dcl_T$-dimensional wim-constructible extension of models of $T_\\mathrm{convex}$ have the gne-property;\n \\item\\label{introthmenum:gne_res_1} all 1-$\\dcl_T$-dimensional res-constructible extensions of models of $T_\\mathrm{convex}$ have the gne property;\n \\item\\label{introthmenum:gne_wim} all wim-constructible extensions of models of $T_\\mathrm{convex}$ have the gne property;\n \\item\\label{introthmenum:gne_res} all res-constructible extensions of models of $T_\\mathrm{convex}$ have the gne property.\n \\end{enumerate}", + "theorem_type": [ + "Biconditional or Equivalence", + "Universal–Existential" + ], + "mcq": { + "question": "Let $T$ be an exponential o-minimal theory, and let $T_{\\mathrm{convex}}$ be the complete theory of models $(\\mathbb E,\\mathcal O)$ of $T$ expanded by a predicate for a nontrivial $T$-convex valuation subring $\\mathcal O$. For a $T$-definable function $f(\\bar x,t)$, write\n\\[\n\\dagger_n f(\\bar x,t)=\\frac{\\partial f(\\bar x,t)/\\partial t}{f(\\bar x,t)}\n\\]\nwhen defined and $\\infty$ otherwise, and let\n\\[\nG_m^f(\\bar x,t):=\\min\\{\\,|(\\dagger_n^k f)(\\bar x,t)|:k\\le m\\,\\}.\n\\]\nCall $T$ transserial if for every $n$ and every $(n+1)$-ary $T$-definable $f$ there exists $m\\in\\mathbb N$ such that\n\\[\nT_{\\mathrm{convex}}\\models \\forall \\bar x\\,\\exists r\\in\\mathcal O\\,\\forall t\\in\\mathcal O\\,(t>r\\to G_m^f(\\bar x,t)<1).\n\\]\nA generalized nested exponential (gne) over $\\mathbb E$ is a finite composition of translations by elements of $\\mathbb E$, sign changes, and exponentials. An elementary extension $(\\mathbb E,\\mathcal O)\\preceq(\\mathbb E_*,\\mathcal O_*)$ has the gne-property if for every $y\\in\\mathbb E_*$ there exist an $\\mathbb E$-definable gne $g$ and some $z\\in\\mathbb E_*$ with $g(z)=y$ and $\\mathbf v(z-\\mathbb E)\\subseteq \\mathbf v(\\mathbb E)$. A res-constructible extension is one admitting a $\\operatorname{dcl}_T$-basis over $\\mathbb E$ whose residues are $\\operatorname{dcl}_T$-independent over the base residue field; a wim-constructible extension is one admitting an ordinal-indexed $\\operatorname{dcl}_T$-basis such that each new element is a pseudolimit of a pseudo-Cauchy sequence over the previously generated structure with no pseudolimit there; and 1-$\\operatorname{dcl}_T$-dimensional means $\\operatorname{dcl}_T$-dimension $1$ over the base. If $b$ lies in an elementary extension and realizes the cut $\\mathcal O\\mathcal O}$, let $\\mathbb E\\langle b\\rangle=\\operatorname{dcl}_T(\\mathbb E b)$, and write $y\\equiv_{\\mathbb E} g(b)$ to mean that $y$ and $g(b)$ have the same type over $\\mathbb E$. Which statement is valid?", + "correct_choice": { + "label": "A", + "text": "The following are equivalent for $T$: (1) $T$ is transserial; (2) $T$ is exponentially bounded and for every $(\\mathbb E,\\mathcal O)\\models T_{\\mathrm{convex}}$, every $b$ realizing $\\mathcal O\\mathcal O}$, and every $y\\in\\mathbb E\\langle b\\rangle$, there exists an $\\mathbb E$-definable gne $g$ such that $y\\equiv_{\\mathbb E} g(b)$; (3) every 1-$\\operatorname{dcl}_T$-dimensional wim-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; (4) every 1-$\\operatorname{dcl}_T$-dimensional res-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; (5) every wim-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; and (6) every res-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property." + }, + "choices": [ + { + "label": "B", + "text": "The following are equivalent for $T$: (1) $T$ is transserial; (2) for every $(\\mathbb E,\\mathcal O)\\models T_{\\mathrm{convex}}$, every $b$ realizing $\\mathcal O\\mathcal O}$, and every $y\\in\\mathbb E\\langle b\\rangle$, there exists an $\\mathbb E$-definable gne $g$ such that $y\\equiv_{\\mathbb E} g(b)$; (3) every 1-$\\operatorname{dcl}_T$-dimensional wim-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; (4) every 1-$\\operatorname{dcl}_T$-dimensional res-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; (5) every wim-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; and (6) every res-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property." + }, + { + "label": "C", + "text": "If $T$ is transserial, then every wim-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property, and every res-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property." + }, + { + "label": "D", + "text": "The following are equivalent for $T$: (1) $T$ is transserial; (2) $T$ is exponentially bounded and for every $(\\mathbb E,\\mathcal O)\\models T_{\\mathrm{convex}}$, every $b$ realizing $\\mathcal O\\mathcal O}$, and every $y\\in\\mathbb E\\langle b\\rangle$, there exists a single $\\mathbb E$-definable gne $g$ depending only on $b$ such that $y\\equiv_{\\mathbb E} g(b)$ for all such $y$; (3) every 1-$\\operatorname{dcl}_T$-dimensional wim-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; (4) every 1-$\\operatorname{dcl}_T$-dimensional res-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; (5) every wim-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; and (6) every res-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property." + }, + { + "label": "E", + "text": "The following are equivalent for $T$: (1) $T$ is transserial; (2) $T$ is exponentially bounded and for every $(\\mathbb E,\\mathcal O)\\models T_{\\mathrm{convex}}$, every $b$ realizing $\\mathcal O\\mathcal O}$, and every $y\\in\\mathbb E\\langle b\\rangle$, there exists an $\\mathbb E$-definable gne $g$ such that $y=g(b)$; (3) every 1-$\\operatorname{dcl}_T$-dimensional wim-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; (4) every 1-$\\operatorname{dcl}_T$-dimensional res-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; (5) every wim-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property; and (6) every res-constructible elementary extension of models of $T_{\\mathrm{convex}}$ has the gne-property." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "B" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "characteristic", + "tampered_component": "exponential_boundedness_hypothesis_in_clause_2", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "equivalence_replaced_by_forward_implication_only", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "existential_dependence_of_gne_on_y", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "type_equivalence_replaced_by_actual_equality", + "template_used": "stronger_trap" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives definitions and setup but does not explicitly reveal the theorem or directly signal choice A. The correct answer is not stated or trivially inferable from the wording alone." + }, + "TAS": { + "score": 1, + "justification": "The item is not a direct restatement of a definition in the stem, but it is very close to asking for the exact formulation of a theorem. It mainly tests recognition of the right equivalence statement rather than deriving a new conclusion." + }, + "GPS": { + "score": 1, + "justification": "There is some reasoning pressure because the choices differ by subtle logical modifications: omitted hypotheses, weakened implication, altered quantifier dependence, and equality versus type-equivalence. However, the task is still largely theorem-recall/statement-discrimination rather than genuinely generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong and mathematically meaningful. Each reflects a plausible failure mode: dropping exponential boundedness, replacing equivalence by one implication, making the gne uniform in the wrong way, or strengthening type-equivalence to equality." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise recall of a theorem statement than toward deep generative reasoning." + } + }, + { + "id": "2511.07383v1", + "paper_link": "http://arxiv.org/abs/2511.07383v1", + "theorems_cnt": 5, + "theorem": { + "env_name": "introthm", + "content": "\\label{main1}\nLet $G$ be a group and let $N$ be a normal subgroup of $G$. Suppose ${\\rm Irr} (G/N)=\\{\\beta_{1}, \\cdots, \\beta_{n}\\}$ and $\\chi\\in {\\rm Irr} (G)$, and suppose that the $\\beta_{1}\\chi, \\cdots, \\beta_{n}\\chi$ are distinct and irreducible. Then $\\chi_{N}$ is an irreducible character of $N$.", + "start_pos": 5840, + "end_pos": 6177, + "label": "main1" + }, + "ref_dict": { + "main1": "\\begin{introthm} \\label{main1}\nLet $G$ be a group and let $N$ be a normal subgroup of $G$. Suppose ${\\rm Irr} (G/N)=\\{\\beta_{1}, \\cdots, \\beta_{n}\\}$ and $\\chi\\in {\\rm Irr} (G)$, and suppose that the $\\beta_{1}\\chi, \\cdots, \\beta_{n}\\chi$ are distinct and irreducible. Then $\\chi_{N}$ is an irreducible character of $N$.\n\\end{introthm}" + }, + "pre_theorem_intro_text_len": 840, + "pre_theorem_intro_text": "In this paper, all groups are finite. Take $G$ to be a group, $p$ to be a prime, and $\\pi$ to be a set of primes. We write ${\\rm Irr} (G)$ for the set of irreducible characters of $G$, ${\\rm IBr} (G)$ for the set irreducible ($p$-)Brauer characters of $G$, and ${\\rm I}_\\pi (G)$ for the set of irreducible $\\pi$-partial characters of $G$.\n\nGallagher's theorem \\cite[Corollary 6.17]{Isaacs1976} can be stated as follows: let $N$ be a normal subgroup of $G$ and let $\\chi\\in {\\rm Irr} (G)$ be such that $\\chi_{N} = \\theta \\in {\\rm Irr} (N)$, where $\\chi_{N}$ is the restriction of $\\chi$ to $N$. Then the characters $\\beta\\chi$ for $\\beta\\in {\\rm Irr} (G/N)$ are irreducible and distinct for distinct $\\beta$ and are all of the irreducible constituents of $\\theta^{G}$.\n\nThe first goal in this paper is to prove the converse of this theorem:", + "context": "In this paper, all groups are finite. Take $G$ to be a group, $p$ to be a prime, and $\\pi$ to be a set of primes. We write ${\\rm Irr} (G)$ for the set of irreducible characters of $G$, ${\\rm IBr} (G)$ for the set irreducible ($p$-)Brauer characters of $G$, and ${\\rm I}_\\pi (G)$ for the set of irreducible $\\pi$-partial characters of $G$.\n\nGallagher's theorem \\cite[Corollary 6.17]{Isaacs1976} can be stated as follows: let $N$ be a normal subgroup of $G$ and let $\\chi\\in {\\rm Irr} (G)$ be such that $\\chi_{N} = \\theta \\in {\\rm Irr} (N)$, where $\\chi_{N}$ is the restriction of $\\chi$ to $N$. Then the characters $\\beta\\chi$ for $\\beta\\in {\\rm Irr} (G/N)$ are irreducible and distinct for distinct $\\beta$ and are all of the irreducible constituents of $\\theta^{G}$.\n\nThe first goal in this paper is to prove the converse of this theorem:", + "full_context": "In this paper, all groups are finite. Take $G$ to be a group, $p$ to be a prime, and $\\pi$ to be a set of primes. We write ${\\rm Irr} (G)$ for the set of irreducible characters of $G$, ${\\rm IBr} (G)$ for the set irreducible ($p$-)Brauer characters of $G$, and ${\\rm I}_\\pi (G)$ for the set of irreducible $\\pi$-partial characters of $G$.\n\nGallagher's theorem \\cite[Corollary 6.17]{Isaacs1976} can be stated as follows: let $N$ be a normal subgroup of $G$ and let $\\chi\\in {\\rm Irr} (G)$ be such that $\\chi_{N} = \\theta \\in {\\rm Irr} (N)$, where $\\chi_{N}$ is the restriction of $\\chi$ to $N$. Then the characters $\\beta\\chi$ for $\\beta\\in {\\rm Irr} (G/N)$ are irreducible and distinct for distinct $\\beta$ and are all of the irreducible constituents of $\\theta^{G}$.\n\nThe first goal in this paper is to prove the converse of this theorem:\n\n\\begin{abstract}\nLet $G$ be a finite group. Suppose $N$ is a normal subgroup of $G$. Recall that Gallagher's theorem states that if $\\chi \\in {\\rm Irr} (G)$ satisfies $\\chi_N$ is irreducible, then $\\chi \\beta$ is irreducible and distinct for all $\\beta \\in {\\rm Irr} (G/N)$. Furthermore, if $\\theta = \\chi_N$, then these are all of the irreducible constituents of $\\theta^G$. We prove that the converse of this theorem holds. We also prove that a partial converse of the Brauer version of this theorem holds. Finally, we prove that an analog of Gallagher's theorem holds for Isaacs' $\\pi$-partial characters and that a partial converse of that theorem is true.\n\\end{abstract}\n\nThe first goal in this paper is to prove the converse of this theorem:\n\nWe will see that we will obtain Theorem \\ref{main1} as a corollary to the version for the Brauer character version of the theorem. The Brauer character version of Gallagher's theorem \\cite[Corollary 8.20]{Navarro1998} is as follows: Let $N\\lhd G$ and let $\\eta\\in \\IBr (G)$. If $\\eta_{N} = \\theta \\in \\IBr (N)$, then the characters $\\beta\\eta$ for $\\beta\\in \\IBr(G/N)$ are irreducible, distinct for distinct $\\beta$ and are all of the irreducible constituents of $\\theta^{G}$. Now we consider a partial converse of the Brauer character version of Gallagher's theorem.\n\n\\begin{introthm} \\label{main2}\nLet $G$ be a group, let $N$ be a normal subgroup of $G$, and let $p$ be a prime. Suppose $\\eta\\in {\\rm IBr}(G)$ and ${\\rm IBr}(G/N)=\\{\\beta_{1}, \\cdots, \\beta_{n}\\}$, and suppose that $p\\nmid |G/N|$ and the $\\beta_{1}\\eta, \\cdots, \\beta_{n}\\eta$ are distinct and irreducible. Then $\\eta_{N}$ is an irreducible Brauer character of $N$.\n\\end{introthm}\n\n\\begin{cor} \\label{bpi Gall,2inpi,|G|odd}\nLet $\\pi$ be a set of primes, let $G$ be a $\\pi$-separable group, and let $N$ be a normal subgroup of $G$. Suppose $2 \\in \\pi$ or $|G|$ is odd. Suppose there exists a character $\\chi \\in {\\rm B}_{\\pi} (G)$ so that $\\theta = \\chi_N \\in \\irr (N)$. Then the map $\\beta \\mapsto \\beta\\chi$ is a bijection from ${\\rm B}_{\\pi} (G/N)$ to ${\\rm B}_\\pi (G \\mid \\theta)$.\n\\end{cor}\n\nIn fact, the partial converse of Gallagher's theorem will apply for sets of lifts of the $\\pi$-partial characters. To see this we make the following definition we again only need that we have subsets of $\\irr (G)$ and $\\irr (N)$ that are in bijection with ${\\rm I}_\\pi (G)$ and ${\\rm I}_\\pi (N)$, and that irreducible constituents lie in this set. Thus, we prove the result in this generality so that it could be applied to any set of lifts. \n\\begin{cor}\\label{lifts conv}\nLet $\\pi$ be a set of primes, let $G$ be a $\\pi$-separable group, and let $N$ be a normal subgroup of $G$ so that $G/N$ is a $\\pi$-group. Assume there exist subsets $X_\\pi (G) \\subseteq \\irr (G)$ and $X_\\pi (N) \\subseteq \\irr (N)$ so that the map $\\theta \\mapsto \\theta^o$ is a bijection from $X_\\pi (G)$ to ${\\rm I}_\\pi (G)$ and from $X_\\pi (N)$ to ${\\rm I}_\\pi (N)$, and assume for $\\chi \\in X_\\pi (G)$ that the constituents of $\\chi_N$ lie in $X_\\pi (N)$. If there exist $\\chi \\in X_\\pi (G)$ and $\\irr (G/N) = \\{ \\gamma_1 = 1, \\dots, \\gamma_n \\}$ that satisfy $(\\gamma_1 \\chi)^o, \\dots, (\\gamma_n \\chi)^o$ are irreducible and distinct, then $\\chi_N \\in X_\\pi (N)$.\n\\end{cor}\n\n\\begin{thm}\nLet $\\pi$ be a set of primes and let $G$ be a $\\pi$-separable group. Let $N$ a normal subgroup of $G$. Let $\\eta \\in {\\rm I}_\\pi (G)$ and ${\\rm I}_{\\pi} (G/N) = \\{\\beta_{1}, \\cdots, \\beta_{n}\\}$. Suppose that $|G/N|$ is divisible only by primes in $\\pi$ and that the $\\{\\beta_{1}\\eta, \\cdots, \\beta_{n}\\eta\\}$ are distinct and irreducible. Then $\\eta_{N}$ is an irreducible $\\pi$-partial character of $N$.\n\\end{thm}\n\n\\begin{thm}\\label{bpi conv}\n Let $\\pi$ be a set of primes, let $G$ be a $\\pi$-separable group, and let $N$ be a normal subgroup of $G$. Suppose there exist characters $\\chi \\in {\\rm B}_{\\pi} (G)$ and ${\\rm B}_{\\pi} (G/N) = \\{\\beta_1, \\dots, \\beta_n \\}$. Assume that $G/N$ is a $\\pi$-group and the $\\{\\beta_1 \\chi, \\dots, \\beta_n \\chi\\}$ are distinct and irreducible Then $\\chi_N$ is irreducible.\n\\end{thm}\n\n\\begin{introthm} \\label{main1}\nLet $G$ be a group and let $N$ be a normal subgroup of $G$. Suppose ${\\rm Irr} (G/N)=\\{\\beta_{1}, \\cdots, \\beta_{n}\\}$ and $\\chi\\in {\\rm Irr} (G)$, and suppose that the $\\beta_{1}\\chi, \\cdots, \\beta_{n}\\chi$ are distinct and irreducible. Then $\\chi_{N}$ is an irreducible character of $N$.\n\\end{introthm}", + "post_theorem_intro_text_len": 3452, + "post_theorem_intro_text": "We will see that we will obtain Theorem \\ref{main1} as a corollary to the version for the Brauer character version of the theorem. The Brauer character version of Gallagher's theorem \\cite[Corollary 8.20]{Navarro1998} is as follows: Let $N\\lhd G$ and let $\\eta\\in \\IBr (G)$. If $\\eta_{N} = \\theta \\in \\IBr (N)$, then the characters $\\beta\\eta$ for $\\beta\\in \\IBr(G/N)$ are irreducible, distinct for distinct $\\beta$ and are all of the irreducible constituents of $\\theta^{G}$. Now we consider a partial converse of the Brauer character version of Gallagher's theorem.\n\n\\begin{introthm} \\label{main2}\nLet $G$ be a group, let $N$ be a normal subgroup of $G$, and let $p$ be a prime. Suppose $\\eta\\in {\\rm IBr}(G)$ and ${\\rm IBr}(G/N)=\\{\\beta_{1}, \\cdots, \\beta_{n}\\}$, and suppose that $p\\nmid |G/N|$ and the $\\beta_{1}\\eta, \\cdots, \\beta_{n}\\eta$ are distinct and irreducible. Then $\\eta_{N}$ is an irreducible Brauer character of $N$.\n\\end{introthm}\n\nWe next consider Isaacs' $\\pi$-theory, where $\\pi$ is a set of primes. For a $\\pi$-separable group $G$, Isaacs has defined an analog of Brauer characters that are defined on the $\\pi$-elements of $G$ that are called the $\\pi$-partial characters of $G$. We will write ${\\rm I}_\\pi (G)$ for the set of irreducible $\\pi$-partial characters of $G$. We will review the points of $\\pi$-theory we need in Sections \\ref{secn: pi theory} and \\ref{secn: nucl}. Interestingly, while Isaacs proved analogs of many other results for $\\pi$-partial characters, he does not seem to have proved an analog of Gallagher's theorem. We do that next.\n\n\\begin{introthm}\\label{ipi Gall,gen}\nLet $\\pi$ be a set of primes, let $G$ be a $\\pi$-separable group, and let $N$ be a normal subgroup of $G$. Suppose there exists a character $\\zeta \\in {\\rm I}_{\\pi} (G)$ so that $\\xi = \\zeta_N \\in {\\rm I}_\\pi (N)$. Then the map $\\kappa \\mapsto \\kappa\\zeta$ is a bijection from ${\\rm I}_{\\pi} (G/N)$ to ${\\rm I}_\\pi (G \\mid \\xi)$.\n\\end{introthm}\n\nNow, in \\cite{Isaacs1976}, there is a generalization of Gallagher's theorem (Theorem 6.16 of \\cite{Isaacs1976}) of which Gallaher's theorem is obtained as a corollary. We have not been able to obtain an analog of this generalization in cases, but we can obtain an analog when we add the extra assumption that $2 \\in \\pi$ or $|G|$ is odd.\n\n\\begin{introthm} \\label{ipi pre,2inpi,|G|odd}\nLet $\\pi$ be a set of primes, let $G$ be a $\\pi$-separable group, and let $N$ be a normal subgroup of $G$. Suppose $2 \\in \\pi$ or $|G|$ is odd. Assume there exist partial characters $\\eta, \\xi \\in {\\rm I}_\\pi (N)$ so that $\\eta$ is $G$-invariant, $\\xi$ extends to $\\zeta \\in {\\rm I}_\\pi (G)$ and $\\eta \\xi \\in {\\rm I}_{\\pi} (N)$. Then the map $\\kappa \\mapsto \\kappa \\zeta$ is a bijection from ${\\rm I}_\\pi (G \\mid \\eta) \\rightarrow {\\rm I}_\\pi (G \\mid \\eta \\xi)$.\n\\end{introthm}\n\nWe close with a partial converse for the analog of Gallagher's theorem for $\\pi$-partial characters. We will show that it is not possible to have a full converse in this case.\n\n\\begin{introthm}\\label{ipi conv}\nLet $\\pi$ be a set of primes, let $G$ be a $\\pi$-separable group, and let $N$ be a normal subgroup of $G$.\nSuppose there exists partial characters $\\zeta \\in {\\rm I}_{\\pi} (G)$ and ${\\rm I}_{\\pi} (G/N) = \\{\\kappa_1, \\dots, \\kappa_n \\}$. Assume that $G/N$ is a $\\pi$-group and the $\\kappa_1 \\zeta, \\dots, \\kappa_n \\zeta $ are distinct and irreducible Then $\\zeta_N$ is irreducible.\n\\end{introthm}", + "sketch": "We will see that we will obtain Theorem~\\ref{main1} as a corollary to the version for the Brauer character version of the theorem. In particular, the introduction cites the Brauer character version of Gallagher's theorem: if $\\eta\\in \\IBr(G)$ and $\\eta_N=\\theta\\in\\IBr(N)$, then the characters $\\beta\\eta$ for $\\beta\\in \\IBr(G/N)$ are irreducible, distinct for distinct $\\beta$, and are all of the irreducible constituents of $\\theta^G$; then it says: \"Now we consider a partial converse\" (stated as Theorem~\\ref{main2}).", + "expanded_sketch": "We will see that we will obtain the main theorem as a corollary to the version for the Brauer character version of the theorem. In particular, the introduction cites the Brauer character version of Gallagher's theorem: if $\\eta\\in \\IBr(G)$ and $\\eta_N=\\theta\\in\\IBr(N)$, then the characters $\\beta\\eta$ for $\\beta\\in \\IBr(G/N)$ are irreducible, distinct for distinct $\\beta$, and are all of the irreducible constituents of $\\theta^G$; then it says: \"Now we consider a partial converse\" (stated as Theorem~\\ref{main2}).,", + "expanded_theorem": "\\label{main1}\nLet $G$ be a group and let $N$ be a normal subgroup of $G$. Suppose ${\\rm Irr} (G/N)=\\{\\beta_{1}, \\cdots, \\beta_{n}\\}$ and $\\chi\\in {\\rm Irr} (G)$, and suppose that the $\\beta_{1}\\chi, \\cdots, \\beta_{n}\\chi$ are distinct and irreducible. Then $\\chi_{N}$ is an irreducible character of $N$.", + "theorem_type": "unknown", + "mcq": { + "question": "Let $G$ be a finite group and $N\\trianglelefteq G$. Write ${\\rm Irr}(H)$ for the set of irreducible complex characters of a finite group $H$. Suppose ${\\rm Irr}(G/N)=\\{\\beta_1,\\dots,\\beta_n\\}$, let $\\chi\\in {\\rm Irr}(G)$, and assume that the characters $\\beta_1\\chi,\\dots,\\beta_n\\chi$ on $G$ (where each $\\beta_i$ is inflated from $G/N$ to $G$ and then multiplied pointwise with $\\chi$) are pairwise distinct and irreducible. Which conclusion holds for the restriction $\\chi_N$ of $\\chi$ to $N$?", + "correct_choice": { + "label": "A", + "text": "The restricted character $\\chi_N$ is an irreducible character of $N$." + }, + "choices": [ + { + "label": "B", + "text": "The restricted character $\\chi_N$ is a sum of pairwise distinct irreducible characters of $N$, each occurring with multiplicity $1$." + }, + { + "label": "C", + "text": "The restricted character $\\chi_N$ has at least one irreducible constituent of $N$." + }, + { + "label": "D", + "text": "The induced character $(\\chi_N)^G$ has irreducible constituents exactly $\\beta_1\\chi,\\dots,\\beta_n\\chi$, each occurring with multiplicity $1$." + }, + { + "label": "E", + "text": "For every irreducible constituent $\\theta$ of $\\chi_N$, the characters $\\beta_1\\chi,\\dots,\\beta_n\\chi$ are precisely the irreducible constituents of $\\theta^G$." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "irreducible-restriction conclusion weakened to multiplicity-free reducibility", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped irreducibility of the whole restriction, keeping only existence of an irreducible constituent", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "converse conclusion replaced by full Gallagher constituent description for induction", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "hidden dependence on choosing the unique constituent of $\\chi_N$; falsified by quantifying over every constituent", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly reveal the correct conclusion about \u001d\u0016_N. It gives a nontrivial hypothesis about the twists \u001d\u0016_i\u001d\u0016 and asks for a consequence, so there is no direct answer leakage." + }, + "TAS": { + "score": 1, + "justification": "The item is very close to a standard character-theoretic result: the hypothesis is essentially tailored to force the conclusion that \u001d\u0016_N is irreducible. So it is not a literal restatement, but it is only a mild reformulation of a known theorem." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is required to distinguish the exact conclusion from stronger or weaker nearby statements, especially against options D and E. However, for a student who recognizes the theorem, the answer is largely pattern-matching rather than deep generative reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and reflect common failure modes: weakening to a trivially true statement (C), confusing irreducibility with multiplicity-free reducibility (B), and overstrengthening via induced-character conclusions (D, E). They are distinct and well aligned with the topic." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with good distractors and no answer leakage, but it is quite close to a standard theorem statement, so it only moderately tests genuine generative reasoning." + } + }, + { + "id": "2511.07058v1", + "paper_link": "http://arxiv.org/abs/2511.07058v1", + "theorems_cnt": 6, + "theorem": { + "env_name": "lemma", + "content": "\\label{boundedind}\n Let $G$ be a definable group of finite dimension and $\\{H_i\\}_{i\\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\\omega$ such that there is no $J=\\{j_1,...,j_n\\}\\subseteq I$ of cardinality $n$ such that $|\\bigcap_{i=1}^kH_{j_i}:\\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\\leq n-1$.", + "start_pos": 9114, + "end_pos": 9493, + "label": "boundedind" + }, + "ref_dict": { + "boundedind": "\\begin{lemma}\\label{boundedind}\n Let $G$ be a definable group of finite dimension and $\\{H_i\\}_{i\\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\\omega$ such that there is no $J=\\{j_1,...,j_n\\}\\subseteq I$ of cardinality $n$ such that $|\\bigcap_{i=1}^kH_{j_i}:\\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\\leq n-1$.\n\\end{lemma}" + }, + "pre_theorem_intro_text_len": 6977, + "pre_theorem_intro_text": "The present work characterizes, under the model-theoretic hypothesis of finite-dimensionality,\nirreducible bi-modules, i.e., abelian groups together with two commuting subrings of endomorphisms, and which are \"irreducible\" for the bi-action.\\\\\nThis result is well-known for groups of finite Morley rank \\cite{zilber1984some}. It is also known for $o$-minimal theories (see, for example \\cite{peterzil2000simple}). The main problem is that the two proofs use techniques proper to the two families of theories involved (in the finite Morley rank case, the indecomposability theorem is necessary). A natural question arises: to which generality can we extend the linearization?\\\\\nAn interesting family of theories is finite-dimensional theories, in the sense of \\cite{wagner2020dimensional}.\n\\begin{defn}\n A theory $T$ is said to be \\emph{finite-dimensional} if there exists a function $\\operatorname{dim}$ from the class of all interpretable subset in any model $\\mathcal{M}$ of $T$ into $\\omega\\cup\\{-\\infty\\}$ such that for any $\\phi(x,y)$ formula, $X,Y$ interpretable sets in $T$ and $f$ interpretable function from $X$ to $Y$,\n\\begin{itemize}\n \\item If $a,a'$ have same type over $\\emptyset$, $\\operatorname{dim}(\\phi(x,a))=\\operatorname{dim}(\\phi(x,a'))$ \n \\item $\\operatorname{dim}(\\emptyset)=-\\infty$ and $\\operatorname{dim}(X)=0$ if and only if $X$ is finite;\n \\item $\\operatorname{dim}(X\\cup Y)=\\max\\{\\operatorname{dim}(X),\\operatorname{dim}(Y)\\}$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\geq k$ if for any $y\\in Y$, then $\\operatorname{dim}(X)\\geq \\operatorname{dim}(Y)+k$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\leq k$ for any $y\\in Y$, then $\\operatorname{dim}(X)\\leq \\operatorname{dim}(Y)+k$.\n\\end{itemize}\n\\end{defn}\nExamples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\\text{\\thorn}}$-rank.\\\\\n A first proof of Zilber's Field Theorem in this context has been given by Deloro in \\cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\\\\n In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\\phi$ of an abelian group $A$ is a subgroup of $A\\times A$ such that $\\pi_1(\\phi)=A$ and $\\{a\\in A:\\ (0,a)\\in \\phi\\}$ is finite. The latter subgroup is called the \\emph{katakernel} of $\\phi$. A \\emph{quasi-endomorphism} $\\phi$ is a subgroup of $A$ such that $\\pi_1(\\phi)$, that is called the domain of $\\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\\\\nEndogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\\Gamma\\leq \\operatorname{End}(G)$. Assume that $H$ and $H_1$ are \\emph{$\\Gamma$-almost invariant} \\hbox{i.e.} $\\gamma(H)\\cap H$ is of finite index in $\\gamma(H)$ for any $\\gamma\\in \\Gamma $. Then, $\\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\\gamma$ equal to $\\gamma(H)+H/H$) and $\\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\\{h\\in H:\\ \\gamma(h)\\in H\\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\\Gamma$-almost invariant, the connected component $H^0$ is $\\Gamma$-invariant and the action of $\\Gamma$ on $G/H^0$ is again by endomorphisms.\\\\\nThe article follows the ideas of \\cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\\\\nIn the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\\\\nThen, we proceed to prove the following result.\n\\begin{thmA}\nIn a finite-dimensional theory, let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that:\n\\begin{itemize}\n \\item $\\Gamma,\\Delta$ sharply commute;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa.\n\\end{itemize}\n There is a finite $\\Gamma$ and $\\Delta$-invariant subgroup $F$ such that $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act by endomorphisms on $A/F$.\n\\end{thmA}\nMoreover, $F$ will be given explicitly.\\\\\nIn the second section, we verify the easy case in which both $\\Gamma$ and $\\Delta$ act by endogenies with finite kernel or finite image.\\\\\nThe third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\\\\nThe main result of the fifth section is the following linearization theorem.\n\\begin{thmB}\n Let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that \n \\begin{itemize}\n \\item $C^{\\#}(\\Gamma)=\\Delta$ and $C^{\\#}(\\Delta)=\\Gamma$;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa;\n \\item $A_0=\\Kat(\\Gamma,\\Delta)$.\n \\end{itemize}\n Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\\big(\\Gamma/{\\sim}\\big)\\cap \\big(\\Delta/{\\sim}\\big)$ and $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act $K$-linearly on $A/A_0$.\n\\end{thmB}\n\\textbf{Remark}:\\\\\nThese two theorems are not direct extensions of the Theorems in \\cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \\emph{absolutely minimal} \\hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\\\\nIn the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\\Gamma,\\Delta$ is of quasi-endomorphisms and the other of endogenies.\\\\\nIn section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.\n\\subsection{Finite-dimensional groups}\nIn this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \\ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.", + "context": "mptyset)=-\\infty$ and $\\operatorname{dim}(X)=0$ if and only if $X$ is finite;\n \\item $\\operatorname{dim}(X\\cup Y)=\\max\\{\\operatorname{dim}(X),\\operatorname{dim}(Y)\\}$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\geq k$ if for any $y\\in Y$, then $\\operatorname{dim}(X)\\geq \\operatorname{dim}(Y)+k$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\leq k$ for any $y\\in Y$, then $\\operatorname{dim}(X)\\leq \\operatorname{dim}(Y)+k$.\n\\end{itemize}\n\\end{defn}\nExamples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\\text{\\thorn}}$-rank.\\\\\n A first proof of Zilber's Field Theorem in this context has been given by Deloro in \\cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\\\\n In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\\phi$ of an abelian group $A$ is a subgroup of $A\\times A$ such that $\\pi_1(\\phi)=A$ and $\\{a\\in A:\\ (0,a)\\in \\phi\\}$ is finite. The latter subgroup is called the \\emph{katakernel} of $\\phi$. A \\emph{quasi-endomorphism} $\\phi$ is a subgroup of $A$ such that $\\pi_1(\\phi)$, that is called the domain of $\\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\\\\nEndogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\\Gamma\\leq \\operatorname{End}(G)$. Assume that $H$ and $H_1$ are \\emph{$\\Gamma$-almost invariant} \\hbox{i.e.} $\\gamma(H)\\cap H$ is of finite index in $\\gamma(H)$ for any $\\gamma\\in \\Gamma $. Then, $\\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\\gamma$ equal to $\\gamma(H)+H/H$) and $\\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\\{h\\in H:\\ \\gamma(h)\\in H\\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\\Gamma$-almost invariant, the connected component $H^0$ is $\\Gamma$-invariant and the action of $\\Gamma$ on $G/H^0$ is again by endomorphisms.\\\\\nThe article follows the ideas of \\cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\\\\nIn the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\\\\nThen, we proceed to prove the following result.\n\\begin{thmA}\nIn a finite-dimensional theory, let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that:\n\\begin{itemize}\n \\item $\\Gamma,\\Delta$ sharply commute;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa.\n\\end{itemize}\n There is a finite $\\Gamma$ and $\\Delta$-invariant subgroup $F$ such that $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act by endomorphisms on $A/F$.\n\\end{thmA}\nMoreover, $F$ will be given explicitly.\\\\\nIn the second section, we verify the easy case in which both $\\Gamma$ and $\\Delta$ act by endogenies with finite kernel or finite image.\\\\\nThe third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\\\\nThe main result of the fifth section is the following linearization theorem.\n\\begin{thmB}\n Let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that \n \\begin{itemize}\n \\item $C^{\\#}(\\Gamma)=\\Delta$ and $C^{\\#}(\\Delta)=\\Gamma$;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa;\n \\item $A_0=\\Kat(\\Gamma,\\Delta)$.\n \\end{itemize}\n Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\\big(\\Gamma/{\\sim}\\big)\\cap \\big(\\Delta/{\\sim}\\big)$ and $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act $K$-linearly on $A/A_0$.\n\\end{thmB}\n\\textbf{Remark}:\\\\\nThese two theorems are not direct extensions of the Theorems in \\cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \\emph{absolutely minimal} \\hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\\\\nIn the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\\Gamma,\\Delta$ is of quasi-endomorphisms and the other of endogenies.\\\\\nIn section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.\n\\subsection{Finite-dimensional groups}\nIn this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \\ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.\n\n\\begin{lemma}\\label{boundedind}\n Let $G$ be a definable group of finite dimension and $\\{H_i\\}_{i\\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\\omega$ such that there is no $J=\\{j_1,...,j_n\\}\\subseteq I$ of cardinality $n$ such that $|\\bigcap_{i=1}^kH_{j_i}:\\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\\leq n-1$.\n\\end{lemma}", + "full_context": "mptyset)=-\\infty$ and $\\operatorname{dim}(X)=0$ if and only if $X$ is finite;\n \\item $\\operatorname{dim}(X\\cup Y)=\\max\\{\\operatorname{dim}(X),\\operatorname{dim}(Y)\\}$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\geq k$ if for any $y\\in Y$, then $\\operatorname{dim}(X)\\geq \\operatorname{dim}(Y)+k$;\n \\item If $\\operatorname{dim}(f^{-1}(y))\\leq k$ for any $y\\in Y$, then $\\operatorname{dim}(X)\\leq \\operatorname{dim}(Y)+k$.\n\\end{itemize}\n\\end{defn}\nExamples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\\text{\\thorn}}$-rank.\\\\\n A first proof of Zilber's Field Theorem in this context has been given by Deloro in \\cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\\\\n In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\\phi$ of an abelian group $A$ is a subgroup of $A\\times A$ such that $\\pi_1(\\phi)=A$ and $\\{a\\in A:\\ (0,a)\\in \\phi\\}$ is finite. The latter subgroup is called the \\emph{katakernel} of $\\phi$. A \\emph{quasi-endomorphism} $\\phi$ is a subgroup of $A$ such that $\\pi_1(\\phi)$, that is called the domain of $\\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\\\\nEndogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\\Gamma\\leq \\operatorname{End}(G)$. Assume that $H$ and $H_1$ are \\emph{$\\Gamma$-almost invariant} \\hbox{i.e.} $\\gamma(H)\\cap H$ is of finite index in $\\gamma(H)$ for any $\\gamma\\in \\Gamma $. Then, $\\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\\gamma$ equal to $\\gamma(H)+H/H$) and $\\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\\{h\\in H:\\ \\gamma(h)\\in H\\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\\Gamma$-almost invariant, the connected component $H^0$ is $\\Gamma$-invariant and the action of $\\Gamma$ on $G/H^0$ is again by endomorphisms.\\\\\nThe article follows the ideas of \\cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\\\\nIn the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\\\\nThen, we proceed to prove the following result.\n\\begin{thmA}\nIn a finite-dimensional theory, let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that:\n\\begin{itemize}\n \\item $\\Gamma,\\Delta$ sharply commute;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa.\n\\end{itemize}\n There is a finite $\\Gamma$ and $\\Delta$-invariant subgroup $F$ such that $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act by endomorphisms on $A/F$.\n\\end{thmA}\nMoreover, $F$ will be given explicitly.\\\\\nIn the second section, we verify the easy case in which both $\\Gamma$ and $\\Delta$ act by endogenies with finite kernel or finite image.\\\\\nThe third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\\\\nThe main result of the fifth section is the following linearization theorem.\n\\begin{thmB}\n Let $A$ be an abelian definable group and $\\Gamma,\\Delta$ two invariant pre-rings of definable endogenies such that \n \\begin{itemize}\n \\item $C^{\\#}(\\Gamma)=\\Delta$ and $C^{\\#}(\\Delta)=\\Gamma$;\n \\item $A$ is absolutely $(\\Gamma,\\Delta)$-minimal;\n \\item $\\Gamma$ is essentially unbounded and $\\Delta$ is essentially infinite or vice-versa;\n \\item $A_0=\\Kat(\\Gamma,\\Delta)$.\n \\end{itemize}\n Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\\big(\\Gamma/{\\sim}\\big)\\cap \\big(\\Delta/{\\sim}\\big)$ and $\\Gamma/{\\sim}$ and $\\Delta/{\\sim}$ act $K$-linearly on $A/A_0$.\n\\end{thmB}\n\\textbf{Remark}:\\\\\nThese two theorems are not direct extensions of the Theorems in \\cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \\emph{absolutely minimal} \\hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\\\\nIn the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\\Gamma,\\Delta$ is of quasi-endomorphisms and the other of endogenies.\\\\\nIn section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.\n\\subsection{Finite-dimensional groups}\nIn this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \\ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.\n\n\\begin{lemma}\\label{boundedind}\n Let $G$ be a definable group of finite dimension and $\\{H_i\\}_{i\\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\\omega$ such that there is no $J=\\{j_1,...,j_n\\}\\subseteq I$ of cardinality $n$ such that $|\\bigcap_{i=1}^kH_{j_i}:\\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\\leq n-1$.\n\\end{lemma}\n\n\\def\\notind#1#2{#1\\setbox0=\\hbox{$#1x$}\\kern\\wd0\n\\hbox to 0pt{\\mathchardef\\nn=12854\\hss$#1\\nn$\\kern1.4\\wd0\\hss}\n\\hbox to 0pt{\\hss$#1\\mid$\\hss}\\lower.9\\ht0 \\hbox to 0pt{\\hss$#1\\smile$\\hss}\\kern\\wd0}\n\n\\end{theorem}\n\\begin{proof}\n We prove the base case. Given $\\Gamma$ by hypothesis essentialy unbounded then it not admits an ascending chain of finite $\\Gamma$-invariant subgroups. Therefore let $A_0$ this subgroup, it contains the katakernel of $\\Delta$, that therefore is finite.\\\\\n Being the katakernel finite, there exists only boundedly many elements for $\\Delta$ or $\\Delta$ is unbounded (and then the proof follows using the previous idea).\\\\\n Then we can take a representative with maximal image and we have to prove that \n\\end{proof}\n\\begin{theorem}", + "post_theorem_intro_text_len": 2223, + "post_theorem_intro_text": "\\begin{proof}\nLet $n=\\dim(G)$. We will work in a sufficiently saturated structure $\\mathfrak{M}$ in which $G$ is definable and assume that $\\phi(x,y)$ is the formula defining the family. Assume, by contrary, that the conclusion is false then for $N=n+2$ and for any $k\\in \\omega$, there exists $g_1,...,g_N$ such that $|\\bigcap_{j\\leq k-1} \\phi(\\mathfrak{M},g_j)/\\bigcap_{j\\leq k} \\phi(\\mathfrak{M},g_j)|\\geq k$. Therefore, the partial type given by the formulas \n$$\\exists a^1_1,...,a^1_k,...,a^N_1,...,a_k^N: a^i_j\\in \\bigcap_{j=1}^i \\phi(\\mathfrak{M},x_j)\\wedge a^i_j{a^i_k}^{-1}\\not\\in \\phi(\\mathfrak{M},x_j)\\wedge \\forall_{i\\leq N}\\ \\phi(\\mathfrak{M},x_i)\\leq G(\\mathfrak{M})$$\nfor all $k<\\omega$, is finitely satisfable. By compactness and saturation, there exist subgroups $\\{H_i\\}_{i\\leq N}$ such that $|\\bigcap_{id\n\\]\nfor every $k\\le n-1$. Equivalently, for any $n$ distinct indices $j_1,\\dots,j_n$, at least one successive index\n\\[\n\\left|\\bigcap_{i=1}^k H_{j_i} : \\bigcap_{i=1}^{k+1} H_{j_i}\\right|\n\\]\nis at most $d$." + }, + "choices": [ + { + "label": "B", + "text": "There exists a natural number $d<\\omega$ such that for every natural number $n<\\omega$, no subset $J=\\{j_1,\\dots,j_n\\}\\subseteq I$ of cardinality $n$ satisfies\n\\[\n\\left|\\bigcap_{i=1}^k H_{j_i} : \\bigcap_{i=1}^{k+1} H_{j_i}\\right|>d\n\\]\nfor every $k\\le n-1$." + }, + { + "label": "C", + "text": "There exists a natural number $n<\\omega$ such that no subset $J=\\{j_1,\\dots,j_n\\}\\subseteq I$ of cardinality $n$ satisfies\n\\[\n\\left|\\bigcap_{i=1}^k H_{j_i} : \\bigcap_{i=1}^{k+1} H_{j_i}\\right|\\text{ is infinite}\n\\]\nfor every $k\\le n-1$." + }, + { + "label": "D", + "text": "There exist natural numbers $n,d<\\omega$ such that for every subset $J=\\{j_1,\\dots,j_n\\}\\subseteq I$ of cardinality $n$ and every $k\\le n-1$,\n\\[\n\\left|\\bigcap_{i=1}^k H_{j_i} : \\bigcap_{i=1}^{k+1} H_{j_i}\\right|\\le d.\n\\]" + }, + { + "label": "E", + "text": "There exist natural numbers $n,d<\\omega$ such that no subset $J=\\{j_1,\\dots,j_n\\}\\subseteq I$ of cardinality $n$ satisfies\n\\[\n\\left|H_{j_k}:H_{j_k}\\cap H_{j_{k+1}}\\right|>d\n\\]\nfor every $k\\le n-1$." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "simultaneous existence of both bounds n and d", + "template_used": "quantifier_dependence" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "replace uniform finite bound by mere finiteness of some drop", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "counting_estimate", + "tampered_component": "at least one bounded drop versus all drops bounded", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "counting_estimate", + "tampered_component": "successive intersections replaced by pairwise intersection indices", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not state or strongly hint at the specific quantified conclusion. It only asks which structural property of successive intersections holds, without leaking the existence/placement of the bounds n and d." + }, + "TAS": { + "score": 1, + "justification": "This is close to a theorem-identification question: under standard hypotheses, one choice reproduces the precise chain-condition statement. However, it is not a pure restatement because the options differ in meaningful ways (quantifier strength, weakening, and altered intersection structure)." + }, + "GPS": { + "score": 1, + "justification": "Selecting the correct option requires moderate reasoning about quantifiers and the difference between 'some successive drop is bounded' versus stronger or weaker variants. Still, it primarily tests recognition of the exact theorem statement rather than substantial derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong: they reflect natural mathematical mistakes such as overstrengthening quantifiers, replacing uniform boundedness by mere infinitude/weakening, and confusing successive intersections with pairwise intersections." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with little answer leakage and high-quality distractors, though it leans more toward precise recall of a known result than deep generative reasoning." + } + }, + { + "id": "2511.02141v1", + "paper_link": "http://arxiv.org/abs/2511.02141v1", + "theorems_cnt": 11, + "theorem": { + "env_name": "thm", + "content": "\\label{XiaFock}\n\t\tWe have \n\t\t\\begin{equation*}\n\t\t\tC^{*} (\\mathcal{WL})=\\mathcal{T}^{(1)}.\n\t\t\\end{equation*}", + "start_pos": 17333, + "end_pos": 17462, + "label": "XiaFock" + }, + "ref_dict": { + "R1": "\\begin{equation}\\label{R1}\n\t\t\t\\langle Bk_{z}, k_{w} \\rangle = \\langle k_{z}, B^{*}k_w \\rangle\n\t\t\\end{equation}", + "XiaFock": "\\begin{thm}\\label{XiaFock}\n\t\tWe have \n\t\t\\begin{equation*}\n\t\t\tC^{*} (\\mathcal{WL})=\\mathcal{T}^{(1)}.\n\t\t\\end{equation*}\n\t\\end{thm}" + }, + "pre_theorem_intro_text_len": 13054, + "pre_theorem_intro_text": "For some \\(\\alpha>0\\), \\(p>0\\) and \\(dV\\) the standard volume measure on \\(\\mathbb{C}^n\\), let \\(L^{p}_{\\alpha}(\\mathbb{C}^n,dV)\\) be the Lebesgue space of measurable functions \\(f\\) on \\(\\mathbb{C}^n\\) such that \n\t\\begin{equation}\\label{Fnorm1}\n\t\t\\left\\lVert f \\right\\rVert^{p}_{\\alpha} := \\left(\\frac{p\\alpha}{2\\pi}\\right)^{n} \\int_{\\mathbb{C}^n} |f(z)e^{-\\frac{\\alpha}{2}|z|^2}|^{p} dV(z) < \\infty~.\n\t\\end{equation}\n\tSimilarly, for \\(\\alpha>0\\) and \\(p=\\infty\\), we denoted by \\(L^{\\infty}_{\\alpha}\\) the space of Lebesgue measurable function \\(f\\) on \\(\\mathbb{C}^n\\) such that \n\t\\begin{equation*}\n\t\t\\left\\lVert f \\right\\rVert_{\\infty,\\alpha} := \\text{esssup}\\{|f(z)|e^{-\\frac{\\alpha|z|^2}{2}} : ~z \\in \\mathbb{C}^n\\}< \\infty ~.\n\t\\end{equation*}\n\tThe classical Fock space \\(F^{p}_{\\alpha}\\) is the space of entire functions on \\(\\mathbb{C}^n\\) which belong to \\( L^{p}_{\\alpha}(\\mathbb{C}^n, dV)\\). Similarly, the Fock space \\(F^{\\infty}_{\\alpha}\\) is the space of entire functions on \\(\\mathbb{C}^n\\) which belong to \\(L^{\\infty}_{\\alpha}\\).\n\n\tLet \\(d\\mu\\) be the Gaussian measure on \\(\\mathbb{C}^n\\), \\(n\\ge 1\\). In terms of the standard volume measure \\(dV\\) on \\(\\mathbb{C}^n\\), it is given by\n\t\\begin{equation*}\n\t\td\\mu(z) = \\pi^{-n} e^{-|z|^2} dV(z)~.\n\t\\end{equation*}\n\n\t The Fock space \\(H^{2}(\\mathbb{C}^n,d\\mu)\\) is defined to be the subspace of the (Hilbert-) Lebesgue space \\(L^{2}(\\mathbb{C}^n,d\\mu)\\) consisting of entire functions. Notice that \\(H^{2}(\\mathbb{C}^n,d\\mu) = F^{2}_{1} \\). The symbol \\(K_z\\) denotes the reproducing kernel and the symbol \\(k_z\\) denotes the normalized reproducing kernel for \\(H^{2}(\\mathbb{C}^n,d\\mu)\\). That is, \n\t\\begin{equation*}\n\t\tK_z(\\zeta) = e^{\\langle \\zeta, z \\rangle}, \\quad k_{z}(\\zeta) = e^{\\langle \\zeta, z\\rangle}e^{-\\frac{|z|^2}{2}}~, \\quad z,\\zeta \\in \\mathbb{C}^n.\n\t\\end{equation*}\n\n\tIn \\cite{xia2015localization}, J. Xia showed in the case of the Bergman space on the unit ball of \\(\\mathbb{C}^n\\) that the norm closure of \\(\\{T_{f}: f \\in L^{\\infty}(B,dv) \\}\\) coincides with the \\(C^{*}\\)-algebra of weakly localized operators. Also, he stated in \\cite[Section 4 ]{xia2015localization} that the analogue of \\cite[Theorem 1.5]{xia2015localization} on the Fock space \\(H^{2}(\\mathbb{C}^n, d\\mu)\\) was true. In this paper, we define the notion of weakly localized operators, state Xia's theorem for the Fock space \\(H^{2}(\\mathbb{C}^n, d\\mu)\\) and provide details of its proof. Further, we present a consequence of this theorem on the compactness of operators on \\(H^{2}(\\mathbb{C}^n,d\\mu)\\). We begin with the following definitions and we state the main theorem, the proof of which will retain our attention in the following sections.\n\n\t\\begin{defn}\n\t\tFor \\(f \\in L^{\\infty}(\\mathbb{C}^n,dV) \\), the \\textbf{Toeplitz operator} \\(T_{f}\\) is defined by the formula\n\t\t\\begin{equation*}\n\t\t\tT_{f}h = P(fh)~,\\quad h \\in H^{2}(\\mathbb{C}^n,d\\mu)~,\n\t\t\\end{equation*}\n\t\twhere \\(P \\colon L^{2}(\\mathbb{C}^n,d\\mu) \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu)\\) is the orthogonal projection.\n\t\\end{defn}\n\n\tThe \\textbf{standard lattice} in \\(\\mathbb{C}^n\\) is denoted by \n\t\\begin{equation*}\n\t\t\\mathbb{Z}^{2n} = \\{(m_{1}+il_{1}, \\dots, m_{n}+il_{n}): m_{1}, l_{1}, \\dots, m_{n},l_{n} \\in \\mathbb{Z}\\}~.\n\t\\end{equation*}\n\tWe fix an orthonormal set \\(\\{e_{u}: u \\in \\mathbb{Z}^{2n}\\}\\) in \\(H^{2}(\\mathbb{C}^n,d\\mu)\\). We let \\(S\\) denote the \\textbf{fundamental unit cube in }\\(\\mathbb{C}^{n}\\). That is,\n\t\\begin{equation*}\n\t\tS = \\{(x_{1}+iy_{1}, \\dots, x_{n}+iy_{n}) : x_{1},y_{1}, \\dots , x_{n},y_{n} \\in [0,1)\\}.\n\t\\end{equation*}\n\tWith \\(\\mathbb{Z}^{2n}\\) and \\(S\\), we have \n\t\\begin{equation*}\n\t\t\\cup_{u \\in \\mathbb{Z}^{2n}} \\{S+u\\} = \\mathbb{C}^{n} = \\cup_{u \\in \\mathbb{Z}^{2n}} \\{u-S\\} ~,\n\t\\end{equation*}\n\twhich is a tiling of the space, meaning that there is no overlap between \\(S+u\\) and \\(S+v\\) for \\(u \\neq v\\) in \\(\\mathbb{Z}^{2n}\\) (resp. between \\(u-S\\) and \\(v-S\\) for \\(u \\neq v \\in \\mathbb{Z}^{2n}\\) ).\n\n\t\\begin{defn}\n\t\tLet \\(\\mathcal{T}^{(1)}\\) denote the norm closure of \\(\\{T_{f}: f\\in L^{\\infty}(\\mathbb{C}^n,dV)\\}\\) in \\(\\mathcal{B}(H^2(\\mathbb{C}^n,d\\mu))\\) with respect to the operator norm. That is \n\t\t\\begin{equation*}\n\t\t\t\\mathcal{T}^{(1)} = \\{B : \\lim_{k \\to \\infty } \\norm{B-T_{b_{k}}}=0 , b_{k} \\in L^{\\infty}(\\mathbb{C}^{n},dV)\\}.\n\t\t\\end{equation*}\n\t\\end{defn}\n\n\t\\begin{defn}\n\t\tWe denote by \\(\\mathcal{S}\\) the linear span of the normalized reproducing kernels \\(k_z\\). A linear operator \\(B : \\mathcal{S} \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu) \\) is said to be \\textbf{admissible} on \\( H^{2}(\\mathbb{C}^n,d\\mu) \\) if there exists a linear operator \\( B^{*} : \\mathcal{S} \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu) \\) such that the duality relation \n\t\t\\begin{equation}\\label{R1}\n\t\t\t\\langle Bk_{z}, k_{w} \\rangle = \\langle k_{z}, B^{*}k_w \\rangle\n\t\t\\end{equation}\n\t\tholds for all \\(z,w \\in \\mathbb{C}^n\\).\t\n\t\\end{defn}\n\tThe inner product here is with respect to \\(d\\mu\\).\n\n\tWe define below sufficiently localized operators following J. Xia and D. Zheng (XZ) in \\cite{xia2013localization}.\n\t\\begin{defn}\n\t\tA bounded linear operator \\(B\\) on \\(H^{2}(\\mathbb{C}^n,d\\mu)\\) is said to be XZ-\\textbf{sufficiently localized} if there exist constants \\(2n<\\beta<\\infty\\) and \\(00\\), \\(p>0\\) and \\(dV\\) the standard volume measure on \\(\\mathbb{C}^n\\), let \\(L^{p}_{\\alpha}(\\mathbb{C}^n,dV)\\) be the Lebesgue space of measurable functions \\(f\\) on \\(\\mathbb{C}^n\\) such that \n \\begin{equation}\\label{Fnorm1}\n \\left\\lVert f \\right\\rVert^{p}_{\\alpha} := \\left(\\frac{p\\alpha}{2\\pi}\\right)^{n} \\int_{\\mathbb{C}^n} |f(z)e^{-\\frac{\\alpha}{2}|z|^2}|^{p} dV(z) < \\infty~.\n \\end{equation}\n Similarly, for \\(\\alpha>0\\) and \\(p=\\infty\\), we denoted by \\(L^{\\infty}_{\\alpha}\\) the space of Lebesgue measurable function \\(f\\) on \\(\\mathbb{C}^n\\) such that \n \\begin{equation*}\n \\left\\lVert f \\right\\rVert_{\\infty,\\alpha} := \\text{esssup}\\{|f(z)|e^{-\\frac{\\alpha|z|^2}{2}} : ~z \\in \\mathbb{C}^n\\}< \\infty ~.\n \\end{equation*}\n The classical Fock space \\(F^{p}_{\\alpha}\\) is the space of entire functions on \\(\\mathbb{C}^n\\) which belong to \\( L^{p}_{\\alpha}(\\mathbb{C}^n, dV)\\). Similarly, the Fock space \\(F^{\\infty}_{\\alpha}\\) is the space of entire functions on \\(\\mathbb{C}^n\\) which belong to \\(L^{\\infty}_{\\alpha}\\).\n\n\\begin{defn}\n We denote by \\(\\mathcal{S}\\) the linear span of the normalized reproducing kernels \\(k_z\\). A linear operator \\(B : \\mathcal{S} \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu) \\) is said to be \\textbf{admissible} on \\( H^{2}(\\mathbb{C}^n,d\\mu) \\) if there exists a linear operator \\( B^{*} : \\mathcal{S} \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu) \\) such that the duality relation \n \\begin{equation}\\label{R1}\n \\langle Bk_{z}, k_{w} \\rangle = \\langle k_{z}, B^{*}k_w \\rangle\n \\end{equation}\n holds for all \\(z,w \\in \\mathbb{C}^n\\). \n \\end{defn}\n The inner product here is with respect to \\(d\\mu\\).\n\n\\begin{exa}\\label{Ex1}\n If \\(f\\) is a \\textbf{bounded measurable function} on \\(\\mathbb{C}^n\\), then there is a positive constant \\(00\\), \\(p>0\\) and \\(dV\\) the standard volume measure on \\(\\mathbb{C}^n\\), let \\(L^{p}_{\\alpha}(\\mathbb{C}^n,dV)\\) be the Lebesgue space of measurable functions \\(f\\) on \\(\\mathbb{C}^n\\) such that \n \\begin{equation}\\label{Fnorm1}\n \\left\\lVert f \\right\\rVert^{p}_{\\alpha} := \\left(\\frac{p\\alpha}{2\\pi}\\right)^{n} \\int_{\\mathbb{C}^n} |f(z)e^{-\\frac{\\alpha}{2}|z|^2}|^{p} dV(z) < \\infty~.\n \\end{equation}\n Similarly, for \\(\\alpha>0\\) and \\(p=\\infty\\), we denoted by \\(L^{\\infty}_{\\alpha}\\) the space of Lebesgue measurable function \\(f\\) on \\(\\mathbb{C}^n\\) such that \n \\begin{equation*}\n \\left\\lVert f \\right\\rVert_{\\infty,\\alpha} := \\text{esssup}\\{|f(z)|e^{-\\frac{\\alpha|z|^2}{2}} : ~z \\in \\mathbb{C}^n\\}< \\infty ~.\n \\end{equation*}\n The classical Fock space \\(F^{p}_{\\alpha}\\) is the space of entire functions on \\(\\mathbb{C}^n\\) which belong to \\( L^{p}_{\\alpha}(\\mathbb{C}^n, dV)\\). Similarly, the Fock space \\(F^{\\infty}_{\\alpha}\\) is the space of entire functions on \\(\\mathbb{C}^n\\) which belong to \\(L^{\\infty}_{\\alpha}\\).\n\n\\begin{defn}\n We denote by \\(\\mathcal{S}\\) the linear span of the normalized reproducing kernels \\(k_z\\). A linear operator \\(B : \\mathcal{S} \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu) \\) is said to be \\textbf{admissible} on \\( H^{2}(\\mathbb{C}^n,d\\mu) \\) if there exists a linear operator \\( B^{*} : \\mathcal{S} \\rightarrow H^{2}(\\mathbb{C}^n,d\\mu) \\) such that the duality relation \n \\begin{equation}\\label{R1}\n \\langle Bk_{z}, k_{w} \\rangle = \\langle k_{z}, B^{*}k_w \\rangle\n \\end{equation}\n holds for all \\(z,w \\in \\mathbb{C}^n\\). \n \\end{defn}\n The inner product here is with respect to \\(d\\mu\\).\n\n\\begin{exa}\\label{Ex1}\n If \\(f\\) is a \\textbf{bounded measurable function} on \\(\\mathbb{C}^n\\), then there is a positive constant \\(0R}{v\\in \\Z^{2n} }} |\\langle B k_{u-z}, k_{v-w} \\rangle| =0 \\quad \\text{ and } \\quad \\lim_{R\\to \\infty} \\sup_{u\\in \\Z^{2n}} \\sum_{\\underset{|u-v|>R}{v\\in \\Z^{2n} }} |\\langle k_{u-z}, Bk_{v-w}\\rangle| = 0~.\n \\end{equation*}\n \\end{lem}\n \\begin{proof}\n By \\cite[Lemma 2.32]{zhu2012}, for any entire function \\(f\\) on \\(\\C^n\\), we have \n \\begin{equation*}\n \\left|f(z)e^{-\\frac{\\alpha}{2}|z|^2}\\right|^p \\le C \\int_{B(z,\\delta)} |f(w)e^{-\\frac{\\alpha}{2}|w|^2}|^p dV(w) \\quad \\text{ for } z \\in \\C^n.\n \\end{equation*}\n Hence for \\(\\alpha=p=1\\) and \\(\\delta\\) small such that the balls \\(\\{B(v-w,\\delta): v \\in \\Z^{2n} \\}\\) are mutually disjoint, we have\n \\begin{equation*}\n |\\langle B k_{u-z}, k_{v-w} \\rangle| = |Bk_{u-z} (v-w) |e^{-\\frac{|v-w|^2}{2}} \\le C \\int_{B(v-w,\\delta)} |Bk_{u-z}(\\zeta) | e^{-\\frac{|\\zeta|^2}{2}} dV(\\zeta).\n \\end{equation*}\n Indeed, for \\(\\delta < \\frac{1}{2} \\), the balls \\(\\{B(v-w,\\delta) : v \\in \\Z^{2n} \\}\\) are mutually disjoint. Otherwise, there would exist \\(v,v' \\in \\Z^{2n} \\) such that \\(v \\neq v' \\), and a point \\(\\xi\\) such that \\(\\xi \\in B(v-w,\\delta) \\cap B(v'-w,\\delta) \\). In other words:\n \\begin{equation*}\n |v-w-\\xi| < \\delta \\quad \\text{ and } \\quad |v'-w - \\xi| < \\delta~.\n \\end{equation*}\n This implies that \n \\begin{equation*}\n |v-v'|=|(v-w-\\xi)-(v'-w-\\xi)|\\le |v-w-\\xi| + |v'-w-\\xi| < \\delta + \\delta= 2 \\delta < 1~.\n \\end{equation*}\n That is \\(|v-v'|<1\\). This contradicts the well-known fact that \\(|v-v'|\\ge 1\\). This result actually implies that there exists \\(N\\in \\N\\) such that each \\(\\zeta\\in \\C^n \\) belongs to at most \\(N\\) balls in \\(\\{B(v-w,\\delta): v\\in \\Z^{2n}\\}\\). That is \\(\\sum_{v \\in \\Z^{2n}} \\chi_{B(v-w,\\delta)}(\\zeta) \\le N \\) for each \\(\\zeta \\in \\C^n \\).\n\n\\begin{proof}[\\textbf{Proof of Proposition \\ref{EwBEzD} }]\n From (\\ref{EwBEz}), we have \n \\begin{equation*}\n E_{w} BE_{z} = \\frac{1}{\\pi^{2n}} \\sum_{u,v \\in \\Z^{2n}} \\langle Bk_{u-z}, k_{v-w} \\rangle k_{v-w} \\otimes k_{u-z}~.\n \\end{equation*}\n Thus for any \\(R>0\\), we can write \\(E_{w}BE_{z} = V_{R} + W_{R}\\), where\n \\begin{equation*}\n V_{R} = \\frac{1}{\\pi^{2n}} \\sum_{\\underset{|u-v|\\le R}{u,v \\in \\Z^{2n}}} \\langle Bk_{u-z}, k_{v-w} \\rangle k_{v-w} \\otimes k_{u-z} \\quad \\text{ and } \n \\end{equation*}\n \\begin{equation*}\n \\quad W_{R} = \\frac{1}{\\pi^{2n}} \\sum_{\\underset{|u-v|>R}{u,v \\in \\Z^{2n}}} \\langle Bk_{u-z}, k_{v-w} \\rangle k_{v-w} \\otimes k_{u-z}~.\n \\end{equation*}\n To complete the proof, it suffices to prove that:\n \\begin{enumerate}[label=(\\alph*)]\n \\item \\label{itm:first1} \\(\\lim_{R\\to \\infty} \\norm{W_{R}}=0\\).\n \\item \\label{itm:second1} \\(V_{R} \\in \\) span(\\(\\mathcal{D}_{0}\\)) for every \\(R>0\\).\n \\end{enumerate}\n\n\\begin{pro}\\label{DoT1}\n We have \\(\\mathcal{D}_{0} \\subset \\mathcal{T}^{(1)}\\).\n \\end{pro}\n To establish the proof of this proposition, we will need the next three propositions.\n \\begin{pro}\\label{YzT1}\n Suppose that \\(\\{c_{u}: u \\in \\Z^{2n}\\}\\) is a bounded set of complex coefficients. Then for each \\(z \\in \\C^{n}\\), the operator \\(Y_{z}\\) defined in (\\ref{Yz}) belongs to \\(\\mathcal{T}^{(1)}\\). \n \\end{pro}\n \\begin{proof}\n \\begin{enumerate}[label=(\\alph*)]\n \\item Let us first show that \\(Y_{0} \\in \\mathcal{T}^{(1)}\\). We have \\(|u-v|\\ge 1\\) for all \\(u \\neq v \\in \\Z^{2n}\\). Hence \\(B(u,\\frac{1}{2}) \\cap B(v,\\frac{1}{2}) = \\emptyset \\) for \\(u\\neq v\\). For each \\(0 < \\varepsilon < \\frac{1}{2}\\), define the operator\n \\begin{equation*}\n A_{\\varepsilon} = \\frac{1}{|B(0,\\varepsilon)|} \\int_{B(0,\\varepsilon)} Y_{z} dV(z) .\n \\end{equation*}\n From Proposition \\ref{proYz}, we have the norm continuity of the map \\(z\\mapsto Y_{z}\\) and it implies that\n \\begin{equation*}\n \\lim_{\\varepsilon \\to 0} \\norm{Y_{0}-A_{\\varepsilon}} = 0.\n \\end{equation*}\n This comes from the fact that\n \\begin{eqnarray*}\n \\norm{Y_{0}-A_{\\varepsilon}} &=& \\norm{\\frac{1}{|B(0,\\varepsilon)|} \\int_{B(0,\\varepsilon)} (Y_{0}-Y_{z}) ~dV(z)} \\le \\frac{1}{|B(0,\\varepsilon)|} \\int_{B(0,\\varepsilon)} \\norm{Y_{0}-Y_{z}} ~dV(z)\n \\end{eqnarray*}\n and \\(\\lim_{z\\to 0}\\norm{Y_{z}-Y_{0}}=0\\) .", + "post_theorem_intro_text_len": 385, + "post_theorem_intro_text": "The organization of this paper is as follows. In Section 2, we will give propositions in the case of the Fock space \\(H^{2}(\\mathbb{C}^n, d\\mu)\\) which are the analogue of those given by Xia in \\cite{xia2015localization} in the Bergman space case of the unit ball. Later, using these propositions in Section 3, we establish the proof of Theorem \\ref{XiaFock} and present a consequence.", + "sketch": "To establish the proof of Theorem~\\ref{XiaFock}, the paper first \"give[s] propositions in the case of the Fock space \\(H^{2}(\\mathbb{C}^n, d\\mu)\\)\" in Section 2, as analogues of results of Xia \\cite{xia2015localization} for the Bergman space of the unit ball. Then, \"using these propositions in Section 3,\" it \"establish[es] the proof of Theorem~\\ref{XiaFock}.\"", + "expanded_sketch": "To establish the proof of the main theorem, the paper first \"give[s] propositions in the case of the Fock space \\(H^{2}(\\mathbb{C}^n, d\\mu)\\)\" in Section 2, as analogues of results of Xia \\cite{xia2015localization} for the Bergman space of the unit ball. Then, \"using these propositions in Section 3,\" it \"establish[es] the proof of the main theorem.\"", + "expanded_theorem": "\\label{XiaFock}\n\t\tWe have \n\t\t\\begin{equation*}\n\t\t\tC^{*} (\\mathcal{WL})=\\mathcal{T}^{(1)}.\n\t\t\\end{equation*}", + "theorem_type": [ + "Classification or Bijection", + "Equivalence" + ], + "mcq": { + "question": "Let $H^{2}(\\mathbb{C}^{n},d\\mu)$ be the classical Fock space, where $d\\mu(z)=\\pi^{-n}e^{-|z|^{2}}\\,dV(z)$. Let $\\mathcal{WL}$ denote the class of weakly localized operators on $H^{2}(\\mathbb{C}^{n},d\\mu)$, and let $C^{*}(\\mathcal{WL})$ be the $C^{*}$-algebra generated by $\\mathcal{WL}$. Let $\\mathcal{T}^{(1)}$ denote the Toeplitz algebra on this Fock space, i.e. the norm-closed algebra generated by Toeplitz operators $T_{f}$ with bounded measurable symbols $f$. Which statement holds about these two operator algebras?", + "correct_choice": { + "label": "A", + "text": "They are equal: $C^{*}(\\mathcal{WL})=\\mathcal{T}^{(1)}$." + }, + "choices": [ + { + "label": "B", + "text": "One only has a strict inclusion $C^{*}(\\mathcal{WL})\\subsetneq \\mathcal{T}^{(1)}$." + }, + { + "label": "C", + "text": "There is a containment $C^{*}(\\mathcal{WL})\\subseteq \\mathcal{T}^{(1)}$." + }, + { + "label": "D", + "text": "They coincide only after taking compact-operator quotients, namely $C^{*}(\\mathcal{WL})/\\mathcal{K}=\\mathcal{T}^{(1)}/\\mathcal{K}$, but not necessarily before quotienting." + }, + { + "label": "E", + "text": "They have the same strong-operator closure, but $C^{*}(\\mathcal{WL})$ and $\\mathcal{T}^{(1)}$ need not be equal as norm-closed algebras." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "two-sided equality_vs_one-sided_generation", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "drop_reverse_inclusion", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "norm_equality_replaced_by_Calkin_equality", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "norm_closure_replaced_by_SOT_closure", + "template_used": "uniformity_effectivity" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the two algebras and asks for their relationship, but it does not state or strongly hint that equality is the correct conclusion." + }, + "TAS": { + "score": 1, + "justification": "This is close to a direct theorem-recall item about the relation between the two algebras. However, it is not purely tautological because the options distinguish equality from weaker containment, strict inclusion, and quotient-level coincidence." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is required to select the strongest valid statement rather than the merely true weaker containment. Still, success depends mostly on recalling the known result rather than generating a substantial argument." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic confusions: settling for one-sided inclusion, mistaking equality for quotient equality, or confusing norm-closed and strong-operator-closed statements." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-recall MCQ with strong distractors and little answer leakage, though it tests recognition of a known result more than deep generative reasoning." + } + }, + { + "id": "2511.20205v1", + "paper_link": "http://arxiv.org/abs/2511.20205v1", + "theorems_cnt": 7, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm-Serrin}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature. Assume that \\(f\\in C(\\mathbb R_+;\\mathbb R_+)\\) satisfies\n\\begin{equation}\\label{cond-Serrin}\n\\int_0^1[f(v)]^{-\\frac{1}{2_*(q)-1}}\\mathrm dv<+\\infty.\n\\end{equation}\nThen any positive supersolution to \\eqref{Meq-gradient} must be constant.", + "start_pos": 106992, + "end_pos": 107420, + "label": "thm-Serrin" + }, + "ref_dict": { + "Meq-q-0": "\\begin{equation}\\label{Meq-q-0}\n-\\Delta u =f(u)\\quad\\text{in }M^n. \n\\end{equation}", + "fig-1": "\\begin{figure}[htbp]\\small\n\\centering\n\\includegraphics[width=0.8\\textwidth]{images/diagram.jpg}\n\\caption{some parameter ranges of equation \\eqref{eq-u^p} in \\(\\mathbb R^n\\) when \\(n=5\\)}\n\\label{fig-1}\n\\end{figure}", + "eq-u^p": "\\begin{equation}\\label{eq-u^p}\n-\\Delta u = u^p|\\nabla u|^q\\quad\\text{in }M^n.\n\\end{equation}", + "thm-subcritical": "\\begin{theorem}\\label{thm-subcritical}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature, and \\(0N,\\quad\\text{for some }\\beta_0\\in(1-q,2^*(q)-1)\\text{ and }N>0,\n\\end{equation}\n\\begin{equation}\nuf'(u)\\leqslant\\alpha_0 f(u),\\quad\\text{for some }\\alpha_0\\in[\\beta_0,2^*(q)-1).\n\\end{equation}\nThen any positive \\(C^3\\) solution to \\eqref{Meq-gradient} must be constant if \\(p<2^*(q)-1\\).\n\\end{theorem}", + "solution": "\\begin{equation}\\label{solution}\nU_{\\lambda,x_0}(x):=\\Big(\\frac{(n+\\frac{q}{1-q})^{\\frac{1}{2-q}}(n-2)^{\\frac{1-q}{2-q}}\\lambda}{1+(\\lambda|x-x_0|)^{\\frac{2-q}{1-q}}}\\Big)^{\\frac{(n-2)(1-q)}{2-q}},\\quad\\lambda>0,~x_0\\in\\mathbb R^n.\n\\end{equation}", + "Meq-q-0-ty": "\\begin{equation}\\label{Meq-q-0-ty}\n-\\Delta u =u^p\\quad\\text{in }M^n.\n\\end{equation}", + "cond-Serrin": "\\begin{equation}\\label{cond-Serrin}\n\\int_0^1[f(v)]^{-\\frac{1}{2_*(q)-1}}\\mathrm dv<+\\infty.\n\\end{equation}", + "solution-q-0": "\\begin{equation}\\label{solution-q-0}\nu(x)=\\Big(\\frac{\\sqrt{n(n-2)}\\lambda}{1+(\\lambda|x-x_0|)^2}\\Big)^{\\frac{n-2}{2}}.\n\\end{equation}", + "Meq-gradient": "\\begin{equation}\\label{Meq-gradient}\n-\\Delta u = f(u)|\\nabla u|^q\\quad\\text{in }M^n,\n\\end{equation}", + "thm-critical": "\\begin{theorem}\\label{thm-critical}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\in\\{3,4,5\\}\\) with nonnegative Ricci curvature, \\(00, 11\\) on a connected geodesically complete Riemannian manifold.\n\nIn the critical case \\(p=\\frac{n+2}{n-2}\\), the equation \\eqref{Meq-q-0-ty} becomes\n\\[-\\Delta u =u^{\\frac{n+2}{n-2}}\\quad\\text{in }M^n, \\]\nwhich is the Euler-Lagrange equation of the Sobolev inequality. The critical case in \\(\\mathbb R^n\\) has been successfully resolved, as can be seen in \\cite{ CGS1989, CL1991,GNN1981}, all positive solutions must be of the form \n\\begin{equation}\\label{solution-q-0}\nu(x)=\\Big(\\frac{\\sqrt{n(n-2)}\\lambda}{1+(\\lambda|x-x_0|)^2}\\Big)^{\\frac{n-2}{2}}.\n\\end{equation}\nOn generally Riemannian manifolds, the classification result turns to the rigidity result: the manifold must be \\(\\mathbb R^n\\) if there exists a nonconstant solution. However, deriving the rigidity result is more challenging. Fogagnolo-Malchiodi-Mazzieri \\cite{FMM2023} got the rigidity result under the assumption \\(u=O(r^{-\\frac{n-2}{2}})\\) at infinity. It is worth mentioning that Catino-Monticelli \\cite{CM2022} obtained the rigidity result and classified it with \\(n=3\\), while \\(n\\geqslant4\\) either under the finite energy condition or under the decay assumption of the solution at infinity. Ciraolo-Farina-Polvara \\cite{CFP2024} extended these results to \\(n=4,5\\) without any finite energy condition by the \\(P\\)-function method. Catino-Monticelli-Roncoroni \\cite{CMR2023}, Ou \\cite{O2025}, and Sun-Wang \\cite{SW2025} extended the classification results on manifolds to \\(m\\)-Laplacian case.\n\nNow, we return to the case with the gradient term. A crucial prototype of semilinear equation \\eqref{Meq-gradient} is the case \\(f(u)=u^p\\), i.e.,\n\\begin{equation}\\label{eq-u^p}\n-\\Delta u = u^p|\\nabla u|^q\\quad\\text{in }M^n.\n\\end{equation}\nThere are three important curves of \\((p,q)\\) for this equation, namely the homogeneous curve \\(A_0(p,q)=0\\), the first critical curve \\(A_1(p,q)=0\\), and the second critical curve \\(A_2(p,q)=0\\), with\n\\[A_0(p,q)=p+q-1,\\quad A_1(p,q)=(n-2)p+(n-1)q-n,\\]\n\\[A_2(p,q)=(n-2)p+(n-1)q-\\Big(n+\\frac{2-q}{1-q}\\Big),\\quad0\\leqslant q<1.\\]\nAs shown in Figure \\ref{fig-1}, we mark these three curves with \\(A_0\\), \\(A_1\\), and \\(A_2\\) respectively. We introduce\n\\[2_*(q)=\\frac{2-q}{n-2}(n-1),\\quad 2^*(q)=\\frac{2-q}{n-2}(n+\\frac{q}{1-q}),\\]\nthus the first subcritical range \\(A_1(p,q)<0\\) is equivalent with \\(p<2_*(q)-1\\), and the second subcritical range \\(A_2(p,q)<0\\) is equivalent with \\(p<2^*(q)-1\\) when \\(0\\leqslant q<1\\).\n\nIn the special case \\(p=0\\), equation \\eqref{eq-u^p} becomes \\(-\\Delta u=|\\nabla u|^q\\), called as the Hamilton-Jacobi equation. For this equation, Lions \\cite{L1985} obtained a Liouville theorem for \\(q>1\\) in \\(\\mathbb R^n\\). Bidaut-V\\'{e}ron, Huidobro, and V\\'{e}ron \\cite{BV-GH-V2014} established the gradient estimate to \\(m\\)-Laplacian case and obtained some Liouville theorems on complete noncompact manifolds, which satisfy a lower bound estimate on the Ricci curvature and sectional curvature.\n\nIn the Euclidean case, there has already been a great deal of work on equation \\eqref{eq-u^p}. In the first subcritical case \\(p<2_*(q)-1\\), any supersolution to \\eqref{eq-u^p} must be constant, see \\cite{BP-GM-Q2016, BV-GH-V2019, CHZ2022, CM1997, F2009, MP2001}. Those past proofs are usually composed of three cases split by the homogeneous curve \\(A_0(p,q)=0\\), which is avoided in our proof of Theorem \\ref{thm-Serrin}. Besides, for \\(q\\geqslant2\\) and \\(p\\geqslant0\\), Filippucci-Pucci-Souple \\cite{FPS2020} obtained that any positive bounded solution to equation \\eqref{eq-u^p} is constant. Later, Bidaut-V\\'{e}ron \\cite{BV2021} extended this result to the \\(m\\)-Laplace equation for \\(q\\geqslant m\\) and established a Liouville theorem with \\(p\\geqslant 0\\).\n\nNow we concentrate on the second subcritical range \\(p<2^*(q)-1\\) with \\(0\\leqslant q<1\\) for equation \\eqref{eq-u^p} in \\(\\mathbb R^n\\). For \\(00\\), Bidaut-V\\'{e}ron, Huidobro, and V\\'{e}ron \\cite{BV-GH-V2019} conjectured that any solution to \\eqref{eq-u^p} must be constant in the second critical range. They gave a positive answer to the left of the curve \\(G(p,q)=0\\) in Figure \\ref{fig-1}, where \n\\[G(p,q)=\\big((n-1)^2q+n-2\\big)p^2+c(q)p-nq^2,\\]\n\\[c(q)=n(n-1)q^2-(n^2+n-1)q-n-2.\\]\nBesides, they also showed that the second critical curve is optimal by proving the existence of a nonconstant solution in the second supercritical range \\(A_2(p,q)>0\\). Using Bernstein's technique, Chang-Hu-Zhang \\cite{CHZ2022} derived a Liouville theorem in \\(m\\)-Laplacian case, and the left of the curve \\(C\\) in Figure \\ref{fig-1} shows their range when \\(m=2\\). An exciting result was obtained by Ma-Wu \\cite{MW2023}. With the help of integral identities, they completely solved the conjecture in \\cite{BV-GH-V2019} when \\(01\\) on a connected geodesically complete Riemannian manifold.\n\nIn the critical case \\(p=\\frac{n+2}{n-2}\\), the equation \\eqref{Meq-q-0-ty} becomes\n\\[-\\Delta u =u^{\\frac{n+2}{n-2}}\\quad\\text{in }M^n, \\]\nwhich is the Euler-Lagrange equation of the Sobolev inequality. The critical case in \\(\\mathbb R^n\\) has been successfully resolved, as can be seen in \\cite{ CGS1989, CL1991,GNN1981}, all positive solutions must be of the form \n\\begin{equation}\\label{solution-q-0}\nu(x)=\\Big(\\frac{\\sqrt{n(n-2)}\\lambda}{1+(\\lambda|x-x_0|)^2}\\Big)^{\\frac{n-2}{2}}.\n\\end{equation}\nOn generally Riemannian manifolds, the classification result turns to the rigidity result: the manifold must be \\(\\mathbb R^n\\) if there exists a nonconstant solution. However, deriving the rigidity result is more challenging. Fogagnolo-Malchiodi-Mazzieri \\cite{FMM2023} got the rigidity result under the assumption \\(u=O(r^{-\\frac{n-2}{2}})\\) at infinity. It is worth mentioning that Catino-Monticelli \\cite{CM2022} obtained the rigidity result and classified it with \\(n=3\\), while \\(n\\geqslant4\\) either under the finite energy condition or under the decay assumption of the solution at infinity. Ciraolo-Farina-Polvara \\cite{CFP2024} extended these results to \\(n=4,5\\) without any finite energy condition by the \\(P\\)-function method. Catino-Monticelli-Roncoroni \\cite{CMR2023}, Ou \\cite{O2025}, and Sun-Wang \\cite{SW2025} extended the classification results on manifolds to \\(m\\)-Laplacian case.\n\nReturn to the Riemannian manifold case. Sun-Xiao-Xu \\cite{SXX2022} obtained some Liouville theorems of equation \\eqref{eq-u^p} under the growth of the geodesical ball and some range \\((p,q)\\in\\mathbb R^2\\) on a complete, noncompact Riemannian manifold. He-Hu-Wang \\cite{HHW2023} obtained that any \\(C^1\\) smooth positive solution of equation \\eqref{eq-u^p} is a constant when \\(p+q<\\frac{n+3}{n-1}\\) on complete noncompact Riemannian manifold with nonnegative Ricci curvature. More generally, both \\cite{HHW2023} and \\cite{SXX2022} studied \\(m\\)-Laplacian case.\n\n\\begin{remark}\nBy an approximation argument, the test function \\(\\varphi\\) may be chosen from \\(H^1(M^n)\\) with compact support.\n\\end{remark}\n\nUnder an integrable condition on \\(f\\), we obtain the following Liouville theorem by Serrin's technique.\n\n\\begin{equation}\\label{Meq-q-0}\n-\\Delta u =f(u)\\quad\\text{in }M^n. \n\\end{equation}\n\n\\begin{equation}\\label{eq-u^p}\n-\\Delta u = u^p|\\nabla u|^q\\quad\\text{in }M^n.\n\\end{equation}", + "full_context": "\\label{sec-introduction}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature. \nConsider a class of semilinear equations with a gradient term \n\\begin{equation}\\label{Meq-gradient}\n-\\Delta u = f(u)|\\nabla u|^q\\quad\\text{in }M^n,\n\\end{equation}\nwhere \\(\\Delta\\) is the Laplace-Beltrami operator, \\(\\nabla\\) is the gradient operator, and \\(f:\\mathbb R_+\\to\\mathbb R_+\\) is continuous.\nThe equation \\eqref{Meq-gradient} has a strong physical background. Such as a qualitative mathematical model studying the groundwater flow in a water-absorbing fissured porous rock in one spatial dimension (see e.g., \\cite{BBCP2000} ), the porous media equation (see e.g., \\cite{P2018}).\n\nA typical case for equation \\eqref{Meq-q-0} is\n\\begin{equation}\\label{Meq-q-0-ty}\n-\\Delta u =u^p\\quad\\text{in }M^n.\n\\end{equation}\nThis equation is the famous Lane-Emden equation, which is related to the Yamabe problem. Gidas-Spruck \\cite{GS1981} used differential identities to derive a Liouville theorem, namely, there is no positive solution when \\(11\\) on a connected geodesically complete Riemannian manifold.\n\nIn the critical case \\(p=\\frac{n+2}{n-2}\\), the equation \\eqref{Meq-q-0-ty} becomes\n\\[-\\Delta u =u^{\\frac{n+2}{n-2}}\\quad\\text{in }M^n, \\]\nwhich is the Euler-Lagrange equation of the Sobolev inequality. The critical case in \\(\\mathbb R^n\\) has been successfully resolved, as can be seen in \\cite{ CGS1989, CL1991,GNN1981}, all positive solutions must be of the form \n\\begin{equation}\\label{solution-q-0}\nu(x)=\\Big(\\frac{\\sqrt{n(n-2)}\\lambda}{1+(\\lambda|x-x_0|)^2}\\Big)^{\\frac{n-2}{2}}.\n\\end{equation}\nOn generally Riemannian manifolds, the classification result turns to the rigidity result: the manifold must be \\(\\mathbb R^n\\) if there exists a nonconstant solution. However, deriving the rigidity result is more challenging. Fogagnolo-Malchiodi-Mazzieri \\cite{FMM2023} got the rigidity result under the assumption \\(u=O(r^{-\\frac{n-2}{2}})\\) at infinity. It is worth mentioning that Catino-Monticelli \\cite{CM2022} obtained the rigidity result and classified it with \\(n=3\\), while \\(n\\geqslant4\\) either under the finite energy condition or under the decay assumption of the solution at infinity. Ciraolo-Farina-Polvara \\cite{CFP2024} extended these results to \\(n=4,5\\) without any finite energy condition by the \\(P\\)-function method. Catino-Monticelli-Roncoroni \\cite{CMR2023}, Ou \\cite{O2025}, and Sun-Wang \\cite{SW2025} extended the classification results on manifolds to \\(m\\)-Laplacian case.\n\nReturn to the Riemannian manifold case. Sun-Xiao-Xu \\cite{SXX2022} obtained some Liouville theorems of equation \\eqref{eq-u^p} under the growth of the geodesical ball and some range \\((p,q)\\in\\mathbb R^2\\) on a complete, noncompact Riemannian manifold. He-Hu-Wang \\cite{HHW2023} obtained that any \\(C^1\\) smooth positive solution of equation \\eqref{eq-u^p} is a constant when \\(p+q<\\frac{n+3}{n-1}\\) on complete noncompact Riemannian manifold with nonnegative Ricci curvature. More generally, both \\cite{HHW2023} and \\cite{SXX2022} studied \\(m\\)-Laplacian case.\n\n\\begin{remark}\nBy an approximation argument, the test function \\(\\varphi\\) may be chosen from \\(H^1(M^n)\\) with compact support.\n\\end{remark}\n\nUnder an integrable condition on \\(f\\), we obtain the following Liouville theorem by Serrin's technique.\n\n\\begin{equation}\\label{Meq-q-0}\n-\\Delta u =f(u)\\quad\\text{in }M^n. \n\\end{equation}\n\n\\begin{equation}\\label{eq-u^p}\n-\\Delta u = u^p|\\nabla u|^q\\quad\\text{in }M^n.\n\\end{equation}\n\nIn both cases, \\(\\mathbf E=0\\) holds in \\(Z^c\\) by letting \\(R\\to\\infty\\). For \\(x\\in\\mathring Z\\), \\(\\nabla u(x)=0\\) yields \\(\\mathbf E=0\\). By the continuity of \\(\\mathbf E\\), we immediately conclude that \\(\\mathbf E=0\\) on \\(M\\). Together with Lemma \\ref{critical-thm-pf}, the proof is complete. \n\\qed\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000sections/intro.tex\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u00000000664\u00000000000\u00000000000\u000000000043630\u000015111220774\u0000013270\u0000 0\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000ustar \u0000root\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000root\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\\section{Introduction}\\label{sec-introduction}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature. \nConsider a class of semilinear equations with a gradient term \n\\begin{equation}\\label{Meq-gradient}\n-\\Delta u = f(u)|\\nabla u|^q\\quad\\text{in }M^n,\n\\end{equation}\nwhere \\(\\Delta\\) is the Laplace-Beltrami operator, \\(\\nabla\\) is the gradient operator, and \\(f:\\mathbb R_+\\to\\mathbb R_+\\) is continuous.\nThe equation \\eqref{Meq-gradient} has a strong physical background. Such as a qualitative mathematical model studying the groundwater flow in a water-absorbing fissured porous rock in one spatial dimension (see e.g., \\cite{BBCP2000} ), the porous media equation (see e.g., \\cite{P2018}).\n\nUnder an integrable condition on \\(f\\), we obtain the following Liouville theorem by Serrin's technique.\n\nWhen \\(\\liminf_{v\\to0^+}f(v)>0\\), the condition \\eqref{cond-Serrin} trivially holds.\n\n\\begin{corollary}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature. Assume that \\(f\\in C(\\mathbb R_+;\\mathbb R_+)\\) satisfies\n\\[\\liminf_{v\\to0^+}f(v)>0.\\]\nThen any positive supersolution to \\eqref{Meq-gradient} must be constant.\n\\end{corollary}\n\n\\begin{corollary}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature. Then any positive supersolution to \\eqref{eq-u^p} must be constant if \\(p<2_*(q)-1\\),\n\\end{corollary}\n\n\\begin{theorem}\\label{thm-subcritical}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature, and \\(0N,\\quad\\text{for some }\\beta_0\\in(1-q,2^*(q)-1)\\text{ and }N>0,\n\\end{equation}\n\\begin{equation}\nuf'(u)\\leqslant\\alpha_0 f(u),\\quad\\text{for some }\\alpha_0\\in[\\beta_0,2^*(q)-1).\n\\end{equation}\nThen any positive \\(C^3\\) solution to \\eqref{Meq-gradient} must be constant if \\(p<2^*(q)-1\\).\n\\end{theorem}\n\n\\begin{corollary}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature, and \\(00,~x_0\\in\\mathbb R^n.\n\\end{equation}", + "post_theorem_intro_text_len": 7972, + "post_theorem_intro_text": "When \\(\\liminf_{v\\to0^+}f(v)>0\\), the condition \\eqref{cond-Serrin} trivially holds.\n\n\\begin{corollary}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature. Assume that \\(f\\in C(\\mathbb R_+;\\mathbb R_+)\\) satisfies\n\\[\\liminf_{v\\to0^+}f(v)>0.\\]\nThen any positive supersolution to \\eqref{Meq-gradient} must be constant.\n\\end{corollary}\n\nTheorem \\ref{thm-Serrin} recovers the first subcritical range \\(p<2_*(q)-1\\) by taking \\(f(u)=u^p\\).\n\n\\begin{corollary}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature. Then any positive supersolution to \\eqref{eq-u^p} must be constant if \\(p<2_*(q)-1\\),\n\\end{corollary}\n\nFor solutions to \\eqref{Meq-gradient}, we concentrate on the cases \\(0N,\\quad\\text{for some }\\beta_0\\in(1-q,2^*(q)-1)\\text{ and }N>0,\n\\end{equation}\n\\begin{equation}\nuf'(u)\\leqslant\\alpha_0 f(u),\\quad\\text{for some }\\alpha_0\\in[\\beta_0,2^*(q)-1).\n\\end{equation}\nThen any positive \\(C^3\\) solution to \\eqref{Meq-gradient} must be constant if \\(p<2^*(q)-1\\).\n\\end{theorem}\n\nTheorem \\ref{thm-subcritical} recovers the second subcritical range \\(p<2^*(q)-1\\) by taking \\(f(u)=u^p\\).\n\n\\begin{corollary}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature, and \\(00,~K=K(n,q)>0.\\]\nBy inserting them into equation \\eqref{Meq-gradient} with \\(f(u) = u^{2^*(q)-1}\\), we rewrite the solutions as\n\\begin{equation}\\label{solution}\nU_{\\lambda,x_0}(x):=\\Big(\\frac{(n+\\frac{q}{1-q})^{\\frac{1}{2-q}}(n-2)^{\\frac{1-q}{2-q}}\\lambda}{1+(\\lambda|x-x_0|)^{\\frac{2-q}{1-q}}}\\Big)^{\\frac{(n-2)(1-q)}{2-q}},\\quad\\lambda>0,~x_0\\in\\mathbb R^n.\n\\end{equation}\nFor \\(q=0\\), the solution \\eqref{solution} is equivalent to \\eqref{solution-q-0}. The rigidity of manifolds has been rigorously established in low-dimensional settings for \\(q=0\\), but for \\(01\\), \\(\\varepsilon>0\\) small enough, and \\(C>0\\) are constants independent of \\(R\\). Let \\(B_R\\) be the geodesic ball centered at some fixed point with radius \\(R\\). Unless otherwise specified, we employ the summation convention for repeated indices from \\(1\\) to \\(n\\). In proofs involving manifolds, we choose a local frame, \\(\\varphi_i\\) denotes the covariant derivative of the function \\(\\varphi\\). Let \\(\\eta\\) be a smooth cutoff function supported in \\(B_{2R}\\) satisfying \\(\\eta\\equiv 1\\) in \\(B_R\\), \\(0\\leqslant\\eta\\leqslant 1\\), and \\(|\\nabla\\eta|\\leqslant CR^{-1}\\).\n\nThis paper is structured as follows. In Section 2, we prove Theorem \\ref{thm-Serrin} by using Serrin's technique with a delicate test function \\(b(u)\\eta^n\\) and a three-term Young's inequality. In Section 3, using the invariant tensor technique, we establish a key differential identity and necessary propositions to prove Theorem \\ref{thm-subcritical}. With the help of this differential identity, the proof of Theorem \\ref{thm-critical}, classification of solutions in the critical case, is finished in Section 4.\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000sections/Serrin.tex\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u00000000664\u00000000000\u00000000000\u000000000004110\u000015111220774\u0000013365\u0000 0\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000ustar \u0000root\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000root\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000", + "sketch": "To prove Theorem~\\ref{thm-Serrin}, the paper uses “Serrin's technique with a delicate test function $b(u)\\eta^n$ and a three-term Young's inequality” (carried out in Section 2).", + "expanded_sketch": "To prove the main theorem, the paper uses “Serrin's technique with a delicate test function $b(u)\\eta^n$ and a three-term Young's inequality” (carried out later).", + "expanded_theorem": "\\label{thm-Serrin}\nLet \\((M^n, g)\\) be a complete, connected, and noncompact Riemannian manifold of dimension \\(n\\geqslant3\\) with nonnegative Ricci curvature. Assume that \\(f\\in C(\\mathbb R_+;\\mathbb R_+)\\) satisfies\n\\begin{equation}\\label{cond-Serrin}\n\\int_0^1[f(v)]^{-\\frac{1}{2_*(q)-1}}\\mathrm dv<+\\infty.\n\\end{equation}\nThen any positive supersolution to \\begin{equation}\\label{Meq-gradient}\n-\\Delta u = f(u)|\\nabla u|^q\\quad\\text{in }M^n,\n\\end{equation}\nmust be constant.,", + "theorem_type": [ + "Implication", + "Universal" + ], + "mcq": { + "question": "Let \\((M^n,g)\\) be a complete, connected, noncompact Riemannian manifold of dimension \\(n\\ge 3\\) with nonnegative Ricci curvature. Let \\(f\\in C(\\mathbb R_+;\\mathbb R_+)\\) satisfy\n\\[\n\\int_0^1 [f(v)]^{-\\frac{1}{2_*(q)-1}}\\,dv<\\infty,\n\\]\nand consider the equation with gradient term\n\\[\n-\\Delta u=f(u)|\\nabla u|^q\\quad \\text{in }M^n,\n\\]\nwhere \\(\\Delta\\) is the Laplace--Beltrami operator and \\(\\nabla u\\) is the gradient of \\(u\\). If \\(u\\) is a positive supersolution, meaning \\(u>0\\) and \\(-\\Delta u\\ge f(u)|\\nabla u|^q\\) on \\(M^n\\), which conclusion holds?", + "correct_choice": { + "label": "A", + "text": "Every positive supersolution \\(u\\) is constant on \\(M^n\\)." + }, + "choices": [ + { + "label": "B", + "text": "Every positive supersolution \\(u\\) is constant on \\(M^n\\) provided additionally that \\(00\\) such that every positive supersolution \\(u\\) satisfies \\(u\\equiv C\\) on \\(M^n\\)." + }, + { + "label": "E", + "text": "Every positive \\(C^1\\) solution \\(u\\) of \\(-\\Delta u=f(u)|\\nabla u|^q\\) on \\(M^n\\) is constant, but the same conclusion is not asserted for positive supersolutions satisfying only \\(-\\Delta u\\ge f(u)|\\nabla u|^q\\)." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "no subcritical range restriction on q in the Serrin argument", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped the full class of positive supersolutions to the smaller class of bounded positive supersolutions", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "constancy depends on the individual supersolution, not a universal constant determined only by the data", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "Serrin test-function method is stated for supersolutions, not merely for exact C^1 solutions", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly reveal the answer. It states the hypotheses and asks for the resulting conclusion, without directly embedding the exact claim of choice A." + }, + "TAS": { + "score": 1, + "justification": "This is close to a theorem-recall item: the correct option is essentially the Liouville-type conclusion under the stated hypotheses. However, the alternatives introduce subtle variants in quantifiers, regularity, and parameter restrictions, so it is not a pure verbatim restatement." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish the exact theorem statement from weaker true or plausibly false variants, but the item mainly tests recognition/recall rather than substantial derivation or synthesis." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target common errors: adding unnecessary restrictions on q, weakening the conclusion to bounded supersolutions, confusing constancy with a universal constant, and restricting from supersolutions to exact solutions." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with strong distractors and little answer leakage, but it remains fairly close to direct theorem recall rather than a deeply generative reasoning task." + } + }, + { + "id": "2511.19744v1", + "paper_link": "http://arxiv.org/abs/2511.19744v1", + "theorems_cnt": 5, + "theorem": { + "env_name": "thm", + "content": "\\label{thm:main}\n Assume \\cref{conj:main}. If $n$ is a positive integer, then $t(n)\\geq 1$. If $5\\nmid n$, then $t(n)\\geq 2$. If $3\\mid n$, then $t(n)\\geq 3$.", + "start_pos": 271680, + "end_pos": 271862, + "label": "thm:main" + }, + "ref_dict": { + "table:toda sets": "\\begin{table}[t]\n\\centering\n\\caption{Sets of Toda primes}\\label{table:toda sets}\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n1 & $3,5$\\\\\n2 & $3,5$\\\\\n3 & $5,7,13$\\\\\n4 & $3,5,17$\\\\\n5 & $3,11$\\\\\n6 & $5,7,13$\\\\\n7 & $3,5,29$\\\\\n8 & $3,5,17$\\\\\n9 & $5,7,13,19,37$\\\\\n10 & $3,11,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n11 & $3,5,23$\\\\\n12 & $5,7,13,17$\\\\\n13 & $3,5,53$\\\\\n14 & $3,5,29$\\\\\n15 & $7,11,13,31,61$\\\\\n16 & $3,5,17$\\\\\n17 & $3,5$\\\\\n18 & $5,7,13,19,37,73$\\\\\n19 & $3,5$\\\\\n20 & $3,11,17,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n21 & $5,13,29,43$\\\\\n22 & $3,5,23,89$\\\\\n23 & $3,5,47$\\\\\n24 & $5,7,13,17,97$\\\\\n25 & $3,11,101$\\\\\n26 & $3,5,53$\\\\\n27 & $5,7,13,19,37,109$\\\\\n28 & $3,5,17,29,113$\\\\\n29 & $3,5,59$\\\\\n30 & $7,11,13,31,41,61$\\\\\n\\hline\n\\end{tabular}\n\\end{table}", + "lem:t(n)=3": "\\begin{lem}\\label{lem:t(n)=3}\n Assume \\cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\\{5,7,13\\}$.\n\\end{lem}", + "ques:at least one": "\\begin{ques}[Mikhailov]\\label{ques:at least one}\n Does every positive integer have a Toda prime?\n\\end{ques}", + "fig:tn": "\\begin{figure}[p]\n \\includegraphics[width=.75\\linewidth]{tn.png}\n \\caption{$t(n)$ for $n\\leq 100000$}\\label{fig:tn}\n\\end{figure}", + "thm:main": "\\begin{thm}\\label{thm:main}\n Assume \\cref{conj:main}. If $n$ is a positive integer, then $t(n)\\geq 1$. If $5\\nmid n$, then $t(n)\\geq 2$. If $3\\mid n$, then $t(n)\\geq 3$.\n\\end{thm}", + "conj:main": "\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}" + }, + "pre_theorem_intro_text_len": 3502, + "pre_theorem_intro_text": "The fourth stable homotopy group of spheres is trivial, meaning that $\\pi_{n+4}(S^n)=0$ for all $n>5$. In contrast to this, it is a theorem that $S^m$ has no trivial higher homotopy groups when $m\\in\\{2,3,4,5\\}$, as we will briefly explain.\n\nCurtis proved that $\\pi_n(S^4)\\neq 0$ for all $n\\geq 4$ \\cite{Cur69}. Curtis also proved that $\\pi_n(S^2)\\neq 0$ for all $n\\not\\equiv 1\\mod 8$. These same results were obtained (via different methods) by Mimura, Mori, and Oda \\cite{MMO75}. The proof that $\\pi_n(S^5)\\neq 0$ for all $n\\geq 5$ was given by Mori \\cite{Mor75} and Mahowald \\cite{Mah75,Mah82}.\n\nSince $\\pi_n(S^2)\\cong\\pi_n(S^3)$ for all $n\\geq 3$, the remaining case was $\\pi_n(S^3)$ with $n\\equiv 1\\mod 8$. This last case was proved by Gray \\cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \\cite{IMW16} using different methods. In \\emph{op.~cit.}, the authors note that \\cite[Theorem 5.2(ii)]{Tod66} implies that\n\\[\\mathbb{Z}/p\\subseteq\\pi_{2(p-1)k+1}(S^3)\\]\nwhenever $\\gcd(p,k)=1$ \\cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\\gcd(p,k)=1$ and $4n=(p-1)k$, then $\\pi_n(S^3)\\neq 0$ for all $n\\equiv 1\\mod 8$. This leads one to the following definition.\n\n\\begin{defn}\n Let $n$ be an integer. A \\emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\\mid 4n$ and $\\gcd(p,\\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \\cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \\cref{fig:tn}).\n\\end{defn}\n\n\\begin{table}[t]\n\\centering\n\\caption{Sets of Toda primes}\\label{table:toda sets}\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n1 & $3,5$\\\\\n2 & $3,5$\\\\\n3 & $5,7,13$\\\\\n4 & $3,5,17$\\\\\n5 & $3,11$\\\\\n6 & $5,7,13$\\\\\n7 & $3,5,29$\\\\\n8 & $3,5,17$\\\\\n9 & $5,7,13,19,37$\\\\\n10 & $3,11,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n11 & $3,5,23$\\\\\n12 & $5,7,13,17$\\\\\n13 & $3,5,53$\\\\\n14 & $3,5,29$\\\\\n15 & $7,11,13,31,61$\\\\\n16 & $3,5,17$\\\\\n17 & $3,5$\\\\\n18 & $5,7,13,19,37,73$\\\\\n19 & $3,5$\\\\\n20 & $3,11,17,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n21 & $5,13,29,43$\\\\\n22 & $3,5,23,89$\\\\\n23 & $3,5,47$\\\\\n24 & $5,7,13,17,97$\\\\\n25 & $3,11,101$\\\\\n26 & $3,5,53$\\\\\n27 & $5,7,13,19,37,109$\\\\\n28 & $3,5,17,29,113$\\\\\n29 & $3,5,59$\\\\\n30 & $7,11,13,31,41,61$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nIf every positive integer has a Toda prime, then one can greatly simplify the proof that $\\pi_n(S^3)\\neq 0$ for $n\\geq 3$. This was asked on MathOverflow \\cite{MO} (and attributed to Roman Mikhailov) several years ago.\n\n\\begin{ques}[Mikhailov]\\label{ques:at least one}\n Does every positive integer have a Toda prime?\n\\end{ques}\n\nIn fact, it appears that every positive integer has at least two Toda primes.\n\n\\begin{conj}\\label{conj:at least two}\n If $n$ is a positive integer, then $t(n)\\geq 2$.\n\\end{conj}\n\nWe tried to answer \\cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\\cref{conj:main}). We will give some heuristic evidence for this conjecture in \\cref{sec:heuristic}.\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}", + "context": "Since $\\pi_n(S^2)\\cong\\pi_n(S^3)$ for all $n\\geq 3$, the remaining case was $\\pi_n(S^3)$ with $n\\equiv 1\\mod 8$. This last case was proved by Gray \\cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \\cite{IMW16} using different methods. In \\emph{op.~cit.}, the authors note that \\cite[Theorem 5.2(ii)]{Tod66} implies that\n\\[\\mathbb{Z}/p\\subseteq\\pi_{2(p-1)k+1}(S^3)\\]\nwhenever $\\gcd(p,k)=1$ \\cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\\gcd(p,k)=1$ and $4n=(p-1)k$, then $\\pi_n(S^3)\\neq 0$ for all $n\\equiv 1\\mod 8$. This leads one to the following definition.\n\n\\begin{defn}\n Let $n$ be an integer. A \\emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\\mid 4n$ and $\\gcd(p,\\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \\cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \\cref{fig:tn}).\n\\end{defn}\n\nIf every positive integer has a Toda prime, then one can greatly simplify the proof that $\\pi_n(S^3)\\neq 0$ for $n\\geq 3$. This was asked on MathOverflow \\cite{MO} (and attributed to Roman Mikhailov) several years ago.\n\n\\begin{conj}\\label{conj:at least two}\n If $n$ is a positive integer, then $t(n)\\geq 2$.\n\\end{conj}\n\nWe tried to answer \\cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\\cref{conj:main}). We will give some heuristic evidence for this conjecture in \\cref{sec:heuristic}.\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{figure}[p]\n \\includegraphics[width=.75\\linewidth]{tn.png}\n \\caption{$t(n)$ for $n\\leq 100000$}\\label{fig:tn}\n\\end{figure}\n\n\\begin{ques}[Mikhailov]\\label{ques:at least one}\n Does every positive integer have a Toda prime?\n\\end{ques}\n\n\\begin{table}[t]\n\\centering\n\\caption{Sets of Toda primes}\\label{table:toda sets}\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n1 & $3,5$\\\\\n2 & $3,5$\\\\\n3 & $5,7,13$\\\\\n4 & $3,5,17$\\\\\n5 & $3,11$\\\\\n6 & $5,7,13$\\\\\n7 & $3,5,29$\\\\\n8 & $3,5,17$\\\\\n9 & $5,7,13,19,37$\\\\\n10 & $3,11,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n11 & $3,5,23$\\\\\n12 & $5,7,13,17$\\\\\n13 & $3,5,53$\\\\\n14 & $3,5,29$\\\\\n15 & $7,11,13,31,61$\\\\\n16 & $3,5,17$\\\\\n17 & $3,5$\\\\\n18 & $5,7,13,19,37,73$\\\\\n19 & $3,5$\\\\\n20 & $3,11,17,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n21 & $5,13,29,43$\\\\\n22 & $3,5,23,89$\\\\\n23 & $3,5,47$\\\\\n24 & $5,7,13,17,97$\\\\\n25 & $3,11,101$\\\\\n26 & $3,5,53$\\\\\n27 & $5,7,13,19,37,109$\\\\\n28 & $3,5,17,29,113$\\\\\n29 & $3,5,59$\\\\\n30 & $7,11,13,31,41,61$\\\\\n\\hline\n\\end{tabular}\n\\end{table}", + "full_context": "Since $\\pi_n(S^2)\\cong\\pi_n(S^3)$ for all $n\\geq 3$, the remaining case was $\\pi_n(S^3)$ with $n\\equiv 1\\mod 8$. This last case was proved by Gray \\cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \\cite{IMW16} using different methods. In \\emph{op.~cit.}, the authors note that \\cite[Theorem 5.2(ii)]{Tod66} implies that\n\\[\\mathbb{Z}/p\\subseteq\\pi_{2(p-1)k+1}(S^3)\\]\nwhenever $\\gcd(p,k)=1$ \\cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\\gcd(p,k)=1$ and $4n=(p-1)k$, then $\\pi_n(S^3)\\neq 0$ for all $n\\equiv 1\\mod 8$. This leads one to the following definition.\n\n\\begin{defn}\n Let $n$ be an integer. A \\emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\\mid 4n$ and $\\gcd(p,\\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \\cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \\cref{fig:tn}).\n\\end{defn}\n\nIf every positive integer has a Toda prime, then one can greatly simplify the proof that $\\pi_n(S^3)\\neq 0$ for $n\\geq 3$. This was asked on MathOverflow \\cite{MO} (and attributed to Roman Mikhailov) several years ago.\n\n\\begin{conj}\\label{conj:at least two}\n If $n$ is a positive integer, then $t(n)\\geq 2$.\n\\end{conj}\n\nWe tried to answer \\cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\\cref{conj:main}). We will give some heuristic evidence for this conjecture in \\cref{sec:heuristic}.\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{figure}[p]\n \\includegraphics[width=.75\\linewidth]{tn.png}\n \\caption{$t(n)$ for $n\\leq 100000$}\\label{fig:tn}\n\\end{figure}\n\n\\begin{ques}[Mikhailov]\\label{ques:at least one}\n Does every positive integer have a Toda prime?\n\\end{ques}\n\n\\begin{table}[t]\n\\centering\n\\caption{Sets of Toda primes}\\label{table:toda sets}\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n1 & $3,5$\\\\\n2 & $3,5$\\\\\n3 & $5,7,13$\\\\\n4 & $3,5,17$\\\\\n5 & $3,11$\\\\\n6 & $5,7,13$\\\\\n7 & $3,5,29$\\\\\n8 & $3,5,17$\\\\\n9 & $5,7,13,19,37$\\\\\n10 & $3,11,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n11 & $3,5,23$\\\\\n12 & $5,7,13,17$\\\\\n13 & $3,5,53$\\\\\n14 & $3,5,29$\\\\\n15 & $7,11,13,31,61$\\\\\n16 & $3,5,17$\\\\\n17 & $3,5$\\\\\n18 & $5,7,13,19,37,73$\\\\\n19 & $3,5$\\\\\n20 & $3,11,17,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n21 & $5,13,29,43$\\\\\n22 & $3,5,23,89$\\\\\n23 & $3,5,47$\\\\\n24 & $5,7,13,17,97$\\\\\n25 & $3,11,101$\\\\\n26 & $3,5,53$\\\\\n27 & $5,7,13,19,37,109$\\\\\n28 & $3,5,17,29,113$\\\\\n29 & $3,5,59$\\\\\n30 & $7,11,13,31,41,61$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\nIn \\cref{sec:lemmas}, we state and prove a few simple lemmas. We prove \\cref{thm:main} in \\cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \\cref{lem:t(n)=3}. We conclude with \\cref{sec:observations}, where we pose a couple questions that arose while working on this project.\n\n\\begin{lem}\\label{lem:not 5 or not 3 and 5}\n Let $p\\in\\{3,5\\}$. If $p\\nmid n$, then $p\\in T(n)$. In particular:\n \\begin{itemize}\n \\item If $5\\nmid n$, then $t(n)\\geq 1$.\n \\item If $3,5\\nmid n$, then $t(n)\\geq 2$.\n \\end{itemize}\n\\end{lem}\n\\begin{proof}\n Let $p\\in\\{3,5\\}$. Then $p-1\\mid 4n$, and $\\gcd(4n,p)=1$ by assumption. Thus $\\gcd(\\frac{4n}{p-1},p)=1$, so $p\\in T(n)$.\n\\end{proof}\n\n\\begin{lem}\\label{lem:t(n)=3}\n Assume \\cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\\{5,7,13\\}$.\n\\end{lem}\n\\begin{proof}\n We will induct on $\\omega(n)$. Our base cases will consist of $\\omega(n)\\leq 4$. Note that if $p\\nmid n$ for each $p\\in\\{5,7,13\\}$, then $T(n)\\supseteq\\{5,7,13\\}$. In particular, we may restrict our attention to multiples of these three primes. Moreover, $t(ap)\\geq t(a)$ for any prime $p\\not\\in T(a)$ by \\cref{cor:multiply by prime}, so we may assume that every prime factor $p\\mid n$ is a Toda prime of some divisor of $n$.\n \\begin{itemize}\n \\item The case of $\\omega(n)=1$ is just the calculation $T(3)=\\{5,7,13\\}$.\n \\item For $\\omega(n)=2$, we just need to compute $t(15)=t(39)=5$ and $t(21)=4$.\n \\item For $\\omega(n)=3$, we first compute $t(3\\cdot 5\\cdot 7)=9$ and $t(3\\cdot 5\\cdot 13)=t(3\\cdot 7\\cdot 13)=8$. It remains to compute, for each $p\\in\\{5,7,13\\}$, the Toda primes of $3pq$ for each $q\\in T(3p)$. Using the code provided in \\cref{sec:code}, we find that $t(3pq)\\geq 4$ for all such $p,q$.\n \\item For $\\omega(n)=4$, we first compute $t(3\\cdot 5\\cdot 7\\cdot 13)=16$. For the remaining computations in this case, we use the code in \\cref{sec:code}.\n \\begin{itemize}\n \\item If $\\{p,q\\}\\subseteq\\{5,7,13\\}$ and $r\\in T(3pq)$, then $t(3pqr)\\geq 9$.\n \\item If $p\\in\\{5,7,13\\}$ and $\\{q,r\\}\\subseteq T(3p)$, then $t(3pqr)\\geq 7$.\n \\item If $p\\in\\{5,7,13\\}$, $q\\in T(3p)$, and $r\\in T(3pq)$, then $t(3pqr)\\geq 5$.\n \\end{itemize}\n \\end{itemize}\n\n\\begin{cor}\\label{prop:divisible by 3}\n Assume \\cref{conj:main}. If $3\\mid n$, then $t(n)\\geq 3$.\n\\end{cor}\n\\begin{proof}\n By \\cref{cor:multiply by divisor}, we may assume that $n$ is square-free. By \\cref{cor:odd}, we may further assume that $n$ is odd. The result now follows from \\cref{lem:t(n)=3}.\n\\end{proof}\n\nFirstly, if there exists $q'\\in T(3p)-\\{5,7,13,q\\}$ such that $q'\\nmid n$, then $\\{5,7,13,q,q'\\}-\\{p\\}\\subseteq T(n)$, and we are done. In fact, the Toda primes of $n$ are precisely those primes among $\\{2d+1:d\\mid 2n\\}$ that are not factors of $n$. Thus if\n\\begin{equation}\\label{eq:heuristic}\n\\{2d+1\\text{ prime}:d\\mid 2n\\}-(\\Omega(n)\\cup\\{5,7,13,q\\})\n\\end{equation}\nis non-empty, then $t(n)\\geq 4$. Our heuristic for \\cref{conj:main} is that the set $\\{2d+1:d\\mid 2n\\}$ consists of $2^{\\omega(n)+1}$ elements, while $\\Omega(n)\\cup\\{5,7,13,q\\}$ consists of $\\omega(n)+3$ elements.\n\n\\begin{prop}\\label{prop:t(p)}\nAssume $p\\geq 7$ is a prime. Let $\\vphi$ denote the totient function. If $\\vphi(x)=4p$ for some integer $x$, then $T(p)=\\{3,5,2p+1\\}$ or $\\{3,5,4p+1\\}$. Otherwise, $T(p)=\\{3,5\\}$.\n\\end{prop}\n\\begin{proof}\n One can directly check that $3,5\\in T(p)$ for all primes greater than 5. Now by Euler's product formula, we have $\\vphi(x)=p_1^{e_1-1}(p_1-1)\\cdots p_m^{e_m-1}(p_m-1)$, where $x=\\prod_{i=1}^m p_i^{e_i}$ is the prime factorization of $x$. It follows that there exists $x$ such that $\\vphi(x)=4p$ if and only if one of the following cases holds:\n \\begin{enumerate}[(i)]\n \\item $x=2^2\\cdot q$, where $q$ is an odd prime such that $q-1=2p$. In this case, $q$ is a Toda prime of $p$ with $\\frac{4p}{q-1}=2$.\n \\item $x=2^r\\cdot 3\\cdot q$, where $r\\in\\{0,1\\}$ and $q$ is an odd prime such that $q-1=2p$. In this case, $q$ is a Toda prime of $p$ with $\\frac{4p}{q-1}=2$.\n \\item $x=2^r\\cdot q$, where $r\\in\\{0,1\\}$ and $q$ is an odd prime such that $q-1=4p$. In this case, $q$ is a Toda prime of $p$ with $\\frac{4p}{q-1}=1$.\n \\item $x=2^r\\cdot 5^2$, where $r\\in\\{0,1\\}$ (in which case $p=5$). This case is not relevant for this lemma, as we have assumed $p\\geq 7$.\n \\end{enumerate}\n It remains to show that no other primes can be the Toda prime of $p$. To this end, let $q>5$ be a Toda prime of $p$. Then $q-1\\mid 4p$, so we either have $q-1=4p$ or $q-1=2p$ (as $q-1$ is even and $p$ is odd). The existence of such a $q$ gives us a solution to $\\vphi(x)=4p$ as outlined in cases (i), (ii), and (iii).\n\n\\begin{lem}\\label{lem:strategy for denoms}\n Let $d$ be a Bernoulli denominator with $F(d)=4a$ for some integer $a$. If $\\{2pi+1:i\\mid 2a\\}$ contains a prime number for each $p\\in T(a)$, then \\cref{conj:general bernoulli} \\eqref{conj:toda for bernoulli} holds for this Bernoulli denominator.\n\\end{lem}\n\\begin{proof}\n We know that $p-1\\mid 4a$ with $\\gcd(\\frac{4a}{p-1},p)=1$ for all $p\\in T(a)$. Thus $p-1\\mid 4am$, and we have $\\gcd(\\frac{4am}{p-1},p)=1$ if and only if $p\\nmid m$. It therefore suffices to show that if $p\\mid m$ for some $p\\in T(a)$, then $D_{4am}>D_{4a}$.\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{lem}\\label{lem:t(n)=3}\n Assume \\cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\\{5,7,13\\}$.\n\\end{lem}\n\n\\begin{thm}\\label{thm:main}\n Assume \\cref{conj:main}. If $n$ is a positive integer, then $t(n)\\geq 1$. If $5\\nmid n$, then $t(n)\\geq 2$. If $3\\mid n$, then $t(n)\\geq 3$.\n\\end{thm}", + "post_theorem_intro_text_len": 541, + "post_theorem_intro_text": "In \\cref{sec:lemmas}, we state and prove a few simple lemmas. We prove \\cref{thm:main} in \\cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \\cref{lem:t(n)=3}. We conclude with \\cref{sec:observations}, where we pose a couple questions that arose while working on this project.\n\n\\subsection*{Acknowledgements}\nWe thank Nick Andersen, Pace Nielsen, and Kyle Pratt for helpful conversations. The author was partially supported by the NSF (DMS-2502365) and the Simons Foundation.", + "sketch": "We prove \\cref{thm:main} in \\cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \\cref{lem:t(n)=3}.", + "expanded_sketch": "No expanded sketch found.", + "expanded_theorem": "\\label{thm:main}\nAssume\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\nIf $n$ is a positive integer, then $t(n)\\geq 1$. If $5\\nmid n$, then $t(n)\\geq 2$. If $3\\mid n$, then $t(n)\\geq 3$.,\n", + "theorem_type": [ + "Implication", + "Inequality or Bound" + ], + "mcq": { + "question": "For a positive integer m, let T(m) be the set of odd primes p such that p-1 divides 4m and \\(\\gcd\\!\\left(p,\\frac{4m}{p-1}\\right)=1\\), and let \\(t(m)=|T(m)|\\). Assume the following statement is true: whenever m is an odd, square-free multiple of 3, there exists some \\(p\\in\\{5,7,13\\}\\) with \\(p\\mid m\\), no prime \\(r\\in\\{5,7,13\\}\\setminus\\{p\\}\\) divides m, and there exists \\(q\\in T(3p)\\setminus\\{5,7,13\\}\\) such that \\(q\\nmid m\\), then \\(t(m)\\ge 4\\). Under this assumption, which statement about a positive integer n holds?", + "correct_choice": { + "label": "A", + "text": "For every positive integer n, \\(t(n)\\ge 1\\). In addition, if \\(5\\nmid n\\), then \\(t(n)\\ge 2\\); and if \\(3\\mid n\\), then \\(t(n)\\ge 3\\)." + }, + "choices": [ + { + "label": "B", + "text": "For every positive integer n, \\(t(n)\\ge 1\\). In addition, if \\(5\\nmid n\\), then \\(t(n)\\ge 2\\); and if \\(3\\nmid n\\), then \\(t(n)\\ge 3\\)." + }, + { + "label": "C", + "text": "For every positive integer n, \\(t(n)\\ge 1\\). In addition, if \\(5\\nmid n\\), then \\(t(n)\\ge 2\\)." + }, + { + "label": "D", + "text": "For every positive integer n, \\(t(n)\\ge 2\\). In addition, if \\(5\\nmid n\\), then \\(t(n)\\ge 3\\); and if \\(3\\mid n\\), then \\(t(n)\\ge 4\\)." + }, + { + "label": "E", + "text": "For every positive integer n, \\(t(n)\\ge 1\\). In addition, if \\(3\\nmid n\\), then \\(t(n)\\ge 2\\); and if \\(3\\mid n\\), then \\(t(n)\\ge 3\\)." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "divisibility condition on 3 in the final clause", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped the strongest conclusion \\(3\\mid n\\Rightarrow t(n)\\ge 3\\)", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "shifted each lower bound up by one", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "replaced the specific hypothesis \\(5\\nmid n\\) by the unrelated \\(3\\nmid n\\)", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives a technical assumed implication about special integers m, but it does not explicitly state the broader conclusion in choice A. There is no direct wording overlap that reveals the correct option." + }, + "TAS": { + "score": 2, + "justification": "The question is not a restatement of the assumption. It asks the test-taker to infer a global conclusion about t(n) from a conditional statement on a restricted class of m, so the correct answer is not tautologically built into the stem." + }, + "GPS": { + "score": 2, + "justification": "Selecting the correct option requires distinguishing between a strongest valid conclusion, a weaker true-looking alternative, and several subtle overgeneralizations or condition swaps. This creates real pressure to reason about how the assumption propagates to statements about arbitrary n." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and target common failure modes: reversing a divisibility condition, choosing a weaker-but-safe statement, and overstrengthening all bounds. They are distinct and mathematically aligned with likely errors." + }, + "total_score": 8, + "overall_assessment": "A strong MCQ on these criteria: it avoids answer leakage, is clearly non-tautological, and uses high-quality distractors that force genuine comparative reasoning, though the stem is quite technically dense." + } + }, + { + "id": "2511.19357v1", + "paper_link": "http://arxiv.org/abs/2511.19357v1", + "theorems_cnt": 4, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:sob-pull-back}\nLet $n,m,Q\\ge 1$ be natural numbers, $U\\subset \\mathbb{R}^m$ open, and $f\\in W_{loc}^{1,p}(f(\\Omega),\\mathcal A_Q(\\mathbb{R}^n))$. If $\\omega$ is a $S_Q$-invariant smooth $k$-form on $(\\mathbb{R}^n)^Q$, and either $k+1\\le p$ of $\\mathrm {d} \\omega=0$ and $k\\le p$, then $\\omega$ has a well-defined pull-back $f^*\\omega \\in L^{p/k}_{loc}(U,\\bigwedge^k\\mathbb{R}^m)$ and moreover\n\\begin{align*}\n \\mathrm {d}(f^*\\omega)=f^*(\\mathrm {d}\\omega)\n\\end{align*}\nweakly.", + "start_pos": 131187, + "end_pos": 131649, + "label": "thm:sob-pull-back" + }, + "ref_dict": { + "prop:ball-meas-upper-bound": "\\begin{proposition}\\label{prop:ball-meas-upper-bound}\nFor any compact set $K\\subset \\Omega_f$ there exists $r_K>0$ such that\n\\begin{align*}\n \\Ha^n(B_{\\Omega_f}(z,r))\\le \\omega_nd^{n/2}K_IK_O r^n,\\quad z\\in K,\\ rn$ such that $\\minv f\\in W_{loc}^{1,p}(f(\\Omega),\\mathcal A_d(\\Omega))$.\n\\end{theorem}", + "thm:geom-QC": "\\begin{theorem}\\label{thm:geom-QC}\nLet $f:\\Omega\\to \\R^n$ be a proper quasiregular map of finite degree $d$, and equip $\\Omega_f=\\minv f(f(\\Omega))$ with the metric from $\\mathcal A_d(\\R^n)$ and the Hausdorff $n$-measure on $\\Omega_f$. \n\\begin{itemize}\n \\item[(1)] $\\Omega_f$ is $n$-rectifiable, upper Ahlfors $n$-regular and satisfies the infinitesimal $n$-Poincar\\'e inequality;\n \\item[(2)] $\\minv f:f(\\Omega)\\to \\Omega_f$ is geometrically quasiconformal. More precisely, \n\\begin{align*}\n \\frac 1{K_IK_O}\\Mod_n\\Gamma\\le \\Mod_n\\minv f(\\Gamma)\\le K_IK_O\\Mod_n\\Gamma\n\\end{align*}\nfir any path family $\\Gamma $ in $f(\\Omega)$.\n\\end{itemize}\n\\end{theorem}", + "def:multi-valued-inv": "\\begin{definition}\\label{def:multi-valued-inv}\nLet $f:X\\to Y$ be a proper branched cover of finite degree $d>0$ between topological $n$-manifolds. The multi-valued inverse $\\minv f:f(X)\\to \\mathcal A_d(X)$ of $f$ is defined by\n\\begin{align*}\n\\minv f(y)=\\sum_{x\\in f\\inv(y)}\\iota(f,x)\\bb x \n\\end{align*}\n\\end{definition}", + "thm:QR-curve": "\\begin{theorem}\\label{thm:QR-curve}\nLet $f:\\Omega\\to \\R^n$ be a proper quasiregular map of finite degree $d$. Then the multivalued inverse $\\minv{f}:f(\\Omega)\\to \\mathcal A_d(\\Omega)$ is an $\\omega$-quasiregular curve, where $\\omega=\\operatorname{tr}(\\operatorname{vol}_{\\R^n})$ is given by \\eqref{eq:natural-n-form}. More precisely, $\\minv f\\in W_{loc}^{1,n}(f(\\Omega),\\mathcal A_d(\\R^n))$, and \n\\[\n\\|\\omega\\|\\circ \\minv f\\ |D\\minv f|^n\\le d^{n/2-1}K_I(f)\\star \\minv f^*\\omega\\quad \\textrm{almost everywhere on }\\Omega.\n\\]\n\\end{theorem}", + "thm:wug": "\\begin{theorem}\\label{thm:wug}\nFor $\\Mod_n$-a.e. curve $\\gamma$ in $f(\\Omega)$, $\\minv f\\circ\\gamma$ is an absolutely continuous curve in $\\mathcal{A}_d(\\Omega)$, and satisfies\n\\begin{align}\\label{eq:ug-ineq}\nH(\\gamma_t)|\\gamma_t'|\\le |(\\minv f\\circ\\gamma)'_t|\\le (K_IK_O)^{1/n} H(\\gamma_t)|\\gamma_t'|\n\\end{align}\na.e. $t$, where\n\\begin{align}\\label{eq:H}\n H(y):=\\big(f_\\ast(\\|Df\\|^{-2})(y)\\big)^{1/2},\\quad y\\in f(\\Omega).\n\\end{align}\n\\end{theorem}", + "eq:natural-n-form": "\\begin{align}\\label{eq:natural-n-form}\n\\operatorname{tr}(\\operatorname{vol}_{\\R^n}):=\\sum_j^d P_j^*(\\operatorname{vol}_{\\R^n})\n\\end{align}", + "lem:homeo": "\\begin{lemma}\\label{lem:homeo}\nThe generalized inverse $\\minv f:f(X)\\to \\mathcal A_d(X)$ is a homeomorphism onto its image.\n\\end{lemma}", + "lem:barycenter": "\\begin{lemma}\\label{lem:barycenter}\nThe barycenter map \\eqref{eq:barycenter} is $1/\\sqrt d$-Lipschitz.\n\\end{lemma}", + "cor:QR-curve": "\\begin{corollary}\\label{cor:QR-curve}\nLet $f:\\Omega\\to \\R^n$ be a proper quasiregular map of finite degree $d$. Then there exists $p>n$ such that the generalized inverse \n\\[\ng:f(\\Omega)\\to \\R^n,\\quad g(y):=\\sum_{x\\in f\\inv(y)}\\iota(f,x)x\n\\]\nbelongs to $W^{1,p}_{loc}(f(\\Omega),\\R^n)$.\n\\end{corollary}", + "eq:QR-curve": "\\begin{align}\\label{eq:QR-curve}\n\\|\\omega\\|\\circ f\\|Df\\|^n\\le K\\star f^*\\omega\\quad \\textrm{ almost everywhere}.\n\\end{align}", + "thm:lip-pullback-flat-form": "\\begin{theorem}\\label{thm:lip-pullback-flat-form}\nSuppose $U\\subset\\R^m$ is an open set, $f:U\\to\\mathcal A_d(\\R^n)$ is Lipschitz and $\\omega\\in \\Omega^k(\\mathcal A_d(\\R^n))$. Then $\\ud(f^*\\omega)=f^*(\\ud\\omega)$ weakly. In particular, $f^*\\omega$ is a flat form on $U$ in the sense of Whitney. \n\\end{theorem}", + "eq:inf-n-PI": "\\begin{align}\\label{eq:inf-n-PI}\n|Dg|_p=\\Lip g\\quad \\mu\\textrm{-a.e. on }X\n\\end{align}", + "thm:sob-pull-back": "\\begin{theorem}\\label{thm:sob-pull-back}\nLet $n,m,Q\\ge 1$ be natural numbers, $U\\subset \\R^m$ open, and $f\\in W_{loc}^{1,p}(f(\\Omega),\\mathcal A_Q(\\R^n))$. If $\\omega$ is a $S_Q$-invariant smooth $k$-form on $(\\R^n)^Q$, and either $k+1\\le p$ of $\\ud \\omega=0$ and $k\\le p$, then $\\omega$ has a well-defined pull-back $f^*\\omega \\in L^{p/k}_{loc}(U,\\bigwedge^k\\R^m)$ and moreover\n\\begin{align*}\n \\ud(f^*\\omega)=f^*(\\ud\\omega)\n\\end{align*}\nweakly.\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 6951, + "pre_theorem_intro_text": "\\subsection{Overview}\nA continuous map $f\\in W^{1,n}_{loc}(M^n,N^k)$ between oriented Riemannian manifolds with $n\\le k$ is a $K$-quasiregular $\\omega$-curve, if \n\\begin{align}\\label{eq:QR-curve}\n\\|\\omega\\|\\circ f\\|Df\\|^n\\le K\\star f^*\\omega\\quad \\textrm{ almost everywhere}.\n\\end{align}\nHere, $\\omega$ is a (closed) $n$-form on $N$, $\\star f^*\\omega$ is the Hodge-star dual of $f^*\\omega$, and $\\|\\omega\\|$ is the pointwise comass of $\\omega$. When $k=n$ and $\\omega$ is the volume form on $N$, \\eqref{eq:QR-curve} recovers the notion of quasiregular (QR) maps, written more classically as \n\\begin{align}\\label{eq:QR}\n\\|Df\\|^n\\le K\\mathbf{J} f\\quad\\textrm{ almost everywhere},\n\\end{align}\nwhere $\\mathbf{J} f=\\det(Df)$ and $Df$ is the weak differential of $f$. Quasiregular maps have a rich theory \\cite{ric93,HKM06}, and enjoy many of the same analytic properties as quasiconformal (QC) maps. While they are not necessarily homeomorphisms, a seminal result of Reshetnyak \\cite{res67} states that QR maps are open and discrete, and thus in particular locally injective outside a singular branch set of topological dimension at most $n-2$ \\cite{vai66}. We call an open discrete map a branched cover.\n\nQuasiregular curves (QR curves) in turn were introduced by Pankka \\cite{pankka20}, and they generalize quasiregular maps by allowing the dimension of the target to be higher than that of the domain; for example pseudoholomorhic curves \\cite{gro85} are QR curves. Much of the classical theory of QR maps extends to QR curves \\cite{pankka20,pan-onn21}, with one notable exception: Reshetnyak's theorem. Namely, QR curves need not be branched covers. Indeed, if $k>n$, a QR curve cannot be open, while the more non-trivial failure of discreteness follows from an example in \\cite{IVV02}. \n\nCompared to QR maps, one of the challenges in QR curves is understanding how the choice of $\\omega$ reflects the behaviour of $f$, see for example \\cite{hei-pan-pry23,iko-pan24} for some rigidity results and connections with calibrations. \n\n\\bigskip\\noindent In this paper, we identify a fruther connection between QR maps and QR curves: we construct a multi-valued inverse $\\minv f$ for a QR map $f:\\Omega\\to \\mathbb{R}^n$ of finite degree $d>0$, and show that $\\minv f$ is naturally a QR curve (see Theorem \\ref{thm:QR-curve}). In particular, QR maps have a local (multi-valued) inverse across the singular branch set (and not only outside it) with suitable Sobolev regularity. Earlier, related constructions include a push-forward operator \\cite[p. 263]{HKM06}, \\cite[Section 4]{teripekka} and a generalized local inverse \\cite[Chapter II.5]{ric93}, \\cite[Section 8]{onn-raj09}, both of which are important for obtaining modulus inequalities.\n\nOur approach is based on Almgren's space of unordered tuples \\cite{del11} (also known as the symmetric product, cf. \\cite[Chapter 4.K]{hat02}), and brings a new perspective to quasiregular theory; together with a pull-back theory by multi-valued maps developed in this paper (see Theorems \\ref{thm:sob-pull-back} and \\ref{thm:lip-pullback-flat-form}) we obtain e.g. higher integrability properties of the generalized inverse \\cite[Chapter II.5]{ric93} (see Corollary \\ref{cor:QR-curve} and Theorem \\ref{thm:higher-integrability}). We describe the construction and briefly discuss some its advantages over alternative approaches below.\n\n\\subsection{Multi-valued inverse}\n\nGiven a non-injective (proper) branched cover $f:\\Omega\\to \\mathbb{R}^n$, there are several natural ways to interpret the set valued ``inverse'' map $f(\\Omega)\\ni y\\mapsto f^{-1}(y)$. One is to consider the natural factorization $f=\\bar f\\circ P_f$, where $P:\\Omega\\to \\widetilde\\Omega_f:=\\Omega/\\sim$ is the projection onto the quotient space given by the equivalence relation $x\\sim y$ if $f(x)=f(y)$, and $\\bar f:\\widetilde \\Omega_f\\to f(\\Omega)$ the induced map, which in this case is a homeomorphism and thus invertible. Another approach is to view the set valued inverse as a map $f(\\Omega)\\to \\mathcal{H}(\\Omega)$ into the Hausdorff space of non-empty compact subsets of $\\Omega$. A drawback of the first approach is that the target space of the inverse depends on $f$ and, more crucially, that $\\bar f$ need not be geometrically quasiconformal unless $f$ is a BLD-map. The second approach, on the other hand, does not see the multiplicities in the preimage, and $\\mathcal{H}(\\Omega)$ does not carry any natural differential forms, nor any bi-Lipschitz embedding in Euclidean space (cf. \\cite[Theorem 2.1]{del11}).\n\nInstead, we give a construction based on Almgren's space and local degree theory. Let $Q\\ge 1$ be a natural number. The Almgren space $\\mathcal A_Q(\\mathbb{R}^n)$ of unordered $Q$-tuples in $\\mathbb{R}^n$ was used by Almgren in his seminal work on regularity of minimal currents \\cite{alm2000}, and was further explored by De Lellis et. al. \\cite{del11}. It is a $nQ$-dimensional smooth orbifold, but not a manifold (unless $Q=1$), defined as\n\\begin{align*}\n\\mathcal A_Q(\\mathbb{R}^n)=(\\mathbb{R}^n)^Q/S_Q,\\quad d_\\mathcal A(\\bb x,\\bb y)=\\inf_{\\sigma\\in S_Q}\\Big(\\sum_j^d|x_j-y_{\\sigma(j)}|^2\\Big)^{1/2},\n\\end{align*}\nwhere $S_Q$ is the permutation group on $Q$ elements acting naturally on $(\\mathbb{R}^n)^Q$. (See Section \\ref{sec:almgren-space} for more details.) \n\n\\bigskip\\noindent The link between the multi-valued inverse and Almgren space is the following simple observation based on local degree theory. Given a proper branched cover $f:\\Omega\\to \\mathbb{R}^n$ of degree $d>0$ from an open set $\\Omega\\subset \\mathbb{R}^n$, the local index $\\iota(f,x)$ is defined for all $x\\in\\Omega$ and satisfies $\\displaystyle \\sum_{x\\in f^{-1}(y)}\\iota(f,x)=d$, $y\\in f(\\Omega)$. Thus, $f^{-1} (y)$ can be regarded as an unordered $d$-tuple in $\\Omega$ (counted with multiplicities) for each $y\\in f(\\Omega)$. The multi-valued inverse $\\minv f$ of the proper branched cover $f:\\Omega\\to \\mathbb{R}^n$ of finite degree $d>0$ is the map\n\\begin{align}\\label{eq:minv}\n\\minv f:f(\\Omega)\\to \\mathcal A_d(\\mathbb{R}^n),\\quad \\minv f(y):= \\sum_{x\\in f^{-1}(y)}\\iota(f,x)\\bb x.\n\\end{align}\nWe refer the reader to Section \\ref{sec:branched-covers} (see in particular Definition \\ref{def:multi-valued-inv}) for the notation and further details used here. \n\n\\subsection{Statement of results}\n\nWhile it is not a manifold, $\\mathcal A_Q(\\mathbb{R}^n)$ nevertheless carries a natural $n$-form given by the trace of the volume form of $\\mathbb{R}^n$; that is, the $n$-form \n\\begin{align}\\label{eq:natural-n-form}\n\\operatorname{tr}(\\operatorname{vol}_{\\mathbb{R}^n}):=\\sum_j^d P_j^*(\\operatorname{vol}_{\\mathbb{R}^n})\n\\end{align}\non $(\\mathbb{R}^n)^Q$ (where $P_j$ is the projection to the $j$th factor) is $S_Q$-invariant: $\\sigma^*\\omega=\\omega$ for all $\\sigma\\in S_Q$. Using $S_Q$-invariance, we develop a pull-back theory of multi-valued maps (see Sections \\ref{sec:diff-forms} and \\ref{sec:pull-back}), which is of independent interest.", + "context": "\\subsection{Overview}\nA continuous map $f\\in W^{1,n}_{loc}(M^n,N^k)$ between oriented Riemannian manifolds with $n\\le k$ is a $K$-quasiregular $\\omega$-curve, if \n\\begin{align}\\label{eq:QR-curve}\n\\|\\omega\\|\\circ f\\|Df\\|^n\\le K\\star f^*\\omega\\quad \\textrm{ almost everywhere}.\n\\end{align}\nHere, $\\omega$ is a (closed) $n$-form on $N$, $\\star f^*\\omega$ is the Hodge-star dual of $f^*\\omega$, and $\\|\\omega\\|$ is the pointwise comass of $\\omega$. When $k=n$ and $\\omega$ is the volume form on $N$, \\eqref{eq:QR-curve} recovers the notion of quasiregular (QR) maps, written more classically as \n\\begin{align}\\label{eq:QR}\n\\|Df\\|^n\\le K\\mathbf{J} f\\quad\\textrm{ almost everywhere},\n\\end{align}\nwhere $\\mathbf{J} f=\\det(Df)$ and $Df$ is the weak differential of $f$. Quasiregular maps have a rich theory \\cite{ric93,HKM06}, and enjoy many of the same analytic properties as quasiconformal (QC) maps. While they are not necessarily homeomorphisms, a seminal result of Reshetnyak \\cite{res67} states that QR maps are open and discrete, and thus in particular locally injective outside a singular branch set of topological dimension at most $n-2$ \\cite{vai66}. We call an open discrete map a branched cover.\n\nGiven a non-injective (proper) branched cover $f:\\Omega\\to \\mathbb{R}^n$, there are several natural ways to interpret the set valued ``inverse'' map $f(\\Omega)\\ni y\\mapsto f^{-1}(y)$. One is to consider the natural factorization $f=\\bar f\\circ P_f$, where $P:\\Omega\\to \\widetilde\\Omega_f:=\\Omega/\\sim$ is the projection onto the quotient space given by the equivalence relation $x\\sim y$ if $f(x)=f(y)$, and $\\bar f:\\widetilde \\Omega_f\\to f(\\Omega)$ the induced map, which in this case is a homeomorphism and thus invertible. Another approach is to view the set valued inverse as a map $f(\\Omega)\\to \\mathcal{H}(\\Omega)$ into the Hausdorff space of non-empty compact subsets of $\\Omega$. A drawback of the first approach is that the target space of the inverse depends on $f$ and, more crucially, that $\\bar f$ need not be geometrically quasiconformal unless $f$ is a BLD-map. The second approach, on the other hand, does not see the multiplicities in the preimage, and $\\mathcal{H}(\\Omega)$ does not carry any natural differential forms, nor any bi-Lipschitz embedding in Euclidean space (cf. \\cite[Theorem 2.1]{del11}).\n\nInstead, we give a construction based on Almgren's space and local degree theory. Let $Q\\ge 1$ be a natural number. The Almgren space $\\mathcal A_Q(\\mathbb{R}^n)$ of unordered $Q$-tuples in $\\mathbb{R}^n$ was used by Almgren in his seminal work on regularity of minimal currents \\cite{alm2000}, and was further explored by De Lellis et. al. \\cite{del11}. It is a $nQ$-dimensional smooth orbifold, but not a manifold (unless $Q=1$), defined as\n\\begin{align*}\n\\mathcal A_Q(\\mathbb{R}^n)=(\\mathbb{R}^n)^Q/S_Q,\\quad d_\\mathcal A(\\bb x,\\bb y)=\\inf_{\\sigma\\in S_Q}\\Big(\\sum_j^d|x_j-y_{\\sigma(j)}|^2\\Big)^{1/2},\n\\end{align*}\nwhere $S_Q$ is the permutation group on $Q$ elements acting naturally on $(\\mathbb{R}^n)^Q$. (See Section \\ref{sec:almgren-space} for more details.)\n\n\\bigskip\\noindent The link between the multi-valued inverse and Almgren space is the following simple observation based on local degree theory. Given a proper branched cover $f:\\Omega\\to \\mathbb{R}^n$ of degree $d>0$ from an open set $\\Omega\\subset \\mathbb{R}^n$, the local index $\\iota(f,x)$ is defined for all $x\\in\\Omega$ and satisfies $\\displaystyle \\sum_{x\\in f^{-1}(y)}\\iota(f,x)=d$, $y\\in f(\\Omega)$. Thus, $f^{-1} (y)$ can be regarded as an unordered $d$-tuple in $\\Omega$ (counted with multiplicities) for each $y\\in f(\\Omega)$. The multi-valued inverse $\\minv f$ of the proper branched cover $f:\\Omega\\to \\mathbb{R}^n$ of finite degree $d>0$ is the map\n\\begin{align}\\label{eq:minv}\n\\minv f:f(\\Omega)\\to \\mathcal A_d(\\mathbb{R}^n),\\quad \\minv f(y):= \\sum_{x\\in f^{-1}(y)}\\iota(f,x)\\bb x.\n\\end{align}\nWe refer the reader to Section \\ref{sec:branched-covers} (see in particular Definition \\ref{def:multi-valued-inv}) for the notation and further details used here.\n\n\\subsection{Statement of results}\n\nWhile it is not a manifold, $\\mathcal A_Q(\\mathbb{R}^n)$ nevertheless carries a natural $n$-form given by the trace of the volume form of $\\mathbb{R}^n$; that is, the $n$-form \n\\begin{align}\\label{eq:natural-n-form}\n\\operatorname{tr}(\\operatorname{vol}_{\\mathbb{R}^n}):=\\sum_j^d P_j^*(\\operatorname{vol}_{\\mathbb{R}^n})\n\\end{align}\non $(\\mathbb{R}^n)^Q$ (where $P_j$ is the projection to the $j$th factor) is $S_Q$-invariant: $\\sigma^*\\omega=\\omega$ for all $\\sigma\\in S_Q$. Using $S_Q$-invariance, we develop a pull-back theory of multi-valued maps (see Sections \\ref{sec:diff-forms} and \\ref{sec:pull-back}), which is of independent interest.\n\n\\begin{definition}\\label{def:multi-valued-inv}\nLet $f:X\\to Y$ be a proper branched cover of finite degree $d>0$ between topological $n$-manifolds. The multi-valued inverse $\\minv f:f(X)\\to \\mathcal A_d(X)$ of $f$ is defined by\n\\begin{align*}\n\\minv f(y)=\\sum_{x\\in f\\inv(y)}\\iota(f,x)\\bb x \n\\end{align*}\n\\end{definition}", + "full_context": "\\subsection{Overview}\nA continuous map $f\\in W^{1,n}_{loc}(M^n,N^k)$ between oriented Riemannian manifolds with $n\\le k$ is a $K$-quasiregular $\\omega$-curve, if \n\\begin{align}\\label{eq:QR-curve}\n\\|\\omega\\|\\circ f\\|Df\\|^n\\le K\\star f^*\\omega\\quad \\textrm{ almost everywhere}.\n\\end{align}\nHere, $\\omega$ is a (closed) $n$-form on $N$, $\\star f^*\\omega$ is the Hodge-star dual of $f^*\\omega$, and $\\|\\omega\\|$ is the pointwise comass of $\\omega$. When $k=n$ and $\\omega$ is the volume form on $N$, \\eqref{eq:QR-curve} recovers the notion of quasiregular (QR) maps, written more classically as \n\\begin{align}\\label{eq:QR}\n\\|Df\\|^n\\le K\\mathbf{J} f\\quad\\textrm{ almost everywhere},\n\\end{align}\nwhere $\\mathbf{J} f=\\det(Df)$ and $Df$ is the weak differential of $f$. Quasiregular maps have a rich theory \\cite{ric93,HKM06}, and enjoy many of the same analytic properties as quasiconformal (QC) maps. While they are not necessarily homeomorphisms, a seminal result of Reshetnyak \\cite{res67} states that QR maps are open and discrete, and thus in particular locally injective outside a singular branch set of topological dimension at most $n-2$ \\cite{vai66}. We call an open discrete map a branched cover.\n\nGiven a non-injective (proper) branched cover $f:\\Omega\\to \\mathbb{R}^n$, there are several natural ways to interpret the set valued ``inverse'' map $f(\\Omega)\\ni y\\mapsto f^{-1}(y)$. One is to consider the natural factorization $f=\\bar f\\circ P_f$, where $P:\\Omega\\to \\widetilde\\Omega_f:=\\Omega/\\sim$ is the projection onto the quotient space given by the equivalence relation $x\\sim y$ if $f(x)=f(y)$, and $\\bar f:\\widetilde \\Omega_f\\to f(\\Omega)$ the induced map, which in this case is a homeomorphism and thus invertible. Another approach is to view the set valued inverse as a map $f(\\Omega)\\to \\mathcal{H}(\\Omega)$ into the Hausdorff space of non-empty compact subsets of $\\Omega$. A drawback of the first approach is that the target space of the inverse depends on $f$ and, more crucially, that $\\bar f$ need not be geometrically quasiconformal unless $f$ is a BLD-map. The second approach, on the other hand, does not see the multiplicities in the preimage, and $\\mathcal{H}(\\Omega)$ does not carry any natural differential forms, nor any bi-Lipschitz embedding in Euclidean space (cf. \\cite[Theorem 2.1]{del11}).\n\nInstead, we give a construction based on Almgren's space and local degree theory. Let $Q\\ge 1$ be a natural number. The Almgren space $\\mathcal A_Q(\\mathbb{R}^n)$ of unordered $Q$-tuples in $\\mathbb{R}^n$ was used by Almgren in his seminal work on regularity of minimal currents \\cite{alm2000}, and was further explored by De Lellis et. al. \\cite{del11}. It is a $nQ$-dimensional smooth orbifold, but not a manifold (unless $Q=1$), defined as\n\\begin{align*}\n\\mathcal A_Q(\\mathbb{R}^n)=(\\mathbb{R}^n)^Q/S_Q,\\quad d_\\mathcal A(\\bb x,\\bb y)=\\inf_{\\sigma\\in S_Q}\\Big(\\sum_j^d|x_j-y_{\\sigma(j)}|^2\\Big)^{1/2},\n\\end{align*}\nwhere $S_Q$ is the permutation group on $Q$ elements acting naturally on $(\\mathbb{R}^n)^Q$. (See Section \\ref{sec:almgren-space} for more details.)\n\n\\bigskip\\noindent The link between the multi-valued inverse and Almgren space is the following simple observation based on local degree theory. Given a proper branched cover $f:\\Omega\\to \\mathbb{R}^n$ of degree $d>0$ from an open set $\\Omega\\subset \\mathbb{R}^n$, the local index $\\iota(f,x)$ is defined for all $x\\in\\Omega$ and satisfies $\\displaystyle \\sum_{x\\in f^{-1}(y)}\\iota(f,x)=d$, $y\\in f(\\Omega)$. Thus, $f^{-1} (y)$ can be regarded as an unordered $d$-tuple in $\\Omega$ (counted with multiplicities) for each $y\\in f(\\Omega)$. The multi-valued inverse $\\minv f$ of the proper branched cover $f:\\Omega\\to \\mathbb{R}^n$ of finite degree $d>0$ is the map\n\\begin{align}\\label{eq:minv}\n\\minv f:f(\\Omega)\\to \\mathcal A_d(\\mathbb{R}^n),\\quad \\minv f(y):= \\sum_{x\\in f^{-1}(y)}\\iota(f,x)\\bb x.\n\\end{align}\nWe refer the reader to Section \\ref{sec:branched-covers} (see in particular Definition \\ref{def:multi-valued-inv}) for the notation and further details used here.\n\n\\subsection{Statement of results}\n\nWhile it is not a manifold, $\\mathcal A_Q(\\mathbb{R}^n)$ nevertheless carries a natural $n$-form given by the trace of the volume form of $\\mathbb{R}^n$; that is, the $n$-form \n\\begin{align}\\label{eq:natural-n-form}\n\\operatorname{tr}(\\operatorname{vol}_{\\mathbb{R}^n}):=\\sum_j^d P_j^*(\\operatorname{vol}_{\\mathbb{R}^n})\n\\end{align}\non $(\\mathbb{R}^n)^Q$ (where $P_j$ is the projection to the $j$th factor) is $S_Q$-invariant: $\\sigma^*\\omega=\\omega$ for all $\\sigma\\in S_Q$. Using $S_Q$-invariance, we develop a pull-back theory of multi-valued maps (see Sections \\ref{sec:diff-forms} and \\ref{sec:pull-back}), which is of independent interest.\n\n\\begin{definition}\\label{def:multi-valued-inv}\nLet $f:X\\to Y$ be a proper branched cover of finite degree $d>0$ between topological $n$-manifolds. The multi-valued inverse $\\minv f:f(X)\\to \\mathcal A_d(X)$ of $f$ is defined by\n\\begin{align*}\n\\minv f(y)=\\sum_{x\\in f\\inv(y)}\\iota(f,x)\\bb x \n\\end{align*}\n\\end{definition}\n\n\\subsection{Statement of results}\n\nIn particular, if $f:U\\to \\mathcal A_Q(\\R^n)$ is a multi-valued (locally) Lipschitz map, the pull-back $f^*\\omega$ of any $S_Q$-invariant smooth $k$-form is a (locally)flat $k$-form on $U$ in the sense of Whitney (cf. Theorem \\ref{thm:lip-pullback-flat-form}).\n\nAn element $\\omega\\in \\Omega^k(\\mathcal A_d(V))$ is a smooth map $\\omega:V^d\\to \\bigwedge^k(\\R^n)^d$ with \n\\begin{align}\\label{eq:S_d-invariant}\n\\omega_x(v_1,\\ldots,v_k)=\\omega_{\\sigma(x)}(\\sigma v_1,\\ldots,\\sigma v_k),\\quad x\\in V^d,\\ v_1,\\ldots, v_k\\in (\\R^n)^d.\n\\end{align}\nIn particular, for each $\\bb{x}\\in \\mathcal A_d(V)$, the value $\\omega_{\\bb{x}}:=\\omega_{\\sigma(x)}\\circ \\sigma\\in \\bigwedge^k(\\R^n)^d$\nis well-defined and independent of $\\sigma\\in S_d$. We define the comass $\\|\\omega\\|:\\mathcal A_d(V)\\to \\R$ of a $k$-form $\\omega\\in \\Omega^k(\\mathcal A_d(V))$ by \n\\begin{align*}\n\\|\\omega\\|_{\\bb x}=\\|\\omega_{x}\\|,\\quad \\bb x\\in \\mathcal A_d(V).\n\\end{align*}\nThis is well-defined by \\eqref{eq:S_d-invariant}.\n\n\\begin{lemma}\\label{lem:pullback-est}\nSuppose $U\\subset \\R^m$ and $V\\subset \\R^n$ are open sets, $\\omega$ is a smooth $k$-form on $\\mathcal A_d(V)$ and $h:U\\to \\mathcal A_d(V)$ a.e. approximately differentiable. Then $h^*\\omega$ is a measurable $k$-form on $U$ and \n\\begin{align*}\n\\|h^*\\omega\\|_x\\le \\|D_xh\\|_x^k\\|\\omega_{h(x)}\\|\\quad\\textrm{a.e. }x\\in U.\n\\end{align*}\n\\end{lemma}\n\\begin{remark}\nIn particular $h^*\\omega\\in L^{p/k}_{loc}(U)$ if $h\\in W^{1,p}_{loc}(U,\\mathcal A_d(V))$.\n\\end{remark}\n\\begin{proof}\nThe approximate differentials $Dh_1,\\ldots,Dh_d$ are measurable, cf. ***. For all $x\\in U$ where $h$ is approximately differentiable, we have\n\\begin{align*}\n\\|h^*\\omega\\|_x&=\\max\\{\\omega_{h(x)}(Dh(v_1),\\ldots,Dh (v_d)):\\ |v_1|,\\ldots,|v_d|\\le 1\\}\\\\\n&\\le \\|D_xh\\|^k\\|\\omega_{h(x)}\\|,\n\\end{align*}\ncompleting the proof.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:pull-back-of-decomp}\nSuppose $f_i:U\\to \\mathcal A_{d_i}(V)$ is Lipschitz ($i=0,1$), $d=d_0+d_1$, and $\\omega$ is a $S_{d_0}\\times S_{d_1}$-invariant form on $V^d$. Then the pull-back $\\bb{f_0,f_1}^*\\omega$ is a well-defined $k$-form on $U$. Moreover, \n\\begin{align*}\n\\bb{f_0,f_1}^*(\\omega_0\\otimes\\omega_1)=f_0^*\\omega_0\\wedge f_1^*\\omega_1\n\\end{align*}\nwhenever $\\omega_i\\in \\Omega^{k_i}(\\mathcal A_{d_i}(V))$ ($i=0,1$) and $k=k_0+k_1$.\n\\end{lemma}\n\\begin{proof}\nLet $x\\in U$ be such that $f_0,f_1$ are differentiable at $x$, and let $f_i^1(x),\\ldots, f_i^{d_i}(x)$ and $D_xf_i^1,\\ldots,D_xf_i^{d_i}$ be as in Theorem \\ref{thm:multi-valued-rademacher} (i) and (ii). (In particular, note that $f=\\bb{f_0,f_1}$ is differentiable at $x$ and $Df=\\bb{D_xf_0^1,\\ldots,D_xf_0^{d_0},D_xf_1^1,\\ldots,D_xf_1^{d_1}}$.) Given $(\\sigma_0,\\sigma_1)\\in S_{d_0}\\times S_{d_1}$, denote\n\\[\nD_xf^{(\\sigma_0,\\sigma_1)}=(D_xf_0^{\\sigma_0\\inv(1)},\\ldots D_xf_0^{\\sigma_0\\inv(d_0)},D_xf_1^{\\sigma_1\\inv(1)},\\ldots,D_xf_1^{\\sigma_1\\inv(d_1)})\n\\]\nand similarly \n\\[\n(f_0^{\\sigma_0}(x),f_1^{\\sigma_1}(x))=(f_0^{\\sigma_0\\inv(1)}(x),\\ldots,f_0^{\\sigma_0\\inv(d_0)},f_1^{\\sigma_1\\inv(1)},\\ldots,f_1^{\\sigma_1\\inv(d_1)}).\n\\]\nWe define\n\\begin{align*}\n (f^*\\omega)_x(v_1,\\ldots,v_k)=\\omega_{(f_0^{\\sigma_0}(x),f_1^{\\sigma_1}(x))}(D_xf^{(\\sigma_0,\\sigma_1)}v_1,\\ldots,D_xf^{(\\sigma_0,\\sigma_1)}v_k)\n\\end{align*}\nfor $v_1,\\ldots,v_k\\in T_xU=\\R^m$. By the $S_{d_0}\\times S_{d_1}$-invariance of $\\omega$, this expression is independent of $(\\sigma_0,\\sigma_1)$ and gives a well-defined element in $\\bigwedge^k\\R^m$, which we denote by $(f^*\\omega)_x=\\omega_{(f_0(x),f_1(x))}\\circ(D_xf_0,D_xf_1)$.\n\n\\begin{proposition}\\label{prop:flat-form}\nLet $d_0,d_1\\ge 1$ be natural numbers, $d:=d_0+d_1$, and let $U\\subset \\R^m$ be an open set. If $f_i:U\\to \\mathcal A_{d_i}(\\R^n)$ are Lipschitz functions such that $\\ud(f_i^*\\omega)=f_i^*(\\ud\\omega)$ weakly for all $l$-forms $\\omega\\in \\Omega^l(\\mathcal A_{d_i}(\\R^n))$, $l\\le k$ ($i=0,1$), then $\\ud(\\bb{f_0,f_1}^*\\omega)=\\bb{f_0,f_1}^*(\\ud\\omega)$ weakly for all $k$-forms $\\omega\\in \\Omega^k(\\mathcal A_d(\\R^n))$.\n\\end{proposition}\n\\begin{proof}\nDenote $f:=\\bb{f_0,f_1}$, and let $\\omega\\in \\Omega^k(\\mathcal A_d(\\R^n))$. Let \n\\[\n(\\omega_j)\\subset \\bigcup_{r+l=k}\\Omega^l((\\R^n)^{d_0})\\otimes \\Omega^r((\\R^n)^{d_1})\n\\]\nbe a sequence such that $(\\omega_j)_x\\to \\omega_x$ and $(\\ud\\omega_j)_x\\to (\\ud\\omega)_x$ for all $x\\in (\\R^n)^d$ and $\\sup_j\\max\\{\\|\\omega_j\\|_\\infty,\\|\\ud\\omega_j\\|_\\infty\\}<\\infty$. Then\n\\begin{align*}\n\\tilde\\omega_j&:=P_{S_{d_0}\\times S_{d_1}}\\omega_j\\to P_{S_{d_0}\\times S_{d_1}}\\omega=\\omega,\\\\ \n\\ud\\tilde\\omega_j&=P_{S_{d_0}\\times S_{d_1}}(\\ud\\omega_j)\\to P_{S_{d_0}\\times S_{d_1}}(\\ud\\omega)=\\ud\\omega\n\\end{align*}\nand $\\sup_j\\max\\{\\|\\tilde \\omega_j\\|_\\infty,\\|\\ud\\tilde \\omega_j\\|_\\infty\\}<\\infty$, cf. Lemma \\ref{lem:proj-onto-invariant-forms}. By Lemma \\ref{lem:pull-back-of-decomp} each $\\tilde\\omega_j$ is a sum of tensor products $\\omega_0\\otimes\\omega_1$ where $\\omega_i$ is $S_{d_i}$-invariant and $f^*(\\omega_0\\otimes\\omega_1)=(f_0^*\\omega_0)\\wedge(f^*_1\\omega_1)$. Together with the assumption that $f_i^*(\\ud\\omega_i)=\\ud(f^*_i\\omega)$ weakly for $S_{d_i}$-invariant forms $\\omega$, this yields \n\\begin{align*}\nf^*(\\ud(\\omega_0\\otimes\\omega_1))&=f^*(\\ud\\omega_0\\otimes \\omega_1+(-1)^{\\deg\\omega_0}\\omega_0\\otimes\\ud\\omega_1)\\\\\n&=f^*_0(\\ud\\omega_0)\\wedge(f^*_1\\omega_1)+(-1)^{\\deg\\omega_0}(f_0^*\\omega_0)\\wedge (f_1^*(\\ud\\omega_1))\\\\\n&=\\ud(f_0^*\\omega_0)\\wedge(f_1^*\\omega_1)+(-1)^{\\deg\\omega_0}(f_0^*\\omega_0)\\wedge\\ud(f_1^*\\omega_1)\\\\\n&=\\ud(f_0^*\\omega_0\\wedge f_1^*\\omega_1)=\\ud f^*(\\omega_0\\otimes\\omega_1)\n\\end{align*}\nweakly. Consequently $f^*(\\ud\\tilde\\omega_j)=\\ud(f^*\\tilde\\omega_j)$ weakly. Now the pointwise convergence $f^*\\tilde\\omega_j\\to f^*\\omega$, $f^*(\\ud\\tilde\\omega_j)\\to f^*(\\ud\\omega)$ together with the dominated convergence theorem yield\n\\begin{align*}\n\\int_U \\alpha\\wedge f^*(\\ud\\omega)&=\\lim_{j\\to\\infty}\\int_U \\alpha\\wedge f^*(\\ud\\tilde\\omega_j)=(-1)^{k+1}\\lim_{j\\to\\infty}\\int_U\\ud\\alpha\\wedge f^*\\tilde\\omega_j\\\\\n&=(-1)^{k+1}\\int_U\\ud\\alpha\\wedge f^*\\omega\n\\end{align*}\nfor all $\\alpha\\in \\Omega_c^{m-k-1}(U)$. This proves the claim. \n\\end{proof}\n\n\\begin{proof}\nDenote $H(y):=(f_\\ast(|Df|^{-2}))^{1/2}$ and observe that \n\\[\n\\lim_{h\\to 0}\\frac{d_\\mathcal A(\\minv f(y+hv),\\minv f(y))}{|h|}=\\md_y\\minv f(v) \\quad\\textrm{ a.e. }y\\in f(\\Omega)\n\\]\nfor each $v\\in \\mathbb S^{n-1}$. By considering the family $\\Gamma_v=\\{\\gamma\\subset f(\\Omega): \\gamma'=v\\}$, it follows from a Fubini type argument and Theorem \\ref{thm:wug} that \n\\begin{align}\\label{eq:metric-diff-est}\nH(y)\\le \\md_y\\minv f(v)\\le (K_IK_O)^{1/n}H(y)\\quad \\textrm{ for all } v\\in \\mathbb S^{n-1}.\n\\end{align}\nThis implies \\eqref{eq:jacob-vs-H}. The remaining claim follows from \\eqref{eq:metric-diff-est}. Let $\\Gamma$ be a path family in $f(\\Omega)$ and let $\\rho$ be admissible for $\\minv f(\\Gamma)$. If $\\Gamma_0$ is the exceptional family in Theorem \\ref{thm:wug} then $\\Mod_v\\Gamma_0=0$ and, for any $\\gamma\\in \\Gamma\\setminus \\Gamma_1$ we have \n\\begin{align*}\n1\\le \\int_{\\minv f\\circ\\gamma}\\rho\\ud s&=\\int_0^1\\rho(\\minv f(\\gamma_t))|(\\minv f\\circ\\gamma)_t'|\\ud t\\le (K_IK_O)^{1/n}\\int_0^1\\rho\\circ \\minv f(\\gamma_t)H(\\gamma_t)|\\gamma_t'|\\ud t\\\\\n&=(K_IK_O)^{1/n}\\int_\\gamma(\\rho\\circ \\minv f)\\cdot H\\ud s.\n\\end{align*}\nThus $(K_IK_O)^{1/n}(\\rho\\circ \\minv f)H$ is admissible for $\\Gamma\\setminus \\Gamma_0$ implying\n\\begin{align*}\n\\Mod_n\\Gamma&=\\Mod_n\\Gamma\\setminus\\Gamma_0\\le K_IK_O\\int_{f(\\Omega)}\\rho^n\\circ \\minv f H^n\\ud y\\\\\n&\\le K_IK_O\\int_{f(\\Omega)}\\rho^n\\circ \\minv f\\J \\minv f\\ud y=K_IK_O\\int_{\\Omega_f}\\rho^n\\ud\\Ha^n.\n\\end{align*}\nTaking infimum over $\\rho$ yields $\\Mod_n\\Gamma\\le K_IK_O\\Mod_n\\minv f(\\Gamma)$ and completes the proof.\n\\end{proof}", + "post_theorem_intro_text_len": 5908, + "post_theorem_intro_text": "In particular, if $f:U\\to \\mathcal A_Q(\\mathbb{R}^n)$ is a multi-valued (locally) Lipschitz map, the pull-back $f^*\\omega$ of any $S_Q$-invariant smooth $k$-form is a (locally)flat $k$-form on $U$ in the sense of Whitney (cf. Theorem \\ref{thm:lip-pullback-flat-form}). \n\nApplying the definition of pull-back in the QR context we can show that, if $f:\\Omega\\to \\mathbb{R}^n$ is a quasiregular map, the multi-valued inverse is a quasiregular curve.\n\n\\begin{theorem}\\label{thm:QR-curve}\nLet $f:\\Omega\\to \\mathbb{R}^n$ be a proper quasiregular map of finite degree $d$. Then the multivalued inverse $\\accentset{\\leftarrow}{f}:f(\\Omega)\\to \\mathcal A_d(\\Omega)$ is an $\\omega$-quasiregular curve, where $\\omega=\\operatorname{tr}(\\operatorname{vol}_{\\mathbb{R}^n})$ is given by \\eqref{eq:natural-n-form}. More precisely, $\\minv f\\in W_{loc}^{1,n}(f(\\Omega),\\mathcal A_d(\\mathbb{R}^n))$, and \n\\[\n\\|\\omega\\|\\circ \\minv f\\ |D\\minv f|^n\\le d^{n/2-1}K_I(f)\\star \\minv f^*\\omega\\quad \\textrm{almost everywhere on }\\Omega.\n\\]\n\\end{theorem}\n\nTheorems \\ref{thm:sob-pull-back} and \\ref{thm:QR-curve} together with standard arguments found e.g. in \\cite{pan-onn21} yield in particular the higher integrability of the generalized inverse of \\cite[Chapter II.5]{ric93}. \n\\begin{corollary}\\label{cor:QR-curve}\nLet $f:\\Omega\\to \\mathbb{R}^n$ be a proper quasiregular map of finite degree $d$. Then there exists $p>n$ such that the generalized inverse \n\\[\ng:f(\\Omega)\\to \\mathbb{R}^n,\\quad g(y):=\\sum_{x\\in f^{-1}(y)}\\iota(f,x)x\n\\]\nbelongs to $W^{1,p}_{loc}(f(\\Omega),\\mathbb{R}^n)$.\n\\end{corollary}\n\nThe multi-valued inverse can also be seen as a homeomorphism $\\minv f:f(\\Omega)\\to \\Omega_f$ onto its image $\\Omega_f:=\\minv f(f(\\Omega))\\subset \\mathcal A_d(\\mathbb{R}^n)$ (Lemma \\ref{lem:homeo}). Thus, although $\\mathcal A_d(\\mathbb{R}^n)$ is not a manifold, $\\Omega_f$ is a metric $n$-manifold when equipped with the metric from $\\mathcal A_d(\\mathbb{R}^n)$. In the following theorem we obtain that $\\minv f:f(\\Omega)\\to \\Omega_f$ is geometrically quasiconformal.\n\n\\begin{theorem}\\label{thm:geom-QC}\nLet $f:\\Omega\\to \\mathbb{R}^n$ be a proper quasiregular map of finite degree $d$, and equip $\\Omega_f=\\minv f(f(\\Omega))$ with the metric from $\\mathcal A_d(\\mathbb{R}^n)$ and the Hausdorff $n$-measure on $\\Omega_f$. \n\\begin{itemize}\n \\item[(1)] $\\Omega_f$ is $n$-rectifiable, upper Ahlfors $n$-regular and satisfies the infinitesimal $n$-Poincar\\'e inequality;\n \\item[(2)] $\\minv f:f(\\Omega)\\to \\Omega_f$ is geometrically quasiconformal. More precisely, \n\\begin{align*}\n \\frac 1{K_IK_O}\\Mod_n\\Gamma\\le \\Mod_n\\minv f(\\Gamma)\\le K_IK_O\\Mod_n\\Gamma\n\\end{align*}\nfir any path family $\\Gamma $ in $f(\\Omega)$.\n\\end{itemize}\n\\end{theorem}\n\nWe refer to \\eqref{eq:inf-n-PI} in Section \\ref{sec:Omega_f-prop} for the definition of infinitesimal Poincar\\'e inequality and further commentary. We mention here that the infinitesimal Poincar\\'e inequality is not a quantitative condition.\n\n\\subsection{Main ideas and further discussion}\nTheorem \\ref{thm:sob-pull-back} follows by an approximation argument from it Lipschitz counterpart, Theorem \\ref{thm:lip-pullback-flat-form}, whose proof uses induction, decomposition of multi-valued maps (cf. \\cite[Proposition 1.6]{del11}), and a somewhat delicate approximation scheme. We outline the main idea.\n\nIn order to apply induction in the proof of Theorem \\ref{thm:lip-pullback-flat-form}, we will need to approximate a given multi-valued Lipschitz map by maps which are locally decomposable. A given map $f:U\\to \\mathcal A_d(\\mathbb{R}^n)$ fails to be locally decomposable on the set $F:=f^{-1}(\\operatorname{diag}\\mathcal A_d(\\mathbb{R}^n))$. The restriction of $f$ to $F$ agrees with the map $d\\llbracket b(f)\\rrbracket$, where $b:\\mathcal A_d(\\mathbb{R}^n)\\to \\mathbb{R}^n$ is the barycenter map\n\\begin{align}\\label{eq:barycenter}\nb(\\llbracket x_1,\\ldots,x_d\\rrbracket)=\\frac{x_1+\\cdots+x_d}{d},\n\\end{align}\nwhich is $\\frac 1{\\sqrt d}$-Lipschitz, see Lemma \\ref{lem:barycenter} below. The equality $f|_F=d\\llbracket b(f)\\rrbracket|_F$ implies that the differentials of the two functions agree almost everywhere. However, since $F$ is closed and not open, we need an approximation argument for the induction step to work. Our approximation scheme consists of a careful interpolation between $f$ and $d\\llbracket b(f)\\rrbracket$ in a neighbourhood of $F$.\n\nThe main step in the proof of Theorem \\ref{thm:QR-curve} are the Sobolev estimates, which we obtain in Theorem \\ref{thm:wug}. Our argument here uses Vitali's covering theorem, which is valid in high generality, and thus Theorem \\ref{thm:wug} probably holds e.g. in the setting of \\cite{onn-raj09}.\n\nThe proof of Theorem \\ref{thm:geom-QC} employs a similar induction argument as in Theorem \\ref{thm:lip-pullback-flat-form} together with some metric quasiconformal theory. We remark here that a large part of metric quasiconformal theory is based on the Ahlfors regularity of the spaces under consideration. Indeed, in our setting this property would imply the Loewner property of the image set $\\Omega_f:=\\minv f(f(\\Omega)$ by a (deep) result of Semmes \\cite{sem96}, which in turn implies the equivalence of the various notions of quasiconformality in the metric setting \\cite{hei98}. In this paper, we establish the upper Ahlfors $n$-regularity of $\\Omega_f$ (Proposition \\ref{prop:ball-meas-upper-bound}), and an infinitesimal Poincar\\'e inequality. It would be interesting to analyze the geometry of $\\Omega_f$ further, in particular whether $\\Omega_f$ admits local bi-Lipschitz parametrizations. By Almgren's bi-Lipschitz embedding $\\mathcal A_d(\\mathbb{R}^n)\\hookrightarrow \\mathbb{R}^N$, $\\Omega_f$ can be regarded as a subset of Euclidean space, and the existence of bi-Lipschitz parametrizations is linked with the existence of Cartan--Whitney presentations in a suitable Sobolev class \\cite[[Theorem 1.2]{hei11}.", + "sketch": "Theorem~\\ref{thm:sob-pull-back} \"follows by an approximation argument from its Lipschitz counterpart, Theorem~\\ref{thm:lip-pullback-flat-form}, whose proof uses induction, decomposition of multi-valued maps (cf. \\cite[Proposition 1.6]{del11}), and a somewhat delicate approximation scheme.\" The key issue for the induction is that one must \"approximate a given multi-valued Lipschitz map by maps which are locally decomposable.\" A Lipschitz map $f:U\\to \\mathcal A_d(\\mathbb{R}^n)$ \"fails to be locally decomposable on the set $F:=f^{-1}(\\operatorname{diag}\\mathcal A_d(\\mathbb{R}^n))$.\" On $F$, one has that $f$ \"agrees with the map $d\\llbracket b(f)\\rrbracket$, where $b$ is the barycenter map\" $b(\\llbracket x_1,\\ldots,x_d\\rrbracket)=\\frac{x_1+\\cdots+x_d}{d}$, and the equality $f|_F=d\\llbracket b(f)\\rrbracket|_F$ implies \"the differentials of the two functions agree almost everywhere.\" Since $F$ is \"closed and not open,\" an approximation is needed: \"Our approximation scheme consists of a careful interpolation between $f$ and $d\\llbracket b(f)\\rrbracket$ in a neighbourhood of $F$.\" This Lipschitz-level result is then used, via approximation, to obtain the Sobolev pull-back statement of Theorem~\\ref{thm:sob-pull-back}.", + "expanded_sketch": "To prove the main theorem, we proceed by an approximation argument from its Lipschitz counterpart. We first prove the following theorem.\n\n\\begin{theorem}\\label{thm:lip-pullback-flat-form}\nSuppose $U\\subset\\R^m$ is an open set, $f:U\\to\\mathcal A_d(\\R^n)$ is Lipschitz and $\\omega\\in \\Omega^k(\\mathcal A_d(\\R^n))$. Then $\\ud(f^*\\omega)=f^*(\\ud\\omega)$ weakly. In particular, $f^*\\omega$ is a flat form on $U$ in the sense of Whitney. \n\\end{theorem}\n\nIts proof uses induction, decomposition of multi-valued maps (cf. \\cite[Proposition 1.6]{del11}), and a somewhat delicate approximation scheme. The key issue for the induction is that one must \"approximate a given multi-valued Lipschitz map by maps which are locally decomposable.\" A Lipschitz map $f:U\\to \\mathcal A_d(\\mathbb{R}^n)$ \"fails to be locally decomposable on the set $F:=f^{-1}(\\operatorname{diag}\\mathcal A_d(\\mathbb{R}^n))$.\" On $F$, one has that $f$ \"agrees with the map $d\\llbracket b(f)\\rrbracket$, where $b$ is the barycenter map\" $b(\\llbracket x_1,\\ldots,x_d\\rrbracket)=\\frac{x_1+\\cdots+x_d}{d}$, and the equality $f|_F=d\\llbracket b(f)\\rrbracket|_F$ implies \"the differentials of the two functions agree almost everywhere.\" Since $F$ is \"closed and not open,\" an approximation is needed: \"Our approximation scheme consists of a careful interpolation between $f$ and $d\\llbracket b(f)\\rrbracket$ in a neighbourhood of $F$.\" This Lipschitz-level result is then used, via approximation, in establishing the main theorem.", + "expanded_theorem": "\\label{thm:sob-pull-back}\nLet $n,m,Q\\ge 1$ be natural numbers, $U\\subset \\mathbb{R}^m$ open, and $f\\in W_{loc}^{1,p}(f(\\Omega),\\mathcal A_Q(\\mathbb{R}^n))$. If $\\omega$ is a $S_Q$-invariant smooth $k$-form on $(\\mathbb{R}^n)^Q$, and either $k+1\\le p$ of $\\mathrm {d} \\omega=0$ and $k\\le p$, then $\\omega$ has a well-defined pull-back $f^*\\omega \\in L^{p/k}_{loc}(U,\\bigwedge^k\\mathbb{R}^m)$ and moreover\n\\begin{align*}\n \\mathrm {d}(f^*\\omega)=f^*(\\mathrm {d}\\omega)\n\\end{align*}\nweakly.,", + "theorem_type": [ + "Implication", + "Existence" + ], + "mcq": { + "question": "Let $n,m,Q\\ge 1$ be natural numbers, let $U\\subset \\mathbb{R}^m$ be open, and let $f\\in W^{1,p}_{\\mathrm{loc}}(U,\\mathcal A_Q(\\mathbb{R}^n))$, where $\\mathcal A_Q(\\mathbb{R}^n)=(\\mathbb{R}^n)^Q/S_Q$ is the Almgren space of unordered $Q$-tuples and $S_Q$ acts on $(\\mathbb{R}^n)^Q$ by permuting the $Q$ factors. Let $\\omega$ be a smooth $k$-form on $(\\mathbb{R}^n)^Q$ that is $S_Q$-invariant, meaning $\\sigma^*\\omega=\\omega$ for every permutation $\\sigma\\in S_Q$. Assume either (i) $k+1\\le p$, or (ii) $\\mathrm d\\omega=0$ and $k\\le p$. Under these hypotheses, which conclusion about the pull-back of $\\omega$ by $f$ is valid?", + "correct_choice": { + "label": "A", + "text": "The pull-back $f^*\\omega$ is well defined and belongs to $L^{p/k}_{\\mathrm{loc}}(U,\\bigwedge^k\\mathbb{R}^m)$; moreover its weak exterior derivative satisfies\n\\[\n\\mathrm d(f^*\\omega)=f^*(\\mathrm d\\omega)\n\\]\nweakly on $U$." + }, + "choices": [ + { + "label": "B", + "text": "The pull-back $f^*\\omega$ is well defined and belongs to $L^{p/k}_{\\mathrm{loc}}(U,\\bigwedge^k\\mathbb{R}^m)$ whenever $k\\le p$; moreover one always has\n\\[\n\\mathrm d(f^*\\omega)=f^*(\\mathrm d\\omega)\n\\]\nweakly on $U$, without any additional assumption such as $k+1\\le p$ or $\\mathrm d\\omega=0$." + }, + { + "label": "C", + "text": "The pull-back $f^*\\omega$ is well defined and belongs to $L^{p/k}_{\\mathrm{loc}}(U,\\bigwedge^k\\mathbb{R}^m)$ under the stated hypotheses." + }, + { + "label": "D", + "text": "The pull-back $f^*\\omega$ is well defined and belongs to $L^{p/k}_{\\mathrm{loc}}(U,\\bigwedge^k\\mathbb{R}^m)$; moreover, if $\\mathrm d\\omega=0$ and $k\\le p$, then\n\\[\n\\mathrm d(f^*\\omega)=0\n\\]\nweakly on $U$, whereas the identity \\(\\mathrm d(f^*\\omega)=f^*(\\mathrm d\\omega)\\) is guaranteed only in the case $k+1\\le p$." + }, + { + "label": "E", + "text": "For every smooth $k$-form $\\omega$ on $(\\mathbb{R}^n)^Q$, the pull-back $f^*\\omega$ is well defined and belongs to $L^{p/k}_{\\mathrm{loc}}(U,\\bigwedge^k\\mathbb{R}^m)$; if in addition $\\omega$ is $S_Q$-invariant, then one has\n\\[\n\\mathrm d(f^*\\omega)=f^*(\\mathrm d\\omega)\n\\]\nweakly on $U$ under the stated hypotheses." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "integrability threshold k+1<=p versus merely k<=p for non-closed forms", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "differential identity conclusion dropped", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "closed-form case still yields full chain-rule identity, not just vanishing derivative", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "S_Q-invariance is required for the pull-back itself to be well defined on Almgren space", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly state the conclusion. It gives hypotheses and asks for the valid consequence, so the correct answer is not leaked directly." + }, + "TAS": { + "score": 1, + "justification": "The item is quite close to a theorem-recall question: the correct option essentially states the theorem under the given hypotheses. However, it is not purely tautological because the distractors alter thresholds, invariance requirements, and the derivative conclusion." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish the exact chain-rule conclusion from weaker or overgeneralized variants, especially around the conditions k+1<=p versus dω=0 and k<=p. Still, the question mainly tests precise recall of a theorem rather than substantial derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target common mistakes: overextending regularity assumptions, omitting the differential identity, confusing the closed-form case, and ignoring the need for S_Q-invariance for well-definedness on Almgren space." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise theorem recall than deep generative reasoning." + } + }, + { + "id": "2511.17325v1", + "paper_link": "http://arxiv.org/abs/2511.17325v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "theorem", + "content": "\\label{1bdry C^s}\n Suppose $\\Omega$ is a locally $C^{1,1}$ domain, $00.\\]\nChoose $R$ sufficiently large, so that $\\Omega\\subset B_R(0).$ Denote\n\\[\\bar{u}(x):=\\norm{g}_{L^{\\infty}(\\Omega)}\\frac{R^{2s}}{a}\\psi_R(x), ~\\mbox{with}~ \\psi_R(x):={\\psi(\\frac{x}{R})}.\\]\nThen it can be easily verified that $\\bar{u}$ and $-\\bar{u}$ are super-solution and sub-solution of \\eqref{eq1}, respectively. Therefore, \n\\begin{equation*}\n \\norm{u}_{L^{\\infty}(\\Omega)}\\le C\\norm{g}_{L^\\infty(\\Omega)},\n\\end{equation*}\nFrom the above construction, one can see that in order to obtain \\eqref{hd}, it is necessary that $\\Omega$ be bounded.\nWhen $\\Omega$ is an {\\em unbounded domain}, it is evident that the solution $u$ is required to be globally bounded. This requirement can not be fulfilled in the process of employing the blow-up and rescaling argument to obtain a priori estimates for solutions to a corresponding family of nonlinear fractional equations on {\\em unbounded domains with boundaries}.\n\nThis motivates us to establish a local version of the boundary regularity, in which, instesd of global one, only a local ${L^{\\infty}}$ norm of the solution is involved.\n\\begin{theorem}\\label{bdry C^s}\n Suppose $\\Omega$ is a unbounded domain with locally $C^{1,1}$ boundary, $00$ such that $B_\\varepsilon(x)\\subset \\Omega$. Then\n \\begin{align*}\n A_i(x,R) & = c_{n,s}\\int _{B_R\\backslash B_\\varepsilon(x)}\\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy \\\\\n &+c_{n,s}\\int _{B_\\varepsilon(x)}\\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy\n \\end{align*}\n By dominated convergence theorem, the first term converges to 0 as $i\\to\\infty$. For the second term,\n \\[\n \\lim _{i\\to\\infty} c_{n,s}\\int _{B_\\varepsilon(x)}\\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy\\le c_{n,s}\\lim _{i\\to\\infty}[u-u_i]_{C^{2s+\\beta}(B_\\varepsilon(x))}\\varepsilon ^\\beta = 0.\n \\]\n Hence $\\lim _{i\\to\\infty}A_i(x,R)=0$. Therefore \n \\begin{equation}\n \\lim _{R\\to\\infty}\\lim _{i\\to\\infty}A_i(x,R)=0.\n \\end{equation}\n Then the same argument in \\cite{Du2023blowup} implies $\\lim _{i\\to\\infty}F_i(x,R)$ exists and\n \\[\n \\lim _{R\\to\\infty}\\lim _{i\\to\\infty}F_i(x,R) =\\lim _{R\\to\\infty}\\lim _{i\\to\\infty}F_i(0,R) = :b\\ge 0.\n \\]\n\\end{proof}", + "post_theorem_intro_text_len": 3811, + "post_theorem_intro_text": "The idea of the proof is that we divide a given solution into two parts: the potential part and the harmonic part. The regularity for the potential part is obtained by Proposition \\ref{otonserra}. For the harmonic part $h$, we rewrite it in terms of the Poisson representation formula in balls. Using this explicit expression, we first carry out a detailed analysis to derive an $\\alpha$ power order decay near $\\partial\\Omega$,\n\\begin{equation*}\n |h(x)|\\le C\\{\\lVertu\\rVert_{L^\\infty(\\Omega \\cap B_4)}+\\lVertu\\rVert_{\\cL_{2s}}\\}\\operatorname{dist}(x,\\partial \\Omega)^{\\alpha },\n \\end{equation*}\n where $\\alpha = \\min\\{s,1-s\\}$. Combining this decay estimate and the interior regularity result (Theorem \\ref{Holder thm}), we derive the $C^\\alpha$ boundary regularity. Then by an iteration process, we increase the power $\\alpha$ successively until it reaches the desired power $s$.\n\nAs an important application, we establish a priori estimate for the solutions to \\eqref{main2}. We assume that $\\Omega \\subset \\mathbb{R} ^n$ is an unbounded domain with uniformly $C^{1,1}$ boundary, and $f$ satisfies the following condition:\n\\begin{itemize}\n \\item $f(x,t):\\Omega \\times [0,\\infty)\\to\\mathbb{R}$ is uniformly H\\\"older continuous with respect to $x$ and continuous with respect to $t$.\n\\end{itemize}\n\\begin{theorem}\\label{1A1}\nAssume $1}[rr]\\ar[rd]_-{\\pi_1} & & \\mathcal{X}\\setminus \\mathcal{B} \\ar[ld] \\\\\n& \\mathbb{D}.& \n}\n\\]\nThe restriction of the previous embedding to the general fiber recovers $j_0$\nwhile the restriction to the central fiber \nis just the embedding of the open torus into the toric variety $\\pp(1,1,4)$.\n}\n\\end{example}", + "def:ct": "\\begin{definition}\\label{def:ct}\n{\\em \nLet $(X,B)$ be log Calabi--Yau pair of dimension $n$. \nWe say that $(X,B)$ is of {\\em cluster type}\nor that $(X,B)$ is a {\\em cluster type pair} if there exists an embedding \nin codimension one \n\\[\n\\mathbb{G}_m^n \\dashrightarrow X\\setminus B, \n\\]\ni.e., there exists a closed subset of codimension at least two $Z\\subset \\mathbb{G}_m^n$ \nand an embedding $\\mathbb{G}_m^n\\setminus Z \\hookrightarrow X\\setminus B$.\n}\n\\end{definition}" + }, + "pre_theorem_intro_text_len": 4534, + "pre_theorem_intro_text": "In this article, we study degenerations of varieties and pairs. \nWe focus on degenerations with central fiber having klt singularities.\nThese are known as {\\em klt degenerations}. \nThe problem is particularly interesting when the general fiber $\\mathcal{X}_t$ of the degeneration $\\mathcal{X}\\rightarrow \\mathbb{D}$ is a Fano variety.\nIn this case, there are many possible degenerations $\\mathcal{X}_0$ \nand these degenerations are interesting even from the combinatorial perspective. \nFor instance, it is known that the toric degenerations of $\\mathbb{P}^2$ are given by \n\\begin{equation}\\label{eq:markov-triple}\n\\mathbb{P}(a^2,b^2,c^2) \\text{ with $a^2+b^2+c^2=3abc$.}\n\\end{equation} \nThese triples $(a,b,c)\\in \\zz_{\\geq 1}^3$ are known as {\\em Markov triples}. \nThe topic of degenerations of projective surfaces is classic in algebraic geometry. \nIn~\\cite{HP05}, Hacking and Prokhorov proved that any degeneration of $\\mathbb{P}^2$ is indeed a partial smoothing of a toric surface as in~\\eqref{eq:markov-triple}.\nThese surfaces were studied by Manetti and are nowadays known as \n{\\em Manetti surfaces} (see~\\cite{Man91}).\nIn~\\cite{HP10}, Hacking and Prokhorov classify del Pezzo surfaces, with quotient singularities and Picard rank one, admitting $\\mathbb{Q}$-Gorenstein smoothings. \nIn~\\cite{Pro19}, Prokhorov studied log canonical degenerations with Picard rank one of del Pezzo surfaces in $\\mathbb{Q}$-Gorenstein families.\nIn~\\cite{UZ24}, Urzua and the second author explain how to relate \nthe degenerations as in~\\eqref{eq:markov-triple} via birational transformations. \nGiven two degenerations $\\mathcal{X} \\rightarrow \\mathbb{D}$ \nand $\\mathcal{Y}\\rightarrow \\mathbb{D}$ of $\\mathbb{P}^2$ into \n$\\mathcal{X}_0 \\simeq \\mathbb{P}(a,b,c)$\nand $\\mathcal{Y}_0\\simeq \\mathbb{P}(d,e,f)$, \nthe authors explain how to perform small birational modifications to go from \n$\\mathcal{X}\\rightarrow \\mathbb{D}$ to $\\mathcal{Y}\\rightarrow \\mathbb{D}$\nand how these small modifications reflect on the combinatorics of the Markov triples.\nIn higher dimensions, H\\\"oring and Peternell proved that a klt degeneration $X$ of $\\mathbb{P}^n$ is indeed isomorphic to the $n$-dimensional projective space whenever $T_X$ is semistable (see~\\cite{HP24}).\nThey also classify normal degenerations of $\\mathbb{P}^3$ with canonical singularities.\nIn~\\cite{HKW25}, Hausen, Kir\\'aly, and Wrobel explicitly determine all log terminal,\nrational, degenerations of $\\mathbb{P}^2$ that admit non-trivial torus actions.\nIn~\\cite{UZ25}, Urzua and the second author classified all Wahl singularities that appear\nin the degenerations of del Pezzo surfaces of degree $d$,\nextending the work of Manetti and Hacking-Prokhorov in degree $9$.\nRecently in~\\cite{Pen25}, Peng studied special $\\mathbb{G}_m$-degenerations of del Pezzo surfaces $X$\ninduced by log canonical places of pairs $(X,C)$ where $C$ is a nodal curve.\nPeng proved that the space of special valuations of $(X,C)$ is connected and admits a \npartition, which is locally finite, and each interval corresponds to a different $\\mathbb{G}_m$-degeneration of $X$. \n\nThroughout this work, we focus on understanding the degenerations of cluster type\nvarieties and cluster type pairs.\nA {\\em cluster type pair} is a pair $(X,B)$ with mild singularities\nfor which $K_X+B\\sim 0$ and there is an embedding in codimension one \n$\\mathbb{G}_m^n \\dashrightarrow X\\setminus B$ (see Definition~\\ref{def:ct}). \nA {\\em cluster type variety} is a variety $X$ that admits a cluster type pair $(X,B)$ structure. \nIn this case, we call $B$ a cluster type boundary. \nCluster type varieties and pairs can be thought of as a generalization of toric varieties and pairs. \nIndeed, any toric pair is a cluster type pair.\nHowever, the realm of cluster type pairs is much broader. \nA del Pezzo surface of degree $d\\geq 2$ as well as a general del Pezzo\nsurface of degree $d=1$ is cluster type (see, e.g.,~\\cite[Theorem 2.1 and Remark 2.2]{ALP23}). \nThus, all the examples discussed in the introduction are cluster type varieties.\nOur first aim is to understand degenerations of cluster type pairs \nfor which the limit of the cluster type boundary still has reasonable singularities. \nIn the next subsection, we prove that there are not many such degenerations\nfor toric pairs, but quite a lot for cluster type pairs.\n\n\\subsection{Degenerations of cluster type pairs}\n\nOur first theorem states that degenerations of toric pairs\nare finite quotients of toric pairs. We impose some conditions \non the singularities of the central fiber.", + "context": "In this article, we study degenerations of varieties and pairs. \nWe focus on degenerations with central fiber having klt singularities.\nThese are known as {\\em klt degenerations}. \nThe problem is particularly interesting when the general fiber $\\mathcal{X}_t$ of the degeneration $\\mathcal{X}\\rightarrow \\mathbb{D}$ is a Fano variety.\nIn this case, there are many possible degenerations $\\mathcal{X}_0$ \nand these degenerations are interesting even from the combinatorial perspective. \nFor instance, it is known that the toric degenerations of $\\mathbb{P}^2$ are given by \n\\begin{equation}\\label{eq:markov-triple}\n\\mathbb{P}(a^2,b^2,c^2) \\text{ with $a^2+b^2+c^2=3abc$.}\n\\end{equation} \nThese triples $(a,b,c)\\in \\zz_{\\geq 1}^3$ are known as {\\em Markov triples}. \nThe topic of degenerations of projective surfaces is classic in algebraic geometry. \nIn~\\cite{HP05}, Hacking and Prokhorov proved that any degeneration of $\\mathbb{P}^2$ is indeed a partial smoothing of a toric surface as in~\\eqref{eq:markov-triple}.\nThese surfaces were studied by Manetti and are nowadays known as \n{\\em Manetti surfaces} (see~\\cite{Man91}).\nIn~\\cite{HP10}, Hacking and Prokhorov classify del Pezzo surfaces, with quotient singularities and Picard rank one, admitting $\\mathbb{Q}$-Gorenstein smoothings. \nIn~\\cite{Pro19}, Prokhorov studied log canonical degenerations with Picard rank one of del Pezzo surfaces in $\\mathbb{Q}$-Gorenstein families.\nIn~\\cite{UZ24}, Urzua and the second author explain how to relate \nthe degenerations as in~\\eqref{eq:markov-triple} via birational transformations. \nGiven two degenerations $\\mathcal{X} \\rightarrow \\mathbb{D}$ \nand $\\mathcal{Y}\\rightarrow \\mathbb{D}$ of $\\mathbb{P}^2$ into \n$\\mathcal{X}_0 \\simeq \\mathbb{P}(a,b,c)$\nand $\\mathcal{Y}_0\\simeq \\mathbb{P}(d,e,f)$, \nthe authors explain how to perform small birational modifications to go from \n$\\mathcal{X}\\rightarrow \\mathbb{D}$ to $\\mathcal{Y}\\rightarrow \\mathbb{D}$\nand how these small modifications reflect on the combinatorics of the Markov triples.\nIn higher dimensions, H\\\"oring and Peternell proved that a klt degeneration $X$ of $\\mathbb{P}^n$ is indeed isomorphic to the $n$-dimensional projective space whenever $T_X$ is semistable (see~\\cite{HP24}).\nThey also classify normal degenerations of $\\mathbb{P}^3$ with canonical singularities.\nIn~\\cite{HKW25}, Hausen, Kir\\'aly, and Wrobel explicitly determine all log terminal,\nrational, degenerations of $\\mathbb{P}^2$ that admit non-trivial torus actions.\nIn~\\cite{UZ25}, Urzua and the second author classified all Wahl singularities that appear\nin the degenerations of del Pezzo surfaces of degree $d$,\nextending the work of Manetti and Hacking-Prokhorov in degree $9$.\nRecently in~\\cite{Pen25}, Peng studied special $\\mathbb{G}_m$-degenerations of del Pezzo surfaces $X$\ninduced by log canonical places of pairs $(X,C)$ where $C$ is a nodal curve.\nPeng proved that the space of special valuations of $(X,C)$ is connected and admits a \npartition, which is locally finite, and each interval corresponds to a different $\\mathbb{G}_m$-degeneration of $X$.\n\nThroughout this work, we focus on understanding the degenerations of cluster type\nvarieties and cluster type pairs.\nA {\\em cluster type pair} is a pair $(X,B)$ with mild singularities\nfor which $K_X+B\\sim 0$ and there is an embedding in codimension one \n$\\mathbb{G}_m^n \\dashrightarrow X\\setminus B$ (see Definition~\\ref{def:ct}). \nA {\\em cluster type variety} is a variety $X$ that admits a cluster type pair $(X,B)$ structure. \nIn this case, we call $B$ a cluster type boundary. \nCluster type varieties and pairs can be thought of as a generalization of toric varieties and pairs. \nIndeed, any toric pair is a cluster type pair.\nHowever, the realm of cluster type pairs is much broader. \nA del Pezzo surface of degree $d\\geq 2$ as well as a general del Pezzo\nsurface of degree $d=1$ is cluster type (see, e.g.,~\\cite[Theorem 2.1 and Remark 2.2]{ALP23}). \nThus, all the examples discussed in the introduction are cluster type varieties.\nOur first aim is to understand degenerations of cluster type pairs \nfor which the limit of the cluster type boundary still has reasonable singularities. \nIn the next subsection, we prove that there are not many such degenerations\nfor toric pairs, but quite a lot for cluster type pairs.\n\n\\subsection{Degenerations of cluster type pairs}\n\nOur first theorem states that degenerations of toric pairs\nare finite quotients of toric pairs. We impose some conditions \non the singularities of the central fiber.\n\n\\begin{definition}\\label{def:ct}\n{\\em \nLet $(X,B)$ be log Calabi--Yau pair of dimension $n$. \nWe say that $(X,B)$ is of {\\em cluster type}\nor that $(X,B)$ is a {\\em cluster type pair} if there exists an embedding \nin codimension one \n\\[\n\\mathbb{G}_m^n \\dashrightarrow X\\setminus B, \n\\]\ni.e., there exists a closed subset of codimension at least two $Z\\subset \\mathbb{G}_m^n$ \nand an embedding $\\mathbb{G}_m^n\\setminus Z \\hookrightarrow X\\setminus B$.\n}\n\\end{definition}", + "full_context": "In this article, we study degenerations of varieties and pairs. \nWe focus on degenerations with central fiber having klt singularities.\nThese are known as {\\em klt degenerations}. \nThe problem is particularly interesting when the general fiber $\\mathcal{X}_t$ of the degeneration $\\mathcal{X}\\rightarrow \\mathbb{D}$ is a Fano variety.\nIn this case, there are many possible degenerations $\\mathcal{X}_0$ \nand these degenerations are interesting even from the combinatorial perspective. \nFor instance, it is known that the toric degenerations of $\\mathbb{P}^2$ are given by \n\\begin{equation}\\label{eq:markov-triple}\n\\mathbb{P}(a^2,b^2,c^2) \\text{ with $a^2+b^2+c^2=3abc$.}\n\\end{equation} \nThese triples $(a,b,c)\\in \\zz_{\\geq 1}^3$ are known as {\\em Markov triples}. \nThe topic of degenerations of projective surfaces is classic in algebraic geometry. \nIn~\\cite{HP05}, Hacking and Prokhorov proved that any degeneration of $\\mathbb{P}^2$ is indeed a partial smoothing of a toric surface as in~\\eqref{eq:markov-triple}.\nThese surfaces were studied by Manetti and are nowadays known as \n{\\em Manetti surfaces} (see~\\cite{Man91}).\nIn~\\cite{HP10}, Hacking and Prokhorov classify del Pezzo surfaces, with quotient singularities and Picard rank one, admitting $\\mathbb{Q}$-Gorenstein smoothings. \nIn~\\cite{Pro19}, Prokhorov studied log canonical degenerations with Picard rank one of del Pezzo surfaces in $\\mathbb{Q}$-Gorenstein families.\nIn~\\cite{UZ24}, Urzua and the second author explain how to relate \nthe degenerations as in~\\eqref{eq:markov-triple} via birational transformations. \nGiven two degenerations $\\mathcal{X} \\rightarrow \\mathbb{D}$ \nand $\\mathcal{Y}\\rightarrow \\mathbb{D}$ of $\\mathbb{P}^2$ into \n$\\mathcal{X}_0 \\simeq \\mathbb{P}(a,b,c)$\nand $\\mathcal{Y}_0\\simeq \\mathbb{P}(d,e,f)$, \nthe authors explain how to perform small birational modifications to go from \n$\\mathcal{X}\\rightarrow \\mathbb{D}$ to $\\mathcal{Y}\\rightarrow \\mathbb{D}$\nand how these small modifications reflect on the combinatorics of the Markov triples.\nIn higher dimensions, H\\\"oring and Peternell proved that a klt degeneration $X$ of $\\mathbb{P}^n$ is indeed isomorphic to the $n$-dimensional projective space whenever $T_X$ is semistable (see~\\cite{HP24}).\nThey also classify normal degenerations of $\\mathbb{P}^3$ with canonical singularities.\nIn~\\cite{HKW25}, Hausen, Kir\\'aly, and Wrobel explicitly determine all log terminal,\nrational, degenerations of $\\mathbb{P}^2$ that admit non-trivial torus actions.\nIn~\\cite{UZ25}, Urzua and the second author classified all Wahl singularities that appear\nin the degenerations of del Pezzo surfaces of degree $d$,\nextending the work of Manetti and Hacking-Prokhorov in degree $9$.\nRecently in~\\cite{Pen25}, Peng studied special $\\mathbb{G}_m$-degenerations of del Pezzo surfaces $X$\ninduced by log canonical places of pairs $(X,C)$ where $C$ is a nodal curve.\nPeng proved that the space of special valuations of $(X,C)$ is connected and admits a \npartition, which is locally finite, and each interval corresponds to a different $\\mathbb{G}_m$-degeneration of $X$.\n\nThroughout this work, we focus on understanding the degenerations of cluster type\nvarieties and cluster type pairs.\nA {\\em cluster type pair} is a pair $(X,B)$ with mild singularities\nfor which $K_X+B\\sim 0$ and there is an embedding in codimension one \n$\\mathbb{G}_m^n \\dashrightarrow X\\setminus B$ (see Definition~\\ref{def:ct}). \nA {\\em cluster type variety} is a variety $X$ that admits a cluster type pair $(X,B)$ structure. \nIn this case, we call $B$ a cluster type boundary. \nCluster type varieties and pairs can be thought of as a generalization of toric varieties and pairs. \nIndeed, any toric pair is a cluster type pair.\nHowever, the realm of cluster type pairs is much broader. \nA del Pezzo surface of degree $d\\geq 2$ as well as a general del Pezzo\nsurface of degree $d=1$ is cluster type (see, e.g.,~\\cite[Theorem 2.1 and Remark 2.2]{ALP23}). \nThus, all the examples discussed in the introduction are cluster type varieties.\nOur first aim is to understand degenerations of cluster type pairs \nfor which the limit of the cluster type boundary still has reasonable singularities. \nIn the next subsection, we prove that there are not many such degenerations\nfor toric pairs, but quite a lot for cluster type pairs.\n\n\\subsection{Degenerations of cluster type pairs}\n\nOur first theorem states that degenerations of toric pairs\nare finite quotients of toric pairs. We impose some conditions \non the singularities of the central fiber.\n\n\\begin{definition}\\label{def:ct}\n{\\em \nLet $(X,B)$ be log Calabi--Yau pair of dimension $n$. \nWe say that $(X,B)$ is of {\\em cluster type}\nor that $(X,B)$ is a {\\em cluster type pair} if there exists an embedding \nin codimension one \n\\[\n\\mathbb{G}_m^n \\dashrightarrow X\\setminus B, \n\\]\ni.e., there exists a closed subset of codimension at least two $Z\\subset \\mathbb{G}_m^n$ \nand an embedding $\\mathbb{G}_m^n\\setminus Z \\hookrightarrow X\\setminus B$.\n}\n\\end{definition}\n\nOur first theorem states that degenerations of toric pairs\nare finite quotients of toric pairs. We impose some conditions \non the singularities of the central fiber.\n\nIn the previous theorem,\nthe boundary $\\mathcal{B}_0$ on the central fiber $\\mathcal{X}_0$\nis defined via adjunction. \nThis means that $\\mathcal{B}_0$ is defined via the formula.\n\\[\nK_{\\mathcal{X}_0}+\\mathcal{B}_0 = (K_{\\mathcal{X}}+\\mathcal{B}+\\mathcal{X}_0)|_{\\mathcal{X}_0}.\n\\]\nIn Example~\\ref{ex:non-toric-central}, \nwe show that in general $(\\mathcal{X}_0,\\mathcal{B}_0)$ is not a toric pair. Indeed, the central fiber $\\mathcal{X}_0$ does not need to be a rational variety under the assumptions of Theorem~\\ref{thm:toric-deg}.\nOur next theorem gives a similar statement for degenerations of cluster type pairs.\nWe need to impose stronger conditions on the singularities in this case.\nIn the proof of Theorem~\\ref{thm:toric-deg}, we will see that $(\\mathcal{X},\\mathcal{B})\\rightarrow \\mathbb{D}$ is indeed a finite quotient of an isotrivial toric family. Thus, in the toric case, there are not many interesting degenerations for which the degeneration of the toric boundary still has lc singularities.\nNow, we turn to discuss the case of cluster type pairs.\n\n\\begin{theorem}\\label{thm:ct-deg}\nLet $\\pi\\colon \\mathcal{X}\\rightarrow \\mathbb{D}$ be a projective Fano fibration.\nLet $(\\mathcal{X},\\mathcal{X}_0+\\mathcal{B})$ be a log Calabi--Yau pair of index one over $\\mathbb{D}$.\nAssume that $(\\mathcal{X},\\mathcal{X}_0)$ is plt \nand purely terminal on the complement of $\\mathcal{B}$.\nIf $(\\mathcal{X}_t,\\mathcal{B}_t)$ is of cluster type for $t\\in \\mathbb{D}^*$, then \n$(\\mathcal{X}_0,\\mathcal{B}_0)$ is a finite quotient of a cluster type pair.\n\\end{theorem}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:toric-deg}]\nLet $f\\colon \\mathcal{X'}\\rightarrow \\mathcal{X}$ be the finite cover given by Lemma~\\ref{lem:finite-cover}.\nBy Lemma~\\ref{lem:finite-cover}.(2), we have an induced \nfinite cover of log Calabi--Yau pairs $(\\mathcal{X}',\\mathcal{B}')\\rightarrow (\\mathcal{X},\\mathcal{B})$.\nThen, up to shrinking $\\mathbb{D}$ near $\\{0\\}$ we have a commutative diagram \n\\[\n\\xymatrix{\n(\\mathcal{X},\\mathcal{B})\\ar[d]_-{\\pi} & (\\mathcal{X}',\\mathcal{B}') \\ar[l]_-{f}\\ar[d]^-{\\pi'} \\\\ \n\\mathbb{D} & \\mathbb{D}\\ar[l]_-{f_\\mathbb{D}}\n}\n\\]\nwhere $f_\\mathbb{D}$ is simply given by $t\\mapsto t^k$ for some suitable positive integer $k$.\nThe morphism $f'\\colon \\mathcal{X}'\\rightarrow \\mathbb{D}$ is a fibration by Lemma~\\ref{lem:finite-cover}.(3). \nBy assumption $(\\mathcal{X},\\mathcal{X}_0)$ is plt so by Riemann-Hurwitz we conclude that $(\\mathcal{X}',\\mathcal{X}_0')$ is plt as well. In particular, the variety $\\mathcal{X}_0'$ is irreducible.\nBy Lemma~\\ref{lem:finite-cover}.(4), every component of $\\mathcal{B}'_t$ \nis the restriction to $\\mathcal{X}'_t$ of a component of $\\mathcal{B}'$. \nNote that $\\pi$ and $\\pi'$ have the same general log fibers; indeed $f$ is induced by a finite cover of $\\mathbb{D}$ ramified over $\\{0\\}$. \nTherefore, the general fiber $(\\mathcal{X'},\\mathcal{B'})$ is a projective toric variety of dimension $n$ \nand Picard rank $\\rho$. \nShrinking $\\mathbb{D}$ around $\\{0\\}$, we may assume that all the fibers over $\\pi'$ are irreducible.\nTherefore, we conclude that $\\rho(\\mathcal{X}'/\\mathbb{D})\\leq \\rho$.\nOn the other hand, as every component of $\\mathcal{B}'_t$ is the restriction to $\\mathcal{X}'_t$ of a component of $\\mathcal{B}$, \nwe conclude that $\\mathcal{B}$ has at least $n+\\rho$ components. \nThus, we can compute the relative complexity of\nthe log Calabi--Yau pair $(\\mathcal{X}',\\mathcal{B}'+\\mathcal{X}_0')$ over $\\{0\\} \\in \\mathbb{D}$. \nWe obtain\n\\[\nc_{\\{0\\}}(\\mathcal{X}'/\\mathbb{D},\\mathcal{B}'+\\mathcal{X}_0') =\n\\dim \\mathcal{X}' + \\rho(\\mathcal{X}'/\\mathbb{D}) - |\\mathcal{B}'+\\mathcal{X}_0'| \\leq \nn+1+\\rho - (n+\\rho+1).\n\\]\nTherefore, by~\\cite[Theorem 1]{MS21}, we conclude that \n$(\\mathcal{X}',\\mathcal{B}'+\\mathcal{X}'_0)$ is a formally toric morphism near $\\{0\\}$. \nThus, the pair $(\\mathcal{X}'_0,\\mathcal{B}'_0)$ obtained from adjunction\nof $(\\mathcal{X}',\\mathcal{B}'+\\mathcal{X}'_0)$ must be a projective\ntoric log Calabi--Yau pair. \nHenceforth, we have a finite crepant morpshim \nof log Calabi--Yau pairs \n\\[\nf_0\\colon (\\mathcal{X}'_0,\\mathcal{B}'_0) \n\\rightarrow (\\mathcal{X}_0,\\mathcal{B}). \n\\]\nWe conclude that the pair $(\\mathcal{X}_0,\\mathcal{B}_0)$\nis a finite quotient of a toric pair. \n\\end{proof}\n\n\\begin{theorem}\\label{thm:1-comp}\nLet $\\pi\\colon \\mathcal{X}\\rightarrow \\mathbb{D}$ be a Fano projective morphism where $\\mathcal{X}_0$ is reduced in $\\mathcal{X}$ and\n$(\\mathcal{X},\\mathcal{X}_0)$ plt.\nAssume that the following conditions hold for the general fiber $\\mathcal{X}_t$ with $t\\neq 0$:\n\\begin{enumerate}\n\\item $\\mathcal{X}_t$ is a toric surface of Picard rank one;\n\\item ${\\rm mld}(\\mathcal{X}_t)<\\frac{1}{6}$; \n\\item $\\mathcal{X}_t$ does not have singularities in the baskets \n$\\mathcal{F}_1,\\dots,\\mathcal{F}_4,\\mathcal{D}$.\n\\end{enumerate}\nThen, the pair $(\\mathcal{X},\\mathcal{X}_0)$ admits a $1$-complement over $\\mathbb{D}$.\n\\end{theorem}\n\\begin{proof}\nWe break the proof into five steps.\\\\\n\n\\begin{example}\\label{ex:non-toric-central}\n{\\em\nWe give an example of a degeneration of a toric pair where the central fiber is not a toric variety. \nFix $n\\geq 3$ and consider the variety $\\mathcal{Y}:=(\\pp^1)^n \\times \\mathbb{D}$ \nand let $\\pi_2\\colon \\mathcal{Y}\\rightarrow \\mathbb{D}$ be the projection onto the second component. \nLet $B_T$ be the torus-invariant boundary of $(\\pp^1)^n$ \nand $\\mathcal{B}_\\mathcal{Y}:=B_T\\times \\mathbb{D}$.\nTherefore, the pair $(\\mathcal{Y},\\mathbb{B}_{\\mathcal{Y}}+\\mathcal{Y}_0)$ \nis a log Calabi--Yau pair for which every fiber of $\\pi_2$ is isomorphic to\n$((\\pp^1)^n,B_T)$. \nConsider the group $\\zz_2$ acting on $\\mathcal{Y}$ via \n\\begin{align*} \n\\mu & \\colon (\\pp^1)^n\\times \\mathbb{D} \\rightarrow \n(\\pp^1)^n \\times \\mathbb{D} \\\\\n\\mu & \\cdot ([x_1:y_1],[x_2:y_2],\\dots,[x_n:y_n],t) :=\n([y_1:x_1],[y_2:x_2],\\dots,[y_n:x_n],-t).\n\\end{align*} \nThe log Calabi--Yau pair $(\\mathcal{Y},\\mathcal{B}_{\\mathcal{Y}}+\\mathcal{Y}_0)$\nis invariant under the $\\zz_2$-action. \nLet $(\\mathcal{X},\\mathcal{B}+\\mathcal{X}_0)$ be the induced quotient, so\nwe obtain a commutative diagram:\n\\[\n\\xymatrix{\n(\\mathcal{X},\\mathcal{B}+\\mathcal{X}_0) \\ar[d]^-{\\pi} & (\\mathcal{Y},\\mathcal{B}_{\\mathcal{Y}}+\\mathcal{Y}_0) \\ar[d]^-{\\pi_2} \\ar[l]^-{/\\zz_2} \\\\\n\\mathbb{D} & \\mathbb{D} \\ar[l]^-{/\\zz_2}\n}\n\\]\nThe morphism $\\pi$ is a degeneration of the toric log Calabi--Yau pair\n$((\\pp^1)^n,B_T)$ into $(\\mathcal{X}_0,\\mathcal{B}_0)=((\\pp^1)^n,B_T)/\\zz_2$. \nWe have that $\\mathcal{D}(\\mathcal{X}_0,\\mathcal{B}_0)\\simeq_{\\rm PL} \\mathbb{P}_\\rr^{n-1}$. \nThus, the central fiber $(\\mathcal{X}_0,\\mathcal{B}_0)$ is not a toric pair. \n}\n\\end{example}", + "post_theorem_intro_text_len": 7373, + "post_theorem_intro_text": "In the previous theorem,\nthe boundary $\\mathcal{B}_0$ on the central fiber $\\mathcal{X}_0$\nis defined via adjunction. \nThis means that $\\mathcal{B}_0$ is defined via the formula.\n\\[\nK_{\\mathcal{X}_0}+\\mathcal{B}_0 = (K_{\\mathcal{X}}+\\mathcal{B}+\\mathcal{X}_0)|_{\\mathcal{X}_0}.\n\\]\nIn Example~\\ref{ex:non-toric-central}, \nwe show that in general $(\\mathcal{X}_0,\\mathcal{B}_0)$ is not a toric pair. Indeed, the central fiber $\\mathcal{X}_0$ does not need to be a rational variety under the assumptions of Theorem~\\ref{thm:toric-deg}.\nOur next theorem gives a similar statement for degenerations of cluster type pairs.\nWe need to impose stronger conditions on the singularities in this case.\nIn the proof of Theorem~\\ref{thm:toric-deg}, we will see that $(\\mathcal{X},\\mathcal{B})\\rightarrow \\mathbb{D}$ is indeed a finite quotient of an isotrivial toric family. Thus, in the toric case, there are not many interesting degenerations for which the degeneration of the toric boundary still has lc singularities.\nNow, we turn to discuss the case of cluster type pairs.\n\n\\begin{theorem}\\label{thm:ct-deg}\nLet $\\pi\\colon \\mathcal{X}\\rightarrow \\mathbb{D}$ be a projective Fano fibration.\nLet $(\\mathcal{X},\\mathcal{X}_0+\\mathcal{B})$ be a log Calabi--Yau pair of index one over $\\mathbb{D}$.\nAssume that $(\\mathcal{X},\\mathcal{X}_0)$ is plt \nand purely terminal on the complement of $\\mathcal{B}$.\nIf $(\\mathcal{X}_t,\\mathcal{B}_t)$ is of cluster type for $t\\in \\mathbb{D}^*$, then \n$(\\mathcal{X}_0,\\mathcal{B}_0)$ is a finite quotient of a cluster type pair.\n\\end{theorem} \n\nWe refer the reader to Definition~\\ref{def:pt} for the concept of purely terminal pairs.\nIn contrast to the toric case, a cluster type pair may have several interesting degenerations for which the degeneration of the cluster type boundary still has lc singularities. In Example~\\ref{ex:P^2-toric-model-1}, we show that some toric degenerations of $\\mathbb{P}^2$ can be regarded as cluster type degenerations for different embeddings of algebraic tori \n$\\mathbb{G}_m^2 \\hookrightarrow \\mathbb{P}^2\\setminus C$ where $C$ is a nodal cubic.\n\n\\subsection{Degenerations of singular surfaces}\n\nNow, we restrict ourselves to the study of degenerations of singular toric surfaces. \nOne of our aims is to understand how the singularities affect the possible degenerations. \nOur third theorem states that almost every weighted projective plane has no interesting degenerations.\n\n\\begin{theorem}\\label{no-deg-almost}\nFor almost all well-formed triples $(a,b,c)\\in \\zz_{\\geq 1}^3$\nthe weighted projective plane $\\mathbb{P}(a,b,c)$ has no non-trivial\n$\\mathbb{Q}$-Gorenstein klt degenerations. \n\\label{weighteddeg}\n\\end{theorem}\n\nIn the previous theorem, when we write {\\em almost all}, we mean that the statement holds up to a subset $S\\subsetneq \\zz_{\\geq 1}^3$ that has density zero with the natural density endowed from $\\mathbb{Z}^3$.\nWe say that a degeneration is {\\em trivial} if the central fiber is isomorphic to the general fiber. \nHowever, in the setting of Theorem~\\ref{no-deg-almost}, we will prove something stronger; the family is a product near the origin of the disk.\nIn order to prove Theorem~\\ref{weighteddeg}, we will prove Theorem~\\ref{thm:1-comp}, which is a general statement about\nthe existence of complements for degenerations of singular toric surfaces.\nThis means that, under some mild conditions, we prove that given a degeneration \n$\\mathcal{X}\\rightarrow \\mathbb{D}$ of a singular toric surface, \nthere exists some boundary $\\mathcal{B}\\in |-K_{\\mathcal{X}}|$ for which $(\\mathcal{X}_t,\\mathcal{B}_t)$ has log canonical singularities for $t$ near $\\{0\\}\\in \\mathbb{D}$.\nIn the setting of Theorem~\\ref{weighteddeg}, in most cases, we can argue that $(\\mathcal{X}_t,\\mathcal{B}_t)$ is toric for every $t$ and so \nthe statement is similar to that of Theorem~\\ref{thm:toric-deg}, which states that there are no interesting such toric degenerations.\nTheorem~\\ref{thm:1-comp} is rather technical and depends on some meticulous analysis of basket of singularities.\nThe idea of using the theory of complements to understand degenerations of del Pezzo surfaces goes back to Hacking and Prokhorov. \nThe situation becomes a bit more delicate when we allow the general fiber of the degeneration to have singularities. \nWe will argue that for a Markov triple $(a,b,c)\\in \\zz_{\\geq 2}^3$ the triple \n$(a^2,b^2,c^2)$ belongs to the complement of the subset $S\\subsetneq \\zz_{\\geq 0}^3$ of density zero mentioned above.\nThus, we conclude the following corollary.\n\n\\begin{corollary}\\label{cor:markov-no-deg}\nLet $(a,b,c)\\in \\zz_{\\geq 2}^3$ be a Markov triple.\nThen, the weighted projective plane $\\mathbb{P}(a^2,b^2,c^2)$ has no non-trivial $\\mathbb{Q}$-Gorenstein klt degenerations.\n\\end{corollary} \n\nWe note that Corollary~\\ref{cor:markov-no-deg} can also be concluded from the work of Hacking and Prokhorov~\\cite{HP10}. Indeed, every iterated degeneration of $\\mathbb{P}(a^2,b^2,c^2)$ is indeed a degeneration of $\\mathbb{P}^2$.\nIn upcoming work~\\cite{Zun25}, the second author will prove \nsome structural theorems about $\\mathbb{Q}$-Gorenstein klt degenerations\nof Hirzebruch surfaces. \nThis will aim to finish the classification of $\\mathbb{Q}$-Gorenstein\nklt degenerations of minimal smooth rational surfaces. \nThis motivates us to pay particular attention to the weighted projective plane \n$\\mathbb{P}(1,1,n)$. \nIn this direction, using the tools introduced above, we can give\na complete classification of klt degenrations of $\\mathbb{P}(1,1,n)$ with $n\\geq 3$.\n\n\\begin{theorem}\\label{thm:1-1-n}\nLet $\\mathcal{X}\\rightarrow \\mathbb{D}$ be a klt Fano degeneration of $\\mathbb{P}(1,1,n)$ with $n \\geq 3$, \nthen for $\\mathcal{X}_0$ one of the following holds:\n\\begin{enumerate}\n\\item $\\mathcal{X}_0$ is a weighted projective plane, or \n\\item $\\mathcal{X}_0$ is a $\\mathbb{G}_m$-surface which is not toric. \n\\end{enumerate}\nFurthermore, in the second case $\\mathcal{X}_0$ is a $\\mathbb{Q}$-Gorenstein deformation of a weighted projective plane.\n\\end{theorem} \n\nIn Proposition~\\ref{prop:tor-deg}, \nwe give a explicit classification\nof the weighted projective planes\nwhich are $\\mathbb{Q}$-Gorenstein klt degenerations of $\\mathbb{P}(1,1,n)$ with $n\\geq 3$.\n\nThe paper is organized as follows.\nIn Section~\\ref{sec:prelims}, we write some preliminary results regarding cluster type pairs, theory of complements, dual complexes, T-singularities, and Wahl singularities. \nIn Section~\\ref{sec:degen-cluster-type}, we prove Theorem~\\ref{thm:toric-deg} and Theorem~\\ref{thm:ct-deg} regarding degenerations of toric pairs as well as cluster type pairs. \nIn Section~\\ref{sec:complements-degen-klt-surfaces}, we prove some general statements regarding the existence of complements for degenerations of singular toric surfaces of Picard rank one. In this section, we also prove Theorem~\\ref{weighteddeg} regarding the degenerations of weighted projective planes $\\mathbb{P}(a,b,c)$. \nIn Section~\\ref{sec:degen-wps}, we classify the degenerations of weighted projective planes $\\mathbb{P}(1,1,n)$ with $n\\geq 3$. \nFinally, in Section~\\ref{sec:ex-and-quest}, we give some examples and propose some questions for further research.\n\n\\subsection*{Acknowledgements}\n\nThe authors would like to thank Audric Lebovitz, Tomoki Oda, Giancarlo Urzúa, and Jos\\'e Ignacio Y\\'a\\~nez\nfor many discussions related to this article.", + "sketch": "In the proof of Theorem~\\ref{thm:toric-deg}, we will see that $(\\mathcal{X},\\mathcal{B})\\rightarrow \\mathbb{D}$ is indeed a finite quotient of an isotrivial toric family. Thus, in the toric case, there are not many interesting degenerations for which the degeneration of the toric boundary still has lc singularities.", + "expanded_sketch": "In the proof of the main theorem, we will see that $(\\mathcal{X},\\mathcal{B})\\rightarrow \\mathbb{D}$ is indeed a finite quotient of an isotrivial toric family. Thus, in the toric case, there are not many interesting degenerations for which the degeneration of the toric boundary still has lc singularities.,", + "expanded_theorem": "\\label{thm:toric-deg}\nLet $\\pi\\colon \\mathcal{X}\\rightarrow \\mathbb{D}$ be a projective fibration.\nLet $(\\mathcal{X},\\mathcal{X}_0+\\mathcal{B})$ be a log Calabi--Yau pair over $\\mathbb{D}$ for which $(\\mathcal{X},\\mathcal{X}_0)$ is plt. \nIf $(\\mathcal{X}_t,\\mathcal{B}_t)$ is a toric pair\\footnote{Meaning that $\\mathcal{X}_t$ is a projective toric variety and $\\mathcal{B}_t$ is the reduced torus invariant boundary.} for $t \\in \\mathbb{D}^*$,\nthen $(\\mathcal{X}_0,\\mathcal{B}_0)$ is a finite quotient of a toric pair.", + "theorem_type": [ + "Implication", + "Existence" + ], + "mcq": { + "question": "Let \\(\\pi\\colon \\mathcal{X}\\to \\mathbb{D}\\) be a projective fibration, let \\(\\mathcal{X}_0=\\pi^{-1}(0)\\), and let \\((\\mathcal{X},\\mathcal{X}_0+\\mathcal{B})\\) be a log Calabi--Yau pair over \\(\\mathbb{D}\\) (so \\(K_{\\mathcal{X}}+\\mathcal{X}_0+\\mathcal{B}\\) is trivial over \\(\\mathbb{D}\\)). Assume that \\((\\mathcal{X},\\mathcal{X}_0)\\) is plt. For every \\(t\\in \\mathbb{D}^*:=\\mathbb{D}\\setminus\\{0\\}\\), suppose the fiber pair \\((\\mathcal{X}_t,\\mathcal{B}_t)\\) is a toric pair, meaning that \\(\\mathcal{X}_t\\) is a projective toric variety and \\(\\mathcal{B}_t\\) is its reduced torus-invariant boundary. Which conclusion about the central fiber pair \\((\\mathcal{X}_0,\\mathcal{B}_0)\\) holds?", + "correct_choice": { + "label": "A", + "text": "The central fiber pair \\((\\mathcal{X}_0,\\mathcal{B}_0)\\) is a finite quotient of a toric pair; equivalently, there exists a toric pair \\((Y,B_Y)\\), with \\(Y\\) a projective toric variety and \\(B_Y\\) its reduced torus-invariant boundary, together with a finite morphism of pairs \\((Y,B_Y)\\to (\\mathcal{X}_0,\\mathcal{B}_0)\\)." + }, + "choices": [ + { + "label": "B", + "text": "The central fiber pair \\((\\mathcal{X}_0,\\mathcal{B}_0)\\) is itself a toric pair; equivalently, \\(\\mathcal{X}_0\\) is a projective toric variety and \\(\\mathcal{B}_0\\) is its reduced torus-invariant boundary." + }, + { + "label": "C", + "text": "The central fiber pair \\((\\mathcal{X}_0,\\mathcal{B}_0)\\) is birational to a toric pair; that is, there exists a projective toric pair \\((Y,B_Y)\\) and a birational map of pairs \\((Y,B_Y) \\dashrightarrow (\\mathcal{X}_0,\\mathcal{B}_0)\\)." + }, + { + "label": "D", + "text": "There exists a single projective toric pair \\((Y,B_Y)\\) such that, after possibly shrinking \\(\\mathbb{D}\\), every fiber pair \\((\\mathcal{X}_t,\\mathcal{B}_t)\\) for \\(t\\in\\mathbb{D}\\) is isomorphic to \\((Y,B_Y)\\); in particular, the family is isotrivial and the central fiber is a toric pair isomorphic to the general fiber." + }, + { + "label": "E", + "text": "The central fiber pair \\((\\mathcal{X}_0,\\mathcal{B}_0)\\) admits a finite crepant morphism to a toric pair; equivalently, there exists a projective toric pair \\((Y,B_Y)\\) and a finite morphism of pairs \\((\\mathcal{X}_0,\\mathcal{B}_0)\\to (Y,B_Y)\\)." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "finite_quotient_vs_toricity_of_central_fiber", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "replace_finite_quotient_by_birational_to_toric", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "isotrivial_cover_vs_original_family", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "direction_of_finite_crepant_morphism", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly state or strongly hint at the key conclusion that the central fiber is a finite quotient of a toric pair. The relevant distinction only appears in the answer choices." + }, + "TAS": { + "score": 2, + "justification": "Although the question is theorem-driven, it is not a bare restatement: the options force the test-taker to distinguish among nearby conclusions such as toricity, birational toricity, isotriviality, and the direction of a finite morphism." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is required to select the strongest valid conclusion and reject stronger, weaker, or directionally incorrect variants. However, for someone who knows the theorem, the answer is largely recall-based rather than deeply generative." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and well-targeted: one is too strong, one is weaker-but-true-looking, one confuses family-level isotriviality with fiberwise toricity, and one reverses the finite morphism direction." + }, + "total_score": 7, + "overall_assessment": "A strong MCQ with little answer leakage and high-quality distractors; it tests careful theorem discrimination well, though it remains somewhat recall-oriented rather than fully generative." + } + }, + { + "id": "2511.13913v1", + "paper_link": "http://arxiv.org/abs/2511.13913v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "introtheorem", + "content": "\\label{intthm:character}\n If $H$ is a type B or type C Hessenberg space, there exist non-negative integers $a,b \\in \\mathbb{N}$, integers $c,d \\in \\{0,1\\}$, and a subset $I \\subseteq [n]$, each determined by $S(H)$, such that\n \\[\n \\mathfrak{ch}\\left(\\mathrm{L}_{H}\\right)_1 = a\\,\\mathbb{1} + \\sum_{i\\in I} \\fh_i + b\\,\\fh_1 + c\\,\\mathfrak{s} + d\\,\\bm{\\delta} \n \\]\n and \n \\[\n \\mathfrak{ch}\\left(\\mathrm{R}_{H}\\right)_1 = \\bm{\\chi} + \\sum_{i\\in I} \\left(\\fh_i - \\mathbb{1}\\right) + b\\left(\\fh_1-\\mathbb{1}\\right) + c \\left(\n \\mathfrak{s}-\\mathbb{1}\\right) + d\\,\\bm{\\delta},\n \\]\n where \n \\begin{itemize}\n \\item $\\bm{\\chi}$ is the character of the defining representation of $\\mathfrak{W}_n$ on $\\mathbb{C}^n$\n \\item $\\fh_i$ is the character of the action of $\\mathfrak{W}_n$ on the cosets of $\\mathfrak{S}_i\\times \\mathfrak{W}_{n-i}$,\n \\item $\\mathfrak{s}$ is the character of the action of $\\mathfrak{W}_n$ on the cosets of $\\mathfrak{W}_1\\times \\mathfrak{W}_{n-1}$,\n \\item $\\mathbb{1}$ is the character of the trivial representation, and \n \\item $\\bm{\\delta}$ is the character $w\\mapsto (-1)^{\\left| \\Neg(w) \\right|}$ where $\\Neg(w) = \\{w(i) \\mid i \\in [n],\\; w(i) < 0\\}$.\n \\end{itemize}\n Moreover, $c$ and $d$ are never simultaneously $1$, and $d$ is always $0$ in type B.", + "start_pos": 22516, + "end_pos": 23761, + "label": "intthm:character" + }, + "ref_dict": { + "intthm:character": "\\begin{introtheorem}\\label{intthm:character}\n If $H$ is a type B or type C Hessenberg space, there exist non-negative integers $a,b \\in \\mathbb{N}$, integers $c,d \\in \\{0,1\\}$, and a subset $I \\subseteq [n]$, each determined by $S(H)$, such that\n \\[\n \\lrep{H}_1 = a\\,\\mathbb{1} + \\sum_{i\\in I} \\fh_i + b\\,\\fh_1 + c\\,\\fs + d\\,\\bm{\\delta} \n \\]\n and \n \\[\n \\rrep{H}_1 = \\bm{\\chi} + \\sum_{i\\in I} \\left(\\fh_i - \\mathbb{1}\\right) + b\\left(\\fh_1-\\mathbb{1}\\right) + c \\left(\n \\fs-\\mathbb{1}\\right) + d\\,\\bm{\\delta},\n \\]\n where \n \\begin{itemize}\n \\item $\\bm{\\chi}$ is the character of the defining representation of $\\Wn$ on $\\C^n$\n \\item $\\fh_i$ is the character of the action of $\\Wn$ on the cosets of $\\mathfrak{S}_i\\times \\mathfrak{W}_{n-i}$,\n \\item $\\fs$ is the character of the action of $\\Wn$ on the cosets of $\\mathfrak{W}_1\\times \\mathfrak{W}_{n-1}$,\n \\item $\\mathbb{1}$ is the character of the trivial representation, and \n \\item $\\bm{\\delta}$ is the character $w\\mapsto (-1)^{\\abs{\\Neg(w)}}$ where $\\Neg(w) = \\{w(i) \\mid i \\in [n],\\; w(i) < 0\\}$.\n \\end{itemize}\n Moreover, $c$ and $d$ are never simultaneously $1$, and $d$ is always $0$ in type B.\n\\end{introtheorem}", + "prop:permutohedral": "\\begin{proposition}\\label{prop:permutohedral}\n Let $\\bm{\\chi}$ be the character of the defining representation of $\\Wn$ (i.e. the action on $\\C^n$). Let $\\fh_i$ be the character of the action of $\\Wn$ on the cosets of $\\mathfrak{S}_i \\times \\mathfrak{W}_{n-i}$. Let $\\mathbb{1}$ be the character of the trivial representation. Then \n \\[\n \\lrep{\\Delta}_1 = \\sum_{i=1}^n \\fh_i - \\bm{\\chi} \n \\hspace{1cm}\\text{and}\\hspace{1cm}\n \\rrep{\\Delta}_1 = \\sum_{i=1}^n \\fh_i - n\\,\\mathbb{1}.\n \\]\n\\end{proposition}", + "def:itypes": "\\begin{definition}\\label{def:itypes}\nConsider $S(H)$ and $i \\in [n]$. We call $i$\\begin{itemize}\n \\item \\emph{uncovered} if $\\begin{cases} \\{t_{i-1},t_i\\} \\cap S(H) = \\emptyset &\\text{and } i \\neq n-1\\text{, or} \\\\ \\{t_{n-1},t_{n-1},t_{n}\\} \\cap S(H) = \\emptyset &\\text{and } i = n-1\\end{cases}$\n \\item \\emph{surrounded} if $i \\in [n-2]$ and $\\{t_{i-1},t_i\\} \\cap S(H) = \\{t_{i-1}\\}$ and $t_m \\in S(H)\\text{ for some } m>i$,\n \\item \\emph{shaded} if $t_i \\in S(H)$ or if $t_n \\in S(H)$ and $i=n-1$.\n\\end{itemize}\n\\end{definition}", + "thm:character": "\\begin{theorem}\\label{thm:character}\n Say $\\Delta \\subsetneq H$. Let $\\bm{\\chi}$ be the character of the defining representation of $\\Wn$ (i.e. the action on $\\C^n$). Let $\\fh_i$ be the character of the action of $\\Wn$ on the cosets of $\\mathfrak{S}_i\\times \\mathfrak{W}_{n-i}$. Let $\\fs$ be the character of the action of $\\Wn$ on cosets of $\\mathfrak{W}_1 \\times \\mathfrak{W}_{n-1}$. Let $\\mathbb{1}$ be the character of the trivial representation and $\\bm{\\delta}$ be the character $w\\mapsto (-1)^{\\abs{\\Neg(w)}}$. Then \n \\begin{align*}\n \\lrep{H}_1 = \\abs{\\{i \\in [n] \\mid i\\text{ shaded}\\,\\}}\\mathbb{1} &+ \\sum_{i\\text{ uncovered}} \\fh_i \\\\[6pt]&+ \\abs{\\{i \\in [n-2] \\mid i\\text{ surrounded}\\,\\}}\\fh_1 \\\\[6pt]&+ \\begin{cases}\n \\fs &\\text{if $t_n \\notin S(H)$}\\\\\n \\bm{\\delta} &\\text{if }\\{t_{n-1},t_n\\} \\cap S(H) = \\{t_{n}\\}\\\\\n 0 &\\text{otherwise.}\n\\end{cases}\n \\end{align*}\n and \n \\begin{align*}\n \\rrep{H}_1 = \\bm{\\chi} &+ \\sum_{i\\text{ uncovered}} \\left(\\fh_i - \\mathbb{1}\\right) \\\\[6pt]&+ \\abs{\\{i \\in [n-2] \\mid i\\text{ surrounded}\\,\\}}\\left(\\fh_1-\\mathbb{1}\\right) \\\\[6pt]&+ \\begin{cases}\n \\fs-\\mathbb{1} &\\text{if $t_n \\notin S(H)$}\\\\\n \\bm{\\delta} &\\text{if }\\{t_{n-1},t_n\\} \\cap S(H) = \\{t_{n}\\}\\\\\n 0 &\\text{otherwise.}\n\\end{cases}\n \\end{align*}\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 4877, + "pre_theorem_intro_text": "\\label{sec:intro}\nLet $\\mathfrak{W}_n$ be the group of signed permutations, which are permutations $w$ on the set $\\{1,\\ldots,n,-n,\\ldots,-1\\}$ such that $w(-i) = -w(i)$. The group $\\mathfrak{W}_n$ is also the Weyl group of types B and C, and so is associated with two distinct root systems. A Hessenberg space $H$ is a particular subset of roots, which determines a set $S(H)$ of transpositions in $\\mathfrak{W}_n$. Given a Hessenberg space, one can define the \\emph{graded ring of splines} $\\mathcal{M}_{H}$ on $\\mathfrak{W}_n$. This ring has two $\\C[x_\\bullet]\\coloneqq \\C[x_1,\\ldots,x_{n}]$-module structures and a $\\mathfrak{W}_n$-module structure. This paper determines the algebraic and combinatorial properties of the degree-one graded piece of $\\mathcal{M}_{H}$ from the combinatorics of $S(H)$.\\par \nLet $(i,j)$ be the unique transposition in $\\mathfrak{W}_n$ that switches $i$ with $j$ and $-i$ with $-j$ (note $i=-j$ is possible). If $H$ is a Hessenberg space, then the ring of splines is \n \\[\n \\mathcal{M}_{H} = \\bigoplus_{i \\geq 0} \\mathcal{M}_{H}^i \\coloneqq \\left\\{ \\left.\\bm{\\rho} \\in \\prod_{w \\in \\mathfrak{W}_n} \\C[x_\\bullet]\\; \\right|\\; \\bm{\\rho}(w) - \\bm{\\rho}(w(i,j)) \\in \\left\\langle x_{w(i)}-x_{w(j)} \\right\\rangle\\text{ if } (i,j) \\in S(H) \\right\\}\\!,\n \\]\nwith (graded) $\\mathfrak{W}_n$-module structure $w\\cdot \\bm{\\rho}(v) = w\\bm{\\rho}(w^{-1}v)$ and (graded) $\\C[x_\\bullet]$-module structure given by multiplication. \\par \nThe ring $\\mathcal{M}_{H}$ is isomorphic to the $T$-equivariant cohomology of a smooth subvariety of the full flag variety $G/B$ called a \\emph{regular semisimple Hessenberg variety}. The connection between this combinatorial ring and equivariant cohomology is called \\emph{GKM Theory} \\cite{GKM_theory}, and it allows for the study of this cohomology in purely combinatorial and algebraic terms. \\par \nThe $\\mathfrak{W}_n$-module structure on $\\mathcal{M}_{H}$ was first defined as the \\emph{dot action} on equivariant cohomology by Tymoczko in \\cite{tymoczko2008permutation}. There are two natural $\\mathfrak{W}_n$-equivariant quotients, called the left $\\mathrm{L}_{H}$ and right $\\mathrm{R}_{H}$ quotients, of $\\mathcal{M}_{H}$ that are in fact graded $\\mathbb{C}$-vector spaces (the left quotient corresponds to ordinary cohomology). The graded $\\mathfrak{W}_n$-module structure of $\\mathcal{M}_{H}$ induces graded $\\mathfrak{W}_n$-representations on the quotients. The primary goal of this paper is to compute the characters of the degree-one pieces of \n\\[\n\\mathrm{L}_{H} = \\bigoplus_{i\\geq 0} \\left(\\mathrm{L}_{H}\\right)_i \\;\\; \\text{ and }\\;\\;\\mathrm{R}_{H} = \\bigoplus_{i\\geq 0} \\left(\\mathrm{R}_{H}\\right)_i\n\\] from the combinatorial data of $S(H)$. In doing so, we compute the dot action on second (equivariant and ordinary) cohomology for all regular semisimple Hessenberg varieties in types B and C. This generalizes what is known in type A \\cite{chow_linearp,chohonglee_second_cohom,ayzenberg2022second}.\\par \nThese graded representations are of interest to algebraic combinatorists in part because they are the $\\mathfrak{W}_n$-equivalents of very well-studied $\\mathfrak{S}_n$-representations, which have connections to chromatic symmetric functions \\cite{guaypaquet2016shar_wachs_conj,SW2016chromaticquasisymmetric,brosnan_chow_dotactn_is_chromsym} and LLT polynomials \\cite{guaypaquet2016shar_wachs_conj,Ayzenberg2018isospectral, ALEXANDERSSON2018LLTchromsym}. We detail some of these connections in the Appendix. From these works, in type A there are combinatorial formulas to compute the characters of both $\\mathrm{L}_{H}$ and $\\mathrm{R}_{H}$ in at least one basis for the character space of $\\mathfrak{S}_n$. This is \\textbf{not} the case in types B or C.\\par \nFor one specific family of $H$ (those corresponding to simple roots), the character of $\\mathrm{L}_{H}$ was computed by Stembridge \\cite{Stembridge_Permuto} in general type. Stembridge's results are via a connection to \\emph{permutohedra}, as the associated Hessenberg varieties are in fact toric varieties. While it is possible to compute the left and right characters for general $H$ in types B and C, even partial results are rare and rely on sophisticated geometric tools \\cite{BalibanuCrooks_sheaves,PrecupSommers_sheaves}. Formulas are non-combinatorial (and usually non-positive), require the computation of type B/C Green polynomials and Poincar\\'e polynomials of other varieties, and proofs require the geometry of perverse sheaves. \\par \nIn contrast, for the degree-one graded piece, the main result of this paper applies to all Hessenberg spaces $H$ and is entirely computable from the combinatorics of roots and transpositions in $\\mathfrak{W}_n$. Moreover, we compute the degree-one pieces of the characters $\\mathfrak{ch}\\left(\\mathrm{L}_{H}\\right)$ and $\\mathfrak{ch}\\left(\\mathrm{R}_{H}\\right)$ in a positive manner.", + "context": "\\label{sec:intro}\nLet $\\mathfrak{W}_n$ be the group of signed permutations, which are permutations $w$ on the set $\\{1,\\ldots,n,-n,\\ldots,-1\\}$ such that $w(-i) = -w(i)$. The group $\\mathfrak{W}_n$ is also the Weyl group of types B and C, and so is associated with two distinct root systems. A Hessenberg space $H$ is a particular subset of roots, which determines a set $S(H)$ of transpositions in $\\mathfrak{W}_n$. Given a Hessenberg space, one can define the \\emph{graded ring of splines} $\\mathcal{M}_{H}$ on $\\mathfrak{W}_n$. This ring has two $\\C[x_\\bullet]\\coloneqq \\C[x_1,\\ldots,x_{n}]$-module structures and a $\\mathfrak{W}_n$-module structure. This paper determines the algebraic and combinatorial properties of the degree-one graded piece of $\\mathcal{M}_{H}$ from the combinatorics of $S(H)$.\\par \nLet $(i,j)$ be the unique transposition in $\\mathfrak{W}_n$ that switches $i$ with $j$ and $-i$ with $-j$ (note $i=-j$ is possible). If $H$ is a Hessenberg space, then the ring of splines is \n \\[\n \\mathcal{M}_{H} = \\bigoplus_{i \\geq 0} \\mathcal{M}_{H}^i \\coloneqq \\left\\{ \\left.\\bm{\\rho} \\in \\prod_{w \\in \\mathfrak{W}_n} \\C[x_\\bullet]\\; \\right|\\; \\bm{\\rho}(w) - \\bm{\\rho}(w(i,j)) \\in \\left\\langle x_{w(i)}-x_{w(j)} \\right\\rangle\\text{ if } (i,j) \\in S(H) \\right\\}\\!,\n \\]\nwith (graded) $\\mathfrak{W}_n$-module structure $w\\cdot \\bm{\\rho}(v) = w\\bm{\\rho}(w^{-1}v)$ and (graded) $\\C[x_\\bullet]$-module structure given by multiplication. \\par \nThe ring $\\mathcal{M}_{H}$ is isomorphic to the $T$-equivariant cohomology of a smooth subvariety of the full flag variety $G/B$ called a \\emph{regular semisimple Hessenberg variety}. The connection between this combinatorial ring and equivariant cohomology is called \\emph{GKM Theory} \\cite{GKM_theory}, and it allows for the study of this cohomology in purely combinatorial and algebraic terms. \\par \nThe $\\mathfrak{W}_n$-module structure on $\\mathcal{M}_{H}$ was first defined as the \\emph{dot action} on equivariant cohomology by Tymoczko in \\cite{tymoczko2008permutation}. There are two natural $\\mathfrak{W}_n$-equivariant quotients, called the left $\\mathrm{L}_{H}$ and right $\\mathrm{R}_{H}$ quotients, of $\\mathcal{M}_{H}$ that are in fact graded $\\mathbb{C}$-vector spaces (the left quotient corresponds to ordinary cohomology). The graded $\\mathfrak{W}_n$-module structure of $\\mathcal{M}_{H}$ induces graded $\\mathfrak{W}_n$-representations on the quotients. The primary goal of this paper is to compute the characters of the degree-one pieces of \n\\[\n\\mathrm{L}_{H} = \\bigoplus_{i\\geq 0} \\left(\\mathrm{L}_{H}\\right)_i \\;\\; \\text{ and }\\;\\;\\mathrm{R}_{H} = \\bigoplus_{i\\geq 0} \\left(\\mathrm{R}_{H}\\right)_i\n\\] from the combinatorial data of $S(H)$. In doing so, we compute the dot action on second (equivariant and ordinary) cohomology for all regular semisimple Hessenberg varieties in types B and C. This generalizes what is known in type A \\cite{chow_linearp,chohonglee_second_cohom,ayzenberg2022second}.\\par \nThese graded representations are of interest to algebraic combinatorists in part because they are the $\\mathfrak{W}_n$-equivalents of very well-studied $\\mathfrak{S}_n$-representations, which have connections to chromatic symmetric functions \\cite{guaypaquet2016shar_wachs_conj,SW2016chromaticquasisymmetric,brosnan_chow_dotactn_is_chromsym} and LLT polynomials \\cite{guaypaquet2016shar_wachs_conj,Ayzenberg2018isospectral, ALEXANDERSSON2018LLTchromsym}. We detail some of these connections in the Appendix. From these works, in type A there are combinatorial formulas to compute the characters of both $\\mathrm{L}_{H}$ and $\\mathrm{R}_{H}$ in at least one basis for the character space of $\\mathfrak{S}_n$. This is \\textbf{not} the case in types B or C.\\par \nFor one specific family of $H$ (those corresponding to simple roots), the character of $\\mathrm{L}_{H}$ was computed by Stembridge \\cite{Stembridge_Permuto} in general type. Stembridge's results are via a connection to \\emph{permutohedra}, as the associated Hessenberg varieties are in fact toric varieties. While it is possible to compute the left and right characters for general $H$ in types B and C, even partial results are rare and rely on sophisticated geometric tools \\cite{BalibanuCrooks_sheaves,PrecupSommers_sheaves}. Formulas are non-combinatorial (and usually non-positive), require the computation of type B/C Green polynomials and Poincar\\'e polynomials of other varieties, and proofs require the geometry of perverse sheaves. \\par \nIn contrast, for the degree-one graded piece, the main result of this paper applies to all Hessenberg spaces $H$ and is entirely computable from the combinatorics of roots and transpositions in $\\mathfrak{W}_n$. Moreover, we compute the degree-one pieces of the characters $\\mathfrak{ch}\\left(\\mathrm{L}_{H}\\right)$ and $\\mathfrak{ch}\\left(\\mathrm{R}_{H}\\right)$ in a positive manner.", + "full_context": "\\label{sec:intro}\nLet $\\mathfrak{W}_n$ be the group of signed permutations, which are permutations $w$ on the set $\\{1,\\ldots,n,-n,\\ldots,-1\\}$ such that $w(-i) = -w(i)$. The group $\\mathfrak{W}_n$ is also the Weyl group of types B and C, and so is associated with two distinct root systems. A Hessenberg space $H$ is a particular subset of roots, which determines a set $S(H)$ of transpositions in $\\mathfrak{W}_n$. Given a Hessenberg space, one can define the \\emph{graded ring of splines} $\\mathcal{M}_{H}$ on $\\mathfrak{W}_n$. This ring has two $\\C[x_\\bullet]\\coloneqq \\C[x_1,\\ldots,x_{n}]$-module structures and a $\\mathfrak{W}_n$-module structure. This paper determines the algebraic and combinatorial properties of the degree-one graded piece of $\\mathcal{M}_{H}$ from the combinatorics of $S(H)$.\\par \nLet $(i,j)$ be the unique transposition in $\\mathfrak{W}_n$ that switches $i$ with $j$ and $-i$ with $-j$ (note $i=-j$ is possible). If $H$ is a Hessenberg space, then the ring of splines is \n \\[\n \\mathcal{M}_{H} = \\bigoplus_{i \\geq 0} \\mathcal{M}_{H}^i \\coloneqq \\left\\{ \\left.\\bm{\\rho} \\in \\prod_{w \\in \\mathfrak{W}_n} \\C[x_\\bullet]\\; \\right|\\; \\bm{\\rho}(w) - \\bm{\\rho}(w(i,j)) \\in \\left\\langle x_{w(i)}-x_{w(j)} \\right\\rangle\\text{ if } (i,j) \\in S(H) \\right\\}\\!,\n \\]\nwith (graded) $\\mathfrak{W}_n$-module structure $w\\cdot \\bm{\\rho}(v) = w\\bm{\\rho}(w^{-1}v)$ and (graded) $\\C[x_\\bullet]$-module structure given by multiplication. \\par \nThe ring $\\mathcal{M}_{H}$ is isomorphic to the $T$-equivariant cohomology of a smooth subvariety of the full flag variety $G/B$ called a \\emph{regular semisimple Hessenberg variety}. The connection between this combinatorial ring and equivariant cohomology is called \\emph{GKM Theory} \\cite{GKM_theory}, and it allows for the study of this cohomology in purely combinatorial and algebraic terms. \\par \nThe $\\mathfrak{W}_n$-module structure on $\\mathcal{M}_{H}$ was first defined as the \\emph{dot action} on equivariant cohomology by Tymoczko in \\cite{tymoczko2008permutation}. There are two natural $\\mathfrak{W}_n$-equivariant quotients, called the left $\\mathrm{L}_{H}$ and right $\\mathrm{R}_{H}$ quotients, of $\\mathcal{M}_{H}$ that are in fact graded $\\mathbb{C}$-vector spaces (the left quotient corresponds to ordinary cohomology). The graded $\\mathfrak{W}_n$-module structure of $\\mathcal{M}_{H}$ induces graded $\\mathfrak{W}_n$-representations on the quotients. The primary goal of this paper is to compute the characters of the degree-one pieces of \n\\[\n\\mathrm{L}_{H} = \\bigoplus_{i\\geq 0} \\left(\\mathrm{L}_{H}\\right)_i \\;\\; \\text{ and }\\;\\;\\mathrm{R}_{H} = \\bigoplus_{i\\geq 0} \\left(\\mathrm{R}_{H}\\right)_i\n\\] from the combinatorial data of $S(H)$. In doing so, we compute the dot action on second (equivariant and ordinary) cohomology for all regular semisimple Hessenberg varieties in types B and C. This generalizes what is known in type A \\cite{chow_linearp,chohonglee_second_cohom,ayzenberg2022second}.\\par \nThese graded representations are of interest to algebraic combinatorists in part because they are the $\\mathfrak{W}_n$-equivalents of very well-studied $\\mathfrak{S}_n$-representations, which have connections to chromatic symmetric functions \\cite{guaypaquet2016shar_wachs_conj,SW2016chromaticquasisymmetric,brosnan_chow_dotactn_is_chromsym} and LLT polynomials \\cite{guaypaquet2016shar_wachs_conj,Ayzenberg2018isospectral, ALEXANDERSSON2018LLTchromsym}. We detail some of these connections in the Appendix. From these works, in type A there are combinatorial formulas to compute the characters of both $\\mathrm{L}_{H}$ and $\\mathrm{R}_{H}$ in at least one basis for the character space of $\\mathfrak{S}_n$. This is \\textbf{not} the case in types B or C.\\par \nFor one specific family of $H$ (those corresponding to simple roots), the character of $\\mathrm{L}_{H}$ was computed by Stembridge \\cite{Stembridge_Permuto} in general type. Stembridge's results are via a connection to \\emph{permutohedra}, as the associated Hessenberg varieties are in fact toric varieties. While it is possible to compute the left and right characters for general $H$ in types B and C, even partial results are rare and rely on sophisticated geometric tools \\cite{BalibanuCrooks_sheaves,PrecupSommers_sheaves}. Formulas are non-combinatorial (and usually non-positive), require the computation of type B/C Green polynomials and Poincar\\'e polynomials of other varieties, and proofs require the geometry of perverse sheaves. \\par \nIn contrast, for the degree-one graded piece, the main result of this paper applies to all Hessenberg spaces $H$ and is entirely computable from the combinatorics of roots and transpositions in $\\mathfrak{W}_n$. Moreover, we compute the degree-one pieces of the characters $\\mathfrak{ch}\\left(\\mathrm{L}_{H}\\right)$ and $\\mathfrak{ch}\\left(\\mathrm{R}_{H}\\right)$ in a positive manner.\n\n\\setcounter{tocdepth}{1}\n\nOur theorem is the first type B/C result to provide an explicit non-negative expansion for a graded piece of $\\lrep{H}$ or $\\rrep{H}$ in terms of characters of irreducible- and/or permutation-representations for all Hessenberg spaces. It is also the first to compute the characters for any graded piece of $\\lrep{H}$ and $\\rrep{H}$ by directly providing a basis for the representation, and in doing so gives a family of characters with which one can attempt to expand the catalog of results and connections that exist in type A to types B and C.\\par\n\nThe right character follows from the general procedure of ``removing\" from any permutation representation with permutation basis $\\{v_i\\}$ the trivial sub-representation $\\C\\left\\{ \\sum v_i \\right\\}$. In particular, if $\\mathfrak{p}$ is the character of a representation with permutation basis $\\{v_i\\}$, then the character on the complementary subspace $\\C\\{v_i-v_{i+1}\\}$ is $\\mathfrak{p}-\\mathbb{1}$.\n\\end{proof}\nIn type B, $t_n \\in S(H)$ implies that $t_{n-1} \\in S(H)$, and so it is impossible to get the $\\bm{\\delta}$ term from Theorem \\ref{thm:character}. In type C the opposite is true: $t_{n-1} \\in S(H)$ implies that $t_n \\in S(H)$, and so $\\delta$ may appear. This is the ``moreover\" part of Theorem \\ref{intthm:character}.\n\\begin{example}\n The table below gives every $\\lrep{H}_1$ and $\\lrep{H}_1$ for $n=4$.\n\\begin{table}[H]\n\\centering\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabular}{|c|p{2in}|p{2in}|p{0.3in}|}\\hline\n { $t_i \\in S(H)$} & {\\large $\\lrep{H}_1$} & \\large $\\rrep{H}_1$ &\\large dim\\\\\\hline\n $\\emptyset$ & $\\fh_1 +\\fh_2+\\fh_3+\\fh_4-\\bchi$ & $\\fh_1 +\\fh_2+\\fh_3+\\fh_4-4\\triv$ & $76$ \\\\\\hline\n $\\{t_1\\}$ & $\\triv + \\fh_3 + \\fh_4 + \\fs$ &$\\bchi + \\fh_3 + \\fh_4 + \\fs -3\\triv$ & $53$\\\\\\hline\n $\\{t_2\\}$ & $\\triv+\\fh_1+\\fh_4+\\fs $ &$\\bchi+\\fh_1+\\fh_4+\\fs -3\\triv$ &$29$\\\\\\hline\n $\\{t_3\\}$ & $\\triv+\\fh_1+\\fh_2+\\fs $ &$\\bchi+\\fh_1+\\fh_2+\\fs -3\\triv$ &$37$\\\\\\hline\n $\\{t_4\\}$ & $2\\triv+\\fh_1+\\fh_2+\\bm{\\delta}$ &$\\bchi+\\fh_1+\\fh_2+\\bm{\\delta}-2\\triv$ &$35$\\\\\\hline\n $\\{t_1,t_2\\}$ & $2\\triv+\\fh_4+\\fs $ &$\\bchi+\\fh_4+\\fs -2\\triv$ &$22$\\\\\\hline\n $\\{t_1,t_3\\}$ & $2\\triv+\\fh_1+\\fs $ &$\\bchi+\\fh_1+\\fs -2\\triv$ &$14$\\\\\\hline\n $\\{t_1,t_4\\}$ & $3\\triv+\\fh_1 +\\bm{\\delta}$ &$\\bchi+\\fh_1 +\\bm{\\delta}-\\triv$ &$12$\\\\\\hline\n $\\{t_2,t_3\\}$ & $2\\triv+\\fh_1+\\fs $ &$\\bchi+\\fh_1+\\fs -2\\triv$ &$14$\\\\\\hline\n $\\{t_2,t_4\\}$ & $3\\triv+\\fh_1+\\bm{\\delta} $ &$\\bchi+\\fh_1+\\bm{\\delta} -\\triv$ &$12$\\\\\\hline\n $\\{t_3,t_4\\}$ & $2\\triv+\\fh_1+\\fh_2 $ &$\\bchi+\\fh_1+\\fh_2 -2\\triv$ &$34$\\\\\\hline\n $\\{t_1,t_2,t_3\\}$ &$3\\triv+\\fs $ &$\\bchi+\\fs -\\triv$ &$7$\\\\\\hline\n $\\{t_1,t_2,t_4\\}$ &$4\\triv+\\bm{\\delta} $ &$\\bchi+\\bm{\\delta} $ &$5$\\\\\\hline\n $\\{t_1,t_3,t_4\\}$ &$3\\triv+\\fh_1$ &$\\bchi+\\fh_1-\\triv$ &$11$\\\\\\hline\n $\\{t_2,t_3,t_4\\}$ & $3\\triv+\\fh_1 $ &$\\bchi+\\fh_1 -\\triv$ &$11$\\\\\\hline\n $\\{t_1,t_2,t_3,t_4\\}$ & $4\\mathbb{1}$ &$\\bchi$ &$4$\\\\\\hline\n\n\\clearpage \n\\begin{table}[H]\n \\centering\n \\renewcommand{\\arraystretch}{2.5}\n \\begin{tabular}{|p{3.5in}|p{2.3in}|}\\hline\n {\\large Representations or Characters of $\\Wn$} & {\\large Expansions in $\\Lambda_n(x,y)$} \\\\\\hline\n The trivial character $\\mathbb{1}$ & $s_{(n),\\emptyset} = h_{(n),\\emptyset}$ \\\\\\hline\n The sign character $\\mathrm{sgn}$ & $s_{\\emptyset,(1^n)}$ \\\\\\hline\n $\\bm{\\delta}$ where $\\bm{\\delta}(w) = (-1)^{\\abs{\\Neg(w)}}$ & $s_{\\emptyset,(n)} = h_{\\emptyset,(n)}$ \\\\\\hline\n The irreducible character $\\chi^{(\\lambda,\\mu)}$ & $s_{\\lambda,\\mu}$ \\\\\\hline\n $\\mathrm{sgn} \\otimes \\chi^{(\\lambda,\\mu)} $ & $s_{\\mu^t,\\lambda^t}$ \\\\\\hline\n $\\bm{\\delta} \\otimes \\chi^{(\\lambda,\\mu)} $ & $s_{\\mu,\\lambda}$ \\\\\\hline\n The defining representation on $\\C^n$ (so $\\bm{\\chi}$) & $h_{(n-1),(1)}$ \\\\\\hline \n Action on cosets of $\\mathfrak{S}_k\\times \\mathfrak{W}_{n-k}$ (so $\\fh_k$) & $\\ds h_{(n-k),\\emptyset} \\sum_{j=0}^{\\lambda_i} h_{(j),(k-j)}$ \\\\\\hline\n Action on cosets of $\\mathfrak{W}_1\\times \\mathfrak{W}_{n-1}$ (so $\\fs$) & $\\ds h_{(n-1,1),\\emptyset}$ \\\\\\hline\n Action on cosets of $\\mathfrak{S}_\\lambda \\times \\mathfrak{W}_\\mu$ & $\\ds h_{\\mu,\\emptyset} \\prod_{\\lambda_i \\in \\lambda} \\sum_{j=0}^{\\lambda_i} h_{(j),(\\lambda_i-j)}$ \\\\\\hline\n Action on cosets of the type D${}_n$ subgroup & $\\ds h_{(n),\\emptyset} + h_{\\emptyset,(n)} $ \\\\\\hline\n An induced character $\\left(\\chi \\times \\theta\\right)_{\\mathfrak{W}_k \\times \\mathfrak{W}_\\ell}^{\\Wn}$ & $\\Frob_{BC}(\\chi)\\Frob_{BC}(\\theta)$\\\\\\hline\n The induced character $\\left(\\chi^{(\\lambda,\\mu)} \\times \\chi^{(\\alpha,\\beta)}\\right)_{\\mathfrak{W}_k \\times \\mathfrak{W}_\\ell}^{\\Wn}$ & $(s_\\lambda(x)\\cdot s_\\alpha(x))(s_\\mu(y)\\cdot s_\\beta(y))$\\\\\\hline\n The induced character $\\left(\\mathbb{1}\\right)_{\\Sn}^{\\Wn}$ & ${\\ds \\sum_{k=0}^n s_{(k),(n-k)} = \\sum_{k=0}^n h_{(k),(n-k)}}$ \\\\\\hline\n The induced character $\\left(\\mathbb{1} \\times \\bm{\\delta} \\right)_{\\mathfrak{W}_\\lambda \\times \\mathfrak{W}_\\mu}^{\\Wn}$ & $\\ds h_{\\lambda,\\mu} = \\sum_{\\gamma}\\sum_{\\nu} K_{\\gamma,\\lambda}K_{\\nu,\\mu}s_\\gamma(x)s_\\nu(y)$ \\\\\\hline\n \\end{tabular}\\vspace*{4pt}\n \\captionsetup{width=6.2in}\n \\caption{Representations of $\\Wn$ and Frobenius characteristics \\cite{GeissKinch_hyperoctChars,Macdonald_symfuncs, Zelevinsky_reps_classicalgroups, Stembridge_Permuto, Stembridge_guide, Skandera_hyperoct}.}\n \\label{tab:reps}\n\\end{table}\n\\clearpage", + "post_theorem_intro_text_len": 3521, + "post_theorem_intro_text": "Theorem \\ref{intthm:character} is Proposition \\ref{prop:permutohedral} if $H = \\Delta$ and Theorem \\ref{thm:character} otherwise, which use language from Definition \\ref{def:itypes} to describe the coefficients $a,b,c$, and $d$. The representation ring of $\\mathfrak{W}_n$ is isomorphic to the \\emph{type B/C symmetric functions} via the Frobenius characteristic map \\cite{Macdonald_symfuncs}. The tools to translate Theorem \\ref{intthm:character} into type B/C symmetric functions are in the Appendix. There, we make the observation that $\\mathfrak{ch}\\left(\\mathrm{L}_{H}\\right)_1$ is in fact $h_{\\lambda,\\mu}$-positive. This is particularly interesting, as it generalizes the behavior seen in type A, where the left graded character is conjectured to be $h_\\lambda$-positive and is intimately connected to \\emph{chromatic symmetric functions}. This is known as the graded Stanley-Stembridge conjecture \\cite{StanleyStembridge,SW2016chromaticquasisymmetric}, and has been the subject of much research \\cite{STANLEY1995chromsym, Gasharov96, GuayPaquet13, harada2017cohomology, brosnan_chow_dotactn_is_chromsym,Dahlberg19, Abreu_Nigro20}. A proof of the ungraded conjecture was recently given by Hikita \\cite{Hikita_stanstem}.\\par\n\nOur theorem is the first type B/C result to provide an explicit non-negative expansion for a graded piece of $\\mathfrak{ch}\\left(\\mathrm{L}_{H}\\right)$ or $\\mathfrak{ch}\\left(\\mathrm{R}_{H}\\right)$ in terms of characters of irreducible- and/or permutation-representations for all Hessenberg spaces. It is also the first to compute the characters for any graded piece of $\\mathfrak{ch}\\left(\\mathrm{L}_{H}\\right)$ and $\\mathfrak{ch}\\left(\\mathrm{R}_{H}\\right)$ by directly providing a basis for the representation, and in doing so gives a family of characters with which one can attempt to expand the catalog of results and connections that exist in type A to types B and C.\\par \n\nThe paper is structured as follows. Section \\ref{sec:background} provides some of the necessary background on signed permutations, Hessenberg spaces, and splines. Section \\ref{sec:ideals_of_transpositions} shows how to translate from Hessenberg spaces $H$ to sets $S(H)$ of signed transpositions in a manner that unifies some type B and type C calculations. Section \\ref{sec:trivial} reduces the $S(H)$ that one must consider in order to compute $\\mathcal{M}_{H}^1$ to a much smaller collection, one that completely unifies the type B and type C calculations. Section \\ref{sec:one_inversion} determines which elements of $\\mathfrak{W}_n$ have exactly one $H$-inversion for each $H$. Section \\ref{sec:linear_splines} defines several sets of splines contained in $\\mathcal{M}_{H}^1$, provides linear relations between them, and computes the dot action on them. Section \\ref{sec:generators} argues, using the elements from \\S \\ref{sec:one_inversion}, that the splines from \\S \\ref{sec:linear_splines} form a $\\mathbb{C}$-generating set for $\\mathcal{M}_{H}^1$, and uses the linear relations between them to reduce the size of this generating set. Section \\ref{sec:bases_reps} reduces this generating set further in two different ways, one that results in a basis for $\\mathcal{M}_{H}^1$ that is conducive to computing $\\mathfrak{ch}\\left(\\mathrm{L}_{H}\\right)_1$ and one that is conducive to computing $\\mathfrak{ch}\\left(\\mathrm{R}_{H}\\right)_1$. Then, we prove the main theorem. Appendix \\ref{sec:symmetricfunctions} describes how one can translate this result to the language of type B/C symmetric functions.", + "sketch": "Theorem~\\ref{intthm:character} is obtained by giving an explicit basis for the relevant graded piece and computing the dot action on it: the paper “directly provid[es] a basis for the representation,” and thereby computes “the characters for any graded piece of $\\mathfrak{ch}(\\mathrm{L}_{H})$ and $\\mathfrak{ch}(\\mathrm{R}_{H})$” (in particular degree $1$ here). The outlined proof route is:\n\\begin{itemize}\n\\item Translate from Hessenberg spaces $H$ to sets $S(H)$ of signed transpositions (Section~\\ref{sec:ideals_of_transpositions}).\n\\item Reduce the $S(H)$ that must be considered to compute $\\mathcal{M}_{H}^1$ to “a much smaller collection” (Section~\\ref{sec:trivial}).\n\\item Determine which elements of $\\mathfrak{W}_n$ have “exactly one $H$-inversion for each $H$” (Section~\\ref{sec:one_inversion}).\n\\item Define “several sets of splines contained in $\\mathcal{M}_{H}^1$,” give “linear relations between them,” and “compute the dot action on them” (Section~\\ref{sec:linear_splines}).\n\\item Use the one-$H$-inversion elements to argue these splines form a $\\mathbb{C}$-generating set for $\\mathcal{M}_{H}^1$, then use the linear relations to “reduce the size of this generating set” (Section~\\ref{sec:generators}).\n\\item Further reduce to bases “conducive to computing $\\mathfrak{ch}(\\mathrm{L}_{H})_1$” and “conducive to computing $\\mathfrak{ch}(\\mathrm{R}_{H})_1$,” and then “prove the main theorem” (Section~\\ref{sec:bases_reps}).\n\\end{itemize}\nAdditionally, the Appendix develops the translation to type B/C symmetric functions and notes the positivity phenomenon: “we make the observation that $\\mathfrak{ch}(\\mathrm{L}_{H})_1$ is in fact $h_{\\lambda,\\mu}$-positive.”", + "expanded_sketch": "In establishing the main theorem, one gives an explicit basis for the relevant graded piece and computes the dot action on it: the paper “directly provid[es] a basis for the representation,” and thereby computes “the characters for any graded piece of $\\mathfrak{ch}(\\mathrm{L}_{H})$ and $\\mathfrak{ch}(\\mathrm{R}_{H})$” (in particular degree $1$ here). The outlined proof route is:\n\\begin{itemize}\n\\item Translate from Hessenberg spaces $H$ to sets $S(H)$ of signed transpositions.\n\\item Reduce the $S(H)$ that must be considered to compute $\\mathcal{M}_{H}^1$ to “a much smaller collection”.\n\\item Determine which elements of $\\mathfrak{W}_n$ have “exactly one $H$-inversion for each $H$”.\n\\item Define “several sets of splines contained in $\\mathcal{M}_{H}^1$,” give “linear relations between them,” and “compute the dot action on them”.\n\\item Use the one-$H$-inversion elements to argue these splines form a $\\mathbb{C}$-generating set for $\\mathcal{M}_{H}^1$, then use the linear relations to “reduce the size of this generating set”.\n\\item Further reduce to bases “conducive to computing $\\mathfrak{ch}(\\mathrm{L}_{H})_1$” and “conducive to computing $\\mathfrak{ch}(\\mathrm{R}_{H})_1$,” and then prove the main theorem.\n\\end{itemize}\nAdditionally, the Appendix develops the translation to type B/C symmetric functions and notes the positivity phenomenon: “we make the observation that $\\mathfrak{ch}(\\mathrm{L}_{H})_1$ is in fact $h_{\\lambda,\\mu}$-positive.”", + "expanded_theorem": "\\label{intthm:character}\n If $H$ is a type B or type C Hessenberg space, there exist non-negative integers $a,b \\in \\mathbb{N}$, integers $c,d \\in \\{0,1\\}$, and a subset $I \\subseteq [n]$, each determined by $S(H)$, such that\n \\[\n \\mathfrak{ch}\\left(\\mathrm{L}_{H}\\right)_1 = a\\,\\mathbb{1} + \\sum_{i\\in I} \\fh_i + b\\,\\fh_1 + c\\,\\mathfrak{s} + d\\,\\bm{\\delta} \n \\]\n and \n \\[\n \\mathfrak{ch}\\left(\\mathrm{R}_{H}\\right)_1 = \\bm{\\chi} + \\sum_{i\\in I} \\left(\\fh_i - \\mathbb{1}\\right) + b\\left(\\fh_1-\\mathbb{1}\\right) + c \\left(\n \\mathfrak{s}-\\mathbb{1}\\right) + d\\,\\bm{\\delta},\n \\]\n where \n \\begin{itemize}\n \\item $\\bm{\\chi}$ is the character of the defining representation of $\\mathfrak{W}_n$ on $\\mathbb{C}^n$\n \\item $\\fh_i$ is the character of the action of $\\mathfrak{W}_n$ on the cosets of $\\mathfrak{S}_i\\times \\mathfrak{W}_{n-i}$,\n \\item $\\mathfrak{s}$ is the character of the action of $\\mathfrak{W}_n$ on the cosets of $\\mathfrak{W}_1\\times \\mathfrak{W}_{n-1}$,\n \\item $\\mathbb{1}$ is the character of the trivial representation, and \n \\item $\\bm{\\delta}$ is the character $w\\mapsto (-1)^{\\left| \\Neg(w) \\right|}$ where $\\Neg(w) = \\{w(i) \\mid i \\in [n],\\; w(i) < 0\\}$.\n \\end{itemize}\n Moreover, $c$ and $d$ are never simultaneously $1$, and $d$ is always $0$ in type B.,", + "theorem_type": [ + "Implication", + "Existence" + ], + "mcq": { + "question": "Let \\(\\mathfrak{W}_n\\) be the group of signed permutations, i.e. permutations \\(w\\) of \\(\\{1,\\ldots,n,-n,\\ldots,-1\\}\\) satisfying \\(w(-i)=-w(i)\\). Let \\(H\\) be a type B or type C Hessenberg space, and let \\(S(H)\\) be its associated set of transpositions in \\(\\mathfrak{W}_n\\). Let \\((\\mathrm{L}_H)_1\\) and \\((\\mathrm{R}_H)_1\\) denote the degree-one graded pieces of the left and right \\(\\mathfrak{W}_n\\)-representation quotients associated to \\(H\\), and let \\(\\mathfrak{ch}((\\mathrm{L}_H)_1)\\) and \\(\\mathfrak{ch}((\\mathrm{R}_H)_1)\\) be their characters. Here \\(\\mathbb{1}\\) is the trivial character, \\(\\bm{\\chi}\\) is the character of the defining representation of \\(\\mathfrak{W}_n\\) on \\(\\mathbb{C}^n\\), \\(\\mathfrak{h}_i\\) is the character of the action of \\(\\mathfrak{W}_n\\) on the cosets of \\(\\mathfrak{S}_i\\times \\mathfrak{W}_{n-i}\\), \\(\\mathfrak{s}\\) is the character of the action of \\(\\mathfrak{W}_n\\) on the cosets of \\(\\mathfrak{W}_1\\times \\mathfrak{W}_{n-1}\\), and \\(\\bm{\\delta}\\) is the character \\(w\\mapsto (-1)^{|\\Neg(w)|}\\), where \\(\\Neg(w)=\\{w(i)\\mid i\\in[n],\\; w(i)<0\\}\\). Which of the following conclusions about these degree-one characters holds?", + "correct_choice": { + "label": "A", + "text": "There exist non-negative integers \\(a,b\\in\\mathbb{N}\\), integers \\(c,d\\in\\{0,1\\}\\), and a subset \\(I\\subseteq [n]\\), each determined by \\(S(H)\\), such that\n\\[\n\\mathfrak{ch}\\big((\\mathrm{L}_H)_1\\big)=a\\,\\mathbb{1}+\\sum_{i\\in I}\\mathfrak{h}_i+b\\,\\mathfrak{h}_1+c\\,\\mathfrak{s}+d\\,\\bm{\\delta},\n\\]\nand\n\\[\n\\mathfrak{ch}\\big((\\mathrm{R}_H)_1\\big)=\\bm{\\chi}+\\sum_{i\\in I}(\\mathfrak{h}_i-\\mathbb{1})+b(\\mathfrak{h}_1-\\mathbb{1})+c(\\mathfrak{s}-\\mathbb{1})+d\\,\\bm{\\delta}.\n\\]\nMoreover, \\(c\\) and \\(d\\) are never simultaneously equal to \\(1\\), and in type B one always has \\(d=0\\)." + }, + "choices": [ + { + "label": "B", + "text": "There exist non-negative integers \\(a,b\\in\\mathbb{N}\\), integers \\(c,d\\in\\{0,1\\}\\), and a subset \\(I\\subseteq [n]\\), each determined by \\(S(H)\\), such that\n\\[\n\\mathfrak{ch}\\big((\\mathrm{L}_H)_1\\big)=a\\,\\mathbb{1}+\\sum_{i\\in I}\\mathfrak{h}_i+b\\,\\mathfrak{h}_1+c\\,\\mathfrak{s}+d\\,\\bm{\\delta},\n\\]\nand\n\\[\n\\mathfrak{ch}\\big((\\mathrm{R}_H)_1\\big)=\\bm{\\chi}+\\sum_{i\\in I}(\\mathfrak{h}_i-\\mathbb{1})+b(\\mathfrak{h}_1-\\mathbb{1})+c(\\mathfrak{s}-\\mathbb{1})+d\\,\\bm{\\delta}.\n\\]\nMoreover, one always has \\(c=d=0\\) in type B, and in type C the coefficients \\(c\\) and \\(d\\) may both be equal to \\(1\\)." + }, + { + "label": "C", + "text": "There exist non-negative integers \\(a,b\\in\\mathbb{N}\\), integers \\(c,d\\in\\{0,1\\}\\), and a subset \\(I\\subseteq [n]\\), each determined by \\(S(H)\\), such that\n\\[\n\\mathfrak{ch}\\big((\\mathrm{L}_H)_1\\big)=a\\,\\mathbb{1}+\\sum_{i\\in I}\\mathfrak{h}_i+b\\,\\mathfrak{h}_1+c\\,\\mathfrak{s}+d\\,\\bm{\\delta},\n\\]\nand\n\\[\n\\mathfrak{ch}\\big((\\mathrm{R}_H)_1\\big)=\\bm{\\chi}+\\sum_{i\\in I}(\\mathfrak{h}_i-\\mathbb{1})+b(\\mathfrak{h}_1-\\mathbb{1})+c(\\mathfrak{s}-\\mathbb{1})+d\\,\\bm{\\delta}.\n\\]" + }, + { + "label": "D", + "text": "There exist non-negative integers \\(a,b\\in\\mathbb{N}\\), integers \\(c,d\\in\\{0,1\\}\\), and a subset \\(I\\subseteq [n]\\), each determined by \\(S(H)\\), such that\n\\[\n\\mathfrak{ch}\\big((\\mathrm{L}_H)_1\\big)=a\\,\\mathbb{1}+\\sum_{i\\in I}\\mathfrak{h}_i+b\\,\\mathfrak{h}_1+c\\,\\mathfrak{s}+d\\,\\bm{\\delta},\n\\]\nand\n\\[\n\\mathfrak{ch}\\big((\\mathrm{R}_H)_1\\big)=\\bm{\\chi}+\\sum_{i\\in I}(\\mathfrak{h}_i-\\mathbb{1})+b(\\mathfrak{h}_1-\\mathbb{1})+c(\\mathfrak{s}-\\mathbb{1})+d(\\bm{\\delta}-\\mathbb{1}).\n\\]\nMoreover, \\(c\\) and \\(d\\) are never simultaneously equal to \\(1\\), and in type B one always has \\(d=0\\)." + }, + { + "label": "E", + "text": "There exist non-negative integers \\(a,b\\in\\mathbb{N}\\), integers \\(c,d\\in\\{0,1\\}\\), and a subset \\(I\\subseteq [n-1]\\), each determined by \\(S(H)\\), such that\n\\[\n\\mathfrak{ch}\\big((\\mathrm{L}_H)_1\\big)=a\\,\\mathbb{1}+\\sum_{i\\in I}\\mathfrak{h}_i+b\\,\\mathfrak{h}_1+c\\,\\mathfrak{s}+d\\,\\bm{\\delta},\n\\]\nand\n\\[\n\\mathfrak{ch}\\big((\\mathrm{R}_H)_1\\big)=\\bm{\\chi}+\\sum_{i\\in I}(\\mathfrak{h}_i-\\mathbb{1})+b(\\mathfrak{h}_1-\\mathbb{1})+c(\\mathfrak{s}-\\mathbb{1})+d\\,\\bm{\\delta}.\n\\]\nMoreover, \\(c\\) and \\(d\\) are never simultaneously equal to \\(1\\), and in type B one always has \\(d=0\\)." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "characteristic", + "tampered_component": "mutual-exclusion-and-type-B-delta-constraint", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped-final-moreover-restrictions-on-c-and-d", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "exact-right-character-basis-term-for-delta", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "case_split", + "tampered_component": "range-of-index-set-I", + "template_used": "boundary_range" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines notation and context but does not explicitly reveal the correct conclusion. There is no direct textual leakage of the extra conditions that distinguish the correct option." + }, + "TAS": { + "score": 1, + "justification": "The item is essentially a theorem-identification question: the correct choice states a full structural conclusion, and the task is to recognize the exact version. It is not a pure tautology, since alternatives differ in meaningful details, but it is close to restating a known result." + }, + "GPS": { + "score": 1, + "justification": "Selecting the answer requires careful comparison of subtle representation-theoretic details, but mostly tests precise recall/discrimination rather than genuine generative derivation or synthesis." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically targeted: one weakens the theorem, others alter boundary conditions or a specific character term, and they align with realistic failure modes such as dropping constraints or confusing basis terms." + }, + "total_score": 6, + "overall_assessment": "A solid but theorem-recognition-heavy MCQ: little answer leakage and strong distractors, but only moderate success at avoiding tautological recall and inducing genuine generative reasoning." + } + }, + { + "id": "2511.13215v1", + "paper_link": "http://arxiv.org/abs/2511.13215v1", + "theorems_cnt": 10, + "theorem": { + "env_name": "theorem", + "content": "\\label{Thm: main1}\n Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have the following mass-capacity inequality\n $$m(M,g,E)\\geq 2\\mathfrak c(M,g,E).$$\n If the equality holds, then $(M,g)$ is harmonically conformal to $\\mathbb R^n\\setminus S$, where $S$ is a bounded closed subset of $\\mathbb R^n$ with Hausdorff dimension no greater than $\\frac{n-2}{2}$. In particular, we have\n $$\\pi_i(M)=0\\mbox{ for all }1\\leq i\\leq n-1- \\left\\lfloor\\frac{n}{2}\\right\\rfloor.$$", + "start_pos": 27647, + "end_pos": 28222, + "label": "Thm: main1" + }, + "ref_dict": { + "Thm: main1": "\\begin{theorem}\\label{Thm: main1}\n Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have the following mass-capacity inequality\n $$m(M,g,E)\\geq 2\\mathfrak c(M,g,E).$$\n If the equality holds, then $(M,g)$ is harmonically conformal to $\\mathbb R^n\\setminus S$, where $S$ is a bounded closed subset of $\\mathbb R^n$ with Hausdorff dimension no greater than $\\frac{n-2}{2}$. In particular, we have\n $$\\pi_i(M)=0\\mbox{ for all }1\\leq i\\leq n-1- \\left\\lfloor\\frac{n}{2}\\right\\rfloor.$$\n\\end{theorem}", + "Cor: Bray": "\\begin{corollary}[Bray's mass capacity inequality]\\label{Cor: Bray}\n Assume that $(M^n,g)$ is a spin asymptotically flat manifold with only one asymptotically flat end $E$, non-empty boundary $\\Sigma$, and nonnegative scalar curvature. \nAssume further that the boundary $\\Sigma$ is minimal. Then we have\n\\begin{equation}\\label{Eq: Bray's mass-capacity}\n m(M,g,E)\\geq \\mathfrak c(M,g,\\Sigma),\n\\end{equation}\nwhere $\\mathfrak c(M,g,\\Sigma)$ denotes the capacity of $\\Sigma$ in $(M,g)$ given by\n$$\\mathfrak c(M,g,\\Sigma)=\\inf_{\\phi\\in\\mathcal C}\\frac{1}{n(n-2)\\omega_n}\\int_M|\\nabla_g\\phi|^2\\,\\mathrm d\\mu_g,$$\n where $\\mathcal C$ denotes the collection of all smooth functions $\\phi$ with compact support such that $\\phi\\equiv 1$ on $\\Sigma$.\nThe equality holds in \\eqref{Eq: Bray's mass-capacity} if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold.\n\\end{corollary}", + "Eq: Bray's mass-capacity": "\\begin{equation}\\label{Eq: Bray's mass-capacity}\n m(M,g,E)\\geq \\mathfrak c(M,g,\\Sigma),\n\\end{equation}", + "Thm: main2": "\\begin{theorem}\\label{Thm: main2}\n Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with corner, which has nonnegative scalar curvature. Assume that the sum of the mean curvatures of the corner on both sides with respect to the outward unit normal is nonnegative. Then we have the following mass-capacity inequality\n $$m(M,g,E)\\geq 2\\mathfrak c(M,g,E).$$\n If the equality holds, then $(M,g)$ is smooth and harmonically conformal to $\\mathbb R^n\\setminus S$, where $S$ is a bounded closed subset of $\\mathbb R^n$ with Hausdorff dimension no greater than $\\frac{n-2}{2}$. In particular, we have\n $$\\pi_i(M)=0\\mbox{ for all }1\\leq i\\leq n-1- \\left\\lfloor\\frac{n}{2}\\right\\rfloor.$$\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 8182, + "pre_theorem_intro_text": "The investigation of mass constitutes a central topic in general relativity. Notably, the basic concept of mass for isolated gravitational systems was first formulated by Arnowitt-Deser-Misner \\cite{Arnowitt-Deser-Misner}, with such systems modeled as asymptotically flat manifolds. Later, Bartnik \\cite{Bartnik1986} and also Chru\\'sciel \\cite{Chru86} showed that the mass is indeed a geometric quantity of asymptotically flat manifolds. We recall the definition of asymptotically flat manifolds and mass as follows.\n\\begin{definition}\n A complete Riemannian $n$-manifold $(M,g)$ with dimension $n\\geq 3$ is asymptotically flat if there is a compact subset $K\\subset M$ such that\n\\begin{itemize}\n\\item[(i)] the complement $M-K$ consists of finitely many ends $\\{E_l\\}_{l=1}^k$, where each end $E_l$ is diffeomorphic to $\\mathbb R^n-\\bar B_1$ with $$\\bar B_1=\\{x\\in \\mathbb R^n:|x|\\leq 1\\},$$\n\\item[(ii)] the metric $g$ on each end $E_l$ has the expression $g=g_{ij}\\mathrm dx_i\\otimes\\mathrm dx_j$ in the Euclidean coordinate chart, where the metric components satisfy the decay condition\n\\begin{equation*}\\label{Eq: decay}\n|g_{ij}-\\delta_{ij}|+|x||\\partial g_{ij}|+|x|^2|\\partial^2 g_{ij}|=O\\left(|x|^{-\\tau}\\right),\n\\end{equation*}\nas $x\\to\\infty$, where $\\tau$ is a positive constant greater than $\\frac{n-2}{2}$,\n\\item[(iii)] the scalar curvature $R(g)$ belongs to $L^1(M,g)$.\n\\end{itemize}\nFor convenience, each end $E_l$ will be called an asymptotically flat end of $(M,g)$.\n\\end{definition}\n\n\\begin{definition}\nLet $(M,g)$ be an asymptotically flat manifold and $E$ be an asymptotically flat end of $(M,g)$.\n The mass of the end $E$ is defined as\n $$m(M,g,E)=\\frac{1}{2n(n-1)\\omega_n}\\lim_{\\rho\\to+\\infty}\\int_{S_\\rho}(\\partial_j g_{ij}-\\partial_i g_{jj})\\frac{x_i}{|x|}\\,\\mathrm d\\sigma,$$\n where $\\omega_n$ denotes the volume of the Euclidean unit ball $B_1^n$, $S_\\rho$ denotes the coordinate sphere $\\{|x|=\\rho\\}$, and $\\mathrm d\\sigma$ is the Euclidean area element of $S_\\rho$.\n\\end{definition}\n\nMany important works have revealed the source of mass. The positive mass conjecture, with the view that nonnegative energy density contributes to nonnegative mass, states that asymptotically flat manifolds with nonnegative scalar curvature have nonnegative mass on each asymptotically flat end, where the mass vanishes on one end if and only if it is the Euclidean space. This conjecture was finally proved by Schoen-Yau \\cite{SY79,SY17} with the minimal surface method, and also by Witten \\cite{Witten81} with the harmonic spinor method under the extra assumption that the asymptotically flat manifold is spin. Recently, the three-dimensional positive mass theorem was given new proofs by Bray-Kazaras-Khuri-Stern \\cite{BKKS22} using level-set method and also by Agostiniani-Mazzieri-Oronzio \\cite{AMO24} using potential method. With black holes taken into consideration, the Penrose inequality states that if an asymptotically flat manifold with only one asymptotically flat end $E$ has nonnegative scalar curvature and the boundary is an outer-minimizing horizon $\\Sigma$, then we have the inequality\n$$\nm(M,g, E)\\geq \\frac{1}{2}\\left(\\frac{|\\Sigma|_g}{|\\mathbb S^{n-1}|}\\right)^{\\frac{n-2}{n-1}},\n$$\nwhere the equality holds if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold\n$(M_m,g_m)$ with\n$$M_m=\\mathbb R^n\\setminus B_{m/2}\\mbox{ and }g_m=\\left(1+\\frac{m}{2}|x|^{2-n}\\right)^{\\frac{4}{n-2}}g_{euc}.$$\nThe Penrose inequality in dimension three was first proved independently by Huisken-Ilmanen \\cite{HI01} and Bray \\cite{Bray01} using different flow methods. Notice that the potential approach developed by Agostiniani-Mazzieri-Oronzio in \\cite{AMO24} also works for the three-dimensional Penrose ineqality. Based on Bray's original proof, Bray-Lee \\cite{BL09} removed the technical use of the Gauss-Bonnet formula and generalized the Penrose inequality up to dimension seven. In order to prove the Penrose inequality, Bray developed a mass-capacity inequality\n$$ m(M,g,E)\\geq \\mathfrak c(M,g,\\Sigma),$$\nwhere $\\mathfrak c(M,g,\\Sigma)$ denotes the capacity of $\\Sigma$ in $(M,g)$. For more works related to mass-capacity inequalities for asymptotically flat manifolds, the audience can refer to the works \\cite{BM08, Miao24, Miao25} and the references therein.\n\nThe study of mass is closely related to conformal geometry. Based upon the fundamental work of Trudinger \\cite{Trudinger68} and Aubin \\cite{Aubin76}, Schoen ultimately resolved the Yamabe problem by leveraging the positive mass theorem. As another link between conformal geometry and general relativity, Schoen-Yau \\cite{Schoen-Yau} reduced the Liouville theorem in conformal geometry to a consequence of the positive mass theorem for a class of generalized asymptotically flat manifolds, which was recently established in the works \\cite{Lesourd-Unger-Yau,LLU23, Zhu23,BC2005,CZ2024}.\n\nIn this paper, we continue the study of mass inequalities for generalized asymptotically flat manifolds, which are initiated from Schoen-Yau's work \\cite{Schoen-Yau}. Such manifolds are also well-known in \\cite{Lesourd-Unger-Yau} as {\\it asymptotically flat manifolds with arbitrary ends} with the following precise definition.\n\\begin{definition}\n A triple $(M,g,E)$ is said to be a generalized asymptotically flat manifold if \n \\begin{itemize}\n \\item $(M,g)$ is a complete Riemannian $n$-manifold {\\it without boundary}, whose dimension $n$ is no less than three;\n \\item $E$ is an asymptotically flat end of $(M,g)$;\n\\item the scalar curvature $R(g)$ belongs to $L^1(E,g)$.\n \\end{itemize}\n\\end{definition}\n\nThe positive mass theorem for generalized asymptotically flat manifolds was systematically researched in \\cite{Lesourd-Unger-Yau,LLU23, Zhu23,BC2005,CZ2024}. Soon after, the Penrose inequality for generalized asymptotically flat manifolds was initiated by the second-named author \\cite{Zhu2024} to include spatial slices of the {\\it extreme} Reissner-Nordstr\\\"om spacetime into application, where the ultimate goal is to show the following {\\it mass-systole conjecture}:\n\\begin{conjecture}\nLet $(M,g, E)$ be a generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have\n$$\nm(M,g, E)\\geq \\frac{1}{2}\\left(\\frac{\\sys(M,g,E)}{|\\mathbb S^{n-1}|}\\right)^{\\frac{n-2}{n-1}},\n$$\nwhere \n\\begin{equation*}\n\\sys(M,g,E)=\\inf\\left\\{|\\Sigma|_g:\\left.\n\\begin{array}{c}\\text{$\\Sigma$ is a smoothly embedded hypersurface }\\\\\n\\text{homologous to $\\partial E$ in $M$}\n\\end{array}\\right.\\right\\}.\n\\end{equation*}\nMoreover, the equality holds if and only if\n\\begin{itemize}\n \\item $(M,g)$ is isometric to the Euclidean space;\n \\item or there is a strictly outer-minimizing minimal $(n-1)$-sphere $\\Sigma_h$ homologous to $\\partial E$ such that the region outside $\\Sigma_h$ is isometric to the half spatial Schwarzschild manifold with mass $m>0$.\n\\end{itemize}\n\\end{conjecture}\nThe conjecture is now only known to be true when $M$ has the underlying topology $\\mathbb R^3\\setminus\\{O\\}$ concerning the work \\cite{Zhu2024}. As a further exploration, we are going to consider the mass-capacity inequality for generalized asymptotically flat manifolds.\n\nTo present our theorem, we need to introduce the {\\it capacity} for generalized asymptotically flat manifolds. \nGiven any constant $s>1$ we shall use $E_s$ to denote the open subset of $E$ diffeomorphic to $\\mathbb R^n\\setminus \\bar B_s$. \nLet $\\eta$ denote a fixed smooth function satisfying $\\eta\\equiv 0$ in $E_2$ and $\\eta\\equiv 1$ outside $E_1$. \n\\begin{definition}\\label{def:capacity}\n The capacity of a generalized asymptotically flat manifold $(M,g,E)$ is defined to be\n $$\\mathfrak c(M,g,E)=\\inf_{\\phi\\in\\mathcal C}\\frac{1}{n(n-2)\\omega_n}\\int_M|\\nabla_g\\phi|^2\\,\\mathrm d\\mu_g,$$\n where $\\mathcal C$ denotes the collection of all smooth functions $\\phi$ such that $\\phi-\\eta$ has compact support.\n\\end{definition}\n\\begin{remark}\n It is a standard fact that we can relax the test function $\\phi$ to be any function in $W^{1,2}_{loc}(M,g)$ such that $\\phi-\\eta$ has compact support.\n\\end{remark}\n\nIn this paper, we are going to prove the following theorem.", + "context": "Many important works have revealed the source of mass. The positive mass conjecture, with the view that nonnegative energy density contributes to nonnegative mass, states that asymptotically flat manifolds with nonnegative scalar curvature have nonnegative mass on each asymptotically flat end, where the mass vanishes on one end if and only if it is the Euclidean space. This conjecture was finally proved by Schoen-Yau \\cite{SY79,SY17} with the minimal surface method, and also by Witten \\cite{Witten81} with the harmonic spinor method under the extra assumption that the asymptotically flat manifold is spin. Recently, the three-dimensional positive mass theorem was given new proofs by Bray-Kazaras-Khuri-Stern \\cite{BKKS22} using level-set method and also by Agostiniani-Mazzieri-Oronzio \\cite{AMO24} using potential method. With black holes taken into consideration, the Penrose inequality states that if an asymptotically flat manifold with only one asymptotically flat end $E$ has nonnegative scalar curvature and the boundary is an outer-minimizing horizon $\\Sigma$, then we have the inequality\n$$\nm(M,g, E)\\geq \\frac{1}{2}\\left(\\frac{|\\Sigma|_g}{|\\mathbb S^{n-1}|}\\right)^{\\frac{n-2}{n-1}},\n$$\nwhere the equality holds if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold\n$(M_m,g_m)$ with\n$$M_m=\\mathbb R^n\\setminus B_{m/2}\\mbox{ and }g_m=\\left(1+\\frac{m}{2}|x|^{2-n}\\right)^{\\frac{4}{n-2}}g_{euc}.$$\nThe Penrose inequality in dimension three was first proved independently by Huisken-Ilmanen \\cite{HI01} and Bray \\cite{Bray01} using different flow methods. Notice that the potential approach developed by Agostiniani-Mazzieri-Oronzio in \\cite{AMO24} also works for the three-dimensional Penrose ineqality. Based on Bray's original proof, Bray-Lee \\cite{BL09} removed the technical use of the Gauss-Bonnet formula and generalized the Penrose inequality up to dimension seven. In order to prove the Penrose inequality, Bray developed a mass-capacity inequality\n$$ m(M,g,E)\\geq \\mathfrak c(M,g,\\Sigma),$$\nwhere $\\mathfrak c(M,g,\\Sigma)$ denotes the capacity of $\\Sigma$ in $(M,g)$. For more works related to mass-capacity inequalities for asymptotically flat manifolds, the audience can refer to the works \\cite{BM08, Miao24, Miao25} and the references therein.\n\nIn this paper, we continue the study of mass inequalities for generalized asymptotically flat manifolds, which are initiated from Schoen-Yau's work \\cite{Schoen-Yau}. Such manifolds are also well-known in \\cite{Lesourd-Unger-Yau} as {\\it asymptotically flat manifolds with arbitrary ends} with the following precise definition.\n\\begin{definition}\n A triple $(M,g,E)$ is said to be a generalized asymptotically flat manifold if \n \\begin{itemize}\n \\item $(M,g)$ is a complete Riemannian $n$-manifold {\\it without boundary}, whose dimension $n$ is no less than three;\n \\item $E$ is an asymptotically flat end of $(M,g)$;\n\\item the scalar curvature $R(g)$ belongs to $L^1(E,g)$.\n \\end{itemize}\n\\end{definition}\n\nThe positive mass theorem for generalized asymptotically flat manifolds was systematically researched in \\cite{Lesourd-Unger-Yau,LLU23, Zhu23,BC2005,CZ2024}. Soon after, the Penrose inequality for generalized asymptotically flat manifolds was initiated by the second-named author \\cite{Zhu2024} to include spatial slices of the {\\it extreme} Reissner-Nordstr\\\"om spacetime into application, where the ultimate goal is to show the following {\\it mass-systole conjecture}:\n\\begin{conjecture}\nLet $(M,g, E)$ be a generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have\n$$\nm(M,g, E)\\geq \\frac{1}{2}\\left(\\frac{\\sys(M,g,E)}{|\\mathbb S^{n-1}|}\\right)^{\\frac{n-2}{n-1}},\n$$\nwhere \n\\begin{equation*}\n\\sys(M,g,E)=\\inf\\left\\{|\\Sigma|_g:\\left.\n\\begin{array}{c}\\text{$\\Sigma$ is a smoothly embedded hypersurface }\\\\\n\\text{homologous to $\\partial E$ in $M$}\n\\end{array}\\right.\\right\\}.\n\\end{equation*}\nMoreover, the equality holds if and only if\n\\begin{itemize}\n \\item $(M,g)$ is isometric to the Euclidean space;\n \\item or there is a strictly outer-minimizing minimal $(n-1)$-sphere $\\Sigma_h$ homologous to $\\partial E$ such that the region outside $\\Sigma_h$ is isometric to the half spatial Schwarzschild manifold with mass $m>0$.\n\\end{itemize}\n\\end{conjecture}\nThe conjecture is now only known to be true when $M$ has the underlying topology $\\mathbb R^3\\setminus\\{O\\}$ concerning the work \\cite{Zhu2024}. As a further exploration, we are going to consider the mass-capacity inequality for generalized asymptotically flat manifolds.\n\nTo present our theorem, we need to introduce the {\\it capacity} for generalized asymptotically flat manifolds. \nGiven any constant $s>1$ we shall use $E_s$ to denote the open subset of $E$ diffeomorphic to $\\mathbb R^n\\setminus \\bar B_s$. \nLet $\\eta$ denote a fixed smooth function satisfying $\\eta\\equiv 0$ in $E_2$ and $\\eta\\equiv 1$ outside $E_1$. \n\\begin{definition}\\label{def:capacity}\n The capacity of a generalized asymptotically flat manifold $(M,g,E)$ is defined to be\n $$\\mathfrak c(M,g,E)=\\inf_{\\phi\\in\\mathcal C}\\frac{1}{n(n-2)\\omega_n}\\int_M|\\nabla_g\\phi|^2\\,\\mathrm d\\mu_g,$$\n where $\\mathcal C$ denotes the collection of all smooth functions $\\phi$ such that $\\phi-\\eta$ has compact support.\n\\end{definition}\n\\begin{remark}\n It is a standard fact that we can relax the test function $\\phi$ to be any function in $W^{1,2}_{loc}(M,g)$ such that $\\phi-\\eta$ has compact support.\n\\end{remark}\n\nIn this paper, we are going to prove the following theorem.", + "full_context": "Many important works have revealed the source of mass. The positive mass conjecture, with the view that nonnegative energy density contributes to nonnegative mass, states that asymptotically flat manifolds with nonnegative scalar curvature have nonnegative mass on each asymptotically flat end, where the mass vanishes on one end if and only if it is the Euclidean space. This conjecture was finally proved by Schoen-Yau \\cite{SY79,SY17} with the minimal surface method, and also by Witten \\cite{Witten81} with the harmonic spinor method under the extra assumption that the asymptotically flat manifold is spin. Recently, the three-dimensional positive mass theorem was given new proofs by Bray-Kazaras-Khuri-Stern \\cite{BKKS22} using level-set method and also by Agostiniani-Mazzieri-Oronzio \\cite{AMO24} using potential method. With black holes taken into consideration, the Penrose inequality states that if an asymptotically flat manifold with only one asymptotically flat end $E$ has nonnegative scalar curvature and the boundary is an outer-minimizing horizon $\\Sigma$, then we have the inequality\n$$\nm(M,g, E)\\geq \\frac{1}{2}\\left(\\frac{|\\Sigma|_g}{|\\mathbb S^{n-1}|}\\right)^{\\frac{n-2}{n-1}},\n$$\nwhere the equality holds if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold\n$(M_m,g_m)$ with\n$$M_m=\\mathbb R^n\\setminus B_{m/2}\\mbox{ and }g_m=\\left(1+\\frac{m}{2}|x|^{2-n}\\right)^{\\frac{4}{n-2}}g_{euc}.$$\nThe Penrose inequality in dimension three was first proved independently by Huisken-Ilmanen \\cite{HI01} and Bray \\cite{Bray01} using different flow methods. Notice that the potential approach developed by Agostiniani-Mazzieri-Oronzio in \\cite{AMO24} also works for the three-dimensional Penrose ineqality. Based on Bray's original proof, Bray-Lee \\cite{BL09} removed the technical use of the Gauss-Bonnet formula and generalized the Penrose inequality up to dimension seven. In order to prove the Penrose inequality, Bray developed a mass-capacity inequality\n$$ m(M,g,E)\\geq \\mathfrak c(M,g,\\Sigma),$$\nwhere $\\mathfrak c(M,g,\\Sigma)$ denotes the capacity of $\\Sigma$ in $(M,g)$. For more works related to mass-capacity inequalities for asymptotically flat manifolds, the audience can refer to the works \\cite{BM08, Miao24, Miao25} and the references therein.\n\nIn this paper, we continue the study of mass inequalities for generalized asymptotically flat manifolds, which are initiated from Schoen-Yau's work \\cite{Schoen-Yau}. Such manifolds are also well-known in \\cite{Lesourd-Unger-Yau} as {\\it asymptotically flat manifolds with arbitrary ends} with the following precise definition.\n\\begin{definition}\n A triple $(M,g,E)$ is said to be a generalized asymptotically flat manifold if \n \\begin{itemize}\n \\item $(M,g)$ is a complete Riemannian $n$-manifold {\\it without boundary}, whose dimension $n$ is no less than three;\n \\item $E$ is an asymptotically flat end of $(M,g)$;\n\\item the scalar curvature $R(g)$ belongs to $L^1(E,g)$.\n \\end{itemize}\n\\end{definition}\n\nThe positive mass theorem for generalized asymptotically flat manifolds was systematically researched in \\cite{Lesourd-Unger-Yau,LLU23, Zhu23,BC2005,CZ2024}. Soon after, the Penrose inequality for generalized asymptotically flat manifolds was initiated by the second-named author \\cite{Zhu2024} to include spatial slices of the {\\it extreme} Reissner-Nordstr\\\"om spacetime into application, where the ultimate goal is to show the following {\\it mass-systole conjecture}:\n\\begin{conjecture}\nLet $(M,g, E)$ be a generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have\n$$\nm(M,g, E)\\geq \\frac{1}{2}\\left(\\frac{\\sys(M,g,E)}{|\\mathbb S^{n-1}|}\\right)^{\\frac{n-2}{n-1}},\n$$\nwhere \n\\begin{equation*}\n\\sys(M,g,E)=\\inf\\left\\{|\\Sigma|_g:\\left.\n\\begin{array}{c}\\text{$\\Sigma$ is a smoothly embedded hypersurface }\\\\\n\\text{homologous to $\\partial E$ in $M$}\n\\end{array}\\right.\\right\\}.\n\\end{equation*}\nMoreover, the equality holds if and only if\n\\begin{itemize}\n \\item $(M,g)$ is isometric to the Euclidean space;\n \\item or there is a strictly outer-minimizing minimal $(n-1)$-sphere $\\Sigma_h$ homologous to $\\partial E$ such that the region outside $\\Sigma_h$ is isometric to the half spatial Schwarzschild manifold with mass $m>0$.\n\\end{itemize}\n\\end{conjecture}\nThe conjecture is now only known to be true when $M$ has the underlying topology $\\mathbb R^3\\setminus\\{O\\}$ concerning the work \\cite{Zhu2024}. As a further exploration, we are going to consider the mass-capacity inequality for generalized asymptotically flat manifolds.\n\nTo present our theorem, we need to introduce the {\\it capacity} for generalized asymptotically flat manifolds. \nGiven any constant $s>1$ we shall use $E_s$ to denote the open subset of $E$ diffeomorphic to $\\mathbb R^n\\setminus \\bar B_s$. \nLet $\\eta$ denote a fixed smooth function satisfying $\\eta\\equiv 0$ in $E_2$ and $\\eta\\equiv 1$ outside $E_1$. \n\\begin{definition}\\label{def:capacity}\n The capacity of a generalized asymptotically flat manifold $(M,g,E)$ is defined to be\n $$\\mathfrak c(M,g,E)=\\inf_{\\phi\\in\\mathcal C}\\frac{1}{n(n-2)\\omega_n}\\int_M|\\nabla_g\\phi|^2\\,\\mathrm d\\mu_g,$$\n where $\\mathcal C$ denotes the collection of all smooth functions $\\phi$ such that $\\phi-\\eta$ has compact support.\n\\end{definition}\n\\begin{remark}\n It is a standard fact that we can relax the test function $\\phi$ to be any function in $W^{1,2}_{loc}(M,g)$ such that $\\phi-\\eta$ has compact support.\n\\end{remark}\n\nIn this paper, we are going to prove the following theorem.\n\n\\begin{abstract}\nIn the spin case, we can establish a mass-capacity inequality for generalized asymptotically flat manifolds $(M,g,E)$ with nonnegative scalar curvature, where the equality implies that $(M,g)$ is harmonically conformal to $\\mathbb R^n\\setminus S$ for a closed bounded subset $S$ of $\\mathbb R^n$ with Hausdorff dimension no greater than $\\frac{n-2}{2}$.\n\\end{abstract}\n\nTo present our theorem, we need to introduce the {\\it capacity} for generalized asymptotically flat manifolds. \nGiven any constant $s>1$ we shall use $E_s$ to denote the open subset of $E$ diffeomorphic to $\\mathbb R^n\\setminus \\bar B_s$. \nLet $\\eta$ denote a fixed smooth function satisfying $\\eta\\equiv 0$ in $E_2$ and $\\eta\\equiv 1$ outside $E_1$. \n\\begin{definition}\\label{def:capacity}\n The capacity of a generalized asymptotically flat manifold $(M,g,E)$ is defined to be\n $$\\mathfrak c(M,g,E)=\\inf_{\\phi\\in\\mathcal C}\\frac{1}{n(n-2)\\omega_n}\\int_M|\\nabla_g\\phi|^2\\,\\mathrm d\\mu_g,$$\n where $\\mathcal C$ denotes the collection of all smooth functions $\\phi$ such that $\\phi-\\eta$ has compact support.\n\\end{definition}\n\\begin{remark}\n It is a standard fact that we can relax the test function $\\phi$ to be any function in $W^{1,2}_{loc}(M,g)$ such that $\\phi-\\eta$ has compact support.\n\\end{remark}\n\nTo include previous Bray's mass-capacity inequality as a special case, we also extend our main theorem to the setting of Riemannian manifolds with corner defined as follows.\n\\begin{definition}\n A triple $(M,g,\\Sigma)$ will be called a Riemannian manifold with corner if $M$ is a smooth manifold equipped with a continuous metric $g$, and $\\Sigma$ is a closed separating smooth hypersurface $\\Sigma$ in $M$ such that the metric $g$ is actually smooth to boundary in the closure of every component of $M\\setminus \\Sigma$. For convenience, $\\Sigma$ will be called the corner of $(M,g)$.\n\\end{definition}\nIn particular, we can consider generalized asymptotically flat manifolds with corner.\n\\begin{definition}\n A generalized asymptotically flat manifold $(M,g,E)$ will be called a generalized asymptotically flat manifold with corner, if there exists a closed separating smooth hypersurface $\\Sigma$ such that $(M,g,\\Sigma)$ is a Riemannian manifold with corner.\n\\end{definition}\n\nThroughout this paper, we always take the convention that $\\mathbb S^2$ has mean curvature two in $\\mathbb R^3$ with respect to the outward unit normal. With extra mean-curvature condition along the corner, we can prove\n\\begin{theorem}\\label{Thm: main2}\n Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with corner, which has nonnegative scalar curvature. Assume that the sum of the mean curvatures of the corner on both sides with respect to the outward unit normal is nonnegative. Then we have the following mass-capacity inequality\n $$m(M,g,E)\\geq 2\\mathfrak c(M,g,E).$$\n If the equality holds, then $(M,g)$ is smooth and harmonically conformal to $\\mathbb R^n\\setminus S$, where $S$ is a bounded closed subset of $\\mathbb R^n$ with Hausdorff dimension no greater than $\\frac{n-2}{2}$. In particular, we have\n $$\\pi_i(M)=0\\mbox{ for all }1\\leq i\\leq n-1- \\left\\lfloor\\frac{n}{2}\\right\\rfloor.$$\n\\end{theorem}\nAs a direct consequence, we can show\n\\begin{corollary}[Bray's mass capacity inequality]\\label{Cor: Bray}\n Assume that $(M^n,g)$ is a spin asymptotically flat manifold with only one asymptotically flat end $E$, non-empty boundary $\\Sigma$, and nonnegative scalar curvature. \nAssume further that the boundary $\\Sigma$ is minimal. Then we have\n\\begin{equation}\\label{Eq: Bray's mass-capacity}\n m(M,g,E)\\geq \\mathfrak c(M,g,\\Sigma),\n\\end{equation}\nwhere $\\mathfrak c(M,g,\\Sigma)$ denotes the capacity of $\\Sigma$ in $(M,g)$ given by\n$$\\mathfrak c(M,g,\\Sigma)=\\inf_{\\phi\\in\\mathcal C}\\frac{1}{n(n-2)\\omega_n}\\int_M|\\nabla_g\\phi|^2\\,\\mathrm d\\mu_g,$$\n where $\\mathcal C$ denotes the collection of all smooth functions $\\phi$ with compact support such that $\\phi\\equiv 1$ on $\\Sigma$.\nThe equality holds in \\eqref{Eq: Bray's mass-capacity} if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold.\n\\end{corollary}\n\n\\begin{proposition}\\label{Prop: mass capacity}\n If $(M,g)$ has nonnegative scalar curvature, then we have\n $$m(M,g,E)\\geq 2 \\mathfrak c(M,g,E).$$\n\\end{proposition}\n\\begin{proof}\n Given any constant $\\varepsilon>0$ we define\n \\begin{equation}\\label{Eq: approximation metric}\n g_\\varepsilon=u_\\varepsilon^{\\frac{4}{n-2}}g\\mbox{ with } u_\\varepsilon=\\frac{u+\\varepsilon}{1+\\varepsilon},\n \\end{equation}\n where $u$ is the positive harmonic function from Corollary \\ref{Cor: function u}.\n Note that $u_\\varepsilon$ has a positive lower bound, so $(M,g_\\varepsilon)$ remains to be a complete Riemannian manifold without boundary. It follows from the expansion of $u$ in Corollary \\ref{Cor: function u} that $(M,g_\\varepsilon,E)$ remains to be a generalized asymptotically flat manifold and that we have\n$$m(M,g_\\varepsilon,E)=m(M,g,E)-2(1+\\varepsilon)^{-1}\\mathfrak c(M,g,E).$$\n Clearly, $u_\\varepsilon$ is a harmonic function on $(M,g)$ since $u$ is. We see that $(M,g_\\varepsilon)$ has nonnegative scalar curvature. Then it follows from the spin positive mass theorem (see \\cite{BC2005, CZ2024} for instance) for generalized asymptotically flat manifolds that we have\n $$m(M,g_\\varepsilon,E)\\geq 0,$$ which yields\n $$m(M,g,E)\\geq 2(1+\\varepsilon)^{-1}\\mathfrak c(M,g,E).$$\n The proof is now completed by letting $\\varepsilon\\to 0$.\n\\end{proof}\n\\begin{remark}\n If one simply takes the function $u$ to be the conformal factor, then the positive mass theorem may not be applied since the Riemannian manifold after conformal deformation can be incomplete. This explains the reason why we have to use an approximation argument for the mass-capacity inequality.\n\\end{remark}\n\n\\begin{proof}[Proof of Theorem \\ref{Thm: main2} ]\n By repeating the proof of Corollary \\ref{Cor: function u} we are able to construct a positive weakly harmonic $C^{1,\\alpha}$-function $u$ on $(M,g)$ such that we have the expansion \n $$u(x)=1-\\mathfrak c(M,g,E)\\cdot|x|^{2-n}+w,$$\n with $w=O_2(|x|^{2-n-\\tau})$ as $x\\to\\infty$. For any constant $\\varepsilon>0$, the conformal manifold\n $(M,g_\\varepsilon,E)$ with $$g_\\varepsilon=\\left(\\frac{u+\\varepsilon}{1+\\varepsilon}\\right)^{\\frac{4}{n-2}}g$$ is a spin generalized asymptotically flat manifold with corner satisfying the assumption of Proposition \\ref{Prop: spin PMT with corner}. As a consequence, we have\n $$0\\leq m(M,g_\\varepsilon,E)=m(M,g,E)-2(1+\\varepsilon)^{-1}\\mathfrak c(M,g,E).$$\n Letting $\\varepsilon\\to 0$, we obtain the desired mass-capacity inequality\n $$m(M,g,E)\\geq 2\\mathfrak c(M,g,E).$$\n If the equality holds, then the same argument in the proof of Proposition \\ref{Prop: spin PMT with corner} yields that $(M,\\bar g)$ with $\\bar g=u^{\\frac{4}{n-2}}g$ is smooth and flat up to a change of the smooth structure. Notice that $u^{-1}$ is a weakly harmonic function on $(M,\\bar g)$. We conclude that $u$ is smooth and so $g$ is also smooth. Now we return to the smooth case and the desired conclusions come from Theorem \\ref{Thm: main1}.\n\\end{proof}", + "post_theorem_intro_text_len": 3808, + "post_theorem_intro_text": "\\begin{remark}\nIn the non-spin case, the mass-capacity inequality is still true from the exactly same proof assuming the validity of the positive mass theorem. However, the rigidity seems out of reach with current techniques due to the issue of incompleteness.\n\\end{remark}\n\nTo include previous Bray's mass-capacity inequality as a special case, we also extend our main theorem to the setting of Riemannian manifolds with corner defined as follows.\n\\begin{definition}\n A triple $(M,g,\\Sigma)$ will be called a Riemannian manifold with corner if $M$ is a smooth manifold equipped with a continuous metric $g$, and $\\Sigma$ is a closed separating smooth hypersurface $\\Sigma$ in $M$ such that the metric $g$ is actually smooth to boundary in the closure of every component of $M\\setminus \\Sigma$. For convenience, $\\Sigma$ will be called the corner of $(M,g)$.\n\\end{definition}\nIn particular, we can consider generalized asymptotically flat manifolds with corner.\n\\begin{definition}\n A generalized asymptotically flat manifold $(M,g,E)$ will be called a generalized asymptotically flat manifold with corner, if there exists a closed separating smooth hypersurface $\\Sigma$ such that $(M,g,\\Sigma)$ is a Riemannian manifold with corner.\n\\end{definition}\n\nThroughout this paper, we always take the convention that $\\mathbb S^2$ has mean curvature two in $\\mathbb R^3$ with respect to the outward unit normal. With extra mean-curvature condition along the corner, we can prove\n\\begin{theorem}\\label{Thm: main2}\n Assume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with corner, which has nonnegative scalar curvature. Assume that the sum of the mean curvatures of the corner on both sides with respect to the outward unit normal is nonnegative. Then we have the following mass-capacity inequality\n $$m(M,g,E)\\geq 2\\mathfrak c(M,g,E).$$\n If the equality holds, then $(M,g)$ is smooth and harmonically conformal to $\\mathbb R^n\\setminus S$, where $S$ is a bounded closed subset of $\\mathbb R^n$ with Hausdorff dimension no greater than $\\frac{n-2}{2}$. In particular, we have\n $$\\pi_i(M)=0\\mbox{ for all }1\\leq i\\leq n-1- \\left\\lfloor\\frac{n}{2}\\right\\rfloor.$$\n\\end{theorem}\nAs a direct consequence, we can show\n\\begin{corollary}[Bray's mass capacity inequality]\\label{Cor: Bray}\n Assume that $(M^n,g)$ is a spin asymptotically flat manifold with only one asymptotically flat end $E$, non-empty boundary $\\Sigma$, and nonnegative scalar curvature. \nAssume further that the boundary $\\Sigma$ is minimal. Then we have\n\\begin{equation}\\label{Eq: Bray's mass-capacity}\n m(M,g,E)\\geq \\mathfrak c(M,g,\\Sigma),\n\\end{equation}\nwhere $\\mathfrak c(M,g,\\Sigma)$ denotes the capacity of $\\Sigma$ in $(M,g)$ given by\n$$\\mathfrak c(M,g,\\Sigma)=\\inf_{\\phi\\in\\mathcal C}\\frac{1}{n(n-2)\\omega_n}\\int_M|\\nabla_g\\phi|^2\\,\\mathrm d\\mu_g,$$\n where $\\mathcal C$ denotes the collection of all smooth functions $\\phi$ with compact support such that $\\phi\\equiv 1$ on $\\Sigma$.\nThe equality holds in \\eqref{Eq: Bray's mass-capacity} if and only if $(M,g)$ is isometric to the half spatial Schwarzschild manifold.\n\\end{corollary}\n\n\\subsection*{Arrangements}\nThe remaining part of this paper is arranged as follows.\nSections \\ref{Sec: inequality} and \\ref{Sec: rigidity} are devoted to proving Theorem \\ref{Thm: main1}. In Section \\ref{Sec: inequality}, we prove the mass-capacity inequality based on the conformal method. In Section \\ref{Sec: rigidity}, we establish the rigidity in the equality case of the mass-capacity inequality by constructing sufficiently many parallel spinors from an approximation procedure. In Section \\ref{Sec: corner}, we show necessary modifications in the corner case and present the proofs of Theorem \\ref{Thm: main2} and Corollary \\ref{Cor: Bray}.\n\n\\bigskip", + "sketch": "Sections \\ref{Sec: inequality} and \\ref{Sec: rigidity} are devoted to proving Theorem~\\ref{Thm: main1}. In Section \\ref{Sec: inequality}, the mass-capacity inequality is proved \"based on the conformal method.\" In Section \\ref{Sec: rigidity}, the rigidity in the equality case is established \"by constructing sufficiently many parallel spinors from an approximation procedure.\"", + "expanded_sketch": "Next we prove the main theorem. First, we establish the mass-capacity inequality, “based on the conformal method.” Then we establish the rigidity in the equality case, “by constructing sufficiently many parallel spinors from an approximation procedure.”", + "expanded_theorem": "\\label{Thm: main1}\nAssume that $(M^n,g,E)$ is a spin generalized asymptotically flat manifold with nonnegative scalar curvature. Then we have the following mass-capacity inequality\n$$m(M,g,E)\\geq 2\\mathfrak c(M,g,E).$$\nIf the equality holds, then $(M,g)$ is harmonically conformal to $\\mathbb R^n\\setminus S$, where $S$ is a bounded closed subset of $\\mathbb R^n$ with Hausdorff dimension no greater than $\\frac{n-2}{2}$. In particular, we have\n$$\\pi_i(M)=0\\mbox{ for all }1\\leq i\\leq n-1- \\left\\lfloor\\frac{n}{2}\\right\\rfloor.$$\n", + "theorem_type": [ + "Implication", + "Inequality or Bound" + ], + "mcq": { + "question": "Let $(M^n,g,E)$ be a spin generalized asymptotically flat manifold with $n\\ge 3$, meaning that $(M,g)$ is a complete boundaryless Riemannian $n$-manifold, $E$ is an asymptotically flat end of $(M,g)$, and the scalar curvature satisfies $R(g)\\in L^1(E,g)$. Assume moreover that the scalar curvature is nonnegative. Let $m(M,g,E)$ denote the ADM mass of the end $E$, and let $\\mathfrak c(M,g,E)$ denote the capacity of the end $E$ (equivalently, the constant appearing in the asymptotic expansion $u(x)=1-\\mathfrak c(M,g,E)|x|^{2-n}+o(|x|^{2-n})$ on $E$ for the associated positive harmonic function tending to $1$ at infinity). Which of the following conclusions is valid under these hypotheses?", + "correct_choice": { + "label": "A", + "text": "One has the mass-capacity inequality $$m(M,g,E)\\ge 2\\mathfrak c(M,g,E).$$ If equality holds, then $(M,g)$ is harmonically conformal to $\\mathbb R^n\\setminus S$, meaning that for some bounded closed set $S\\subset \\mathbb R^n$ there is a conformal identification with Euclidean space off $S$ whose conformal factor is positive and harmonic; moreover $S$ has Hausdorff dimension at most $\\frac{n-2}{2}$. In particular, $$\\pi_i(M)=0\\quad\\text{for all }1\\le i\\le n-1-\\left\\lfloor\\frac n2\\right\\rfloor.$$" + }, + "choices": [ + { + "label": "B", + "text": "One has the mass-capacity inequality $$m(M,g,E)\\ge \\mathfrak c(M,g,E).$$ If equality holds, then $(M,g)$ is harmonically conformal to $\\mathbb R^n\\setminus S$ for some bounded closed set $S\\subset \\mathbb R^n$ with Hausdorff dimension at most $\\frac{n-2}{2}$, and in particular $$\\pi_i(M)=0\\quad\\text{for all }1\\le i\\le n-1-\\left\\lfloor\\frac n2\\right\\rfloor.$$" + }, + { + "label": "C", + "text": "One has the mass-capacity inequality $$m(M,g,E)\\ge 2\\mathfrak c(M,g,E).$$" + }, + { + "label": "D", + "text": "One has the mass-capacity inequality $$m(M,g,E)\\ge 2\\mathfrak c(M,g,E).$$ If equality holds, then $(M,g)$ is globally conformal to $\\mathbb R^n\\setminus S$ for some bounded closed set $S\\subset \\mathbb R^n$ with Hausdorff dimension at most $\\frac{n-2}{2}$; moreover the conformal factor extends smoothly across $S$. In particular, $$\\pi_i(M)=0\\quad\\text{for all }1\\le i\\le n-1.$$" + }, + { + "label": "E", + "text": "There exists a bounded closed set $S\\subset \\mathbb R^n$ with Hausdorff dimension at most $\\frac{n-2}{2}$, depending only on $n$, such that every spin generalized asymptotically flat manifold $(M^n,g,E)$ with nonnegative scalar curvature and satisfying $$m(M,g,E)=2\\mathfrak c(M,g,E)$$ is harmonically conformal to $\\mathbb R^n\\setminus S$. In particular, $$\\pi_i(M)=0\\quad\\text{for all }1\\le i\\le n-1-\\left\\lfloor\\frac n2\\right\\rfloor.$$" + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "sharp factor 2 from conformal mass formula", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "entire equality-case rigidity and homotopy conclusion dropped", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "equality gives harmonic conformality off a singular set, not smooth extension across it nor full homotopy vanishing", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "dependence of the singular set on the manifold; no universal set S depending only on n", + "template_used": "quantifier_dependence" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem states only the hypotheses and definitions; it does not explicitly reveal the sharp mass-capacity inequality, the equality case, or the topological consequences. There is no direct answer leakage." + }, + "TAS": { + "score": 1, + "justification": "This is very close to a theorem-recall item: the correct choice is essentially the full theorem statement under its exact hypotheses. However, the alternatives differ in sharp constants, rigidity strength, and quantifier structure, so it is not a pure verbatim restatement." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is required to distinguish the strongest valid conclusion from a weaker true statement and from subtly overstated false variants. Still, success depends largely on recalling the theorem rather than generating a conclusion from first principles." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong: one is a weaker true statement, others modify the sharp factor, overstate regularity/topological rigidity, or introduce an incorrect universal quantifier. These are plausible and mathematically meaningful failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it leans toward theorem recognition rather than deep generative reasoning." + } + }, + { + "id": "2511.08099v1", + "paper_link": "http://arxiv.org/abs/2511.08099v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "theorem", + "content": "\\label{main result}\nLet $k$ be a positive integer such that $k\\in \\{7, 8, 10,$ $11,12,13\\}.$ If $\\{a, b, c, d\\}$ is a $D(4)$-quadruple with $b=ka$, then it is regular. In other words, we have $d=d_{\\pm}.$", + "start_pos": 5471, + "end_pos": 5704, + "label": "main result" + }, + "ref_dict": { + "conj": "\\begin{conjecture}\\label{conj}\nAny $D(4)$-quadruple is regular.\n\\end{conjecture}" + }, + "pre_theorem_intro_text_len": 1230, + "pre_theorem_intro_text": "\\label{intr}\n\n\\begin{definition}Let $n\\neq0$ be an integer. We call a set of $m$ distinct positive integers a $D(n)$-$m$-tuple, or $m$-tuple with the property $D(n)$, if the product of any two of its distinct elements increased by $n$ is a perfect square.\n\\end{definition}\nWe research the $n=4$ case, which has many similarities to the classical $n=1$ case. First author and Filipin proved in \\cite{blizfil} the nonexistence of $D(4)$-quintuples.\n\nFor a $D(4)$-triple $\\{a,b,c\\}, a1$ then\n\\begin{align*}\n\\frac{l}{\\log(el)}&< 3.34\\cdot 10^{13}\\cdot\\log^2(8.09c^2),\\quad \\text{with solutions of Type $1$},\\; \\\\\n\\frac{l}{\\log(el)}&< 6.63\\cdot 10^{13}\\cdot\\log^2(8.09c^2), \\quad \\text{with solutions of Type $2$}.\n\\end{align*}\nIf $Q_m=P_l,\\ l\\geq1$ with $c=c_1^-,$ then we get \n\\begin{align*}\n\\frac{m}{\\log(em)}< 13.36\\cdot 10^{13}\\cdot\\log^2(21.3a),\\quad \\text{for}\\; k=7,8,10,11,12,13. \n\\end{align*}\n\\end{proposition}\n\\begin{proof}\n We will demonstrate the proof for the case $c=c_1^-$ since the proof for the other cases closely follows \\cite[Proposition 2.9]{glavni}. We apply Lemma \\ref{proposition-1}\n with $j=3$ and $\\chi=1$ to the linear form \\ref{Lambda'} and take \\[D=4,\\ b_1=m,\\ b_2=-l,\\ b_3=1,\\ \\alpha_1=\\beta,\\ \\alpha_2=\\alpha,\\ \\alpha_3=\\gamma'.\\]\n Since $l\\leq m$, we can take $B=m$. Also, we have\n \\[h(\\alpha_1)=\\frac{1}{2}\\log\\beta,\\ h(\\alpha_2)=\\frac{1}{2}\\log\\alpha.\\]\n Since $\\gamma'=\\gamma^{-1}$, then $h(\\gamma)=h(\\gamma')$ and from \\cite[Proposition 2.9]{glavni} we have\n \\[h(\\gamma)<\\frac{1}{4}\\log\\left[\\frac{2^4r^4c^4(1+\\sqrt{k})^4}{(c-a)^2}\\right].\\]\n We have \\begin{align*}(c_1^--a)^2&=a^2(2\\sqrt{k+\\frac{4}{a^2}}-k)^2>a^2(2\\sqrt{k}-k)^2,\\\\\n r&-1.3901&\\cdot 10^{11}\\cdot16\\cdot2 \\cdot\\log\\beta\\\\&\\cdot2\\log\\alpha\\cdot6\\log(21.3a)\\cdot\\log(4e)\\cdot\\log(em).\\nonumber\n\\end{align}\nFrom $1)$ of Lemma \\ref{bound-Lambda} and using $l\\geq1$, it is easy to conclude that $$m\\log\\beta0$. \n Assume at least one of the following conditions:\n \\begin{enumerate}\n \\item $v_{\\mathfrak{p}}(j_0) \\leq 0$ for $\\mathfrak{p}$ above $2,3$, and $[L:\\bQ]+v_3(d\\Nm_{L/\\bQ}(27j_0+16))$ is odd or $j_0$ has at least one real conjugate in $(-\\frac{16}{27},0)$;\n \\item $v_{\\mathfrak{p}}(j_0) \\geq 0$ for $\\mathfrak{p}$ above $2$, $v_{\\mathfrak{p}}(j_0) \\leq 0$ for $\\mathfrak{p}$ above $3$, and $j_0$ has at least one real conjugate in $(0, \\infty)$;\n \\item $[L:\\bQ]$ is even, $v_{\\mathfrak{p}}(j_0) \\geq 0$ for $\\mathfrak{p}$ above $2,3$, $j_0$ has at least one real conjugate in $(-\\infty, -\\frac{16}{27})\\cup (0, \\infty)$.\n \\end{enumerate} \n Then the Jacobian of $C$ has superspecial reduction at infinitely many primes. \n\\end{theorem}\n\nAn abelian surface with action of a maximal order in a rational quaternion algebra $B$ has either ordinary or superspecial reduction modulo primes not dividing the discriminant of $B$ (\\cite{Clark2003}*{p.70} and \\cite{MR1097624}*{p.23}). It therefore suffices to construct primes of supersingular reduction, for which we follow Elkies' general strategy. Given a genus $2$ curve $C$, we find supersingular reduction of its Jacobian from its intersection with some Heegner cycle $\\cP_D$ of discriminant $D$. The intersection at a prime $p$ is captured by $v_p(P_D(j_0))>0$, where $P_D(x)$ is the integral minimal polynomial of the $j$-invariants of the points in $\\cP_D$, and when this occurs, supersingular reduction at $p$ is detected by $\\left(\\frac{D}{p}\\right) \\neq 1$.\n\nIn each case, we are going to find some discriminant $D$ with a rational prime $p\\notin \\Nm_{L/\\bQ}(S)$ such that $v_p(P_{D}(j_0)) > 0$ and $\\left(\\frac{D}{p}\\right) \\neq 1$, then as in \\cite{MR2414789}, there is a prime $\\fp \\notin S$ above $p$, such that the Jacobian of $C$ has supersingular, and hence superspecial reduction. \n \\begin{enumerate}\n \\item Choose a prime $l$ satisfying the conditions: \n \\begin{itemize}\n \\item $l\\equiv 13 \\pmod{24}$,\n \\item $\\left(\\frac{-l}{q}\\right) = 1$ for every prime $q\\in \\Nm_{L/\\bQ}(S) \\backslash \\{2,3\\}$, \n \\item $P_{-4l}(x)$ has a real root $j(a_i)$ in a subinterval of $(-\\frac{16}{27},0)$ to be specified later.\n \\end{itemize}\n Let $D = -4l$. By \\cite{MR2414789}*{Lemma 6} and \\Cref{lem:red23}, \\begin{align*}\n P_{-4l}(x) &\\equiv x^{h'} \\pmod{4}, \\\\\n P_{-4l}(x) &\\equiv \\pm x^{h'} \\pmod{3}, \\\\\n P_{-4l}(x) &\\equiv (27x+16)S(x)^2 \\pmod{l} \\text{ for some }S(x) \\in\\bZ[x].\n \\end{align*}\n Since $v_{\\mathfrak{p}}(j_0) \\leq 0$ for all $\\mathfrak{p}$ above $2,3$, we can choose $d$ such that\n $d\\prod_{\\sigma\\in T} \\sigma(j_0)$ is integral for any $T\\subset \\Hom(L, \\overline{\\bQ})$ and any prime dividing $d$ lies in $\\Nm_{L/\\bQ}(S)$, then\n $N, dQ\\in \\bN$ and $(N, 6) = 1,\\, (dQ,2) = 1$. \n Suppose $l\\nmid N$, then \\begin{align*}\n \\left(\\frac{-4l}{N}\\right) &= \\left(\\frac{-1}{N}\\right)\\left(\\frac{l}{N}\\right)= \\left(\\frac{-1}{N}\\right)\\left(\\frac{N}{l}\\right) \\\\\n &= \\left(\\frac{-1}{N}\\right)\\left(\\frac{dQ}{l}\\right) = \\left(\\frac{-1}{N}\\right)\\left(\\frac{l}{dQ}\\right)\\\\\n &= \\left(\\frac{-1}{N}\\right)\\left(\\frac{-1}{dQ}\\right) \\left(\\frac{-l}{dQ}\\right) \\\\ &= \\left(\\frac{-1}{s(d\\Nm_{L/\\bQ} j_0)^{h'}}\\right)\\left(\\frac{-1}{s'3^{[L:\\bQ]}d\\Nm_{L/\\bQ}(j_0)}\\right)\\left(\\frac{-l}{3^{v_3(dQ)}}\\right) \\\\\n &= ss'(-1)^{[L:\\bQ]+v_3(dQ)}.\\end{align*} \n If $[L:\\bQ]+v_3(dQ)$ is odd, let \\[j(a_1)\\in\\left(-\\frac{16}{27}, -\\frac{16}{27} + \\min_{1\\leq i \\leq r} \\left|j_i + \\frac{16}{27}\\right|\\right)\\]\n so that $(27j_i + 16) P_{-4l}(j_i) > 0$ for each $1\\leq i\\leq r$. \n If $[L:\\bQ]+v_3(dQ)$ is even, by assumption let $j_t$ be the minimal real conjugate of $j_0$ in $( -\\frac{16}{27} , 0)$, and \\[j(a_1)\\in\\left(j_t, j_{t+1}\\right)\\] so that $(27j_i + 16) P_{-4l}(j_i) > 0$ for each $1\\leq i\\leq r, i\\neq t$ and $(27j_t + 16) P_{-4l}(j_t) < 0$. In either case we have \\[\\left(\\frac{-4l}{N}\\right) = -1\\] and there is a prime divisor $p$ of $N$ such that \\[\\left(\\frac{-4l}{p}\\right) = -1.\\] The conditions on $l$ and $(N,6) = 1$ imply that $p\\notin \\Nm_{L/\\bQ}(S)$ and $v_p(\\Nm_{L/\\bQ}(P_D(j_0))) = v_p(N) > 0$. \n If $l | N$, then since $l\\notin \\Nm_{L/\\bQ}(S)$ by construction, we can choose $p = l$. \n \\item Choose a prime $l$ satifying the conditions:\n \\begin{itemize}\n \\item $l\\equiv 19\\pmod{24}$,\n \\item $\\left(\\frac{q}{l}\\right)=\\left(\\frac{-l}{q}\\right) = 1$ for every prime $q\\in \\Nm_{L/\\bQ}(S) \\backslash \\{2,3\\}$, \n \\item $P_{-l}(x)$ has a real root $j(a_1)$ in a subinterval of $(0, \\infty)$ to be specified later.\n \\end{itemize} Let $D = -l$. By \\Cref{lem:red23}, \\Cref{lem:pair}, and \\cite{MR2414789}*{Lemma 4},\n \\begin{align*}\n P_{-l}(x) &\\equiv 1 \\pmod{2}, \\\\\n P_{-l}(x) &\\equiv \\pm x^{h'} \\pmod{3}, \\\\\n P_{-l}(x) &\\equiv 3(27x+16)S(x)^2 \\pmod{l} \\text{ for some }S(x) \\in\\bZ[x].\n \\end{align*}\n Since $v_{\\mathfrak{p}}(j_0) \\geq 0$ for $\\mathfrak{p}$ above $2$ and $v_{\\mathfrak{p}}(j_0) \\leq 0$ for $\\mathfrak{p}$ above $3$, we can choose $d$ such that\n $d\\prod_{\\sigma\\in T} \\sigma(j_0)$ is integral for any $T\\subset \\Hom(L, \\overline{\\bQ})$ and any prime dividing $d$ lies in $\\Nm_{L/\\bQ}(S)\\backslash\\{2\\}$, then\n $N, dQ\\in \\bN$ and $(N, 6) = 1$ . \n Suppose $l\\nmid N$, then\n \\begin{align*}\n \\left(\\frac{-l}{N}\\right) &= \\left(\\frac{N}{l}\\right) \\\\\n &= \\left(\\frac{s(d\\Nm_{L/\\bQ}(3(27j_0+16)))}{l}\\right)\\\\\n &= \\left(\\frac{ss'3^{[L:\\bQ]}dQ}{l}\\right) \\\\ \n &= ss'(-1)^{[L:\\bQ]+v_3(dQ)+v_2(Q)}\n \\end{align*}\n Let $j_t$ be the minimal real conjugate of $j_0$ in $(0, \\infty)$ and $n$ ($0\\leq n < r$) be the number of real conjugates of $j_0$ in $(-\\frac{16}{27}, 0)$. If $[L:\\bQ]+v_3(dQ)+v_2(Q)$ is odd, let \\[\\twocase{j(a_1)\\in}{(j_t, j_{t+1})}{$n$ is odd}{(0, j_t)}{$n$ is even}\\] so that $\\Nm_{L/\\bQ}(P_{-l}(j_0))\\Nm_{L/\\bQ}(27j_0+16)>0$. If $[L:\\bQ]+v_3(dQ)+v_2(Q)$ is even, let \\[\\twocase{j(a_1)\\in}{(j_t, j_{t+1})}{$n$ is even}{(0, j_t)}{$n$ is odd}\\] so that $\\Nm_{L/\\bQ}(P_{-l}(j_0))\\Nm_{L/\\bQ}(27j_0+16)<0$. In either case we have \\[\\left(\\frac{-l}{N}\\right) = -1\\] and there is a prime divisor $p$ of $N$ such that \\[\\left(\\frac{-l}{p}\\right) = -1.\\] The conditions on $l$ and $(N,6) = 1$ imply that $p\\notin \\Nm_{L/\\bQ}(S)$ and $v_p(\\Nm_{L/\\bQ}(P_D(j_0))) = v_p(N) > 0$. \n If $l | N$, then since $l\\notin \\Nm_{L/\\bQ}(S)$ by construction, we can choose $p = l$.\n\n\\begin{rem} \\label{rem:conclude} One can try to further weaken the local conditions by considering Heegner cycles with different forms of discriminant $D$. \nFor primes $p$ dividing $D$, the unpaired roots of $P_D(x)$ modulo $p$ can be predicted by checking whether a maximal order in a definite quaternion algebra ramified at $2,3,p$ contains two anticommuting CM orders of some given discriminants, as computed in \\cite{MR2704678}*{3.4.2}. One can then impose conditions on the number of prime divisors of $D$ and the congruence class modulo $24$ of primes dividing $D$, so that $P_D(x)$ has no unpaired roots or $P_D(x)$ has a single unpaired root from the same elliptic point modulo each $p\\nmid D$. In addition, one imposes a congruence condition on $D$ so that $\\cP_{D, \\cE}$ avoids intersection with one of $\\cP_{-3, \\cE}$ and $\\cP_{-4, \\cE}$ by \\Cref{lem:avoidintersection}. \nThe local conditions on $j_0$ are determined so that it avoids intersection with the these $\\cP_D$ at $p = 2,3$, and some real conjugate of $j_0$ and some real root of $P_D$ lie in the subinterval corresponding to one geodesic. For $D$ not of the form $-Np$ with $N = 1, 3, 4, 8, 12, 24$, it is possible that $P_D(x)$ has multiple real roots in each subinterval, but we still expect an equidistribution result. \n\\end{rem}\n\\bibliographystyle{amsalpha}\n\\bibliography{references}", + "post_theorem_intro_text_len": 4158, + "post_theorem_intro_text": "More generally, the coarse moduli variety of principally polarized abelian surfaces with potential multiplication by the maximal quaternion order of discriminant $6$ is isomorphic to $\\mathbb{P}^1$, and an arithmetic $j$-function (see \\eqref{eq:j} in \\cref{sec:coord}) is defined in \\cite{MR2423455}. In terms of this coordinate, the assumption at primes $\\mathfrak{p}$ above $2$ and $3$ in \\Cref{thm_specialcase} corresponds to the case $v_{\\mathfrak{p}}(j(C)) = 0$ by \\cite{MR2414789}*{Proposition 2}, and our main theorem can be stated as follows. \n\n\\begin{theorem}\\label{thm_main}\nLet $C$ be a genus $2$ curve with Jacobian that has multiplication by the maximal quaternion order with discriminant $6$, \n and has field of moduli a number field $L$ with at least one real embedding. \n Write $j_0:=j(C)\\in L$ and $\\Nm_{L/\\mathbb{Q}}(j_0) = \\frac{n}{d}$ with $n,d\\in \\mathbb{Z}, (n,d,6) = 1, d>0$. \n Assume at least one of the following conditions:\n \\begin{enumerate}\n \\item $v_{\\mathfrak{p}}(j_0) \\leq 0$ for $\\mathfrak{p}$ above $2,3$, and $[L:\\mathbb{Q}]+v_3(d\\Nm_{L/\\mathbb{Q}}(27j_0+16))$ is odd or $j_0$ has at least one real conjugate in $(-\\frac{16}{27},0)$;\n \\item $v_{\\mathfrak{p}}(j_0) \\geq 0$ for $\\mathfrak{p}$ above $2$, $v_{\\mathfrak{p}}(j_0) \\leq 0$ for $\\mathfrak{p}$ above $3$, and $j_0$ has at least one real conjugate in $(0, \\infty)$;\n \\item $[L:\\mathbb{Q}]$ is even, $v_{\\mathfrak{p}}(j_0) \\geq 0$ for $\\mathfrak{p}$ above $2,3$, $j_0$ has at least one real conjugate in $(-\\infty, -\\frac{16}{27})\\cup (0, \\infty)$.\n \\end{enumerate} \n Then the Jacobian of $C$ has superspecial reduction at infinitely many primes. \n\\end{theorem} \n\nAn abelian surface with action of a maximal order in a rational quaternion algebra $B$ has either ordinary or superspecial reduction modulo primes not dividing the discriminant of $B$ (\\cite{Clark2003}*{p.70} and \\cite{MR1097624}*{p.23}). It therefore suffices to construct primes of supersingular reduction, for which we follow Elkies' general strategy. Given a genus $2$ curve $C$, we find supersingular reduction of its Jacobian from its intersection with some Heegner cycle $\\cP_D$ of discriminant $D$. The intersection at a prime $p$ is captured by $v_p(P_D(j_0))>0$, where $P_D(x)$ is the integral minimal polynomial of the $j$-invariants of the points in $\\cP_D$, and when this occurs, supersingular reduction at $p$ is detected by $\\left(\\frac{D}{p}\\right) \\neq 1$. \n\nAs in \\cite{MR2414789}, we need to consider certain elliptic point in addition to the Heegner cycle in order to pair the roots of $P_D(x)$ modulo primes dividing $D$. We use \\cite{MR2704678} to generalize \\cite{MR2414789} through a more detailed, case-by-case study, where in each case the discriminant $D$ is chosen in a form adapted to the conditions. \nIn characteristic $2$ and $3$, \\cite{MR2414789} shows that the reduction of any Heegner cycle lies in the superspecial locus $\\{j = 0, \\infty\\}$, and the intersection formula of Heegner divisors in \\cite{MR2441697} provides the new input that determines the specific superspecial point for each chosen Heegner cycle. The local conditions on $j_0$ at primes above $2$ and $3$ ensure that it avoids intersection with the chosen Heegner cycles at these primes. \nFor more on the choice of local conditions and Heegner cycles, and for potential further weakenings of these conditions, see \\Cref{rem:conclude}.\n\n\\subsection{Notation and conventions}\n\nAssume the following unless specified otherwise.\n\nLet $B = B_{\\Delta}$ be an indefinite quaternion algebra over $\\mathbb{Q}$ of discriminant $\\Delta$, $\\Lambda = \\Lambda_{\\Delta}$ be a maximal order of $B$, and $\\Lambda^1 = \\Lambda_{\\Delta}^1$ be the group of units in $\\Lambda$ of norm $1$. Fix an element $\\mu \\in \\Lambda_{\\Delta}$ such that $\\mu^2 = -\\Delta$,\\footnote{The element $\\mu$ exists because the field $\\mathbb{Q}(\\sqrt{-\\Delta})$ embeds in $B$ by the local-global principle, and any two maximal orders in $B$ are conjugate to each other by strong approximation.} then the involution $\\alpha \\mapsto \\alpha' = \\mu^{-1} \\bar{\\alpha} \\mu$ is a positive anti-involution on $B$.", + "sketch": "An abelian surface with action of a maximal order in a rational quaternion algebra $B$ has either ordinary or superspecial reduction modulo primes not dividing the discriminant of $B$ (\\cite{Clark2003}*{p.70} and \\cite{MR1097624}*{p.23}), so “it therefore suffices to construct primes of supersingular reduction,” following “Elkies' general strategy.” Given $C$, one finds supersingular reduction of its Jacobian “from its intersection with some Heegner cycle $\\cP_D$ of discriminant $D$.” The intersection at a prime $p$ is “captured by $v_p(P_D(j_0))>0$,” where $P_D(x)$ is “the integral minimal polynomial of the $j$-invariants of the points in $\\cP_D$,” and when this occurs, supersingular reduction at $p$ is “detected by $\\left(\\frac{D}{p}\\right) \\neq 1$.”\n\nAs in \\cite{MR2414789}, one must “consider certain elliptic point in addition to the Heegner cycle in order to pair the roots of $P_D(x)$ modulo primes dividing $D$.” Using \\cite{MR2704678}, the argument proceeds via “a more detailed, case-by-case study,” where “in each case the discriminant $D$ is chosen in a form adapted to the conditions.” In characteristic $2$ and $3$, \\cite{MR2414789} shows “the reduction of any Heegner cycle lies in the superspecial locus $\\{j = 0, \\infty\\}$,” and “the intersection formula of Heegner divisors in \\cite{MR2441697} provides the new input that determines the specific superspecial point for each chosen Heegner cycle.” Finally, “the local conditions on $j_0$ at primes above $2$ and $3$ ensure that it avoids intersection with the chosen Heegner cycles at these primes,” yielding infinitely many primes of superspecial reduction.", + "expanded_sketch": "No expanded sketch found.", + "expanded_theorem": "\\label{thm_specialcase}\n Let $C$ be a genus $2$ curve with Jacobian that has multiplication by the maximal quaternion order with discriminant $6$, \n and has field of moduli a number field $L$ with at least one real embedding. \n Assume $C$ has potentially smooth stable reduction at primes above $2$ and $3$. Then its Jacobian has superspecial reduction at infinitely many primes.", + "theorem_type": [ + "Implication", + "Universal" + ], + "mcq": { + "question": "Let $C$ be a genus $2$ curve whose Jacobian $J(C)$ admits an action of the maximal order in the rational quaternion algebra of discriminant $6$, and suppose the field of moduli of $C$ is a number field $L$ having at least one real embedding. Assume moreover that for every prime of $L$ above $2$ or $3$, the curve $C$ has potentially smooth stable reduction there (that is, after a finite extension, its stable reduction becomes smooth). Under these hypotheses, which statement about the reduction of $J(C)$ holds? Here, saying that $J(C)$ has superspecial reduction at a prime means that its reduction modulo that prime is a superspecial abelian surface.", + "correct_choice": { + "label": "A", + "text": "The Jacobian $J(C)$ has superspecial reduction at infinitely many primes." + }, + "choices": [ + { + "label": "B", + "text": "The Jacobian $J(C)$ has superspecial reduction at all but finitely many primes." + }, + { + "label": "C", + "text": "The Jacobian $J(C)$ has superspecial reduction at at least one prime." + }, + { + "label": "D", + "text": "There exists a finite extension $L'/L$ such that the base change $J(C)_{L'}$ has superspecial reduction at infinitely many primes of $L'$." + }, + { + "label": "E", + "text": "The Jacobian $J(C)$ has infinitely many primes of supersingular reduction." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "finiteness", + "tampered_component": "infinitely_many_vs_density_one", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "finiteness", + "tampered_component": "replace_infinitely_many_by_existence", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "base_field_of_primes", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "property_confusion", + "tampered_component": "superspecial_vs_supersingular", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not state the conclusion outright; it only gives hypotheses and asks for the resulting reduction statement. Defining 'superspecial reduction' clarifies terminology but does not reveal which quantified claim is correct." + }, + "TAS": { + "score": 1, + "justification": "This is close to a theorem-recall item: the hypotheses are highly specific and the correct option matches the natural theorem conclusion. However, the options introduce meaningful quantifier and property variations, so it is not a pure verbatim restatement." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish 'infinitely many' from weaker existence, overly strong density-type claims, base-change variants, and supersingular/superspecial confusion. Still, the task is mainly recognition of the exact theorem-level conclusion rather than deep derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically well targeted: one is too strong (all but finitely many), one is a weaker true-looking statement, one alters the field/quantifier dependence, and one exploits the common superspecial vs. supersingular confusion." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-conclusion MCQ with strong distractors and no direct answer leakage, but it remains fairly close to theorem recall rather than testing substantial generative mathematical reasoning." + } + }, + { + "id": "2511.07680v1", + "paper_link": "http://arxiv.org/abs/2511.07680v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "thm", + "content": "\\label{main_theorem}\n\t\tLet $k$ be a number field containing a primitive $s$-th root of unity. Assume the strong version of Lang's conjecture (Conjecture \\ref{conj1}) holds for varieties defined over $k$. Then the rank $r(J_C(k))$ is uniformly bounded as $J_C$ varies over the Jacobian varieties of all smooth curves $C$ defined over $k$ by equation \\eqref{curve_main} with genus $g_{r,s}(C) \\geq 1$.", + "start_pos": 19965, + "end_pos": 20387, + "label": "main_theorem" + }, + "ref_dict": { + "examples": "\\label{examples}\nIn this section, we provide a concrete example to show how specific high rank elliptic curve of the form \n$y^2=x(x^2+B)$ with a known set of rational points correspond to a single", + "curve_main": "\\begin{equation} \\label{curve_main}\n\t\tC_{a,b}: y^s=x(ax^r+b),\n\t\\end{equation}", + "conj1": "\\begin{conj}[Lang, \\cite{Lang1986, Lang1991}]\n\t\t\\label{conj1}\n\t\tLet $k$ be a number field and $X$ a smooth variety of general type over $k$.\n\t\t\\begin{enumerate}\n\t\t\t\\item[(a)] \\textbf{(Weak)} The set of $k$-rational points $X(k)$ is not Zariski dense in $X$.\n\t\t\t\\item[(b)] \\textbf{(Strong)} There exists a proper Zariski-closed subset $Z \\subset X$ such that for any finite extension $K$ of $k$, the set of $K$-rational points on $X \\setminus Z$ is finite.\n\t\t\\end{enumerate}\n\t\\end{conj}", + "main_theorem": "\\begin{thm} \\label{main_theorem}\n\t\tLet $k$ be a number field containing a primitive $s$-th root of unity. Assume the strong version of Lang's conjecture (Conjecture \\ref{conj1}) holds for varieties defined over $k$. Then the rank $r(J_C(k))$ is uniformly bounded as $J_C$ varies over the Jacobian varieties of all smooth curves $C$ defined over $k$ by equation \\eqref{curve_main} with genus $g_{r,s}(C) \\geq 1$.\n\t\\end{thm}", + "uniformity_theorem": "\\begin{thm} \\label{uniformity_theorem}\n\t\tAssume the weak version of Lang's conjecture (Conjecture \\ref{conj1}) holds over $k$. Let $r \\geq 1, s \\geq 2$ such that $g_{r,s}(C) \\geq 1$, and let $n_0$ be as in Theorem \\ref{finiteness_theorem}. Then for $n \\geq n_0$, there exists a uniform bound $M_0 = M_0(k, r, s, n)$ such that $\\#\\Cc_{\\ba_n}(k) < M_0$ for all valid choices of $\\ba_n$. Furthermore, if $g_{r,s}(C) \\geq 2$, there exists an integer $m_0 > n_0$ such that $\\Cc_{\\ba_m}(k) = \\emptyset$ for all $m \\geq m_0$ and all valid $\\ba_m$. (The strong Lang conjecture implies $M_0$ and $m_0$ can be chosen independently of $k$.)\n\t\\end{thm}", + "finiteness_theorem": "\\begin{thm} \\label{finiteness_theorem}\n\t\tLet $k$ be a number field containing $\\zeta_s$. Let $n_0=4$ if $s=2$ and $n_0=3$ otherwise. For any choice of $\\ba_n = \\{\\alpha_0, \\ldots, \\alpha_n\\}$ consisting of $n+1$ distinct, non-zero elements in $k$ with $\\alpha_i^r \\neq \\alpha_j^r$ for $i \\neq j$, and for integers $r \\geq 1, s \\geq 2$ such that $g_{r,s}(C) \\geq 1$, the set $\\Cc_{\\ba_n}(k)$ is infinite if $n < n_0$ and finite if $n \\geq n_0$. Consequently, for a fixed $\\ba_n$ with $n \\geq n_0$, the rank $r(J_C(k))$ is bounded for $C \\in \\Cc_{\\ba_n}(k)$.\n\t\\end{thm}" + }, + "pre_theorem_intro_text_len": 2856, + "pre_theorem_intro_text": "A central theme in arithmetic geometry concerns the structure of the group of rational points on algebraic varieties defined over number fields. For an abelian variety $A$ over a number field $k$, the celebrated Mordell-Weil theorem asserts that the group $A(k)$ is finitely generated \\cite{Silverman1986}. Thus, $A(k) \\cong A(k)_{\\text{tors}} \\oplus \\mathbb{Z}^r$, where $A(k)_{\\text{tors}}$ is the finite torsion subgroup and $r = \\rk A(k)$ is the Mordell-Weil rank. Understanding the behavior of this rank, particularly for Jacobians $J_C$ of algebraic curves $C$, remains a profound challenge. A fundamental open question asks whether $r(J_C(k))$ is uniformly bounded as $C$ varies over all curves of a fixed genus $g \\geq 1$ defined over $k$.\n\n Even for elliptic curves ($g=1$) over $k=\\mathbb{Q}$, this question is unresolved. While computational evidence reveals curves of high rank, theoretical heuristics suggest boundedness, possibly with rank exceeding 21 being rare \\cite{Park2019}. Deep conjectures in Diophantine geometry provide conditional evidence for rank boundedness. Notably, Lang's conjectures on rational points on varieties of general type \\cite{Lang1986, Lang1991} imply, via the work of Caporaso, Harris, and Mazur \\cite{Caporaso1997, caporaso2022update}, strong uniformity statements about the number of rational points on curves. Furthermore, Pasten demonstrated that conjectures on Diophantine approximation imply Honda's conjecture on rank boundedness for elliptic curves \\cite{Pasten2019}. The link between point counts and rank was solidified by Dimitrov, Gao, and Habegger \\cite{Dimitrov2021}, making uniform point boundedness equivalent to uniform rank boundedness.\n\n An alternative approach, pioneered by Yamagishi \\cite{Yamagishi2003} for the family $y^2=ax^4+bx^2+c$, employs geometric methods. By constructing parameter spaces for curves with multiple rational points and demonstrating that these spaces (or relevant subvarieties) become varieties of general type, one can invoke Lang's conjectures to deduce finiteness and uniformity properties, ultimately leading to rank bounds. This geometric strategy was adapted by the author in \\cite{Salami2025toappear} to study the uniformity of Mordell-Weil rank of Jacobian variety of the family of curves\n $ y^s = ax^r+b$.\n\n This paper applies the same geometric methodology to investigate the closely related two-parameter family of curves defined by the affine equation\n\t\\begin{equation} \\label{curve_main}\n\t\tC_{a,b}: y^s=x(ax^r+b),\n\t\\end{equation}\n\twhere $r \\geq 1, s \\geq 2$ are fixed integers, and $k$ is a number field containing $\\zeta_s$, a primitive $s$-th root of unity. We focus on smooth curves $C_{a,b}$ with genus $g_{r,s}(C_{a,b}) \\geq 1$. Our main result, conditional on Lang's conjecture, establishes uniform rank boundedness for this family.", + "context": "A central theme in arithmetic geometry concerns the structure of the group of rational points on algebraic varieties defined over number fields. For an abelian variety $A$ over a number field $k$, the celebrated Mordell-Weil theorem asserts that the group $A(k)$ is finitely generated \\cite{Silverman1986}. Thus, $A(k) \\cong A(k)_{\\text{tors}} \\oplus \\mathbb{Z}^r$, where $A(k)_{\\text{tors}}$ is the finite torsion subgroup and $r = \\rk A(k)$ is the Mordell-Weil rank. Understanding the behavior of this rank, particularly for Jacobians $J_C$ of algebraic curves $C$, remains a profound challenge. A fundamental open question asks whether $r(J_C(k))$ is uniformly bounded as $C$ varies over all curves of a fixed genus $g \\geq 1$ defined over $k$.\n\nEven for elliptic curves ($g=1$) over $k=\\mathbb{Q}$, this question is unresolved. While computational evidence reveals curves of high rank, theoretical heuristics suggest boundedness, possibly with rank exceeding 21 being rare \\cite{Park2019}. Deep conjectures in Diophantine geometry provide conditional evidence for rank boundedness. Notably, Lang's conjectures on rational points on varieties of general type \\cite{Lang1986, Lang1991} imply, via the work of Caporaso, Harris, and Mazur \\cite{Caporaso1997, caporaso2022update}, strong uniformity statements about the number of rational points on curves. Furthermore, Pasten demonstrated that conjectures on Diophantine approximation imply Honda's conjecture on rank boundedness for elliptic curves \\cite{Pasten2019}. The link between point counts and rank was solidified by Dimitrov, Gao, and Habegger \\cite{Dimitrov2021}, making uniform point boundedness equivalent to uniform rank boundedness.\n\nAn alternative approach, pioneered by Yamagishi \\cite{Yamagishi2003} for the family $y^2=ax^4+bx^2+c$, employs geometric methods. By constructing parameter spaces for curves with multiple rational points and demonstrating that these spaces (or relevant subvarieties) become varieties of general type, one can invoke Lang's conjectures to deduce finiteness and uniformity properties, ultimately leading to rank bounds. This geometric strategy was adapted by the author in \\cite{Salami2025toappear} to study the uniformity of Mordell-Weil rank of Jacobian variety of the family of curves\n $ y^s = ax^r+b$.\n\nThis paper applies the same geometric methodology to investigate the closely related two-parameter family of curves defined by the affine equation\n \\begin{equation} \\label{curve_main}\n C_{a,b}: y^s=x(ax^r+b),\n \\end{equation}\n where $r \\geq 1, s \\geq 2$ are fixed integers, and $k$ is a number field containing $\\zeta_s$, a primitive $s$-th root of unity. We focus on smooth curves $C_{a,b}$ with genus $g_{r,s}(C_{a,b}) \\geq 1$. Our main result, conditional on Lang's conjecture, establishes uniform rank boundedness for this family.\n\n\\begin{conj}[Lang, \\cite{Lang1986, Lang1991}]\n\t\t\\label{conj1}\n\t\tLet $k$ be a number field and $X$ a smooth variety of general type over $k$.\n\t\t\\begin{enumerate}\n\t\t\t\\item[(a)] \\textbf{(Weak)} The set of $k$-rational points $X(k)$ is not Zariski dense in $X$.\n\t\t\t\\item[(b)] \\textbf{(Strong)} There exists a proper Zariski-closed subset $Z \\subset X$ such that for any finite extension $K$ of $k$, the set of $K$-rational points on $X \\setminus Z$ is finite.\n\t\t\\end{enumerate}\n\t\\end{conj}\n\n\\begin{equation} \\label{curve_main}\n\t\tC_{a,b}: y^s=x(ax^r+b),\n\t\\end{equation}", + "full_context": "A central theme in arithmetic geometry concerns the structure of the group of rational points on algebraic varieties defined over number fields. For an abelian variety $A$ over a number field $k$, the celebrated Mordell-Weil theorem asserts that the group $A(k)$ is finitely generated \\cite{Silverman1986}. Thus, $A(k) \\cong A(k)_{\\text{tors}} \\oplus \\mathbb{Z}^r$, where $A(k)_{\\text{tors}}$ is the finite torsion subgroup and $r = \\rk A(k)$ is the Mordell-Weil rank. Understanding the behavior of this rank, particularly for Jacobians $J_C$ of algebraic curves $C$, remains a profound challenge. A fundamental open question asks whether $r(J_C(k))$ is uniformly bounded as $C$ varies over all curves of a fixed genus $g \\geq 1$ defined over $k$.\n\nEven for elliptic curves ($g=1$) over $k=\\mathbb{Q}$, this question is unresolved. While computational evidence reveals curves of high rank, theoretical heuristics suggest boundedness, possibly with rank exceeding 21 being rare \\cite{Park2019}. Deep conjectures in Diophantine geometry provide conditional evidence for rank boundedness. Notably, Lang's conjectures on rational points on varieties of general type \\cite{Lang1986, Lang1991} imply, via the work of Caporaso, Harris, and Mazur \\cite{Caporaso1997, caporaso2022update}, strong uniformity statements about the number of rational points on curves. Furthermore, Pasten demonstrated that conjectures on Diophantine approximation imply Honda's conjecture on rank boundedness for elliptic curves \\cite{Pasten2019}. The link between point counts and rank was solidified by Dimitrov, Gao, and Habegger \\cite{Dimitrov2021}, making uniform point boundedness equivalent to uniform rank boundedness.\n\nAn alternative approach, pioneered by Yamagishi \\cite{Yamagishi2003} for the family $y^2=ax^4+bx^2+c$, employs geometric methods. By constructing parameter spaces for curves with multiple rational points and demonstrating that these spaces (or relevant subvarieties) become varieties of general type, one can invoke Lang's conjectures to deduce finiteness and uniformity properties, ultimately leading to rank bounds. This geometric strategy was adapted by the author in \\cite{Salami2025toappear} to study the uniformity of Mordell-Weil rank of Jacobian variety of the family of curves\n $ y^s = ax^r+b$.\n\nThis paper applies the same geometric methodology to investigate the closely related two-parameter family of curves defined by the affine equation\n \\begin{equation} \\label{curve_main}\n C_{a,b}: y^s=x(ax^r+b),\n \\end{equation}\n where $r \\geq 1, s \\geq 2$ are fixed integers, and $k$ is a number field containing $\\zeta_s$, a primitive $s$-th root of unity. We focus on smooth curves $C_{a,b}$ with genus $g_{r,s}(C_{a,b}) \\geq 1$. Our main result, conditional on Lang's conjecture, establishes uniform rank boundedness for this family.\n\n\\begin{conj}[Lang, \\cite{Lang1986, Lang1991}]\n\t\t\\label{conj1}\n\t\tLet $k$ be a number field and $X$ a smooth variety of general type over $k$.\n\t\t\\begin{enumerate}\n\t\t\t\\item[(a)] \\textbf{(Weak)} The set of $k$-rational points $X(k)$ is not Zariski dense in $X$.\n\t\t\t\\item[(b)] \\textbf{(Strong)} There exists a proper Zariski-closed subset $Z \\subset X$ such that for any finite extension $K$ of $k$, the set of $K$-rational points on $X \\setminus Z$ is finite.\n\t\t\\end{enumerate}\n\t\\end{conj}\n\n\\begin{equation} \\label{curve_main}\n\t\tC_{a,b}: y^s=x(ax^r+b),\n\t\\end{equation}\n\n\\begin{abstract}\n Let $k$ be a number field. We investigate the Mordell-Weil ranks of Jacobian varieties $J_C$ associated with algebraic curves $C$ of genus $g \\geq 1$ defined by affine equations of the form $y^s=x(ax^r+b)$, where $a, b \\in k$ ($ab \\neq 0$), and $r \\geq 1, s \\geq 2$ are fixed integers. Assuming the strong version of Lang's conjecture concerning rational points on varieties of general type, we establish that the ranks $r(J_C(k))$ are uniformly bounded as $C$ varies within this family. \n Our methodology builds upon the geometric approach employed by H. Yamagishi and subsequently adapted by the author for the family $y^s=ax^r+b$. We construct a parameter space $\\Wc_n$ for curves possessing $n+1$ specified rational points and analyze its birational model $\\Xc_n$, a complete intersection variety. The geometric properties of the fibers of $\\Xc_n \\to \\Sym^{n+1}(\\Pp^1)$, specifically their genus and gonality, are studied. Combining these geometric insights with Faltings' theorem, uniformity conjectures stemming from Lang's work, and recent results connecting rank with the number of rational points, we deduce the main boundedness result. In the case of genus one curves $C$, it \n states that the rank of elliptic curves $y^2=x (x^2+B)$ \n is uniformly bounded subject to the strong version of Lang's conjecture.\n\\end{abstract}\n\nA central theme in arithmetic geometry concerns the structure of the group of rational points on algebraic varieties defined over number fields. For an abelian variety $A$ over a number field $k$, the celebrated Mordell-Weil theorem asserts that the group $A(k)$ is finitely generated \\cite{Silverman1986}. Thus, $A(k) \\cong A(k)_{\\text{tors}} \\oplus \\mathbb{Z}^r$, where $A(k)_{\\text{tors}}$ is the finite torsion subgroup and $r = \\rk A(k)$ is the Mordell-Weil rank. Understanding the behavior of this rank, particularly for Jacobians $J_C$ of algebraic curves $C$, remains a profound challenge. A fundamental open question asks whether $r(J_C(k))$ is uniformly bounded as $C$ varies over all curves of a fixed genus $g \\geq 1$ defined over $k$.\n\nThis paper applies the same geometric methodology to investigate the closely related two-parameter family of curves defined by the affine equation\n \\begin{equation} \\label{curve_main}\n C_{a,b}: y^s=x(ax^r+b),\n \\end{equation}\n where $r \\geq 1, s \\geq 2$ are fixed integers, and $k$ is a number field containing $\\zeta_s$, a primitive $s$-th root of unity. We focus on smooth curves $C_{a,b}$ with genus $g_{r,s}(C_{a,b}) \\geq 1$. Our main result, conditional on Lang's conjecture, establishes uniform rank boundedness for this family.\n\nThe main result and its consequences provide new geometric evidence for rank boundedness conjectures for elliptic curves similar to that given in author's previous work \\cite{Salami2025toappear} \ndifferent from that of given in \\cite{Pasten2019}. \nTo be explicit, let us consider\nthe case $g_{r,s}(C) = 1$ and $k=\\Q$, where the curve $C $ is an elliptic curves of the form $y^2= x(x^2+ b)$ with $\\Z$.\nBy Theorem\\ref{main_theorem}, one can conclude that the rank of elliptic curves \n is uniformly bounded subject to the strong version of Lang's conjecture, see Section \\ref{sec:applications}.\n We note that the record of rank in this family is 14 due to M. Watkins obtained in 2002, according to \\cite{DujellaRankRecord}.\n As a particular case of the above family, is the congruent number elliptic curves. There are only 27 congruent number $N$ for which the rank of corresponding elliptic curves $y^2=x (x^2-N^2)$ is equal to 7. It is also suspected that there does not exist any congruent number elliptic curve of rank 8, see \\cite{Watkins2014}. \nTheorem\\ref{main_theorem} implies the uniformity of rank in family of congruent number elliptic curves, subject to the strong version of Lang's conjecture, see Section \\ref{sec:applications}.\n\n\\begin{thm} \\label{uniformity_theorem}\n Assume the weak version of Lang's conjecture (Conjecture \\ref{conj1}) holds over $k$. Let $r \\geq 1, s \\geq 2$ such that $g_{r,s}(C) \\geq 1$, and let $n_0$ be as in Theorem \\ref{finiteness_theorem}. Then for $n \\geq n_0$, there exists a uniform bound $M_0 = M_0(k, r, s, n)$ such that $\\#\\Cc_{\\ba_n}(k) < M_0$ for all valid choices of $\\ba_n$. Furthermore, if $g_{r,s}(C) \\geq 2$, there exists an integer $m_0 > n_0$ such that $\\Cc_{\\ba_m}(k) = \\emptyset$ for all $m \\geq m_0$ and all valid $\\ba_m$. (The strong Lang conjecture implies $M_0$ and $m_0$ can be chosen independently of $k$.)\n \\end{thm}\n\n\\begin{proof}[Proof of Theorem \\ref{main_theorem}]\n Let $k$ be a number field, $[k:\\Q]=d$. Let $\\mathcal{F}$ be the family of smooth curves $C: y^s=x(ax^r+b)$ over $k$ with genus $g=g_{r,s}(C) \\geq 1$. Assume, for contradiction, that the rank $r(J_C(k))$ is unbounded for $C \\in \\mathcal{F}$.\n Then there exists a sequence of non-isomorphic curves $C_j \\in \\mathcal{F}$ such that $r_j = r(J_{C_j}(k)) \\to \\infty$ as $j \\to \\infty$.\n By the result of Dimitrov, Gao, and Habegger (Theorem \\ref{dgh}), there is a constant $c=c(g,d)$ such that $\\#C_j(k) \\leq c^{1+r_j}$. Since $r_j \\to \\infty$, it follows that $\\#C_j(k) \\to \\infty$.\n\nNow, assume the Strong Lang Conjecture (Conjecture \\ref{conj1}). This implies the strong version of Theorem \\ref{uniformity_theorem}: there exists an integer $m_0$ (depending only on $r, s$, possibly $g$, but not $k$) such that for any $m \\geq m_0$, $\\Cc_{\\ba_m}(K) = \\emptyset$ for any number field $K$ and any valid $\\ba_m \\in \\Uc_{r,m}(K)$. This means no smooth curve in the family \\eqref{curve_main} over any number field $K$ can possess $m+1$ points $(\\alpha_i, \\beta_i)$ where the $\\alpha_i$ are distinct, non-zero, and satisfy $\\alpha_i^r \\neq \\alpha_j^r$, provided $m \\ge m_0$.\n\n\\begin{thm} \n \\label{CN-rank1}\n Assume the strong version of Lang's conjecture (Conjecture \\ref{conj1}) holds. Then the Mordell-Weil ranks \n$r(E_B(\\Q))$ of elliptic curves $E_B: y^2=x (x^2+B)$ are uniformly bounded as $B$ varies over all integers.\n\\end{thm}", + "post_theorem_intro_text_len": 5028, + "post_theorem_intro_text": "The main result and its consequences provide new geometric evidence for rank boundedness conjectures for elliptic curves similar to that given in author's previous work \\cite{Salami2025toappear} \ndifferent from that of given in \\cite{Pasten2019}. \nTo be explicit, let us consider\nthe case $g_{r,s}(C) = 1$ and $k=\\mathbb{Q}$, where the curve $C $ is an elliptic curves of the form $y^2= x(x^2+ b)$ with $\\mathbb{Z}$.\nBy Theorem\\ref{main_theorem}, one can conclude that the rank of elliptic curves \n is uniformly bounded subject to the strong version of Lang's conjecture, see Section \\ref{sec:applications}.\n We note that the record of rank in this family is 14 due to M. Watkins obtained in 2002, according to \\cite{DujellaRankRecord}.\n As a particular case of the above family, is the congruent number elliptic curves. There are only 27 congruent number $N$ for which the rank of corresponding elliptic curves $y^2=x (x^2-N^2)$ is equal to 7. It is also suspected that there does not exist any congruent number elliptic curve of rank 8, see \\cite{Watkins2014}. \nTheorem\\ref{main_theorem} implies the uniformity of rank in family of congruent number elliptic curves, subject to the strong version of Lang's conjecture, see Section \\ref{sec:applications}. \n\n\tThe proof of Theorem\\ref{main_theorem} relies crucially on studying the set $\\Cc_{\\ba_n}(k)$ consisting of smooth curves $C$ in the family \\eqref{curve_main} that pass through a given set of $n+1$ points with $x$-coordinates $\\alpha_i \\in k$ ($i=0,\\dots,n$) where $\\ba_n$ represents the collection $\\{\\alpha_0, \\dots, \\alpha_n\\}$ of distinct, non-zero elements of $k$ satisfying $\\alpha_i^r \\neq \\alpha_j^r$ for $i \\neq j$. We establish the following finiteness and uniformity results for these sets.\n\n\t\\begin{thm} \\label{finiteness_theorem}\n\t\tLet $k$ be a number field containing $\\zeta_s$. Let $n_0=4$ if $s=2$ and $n_0=3$ otherwise. For any choice of $\\ba_n = \\{\\alpha_0, \\ldots, \\alpha_n\\}$ consisting of $n+1$ distinct, non-zero elements in $k$ with $\\alpha_i^r \\neq \\alpha_j^r$ for $i \\neq j$, and for integers $r \\geq 1, s \\geq 2$ such that $g_{r,s}(C) \\geq 1$, the set $\\Cc_{\\ba_n}(k)$ is infinite if $n < n_0$ and finite if $n \\geq n_0$. Consequently, for a fixed $\\ba_n$ with $n \\geq n_0$, the rank $r(J_C(k))$ is bounded for $C \\in \\Cc_{\\ba_n}(k)$.\n\t\\end{thm}\n\nWe use the uniformity results given this non-conditional result to discuss a solution for the problem of arithmetic progression on the points on the elliptic curves $y^2=x (x^2+B)$ as well as the congruent number elliptic curves, in Section~\\ref{sec:applications}.\n\n\tUsing the uniformity theorems on the number of $k$-rational points on curve of genus $\\geq 2$ gibem in \\cite{Caporaso1997}, as\n\tconsequences of weak version of Lang's conjecture, we showed the cardinal number of $\\Cc_{\\ba_n}(k)$ does not depends on the choice of ${\\ba_n}$ and is uniformly bounded and eventually empty for large n.\n\n\t\\begin{thm} \\label{uniformity_theorem}\n\t\tAssume the weak version of Lang's conjecture (Conjecture \\ref{conj1}) holds over $k$. Let $r \\geq 1, s \\geq 2$ such that $g_{r,s}(C) \\geq 1$, and let $n_0$ be as in Theorem \\ref{finiteness_theorem}. Then for $n \\geq n_0$, there exists a uniform bound $M_0 = M_0(k, r, s, n)$ such that $\\#\\Cc_{\\ba_n}(k) < M_0$ for all valid choices of $\\ba_n$. Furthermore, if $g_{r,s}(C) \\geq 2$, there exists an integer $m_0 > n_0$ such that $\\Cc_{\\ba_m}(k) = \\emptyset$ for all $m \\geq m_0$ and all valid $\\ba_m$. (The strong Lang conjecture implies $M_0$ and $m_0$ can be chosen independently of $k$.)\n\t\\end{thm}\n\n\tThe structure of the paper mirrors the logical progression of the proof. Section \\ref{sec_geom_constr} details the construction of the parameter space $\\Wc_n$ using twists and its birational model $\\Xc_n$, a complete intersection variety, establishing their relationship. Section \\ref{sec_fiber_analysis} focuses on the geometry of the fibers $\\Xc_{\\ba_n}$ of $\\Xc_n \\to \\Sym^{n+1}(\\mathbb{P}^1)$, calculating their genus and gonality, analyzing rational points in low-genus cases, and invoking the theory of towers of curves for the general case, integrating necessary background results. Section \\ref{sec_finiteness_uniformity_proofs} provides proofs for Theorems \\ref{finiteness_theorem} and \\ref{uniformity_theorem}, incorporating Faltings' theorem and Lang's conjectures. In Section \\ref{sec_main_theorem_proof}, we prove the main result, Theorem \\ref{main_theorem}, using the uniformity theorem and the work of Dimitrov, Gao, and Habegger.\t\n\t Throughout, we cite analogous arguments from \\cite{Salami2025toappear} where applicable.\n\tIn Section~\\ref{sec:applications}, we have provided consequences of our results for elliptic curves of the form $y^2=x(x^2+B)$ and in particular the congruent number elliptic curves. \n\t Finally, in Section \\ref{examples}, we have provided a computational example of the correspondence between\n\tthe set of curves $\\Cc_{\\ba_n}$ and rational points on the fiber $\\Xc_{\\ba_n}$.", + "sketch": "The post-theorem introduction gives the following proof outline for Theorem\\ref{main_theorem}.\n\n- The proof \\emph{“relies crucially on studying the set $\\Cc_{\\ba_n}(k)$ consisting of smooth curves $C$ in the family \\eqref{curve_main} that pass through a given set of $n+1$ points with $x$-coordinates $\\alpha_i\\in k$”}, where $\\ba_n=\\{\\alpha_0,\\dots,\\alpha_n\\}$ are distinct nonzero elements with $\\alpha_i^r\\neq \\alpha_j^r$ for $i\\neq j$.\n\n- One first \\emph{“establish[es]… finiteness and uniformity results for these sets”} via Theorems \\ref{finiteness_theorem} and \\ref{uniformity_theorem} (finiteness for $n\\ge n_0$; uniform bounds $\\#\\Cc_{\\ba_n}(k) 0$ be a positive real number. Let $\\Phi:(M,g)\\to (S^{n-m}\\times \\T^m \\times [-1,1], g_{S^{n-m}} + g_{\\T^m} + dt^2)$ be a smooth map with the following properties:\n \\begin{itemize}\n \\item $\\Phi$ has non-zero degree.\n \\item $\\Phi(\\partial_{\\pm}M)\\subset S^{n-m}\\times \\T^m \\times\\{\\pm1\\}.$\n \\item $\\pr_{S^{n-m}}\\circ\\Phi:(M,g)\\to (S^{n-m}, g_{S^{n-m}})$ is area non-increasing when $n-m\\geq 3$ and 1-Lipschitz when $n-m=2$.\n \\end{itemize}\nThe followings hold for the bandwidth of $M$:\n\\begin{enumerate}[(i)]\n \\item Suppose that $R_M \\geq (n-m)(n-m-1) + \\sigma$. Then\n \\[ \\textup{dist}_g(\\partial_-M, \\partial_+M) \\leq 2\\pi\\sqrt{\\frac{n}{(n+1)\\sigma}}.\\]\n \\item Suppose that $\\psi$ is a smooth function on $M$ such that \n \\[ -\\Delta_M\\psi -\\frac{1}{2}|D_M\\psi|^2 + \\frac{1}{2}\\Big(R_M - (n-m)(n-m-1) - \\sigma \\Big) \\geq 0.\\]\n Then\n \\[ d(\\partial_-M, \\partial_+M) \\leq \\frac{2\\pi}{\\sqrt{\\sigma}}.\\]\n\\end{enumerate}\n\\end{maintheorem}\n\n\\begin{proposition}\\label{slicing;exist;prop}\n Let $2\\leq n\\leq 7$ and $1\\leq m\\leq n-2$. Let $(M^n, g)$ be a closed, connected Riemannian manifold of dimension $n$ with a smooth map $\\Phi:(M^n,g)\\to (S^{n-m}\\times \\T^m, g_{S^{n-m}} + g_{\\mathbb{T}^{m}})$ of non-zero degree such that $\\pr_{S^{n-m}}\\circ\\Phi :(M^n,g)\\to (S^{n-m}, g_{S^{n-m}})$ is 1-Lipschitz. Then there exists a stable weighted slicing\n \\[ \\Sigma_m\\subset\\Sigma_{m-1}\\subset\\cdots\\subset\\Sigma_1\\subset\\Sigma_0 = M^n\\]\n of order $m$.\n\n\\begin{theorem}\\label{spectral;Llarull;thm}\n Let $N^n$ be a closed, connected spin manifold of dimension $n$. Let $g$ be a Riemannian metric on $N$. Suppose that there exists a smooth function $\\rho >0$ on $N$ such that\n\\begin{align}\\label{spectral;Llarull;assumption}\n 0 \\leq& \\int_{N}\\rho|D_{N}f|^2 - \\frac{1}{2}\\int_{N}(n(n-1) - R_{N})\\rho\\, f^2 - \\int_{N} \\left(\\Delta_{N}\\log\\rho + \\frac{1}{2} |D_{N}\\log\\rho|^2\n \\right) \\rho\\, f^2\n\\end{align}\nfor all $f\\in C^{\\infty}(N)$.\nSuppose that $\\Phi:(N,g)\\to (S^n, g_{S^n})$ is a smooth map with the following properties:\n \\begin{itemize}\n \\item $\\Phi$ has non-zero degree,\n \\item $\\Phi$ is 1-Lipschitz.\n \\end{itemize}\n Then $\\rho$ is a constant on $N$ and $\\Phi$ is a Riemannian isometry.\n\\end{theorem}\n\n\\subsection{Proof of Theorem \\ref{main;thm;2}}\nIn the first step, we construct a $\\mu$-bubble that allows us to apply Theorem \\ref{main;thm;1}. Denote the projection of $\\Phi$ onto the factors by $\\phi:M\\to S^{n-m}\\times \\T^m$ and $\\varphi:M\\to [-1,1]$. By assumption, the map $\\pr_{S^{n-m}}\\circ\\phi = \\pr_{S^{n-m}}\\circ\\Phi$ is 1-Lipschitz. Let $\\Theta$ be a top-form of $S^{n-m}\\times\\T^m$ such that $\\int_{S^{n-m}\\times\\T^m}\\Theta = 1$. Define the pull-back form $\\omega = \\phi^*\\Theta$. By Sard's theorem, we can find a regular value $t_0\\in (-1,1)$ of the map $\\varphi$. Let $\\hat{\\Sigma} = \\varphi^{-1}(t_0)$. Because $M$ is connected and $\\varphi$ is continuous, and by assumption $\\Phi(\\partial_{-}M)\\subset S^{n-m}\\times \\T^m \\times\\{-1\\}$ and $\\Phi(\\partial_{+}M)\\subset S^{n-m}\\times \\T^m \\times\\{1\\}$, we see that $\\varphi(M)$ is a connected subset of $[-1,1]$ containing both $-1$ and $1$. It follows that $\\varphi(M) = [-1,1]$ and $ \\varphi^{-1}(t)$ is non-empty for all regular values $t$. So $\\hat{\\Sigma}$ is a smooth, orientable and embedded hypersurface in $M$. By the coarea formula,\n\\begin{align}\\label{bandwidth;eqn1}\n \\deg(\\Phi) = \\int_M d\\varphi\\wedge\\omega = \\int_{-1}^1\\left(\\int_{\\varphi^{-1}(t)}\\omega\\right) dt\n\\end{align}\nOn the other hand, if $t_1 < t_2$ are two regular values, the Stokes theorem gives\n\\begin{align}\\label{bandwidth;eqn2}\n 0 = \\int_{\\varphi^{-1}([t_1, t_2])} d\\omega = \\int_{\\varphi^{-1}(t_2)}\\omega - \\int_{\\varphi^{-1}(t_1)}\\omega.\n\\end{align}\nPutting (\\ref{bandwidth;eqn1}) and (\\ref{bandwidth;eqn2}) together, we obtain\n\\begin{align}\\label{bandwidth;eqn3}\n \\int_{\\hat{\\Sigma}}\\omega = \\deg(\\Phi).\n\\end{align}\n\nIn conclusion, we have obtained a smooth map $\\Psi:\\Sigma^n\\to S^{n-m}\\times\\T^m$ of non-zero degree such that $\\pr_{S^{n-m}}\\circ\\Psi$ is 1-Lipschitz, and a smooth function $\\log\\rho$ on $\\Sigma$ satisfying\n\\begin{align}\\label{bandwidth;eqn6}\n -\\Delta_{\\Sigma}\\log\\rho -\\frac{1}{2}|D_{\\Sigma}\\log\\rho|^2 + \\frac{1}{2}(R_{\\Sigma} - (n-m)(n-m-1) ) > 0.\n\\end{align}\n Theorem \\ref{main;thm;1} then implies $R_{\\Sigma} = (n-m)(n-m-1)$ and $\\rho$ is a constant function, contradicting to (\\ref{bandwidth;eqn6}). This finishes the proof of Theorem \\ref{main;thm;2}.", + "post_theorem_intro_text_len": 3937, + "post_theorem_intro_text": "Our approach combines stable weighted slicing \\cite{BHJ23} with a spectral Llarull-type argument compatible with the stability inequality. We construct a stable weighted slicing of order $m$, i.e., a nested family of hypersurfaces $\\Sigma_m \\subset \\cdots \\subset \\Sigma_1 \\subset M$\nwith positive weights that record the torus directions. On the bottom slice we use the stability inequality together with the Weitzenböck-Lichnerowicz identity for spinors. Equality forces $\\Sigma_m$ to be a round $S^{n-m}$; we then propagate rigidity upward by foliating with weighted minimizers and applying a slice-by-slice splitting argument. It is worth noting that the local isometry statement in the case $n-m=3$ and $R_g\\ge 6$ was proved by different methods in \\cite{HLS}. Related consequences under higher mapping-degree assumptions was done in \\cite{Tony}, which shows that when $\\pr_{S^{n-m}}\\circ\\Phi:M^n\\to S^{n-m}$ is a fiber bundle, it must be a Riemannian submersion.\\\\\n\nIn the incomplete setting, we extend Gromov's torical band inequality \\cite{Gro18,Gro19} to bands over sphere--torus products. Bands are among the most flexible test-objects for scalar curvature: they detect how lower bounds on scalar curvature constrain the macroscopic separation of boundary components, yielding distance-type obstructions to fill-ins, doubling, and collaring. Recent progress has deepened the connection between scalar curvature and band width; see for instance \\cite{CZ24, HKKZ23, HKKZ25}. For targets of the form $S^{n-m}\\times \\mathbb{T}^m$, the interplay between a positively curved spherical factor and $m$ flat directions raises a natural quantitative question: how far apart can two boundary components be kept under a scalar curvature lower bound that matches the spherical threshold up to a gap $\\sigma>0$? Our results below give sharp-in-scale ($\\sim\\sigma^{-1/2}$) upper bounds, reflecting the model behavior of constant-curvature metrics and extending the torical width control to the mixed sphere-torus regime. In particular, requiring the spherical projection to be area non-increasing prevents macroscopic stretching in the directions where scalar curvature is most constrained, and the band-width estimates quantify this restriction. To that end, we prove:\\\\\n\n\\begin{maintheorem}\\label{main;thm;2}\n\tLet $3\\leq n+1\\leq 7$ and $1\\leq m\\leq n-2$. Let $M^{n+1}$ be an orientable, connected spin manifold of dimension $n$ with non-empty boundary with two connected components $\\partial M = \\partial_-M\\sqcup\\partial_+M$. Let $g$ be a Riemannian metric on $M$. Let $\\sigma > 0$ be a positive real number. Let $\\Phi:(M,g)\\to (S^{n-m}\\times \\mathbb{T}^m \\times [-1,1], g_{S^{n-m}} + g_{\\mathbb{T}^m} + dt^2)$ be a smooth map with the following properties:\n\t\\begin{itemize}\n\t\t\\item $\\Phi$ has non-zero degree.\n\t\t\\item $\\Phi(\\partial_{\\pm}M)\\subset S^{n-m}\\times \\mathbb{T}^m \\times\\{\\pm1\\}.$\n\t\t\\item $\\pr_{S^{n-m}}\\circ\\Phi:(M,g)\\to (S^{n-m}, g_{S^{n-m}})$ is area non-increasing when $n-m\\geq 3$ and 1-Lipschitz when $n-m=2$.\n\t\\end{itemize}\nThe followings hold for the bandwidth of $M$:\n\\begin{enumerate}[(i)]\n \\item Suppose that $R_M \\geq (n-m)(n-m-1) + \\sigma$. Then\n \\[\t\\textup{dist}_g(\\partial_-M, \\partial_+M) \\leq 2\\pi\\sqrt{\\frac{n}{(n+1)\\sigma}}.\\]\n \\item Suppose that $\\psi$ is a smooth function on $M$ such that\t\n\t\\[\t-\\Delta_M\\psi -\\frac{1}{2}|D_M\\psi|^2 + \\frac{1}{2}\\Big(R_M - (n-m)(n-m-1) - \\sigma \\Big) \\geq 0.\\]\n Then\n \\[\td(\\partial_-M, \\partial_+M) \\leq \\frac{2\\pi}{\\sqrt{\\sigma}}.\\]\n\\end{enumerate}\n\\end{maintheorem}\n\n\\bigskip\n\n\\begin{remark}\nFor the sake of exposition we will henceforth work under the stronger hypothesis that\n \\[\\pr_{S^{n-m}}\\circ\\Phi:(M,g)\\to (S^{n-m}, g_{S^{n-m}})\\]\nis 1-Lipschitz. This assumption streamlines the notations used in the proof. All arguments, however, carry over verbatim when $n-m\\geq 3$ if one merely assumes that the map is area non-increasing.\n\\end{remark}", + "sketch": "To prove Theorem~\\ref{main;thm;1}, the approach “combines stable weighted slicing \\cite{BHJ23} with a spectral Llarull-type argument compatible with the stability inequality.” One “construct[s] a stable weighted slicing of order $m$, i.e., a nested family of hypersurfaces $\\Sigma_m \\subset \\cdots \\subset \\Sigma_1 \\subset M$ with positive weights that record the torus directions.” On the bottom slice, one “use[s] the stability inequality together with the Weitzenb\\\"ock-Lichnerowicz identity for spinors.” “Equality forces $\\Sigma_m$ to be a round $S^{n-m}$; we then propagate rigidity upward by foliating with weighted minimizers and applying a slice-by-slice splitting argument.”", + "expanded_sketch": "To prove the main theorem, the approach “combines stable weighted slicing \\cite{BHJ23} with a spectral Llarull-type argument compatible with the stability inequality.” One “construct[s] a stable weighted slicing of order $m$, i.e., a nested family of hypersurfaces $\\Sigma_m \\subset \\cdots \\subset \\Sigma_1 \\subset M$ with positive weights that record the torus directions.” On the bottom slice, one “use[s] the stability inequality together with the Weitzenb\\\"ock-Lichnerowicz identity for spinors.” “Equality forces $\\Sigma_m$ to be a round $S^{n-m}$; we then propagate rigidity upward by foliating with weighted minimizers and applying a slice-by-slice splitting argument.”", + "expanded_theorem": "\\label{main;thm;1}\n\tLet $3\\leq n\\leq 7$ and $1\\leq m\\leq n-2$. Let $M^n$ be a closed, orientable, connected spin manifold of dimension $n$. Let $g$ be a Riemannian metric on $M$. Let $\\psi$ be a smooth function on $M$ such that\t\n\t\\[\t-\\Delta_M\\psi -\\frac{1}{2}|D_M\\psi|^2 + \\frac{1}{2}\\Big(R_M - (n-m)(n-m-1)\\Big) \\geq 0.\\]\nSuppose that $\\Phi:(M,g)\\to (S^{n-m}\\times \\mathbb{T}^m, g_{S^{n-m}} + g_{\\mathbb{T}^m})$ is a smooth map with the following properties:\n\t\\begin{itemize}\n\t\t\\item $\\Phi$ has non-zero degree,\n\t\t\\item $\\pr_{S^{n-m}}\\circ\\Phi:(M,g)\\to (S^{n-m}, g_{S^{n-m}})$ is area non-increasing when $n-m\\geq 3$ and 1-Lipschitz when $n-m=2$.\n\t\\end{itemize}\n\tThen $(M^n, g)$ is isometrically covered by $(S^{n-m}\\times\\mathbb{R}^m,\\, g_{S^{n-m}} + g_{\\mathbb{R}^m})$.,", + "theorem_type": [ + "Implication", + "Universal" + ], + "mcq": { + "question": "Let $3\\le n\\le 7$ and $1\\le m\\le n-2$. Let $M^n$ be a closed, orientable, connected spin $n$-manifold with Riemannian metric $g$, and let $\\psi\\in C^\\infty(M)$ satisfy\n\\[\n-\\Delta_M\\psi-\\frac12|D_M\\psi|^2+\\frac12\\bigl(R_M-(n-m)(n-m-1)\\bigr)\\ge 0.\n\\]\nAssume there is a smooth map\n\\[\n\\Phi:(M,g)\\to \\bigl(S^{n-m}\\times \\mathbb T^m,\\, g_{S^{n-m}}+g_{\\mathbb T^m}\\bigr)\n\\]\nof nonzero degree, where $S^{n-m}$ has its round metric and $\\mathbb T^m$ its flat torus metric, such that the projection onto the sphere factor,\n\\[\n\\operatorname{pr}_{S^{n-m}}\\circ \\Phi:(M,g)\\to (S^{n-m},g_{S^{n-m}}),\n\\]\nis area non-increasing when $n-m\\ge 3$ and is $1$-Lipschitz when $n-m=2$. Under these assumptions, which conclusion about $(M,g)$ holds?", + "correct_choice": { + "label": "A", + "text": "$(M,g)$ is isometrically covered by $\\bigl(S^{n-m}\\times \\mathbb R^m,\\, g_{S^{n-m}}+g_{\\mathbb R^m}\\bigr)$; equivalently, there exists a covering map from $S^{n-m}\\times \\mathbb R^m$ onto $M$ that is a local isometry for the product of the round metric on $S^{n-m}$ and the Euclidean metric on $\\mathbb R^m$." + }, + "choices": [ + { + "label": "B", + "text": "$(M,g)$ is isometric to $\\bigl(S^{n-m}\\times \\mathbb T^m,\\, g_{S^{n-m}}+g_{\\mathbb T^m}\\bigr)$ itself; in particular, the given map $\\Phi$ can be taken to be a Riemannian isometry." + }, + { + "label": "C", + "text": "The universal cover of $(M,g)$ splits isometrically as $\\bigl(S^{n-m}\\times \\mathbb R^m,\\, g_{S^{n-m}}+g_{\\mathbb R^m}\\bigr)$." + }, + { + "label": "D", + "text": "$(M,g)$ is isometrically covered by $\\bigl(S^{n-m}\\times \\mathbb R^m,\\, g_{S^{n-m}}+g_{\\mathbb R^m}\\bigr)$ provided that $\\operatorname{pr}_{S^{n-m}}\\circ\\Phi$ is $1$-Lipschitz; in fact the same conclusion holds without distinguishing the cases $n-m\\ge 3$ and $n-m=2$." + }, + { + "label": "E", + "text": "$(M,g)$ is isometrically covered by $\\bigl(S^{n-m}\\times N^m,\\, g_{S^{n-m}}+h\\bigr)$ for some complete flat $m$-manifold $(N^m,h)$, but one cannot in general conclude that the cover is $S^{n-m}\\times \\mathbb R^m$ with the Euclidean metric on the second factor." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "finiteness", + "tampered_component": "global quotient-versus-cover conclusion", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "replace explicit isometric covering statement by universal-cover splitting", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "characteristic", + "tampered_component": "dimension-dependent hypothesis on spherical projection", + "template_used": "boundary_range" + }, + { + "label": "E", + "sketch_hook_type": "geometric_construction", + "tampered_component": "rigid upward propagation forcing Euclidean torus directions", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly reveal the conclusion, and no choice is singled out by wording alone. It presents hypotheses of a rigidity theorem and asks for the resulting geometric conclusion, so there is no direct answer leakage." + }, + "TAS": { + "score": 1, + "justification": "This is close to a theorem-recall item: the stem essentially states the full hypotheses and asks for the theorem's conclusion. However, it is not completely tautological because the options include stronger, weaker, and hypothesis-tampered variants that must be distinguished." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to reject the stronger claim (B), the weaker-but-true formulation (C), the altered-hypothesis claim (D), and the plausible alternative flat-factor conclusion (E). Still, the item mainly tests precise recall/application of a known theorem rather than substantial derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic failure modes: confusing exactness with a stronger isometry statement, settling for a weaker universal-cover splitting, overlooking the dimension-dependent hypothesis, or weakening the flat factor to an arbitrary flat manifold." + }, + "total_score": 6, + "overall_assessment": "A solid advanced theorem-application MCQ with strong distractors and no answer leakage, but it leans heavily on recall of a specific rigidity theorem rather than deep generative reasoning." + } + }, + { + "id": "2511.03672v1", + "paper_link": "http://arxiv.org/abs/2511.03672v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "MainThm", + "content": "\\label{thm:main}\n Let $(M,g)$ be a closed smooth $(C^\\infty)$ Riemannian manifold without conjugate points with divergence property of geodesic rays and Gromov hyperbolic and residually finite fundamental group.\n Assume that \n $$\n h_\\nu(\\phi)<\\htop(\\phi)\n $$\nfor all non-expansive measures\n$\\nu \\in \\MMM_\\phi(SM)$. Then the geodesic flow has a unique measure $\\mu$ of maximal entropy. \n\n Furthermore, $\\mu(\\mathcal{E}) =1$, $\\mu$ is mixing and fully supported on the unit tangent bundle.", + "start_pos": 17670, + "end_pos": 18185, + "label": "thm:main" + }, + "ref_dict": { + "subsec:hypgroups": "\\begin{proof}\n For a proof of (1), (2), (3a) see \\cite[Lemma III.H 3.2, Lemma III.H 3.3]{BH99}. The assertion (3b) is a consequence\n of (3a) and Lemma \\ref{lem:endpts-suffice}.\n\\end{proof}\nGromov introduced in \\cite{mG87} a natural topology on $\\partial X$ and on $ \\bar X :=X \\cup \\partial X$ such that $ \\bar X $ is compact (see also \\cite[Def. III.H 3.5]{BH99}). \nWe will give a definition in the geometric context in subsection \\ref{subsec:Noconjugatepts-Ghyp}.\n\n\\subsection{Gromov hyperbolic groups}\\label{subsec:hypgroups}\n\\begin{definition}\\label{def:Hypgroups}\nLet $G$ be a finitely generated group and denote by $\\CC(G, S)$ the Cayley graph of $G$ with respect to some finite generating set $S \\subset G$. Then $G$ is called Gromov hyperbolic if $\\CC(G, S)$ equipped with the word metric\nis a $\\delta$-hyperbolic metric space for some $\\delta\\ge 0$\n\\end{definition}\n\\begin{remark}\n\\begin{itemize}\n\\item\nIf $S_1, S_2$ are finite generating sets of $G$ then the identity $\\id: \\CC(G, S_1) \\to \\CC(G, S_2)$ is a quasi isometry.\nIn particular, the definition of Gromov hyperbolicity of a finitely generated group does not depend on the choice of the finite generating set.\n\\item Let $\\Gamma$ be a group acting properly, cocompactly and isometrically on some geodesic metric space $X$.\nThen $\\Gamma$ is finitely generated and for any reference point $x_0 \\in X$ and finite set of generators the orbit map $\\Gamma \\to X$ with\n$\\gamma \\mapsto \\gamma x_0$ extends to a quasi isometry between $\\CC(G, S)$ and $X$ (see e.g. \\cite[Prop. I.8.19]{BH99}).\nIn particular, $\\Gamma$ is Gromov hyperbolic if and only if $X$ is Gromov hyperbolic.\n\\end{itemize}\n\\end{remark}\n\nAssume that $\\Gamma$ is a Gromov hyperbolic group acting properly and cocompactly\nby isometries on proper metric space $X$. Consider for $x\\in X$ and $R>0$ the subset of $\\Gamma$ given by\n$$\n\\Gamma_R(x) = \\{\\gamma \\in \\Gamma \\mid\\; d(x,\\gamma x) \\leq R\\}.\n$$\n Then by a result of Coornaert \\cite{mCo93} there are constants $0 0$ such that\n\\begin{equation}\\label{eqn:co93}\nC_1 e^{hR}\\le \\card \\ \\Gamma_R(x) \\le C_2 e^{hR}\n\\end{equation}\nIf $(X,g)$ is a Riemannian manifold then $h$ is the volume entropy $\\hvol(g)$, i.e.\n$$\nh = \\lim\\limits_{r \\to \\infty} \\frac{\\log \\mathrm{vol} B(p,r)}{r} \n$$\nDefine the translation length of $\\gamma \\in \\Gamma$ by\n \\begin{equation} \\label{e: length}\n\\ell(\\gamma) = \\inf_{x \\in X} d(x,\\gamma x).\n\\end{equation}\nNote that the infimum in\n\\eqref{e: length} is attained\nfor each $\\gamma \\in \\Gamma$ (see\nfor example \\cite[Prop. II.6.10]{BH99}). Furthermore, $\\ell(\\gamma^{-1}) = \\ell(\\gamma)$ and\n$\\ell(\\alpha\\gamma\\alpha^{-1}) = \\ell(\\gamma)$ for\nevery isometry $\\alpha$ of $X$. Therefore the length of a conjugacy class $[\\gamma]$ of $\\gamma$ can be defined as the translation length of a representative of $[\\gamma]$. \nUsing ideas developed in \\cite{gK83} and \\cite{gK97} we proved in \\cite{CK02}: \n\\begin{theorem} \\label{thm:conj-Grhyp}\nLet $\\Gamma$ be a group acting\nproperly and cocompactly by isometries on\na proper geodesic Gromov-hyperbolic\nmetric space $X$. Assume\nthat the Gromov boundary $\\partial X$ of $X$\ncontains more than two points.\n For all $t \\geq 0$, let\n$$\nP(t) = \\card \\{ [\\gamma] \\mid\\; \\gamma \\in \\Gamma \\; \\; \\text{is primitive and } \\; \\; \\ell(\\gamma) \\leq t \\},\n$$\nwhere $ \\gamma \\in \\Gamma$ is called primitive if it cannot be written as a proper power $\\gamma = \\alpha^n$ for some $\\alpha \\in \\Gamma$ and $ n \\ge 2$.\n\n Then there exists\nconstants $A\\ge 1$, $t_0 >0$ and such that\n$$\n\\frac{1}{A} \\frac{ e^{ht} } {t} \\leq P(t) \\leq A e^{ht}\n$$\nfor all $t \\geq t_0$, where $h>0$ is as in \\ref{eqn:co93}.\n\\end{theorem}\n\\begin{remark} \\label{rem:conj-Grhyp}\n\nThis results has the following interpretation in the Riemannian setting.\nAssume that $(M,g)$ is a closed manifold Riemannian manifold with a Gromov hyperbolic fundamental group\n$\\pi_1(M)$. Let $\\wM$ be the universal covering then $M = \\wM/ \\Gamma$ where $\\Gamma$ is the group of covering transformations isomorphic to $\\pi_1(M)$.\nAs mentioned above\n$h$ is the volume entropy $\\hvol(g)$ of $g$ and $P(t)$ is equal to the number of free homotopy classes containing a closed geodesic of period less than $t$.\nBy a result of Manning \\cite{aM79}, the topological entropy $\\htop(\\phi)$ of the geodesic flow $\\phi^t$ on $SM$ is larger than the volume entropy.\n\n\\end{remark}\n\n\\subsection{Closed manifolds without conjugate points and Gromov hyperbolic fundamental groups}\\label{subsec:Noconjugatepts-Ghyp}\nIn the following we will assume that $(M,g)$ is a closed Riemannina manifold without conjugate points and Gromov hyperbolic fundamental\ngroup $\\pi_1(M)$. As pointed out above in this case Freir\\'{e} and Ma\\~n\\'e \\cite{FM82} \nproved that\nthe volume entropy $\\hvol(g)$ is equal to the topological entropy $\\htop(\\phi)$. \n\nFirst we recall the following theorem which is foundational in the theory\nof manifolds without conjugate points. \n\\begin{theorem}(Hadamard-Cartan)\\label{thm: Hadamard-Cartan}\\\\\nLet $(M,g)$ be a complete $n$-dimensional Riemannian manifold with\nno conjugate points. Then for all $p \\in M$ the exponential map\n$\\exp_p: T_pM \\to M$ is a covering map.\nIn particular complete simply connected manifolds without conjugate points are diffeomorph to\n$\\R^n$.\n\\end{theorem}\n\\begin{remark}\n\\begin{enumerate}\n\\item[{\\rm (a)}] A complete Riemannian manifold $(M,g)$ has no conjugate points iff for any pair of points on the universal cover $\\wM$ there is a unique connecting geodesic geodesics with respect to the lifted metric. In particular all geodesics are minimizing.\n\\item[{\\rm (b)}] The topology of those manifolds is to a large extend determined by the\nfundamental group since the contractibility of the universal cover implies that the higher\nhomotopy groups are vanishing, i.e., $\\pi_k(M) = 0$ for $k \\ge 2$.\n\\item[{\\rm (c)}] Manifolds of non-positive sectional curvature form an important subclass of manifolds with no conjugate points. Simply connected complete manifolds of nonpositive curvature are called Hadamard manifolds.\n\n\\end{enumerate}\n\\end{remark}\n\nWe assume that $(M,g)$ satisfies the divergence property, i.e. \n for any pair of geodesics $c_1 \\neq c_2$ in $( \\wM, g)$ with $c_1(0) =c_2(0)$ \nwe have\n\\begin{equation}\\label{eqn:divergence}\n\\lim\\limits_{t \\to \\infty} d(c_1(t), c_2(t)) = \\infty.\n\\end{equation}\n\\begin{remark}\n\\begin{itemize}\n\\item\nBy a result of E. Hopf \\cite{eH48} all closed non-flat surfaces without conjugate points have genus at least two. Since such surfaces \ncarry a metric with negative curvature their fundamental groups are Gromov hyperbolic. Furthermore, Green \\cite{wG56} showed that surfaces without conjugate have the divergence property.\n\\item\nUntil now there is no example of a closed manifold without conjugate points known where the divergence property does not hold. A sufficient condition for the divergence property is the continuity of the stable Jacobi-tensors (see \\cite{ES76}). In particular, this assumption holds if $(M,g)$ has non-positive curvature, or more generally no focal points.\n\\end{itemize}\n\\end{remark}\n The following notion was introduced by Eberlein in \\cite{pEb72} and Eberlein and O'Neill \\cite{EO73}.\n\\begin{definition}\nA simply connected Riemannian manifold $\\wM$ without conjugate points \nis a \\emph{(uniform) visibility manifold} if for every $\\epsilon>0$ there exists $L>0$ such that whenever a geodesic $c\\colon [a, b] \\to\\wM$ stays at distance at least $L$ from some point $p\\in\\wM$, then the angle sustained by $c$ at $p$ is less than $\\epsilon$, that is\n \\begin{equation*}\n \\angle_p(c)=\\sup_{a\\leq s,t\\leq b} \\angle_p(c(s),c(t))<\\epsilon.\n \\end{equation*}\n\\end{definition}\nThe following Theorem is due to Ruggiero \\cite{rR03},\n \\begin{theorem}\\label{thm:visibility}\nLet $(M,g)$ be a closed manifold without conjugate points and Gromov hyperbolic fundamental group. Then $( \\wM, g)$ is a visibility manifold if and only if $(M,g)$ has the divergence property.\n\\end{theorem}\n\\begin{remark}\nUnder the stronger assumption that $(M,g)$ is a closed manifold without conjugate points admitting a background metric of negative curvature this has been proved by Eberlein \\cite{pEb72}.\n\\end{remark}\nSince $\\wM$ is Gromov hyperbolic, Lemma \\ref{lem:GB} and the divergence property implies that the map\n$f_p\\colon S_p\\wM \\to \\ideal$ defined by $f_p(v) = c_v(\\infty)$ is a bijection.\nThe topology (sphere-topology) on $\\partial \\wM$ is defined such that $f_p$\nbecomes a homeomorphism.\nSince for all $q \\in \\wM$ the map $f_q^{-1} f_p \\colon S_p\\wM \\to S_q\\wM$ is a homeomorphism, see \\cite{pEb72},\nthe topology is independent on the reference point $p$.\nThe topologies on $\\partial \\wM$ and $\\wM$\nextend naturally\n to $\\cl (\\wM): = \\wM\\cup \\partial \\wM$\nby requiring that the map\n$\\varphi\\colon B_1(p) = \\{v \\in T_p \\wM: \\|v\\| \\le 1\\} \\to \\cl(\\wM)$\ndefined by\n\\[\n\\varphi(v) = \\begin{cases}\n \\exp_p\\left(\\frac{v}{1-\\|v\\|}\\right) & \\|v\\| < 1\\\\\nf_p(v) & \\|v\\| = 1\n\\end{cases}\n\\]\nis a homeomorphism. This topology, called the cone topology, was introduced by Eberlein\nand O'Neill \\cite{EO73} in the case of Hadamard manifolds and by Eberlein \\cite{pEb72} in the case of visibility manifolds. In\nparticular, $\\cl( \\wM) $ is homeomorphic to a closed ball in\n$\\mathbb{R}^n$. The relative topology on $\\ideal$ coincides with the sphere topology, and the relative topology on $\\wM$ coincides with the topology of $\\wM$.\n\nNote that for the case which we are considering\nthis compactification agrees with the compactification of $\\delta$-hyperbolic spaces do to Gromov \\cite{mG87}.\n\nFor simply connected manifolds $X$ without conjugate points for $v\\in SX$, the limit\n\\[\nb_{v}(q) := \\lim_{t\\to\\infty} \\left( d(q, c_{v}(t)) - t\\right)\n\\]\nexists and is called the \\emph{Busemann function} associated to $v$. In \\cite{gK85} it was shown that Busemann functions are of class $C^{1,1}$ provided the sectional curvature is uniformly bounded from below.\n\\begin{definition}\\label{def:busemann}\nLet $M$ be a closed manifold without conjugate points, divergence property and Gromov hyperbolic fundamental group. \nThen according to Lemma \\ref{lem:GB} and the divergence property, \nfor each $p\\in X$ and $\\xi \\in \\partial X$ there exists a uniquely determined vector $v\\in S_p X$ such that $c_{v}(\\infty) = \\xi$.\nWe call $b_\\xi(q,p) := b_{v}(q)$ the Busemann function based at $\\xi$\nand normalized by $b_\\xi(p,p) =0$.\n\\end{definition}\n\n\\begin{remark}\\label{rem:minmal}\nThe isometric action of $\\Gamma=\\pi_1(M)$ on $\\wM$ extends to a continuous action on $\\ideal$.\nSince by \\cite{pEb72} the geodesic flow is topologically transitive, every $\\Gamma$-orbit in $\\ideal$ is dense, i.e. the action on $\\ideal$ is minimal.\n\\end{remark}\n\n\\begin{proposition}\\label{cor:busemann}\nLet $M$ be a closed manifold without conjugate points, divergence property and Gromov hyperbolic fundamental group. \nThen the following holds\n\\begin{enumerate}\n\\item\nFor $p,q \\in \\wM$ and $\\xi \\in \\ideal$ we have\n$$\n\\lim\\limits_{z \\to \\xi }d(q,z) -d(p,z) = b_p(q, \\xi)\n$$\n\\item\nFor all $p,q,z\\in \\wM$ we have\n$$b_{q}(z,\\xi) = b_p(z,\\xi) - b_p(q,\\xi)$$\nIn particular $b_{q}(z,\\xi) = - b_z(q,\\xi)$\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nFor a proof see e.g. \\cite{CKW21}\n\\end{proof}", + "thm:expansiveMME": "\\begin{MainThm}\\label{thm:expansiveMME}\n Let $(M,g)$ be a closed Riemannian manifold without conjugate points with divergence property and Gromov\n hyperbolic and residually finite fundamental group. Assume that the geodesic flow is entropy expansive at a scale larger than $8 \\delta$, where $\\delta$ is the Gromov hyperbolicity constant of the universal cover. If the expansive set has non-trivial interior then the geodesic flow\n has a unique measure of maximal entropy $\\mu$.\n\nFurthermore, $\\mu(\\EE) =1$, $\\mu$ is mixing and fully supported on the unit tangent bundle.\n\\end{MainThm}", + "thm:main": "\\begin{MainThm}\\label{thm:main}\n Let $(M,g)$ be a closed smooth $(C^\\infty)$ Riemannian manifold without conjugate points with divergence property of geodesic rays and Gromov hyperbolic and residually finite fundamental group.\n Assume that \n $$\n h_\\nu(\\phi)<\\htop(\\phi)\n $$\nfor all non-expansive measures\n$\\nu \\in \\MMM_\\phi(SM)$. Then the geodesic flow has a unique measure $\\mu$ of maximal entropy. \n\n Furthermore, $\\mu(\\EE) =1$, $\\mu$ is mixing and fully supported on the unit tangent bundle.\n\n\\end{MainThm}", + "rem:conj-Grhyp": "\\begin{remark} \\label{rem:conj-Grhyp}\n\nThis results has the following interpretation in the Riemannian setting.\nAssume that $(M,g)$ is a closed manifold Riemannian manifold with a Gromov hyperbolic fundamental group\n$\\pi_1(M)$. Let $\\wM$ be the universal covering then $M = \\wM/ \\Gamma$ where $\\Gamma$ is the group of covering transformations isomorphic to $\\pi_1(M)$.\nAs mentioned above\n$h$ is the volume entropy $\\hvol(g)$ of $g$ and $P(t)$ is equal to the number of free homotopy classes containing a closed geodesic of period less than $t$.\nBy a result of Manning \\cite{aM79}, the topological entropy $\\htop(\\phi)$ of the geodesic flow $\\phi^t$ on $SM$ is larger than the volume entropy.\n\n\\end{remark}", + "thm:conj-Grhyp": "\\begin{theorem} \\label{thm:conj-Grhyp}\nLet $\\Gamma$ be a group acting\nproperly and cocompactly by isometries on\na proper geodesic Gromov-hyperbolic\nmetric space $X$. Assume\nthat the Gromov boundary $\\partial X$ of $X$\ncontains more than two points.\n For all $t \\geq 0$, let\n$$\nP(t) = \\card \\{ [\\gamma] \\mid\\; \\gamma \\in \\Gamma \\; \\; \\text{is primitive and } \\; \\; \\ell(\\gamma) \\leq t \\},\n$$\nwhere $ \\gamma \\in \\Gamma$ is called primitive if it cannot be written as a proper power $\\gamma = \\alpha^n$ for some $\\alpha \\in \\Gamma$ and $ n \\ge 2$.\n\n Then there exists\nconstants $A\\ge 1$, $t_0 >0$ and such that\n$$\n\\frac{1}{A} \\frac{ e^{ht} } {t} \\leq P(t) \\leq A e^{ht}\n$$\nfor all $t \\geq t_0$, where $h>0$ is as in \\ref{eqn:co93}.\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 4203, + "pre_theorem_intro_text": "\\label{sec:intro}\nIf $f: X \\to X$ is a homeomorphism of a compact metric space $X$,\nthe topological entropy $h_{\\mathrm{top}}(f)$ of $f$ is an invariant in topological dynamics measuring the orbit complexity of the dynamical system on an\nexponential scale. On the other hand, measure-theoretic entropy $h_{\\mu}(f)$ is an invariant of \nmeasurable dynamics reflecting the average complexity of the system relative\nto a measure $\\mu$ contained in the space of $f$-invariant Borel probability measure $\\MMM_f(X)$ on $X$. \nThe relation between these notions is provided by the variational principle which implies (see e.g. \\cite{pW82} for details)\n$$\nh_{\\mathrm{top}} (f) = \\sup \\{h_\\mu (f) \\;|\\; \\mu \\in \\MMM_f(X)\\}.\n$$\nA measure $\\mu \\in \\MMM_f(X)$ such that $h_{\\mu}(f) =h_{\\mathrm{top}}(f)$ is called a measure of maximal entropy (MME).\nFor continuous flows $\\phi^t: X \\to X$ topological - and measure theoretic entropy is defined as the entropy of the time 1 map $\\phi := \\phi^1$.\n\nIn the Riemannian setting topological entropy of the geodesic flow is related to large scale geometry.\nIf $(M,g)$ is a closed Riemannian manifold without conjugate points \n(all geodesics are globally minimizing if lifted to the universal cover $\\widetilde{M}$) then by a theorem of Freir\\'{e} and Ma\\~n\\'e \\cite{FM82} the topological entropy $\\htop (\\phi)$ of the geodesic\nflow $\\phi^t: SM \\to SM$ agrees with the volume entropy $\\hvol(g)$. Volume entropy measures the asymptotic volume growth of geodesics balls $B(p,r) \\subset \\widetilde{M}$ in the universal covering $\\widetilde{M}$ of $M$ and is given by\n$$\n\\hvol(g) = \\lim\\limits_{r \\to \\infty} \\frac{\\log \\mathrm{vol} B(p,r)}{r} \n$$\nFor manifolds $(M,g)$ of non-positive sectional curvature, which are special manifolds of no conjugate points, \nan easier proof of the result of Freir\\'{e} and Ma\\~n\\'e were previously given by Manning \\cite{aM79}. However, in contrast to non-positive sectional curvature, no conjugate points \nis not a local condition. Properties like the convexity of distance functions which\n hold for manifolds of non-positive sectional curvature and even no focal points\nare not anymore true for manifolds without conjugate points. Furthermore, the flat strip theorem, important in the study of manifolds \nof non-positive sectional curvature, fail under the assumption of no conjugate points \\cite{kB92}. \n\nIn this paper, we like to study the entropy of geodesic flows on closed Riemannian manifolds $(M, g)$ without conjugate points under some coarse hyperbolicity of the geodesic flow.\nHyperbolicity implies the existence of a non-trivial expansive set $\\mathcal{E} \\subset SM$ consisting of flow lines\nwhich diverge from each other over time. More formally, if $d$ is the distance function induced by the Riemannian metric on $M$, a vector $v \\in SM$ is contained in $\\mathcal{E}$ if there is some\n $\\epsilon >0$ such that for all $w \\in SM $ with $d(c_v(t), c_w(\\mathbb{R})) < \\epsilon$ for all $t \\in \\mathbb{R}$,\nthe geodesics $c_v$, $c_w$ with initial conditions $v$ and $w$ agree up to a time shift. Furthermore, the expansive set\nis invariant under the geodesic flow.\n\nGeodesic flows are called expansive if $SM =\\mathcal{E}$.\nIn case that $(M,g)$ has no conjugate points it was proved by Ruggiero \\cite{rR94} that the expansiveness of the geodesic\nflow \nimplies that the fundamental group $\\pi_1(M)$ is Gromov hyperbolic. Furthermore the divergence property hold, i.e. for each pair of geodesic rays $c_1, c_2: [0, \\infty) \\to \\widetilde{M} $ on the universal cover we have \n$\\sup_{t \\ge 0} d(c_1(t),c_2(t)) = \\infty$. \nWhile negatively curved manifolds have expansive geodesic flows, this is not longer true for a typical manifold without conjugate points or even non-positive curvature. Nevertheless, in many cases the expansive set is not empty.\n\nIn the following we call an invariant Borel probability measure $\\nu \\in \\MMM_\\phi(SM)$ of the geodesic flow $\\phi^t: SM \\to SM$ non-expansive if its weight of the expansive set is vanishing, i.e. $\\nu(\\mathcal{E}) =0$. It is natural to assume that such measures have less complexity and are not of maximal entropy. Using this assumption we can prove the following.", + "context": "\\label{sec:intro}\nIf $f: X \\to X$ is a homeomorphism of a compact metric space $X$,\nthe topological entropy $h_{\\mathrm{top}}(f)$ of $f$ is an invariant in topological dynamics measuring the orbit complexity of the dynamical system on an\nexponential scale. On the other hand, measure-theoretic entropy $h_{\\mu}(f)$ is an invariant of \nmeasurable dynamics reflecting the average complexity of the system relative\nto a measure $\\mu$ contained in the space of $f$-invariant Borel probability measure $\\MMM_f(X)$ on $X$. \nThe relation between these notions is provided by the variational principle which implies (see e.g. \\cite{pW82} for details)\n$$\nh_{\\mathrm{top}} (f) = \\sup \\{h_\\mu (f) \\;|\\; \\mu \\in \\MMM_f(X)\\}.\n$$\nA measure $\\mu \\in \\MMM_f(X)$ such that $h_{\\mu}(f) =h_{\\mathrm{top}}(f)$ is called a measure of maximal entropy (MME).\nFor continuous flows $\\phi^t: X \\to X$ topological - and measure theoretic entropy is defined as the entropy of the time 1 map $\\phi := \\phi^1$.\n\nIn the Riemannian setting topological entropy of the geodesic flow is related to large scale geometry.\nIf $(M,g)$ is a closed Riemannian manifold without conjugate points \n(all geodesics are globally minimizing if lifted to the universal cover $\\widetilde{M}$) then by a theorem of Freir\\'{e} and Ma\\~n\\'e \\cite{FM82} the topological entropy $\\htop (\\phi)$ of the geodesic\nflow $\\phi^t: SM \\to SM$ agrees with the volume entropy $\\hvol(g)$. Volume entropy measures the asymptotic volume growth of geodesics balls $B(p,r) \\subset \\widetilde{M}$ in the universal covering $\\widetilde{M}$ of $M$ and is given by\n$$\n\\hvol(g) = \\lim\\limits_{r \\to \\infty} \\frac{\\log \\mathrm{vol} B(p,r)}{r} \n$$\nFor manifolds $(M,g)$ of non-positive sectional curvature, which are special manifolds of no conjugate points, \nan easier proof of the result of Freir\\'{e} and Ma\\~n\\'e were previously given by Manning \\cite{aM79}. However, in contrast to non-positive sectional curvature, no conjugate points \nis not a local condition. Properties like the convexity of distance functions which\n hold for manifolds of non-positive sectional curvature and even no focal points\nare not anymore true for manifolds without conjugate points. Furthermore, the flat strip theorem, important in the study of manifolds \nof non-positive sectional curvature, fail under the assumption of no conjugate points \\cite{kB92}.\n\nIn this paper, we like to study the entropy of geodesic flows on closed Riemannian manifolds $(M, g)$ without conjugate points under some coarse hyperbolicity of the geodesic flow.\nHyperbolicity implies the existence of a non-trivial expansive set $\\mathcal{E} \\subset SM$ consisting of flow lines\nwhich diverge from each other over time. More formally, if $d$ is the distance function induced by the Riemannian metric on $M$, a vector $v \\in SM$ is contained in $\\mathcal{E}$ if there is some\n $\\epsilon >0$ such that for all $w \\in SM $ with $d(c_v(t), c_w(\\mathbb{R})) < \\epsilon$ for all $t \\in \\mathbb{R}$,\nthe geodesics $c_v$, $c_w$ with initial conditions $v$ and $w$ agree up to a time shift. Furthermore, the expansive set\nis invariant under the geodesic flow.\n\nGeodesic flows are called expansive if $SM =\\mathcal{E}$.\nIn case that $(M,g)$ has no conjugate points it was proved by Ruggiero \\cite{rR94} that the expansiveness of the geodesic\nflow \nimplies that the fundamental group $\\pi_1(M)$ is Gromov hyperbolic. Furthermore the divergence property hold, i.e. for each pair of geodesic rays $c_1, c_2: [0, \\infty) \\to \\widetilde{M} $ on the universal cover we have \n$\\sup_{t \\ge 0} d(c_1(t),c_2(t)) = \\infty$. \nWhile negatively curved manifolds have expansive geodesic flows, this is not longer true for a typical manifold without conjugate points or even non-positive curvature. Nevertheless, in many cases the expansive set is not empty.\n\nIn the following we call an invariant Borel probability measure $\\nu \\in \\MMM_\\phi(SM)$ of the geodesic flow $\\phi^t: SM \\to SM$ non-expansive if its weight of the expansive set is vanishing, i.e. $\\nu(\\mathcal{E}) =0$. It is natural to assume that such measures have less complexity and are not of maximal entropy. Using this assumption we can prove the following.", + "full_context": "\\label{sec:intro}\nIf $f: X \\to X$ is a homeomorphism of a compact metric space $X$,\nthe topological entropy $h_{\\mathrm{top}}(f)$ of $f$ is an invariant in topological dynamics measuring the orbit complexity of the dynamical system on an\nexponential scale. On the other hand, measure-theoretic entropy $h_{\\mu}(f)$ is an invariant of \nmeasurable dynamics reflecting the average complexity of the system relative\nto a measure $\\mu$ contained in the space of $f$-invariant Borel probability measure $\\MMM_f(X)$ on $X$. \nThe relation between these notions is provided by the variational principle which implies (see e.g. \\cite{pW82} for details)\n$$\nh_{\\mathrm{top}} (f) = \\sup \\{h_\\mu (f) \\;|\\; \\mu \\in \\MMM_f(X)\\}.\n$$\nA measure $\\mu \\in \\MMM_f(X)$ such that $h_{\\mu}(f) =h_{\\mathrm{top}}(f)$ is called a measure of maximal entropy (MME).\nFor continuous flows $\\phi^t: X \\to X$ topological - and measure theoretic entropy is defined as the entropy of the time 1 map $\\phi := \\phi^1$.\n\nIn the Riemannian setting topological entropy of the geodesic flow is related to large scale geometry.\nIf $(M,g)$ is a closed Riemannian manifold without conjugate points \n(all geodesics are globally minimizing if lifted to the universal cover $\\widetilde{M}$) then by a theorem of Freir\\'{e} and Ma\\~n\\'e \\cite{FM82} the topological entropy $\\htop (\\phi)$ of the geodesic\nflow $\\phi^t: SM \\to SM$ agrees with the volume entropy $\\hvol(g)$. Volume entropy measures the asymptotic volume growth of geodesics balls $B(p,r) \\subset \\widetilde{M}$ in the universal covering $\\widetilde{M}$ of $M$ and is given by\n$$\n\\hvol(g) = \\lim\\limits_{r \\to \\infty} \\frac{\\log \\mathrm{vol} B(p,r)}{r} \n$$\nFor manifolds $(M,g)$ of non-positive sectional curvature, which are special manifolds of no conjugate points, \nan easier proof of the result of Freir\\'{e} and Ma\\~n\\'e were previously given by Manning \\cite{aM79}. However, in contrast to non-positive sectional curvature, no conjugate points \nis not a local condition. Properties like the convexity of distance functions which\n hold for manifolds of non-positive sectional curvature and even no focal points\nare not anymore true for manifolds without conjugate points. Furthermore, the flat strip theorem, important in the study of manifolds \nof non-positive sectional curvature, fail under the assumption of no conjugate points \\cite{kB92}.\n\nIn this paper, we like to study the entropy of geodesic flows on closed Riemannian manifolds $(M, g)$ without conjugate points under some coarse hyperbolicity of the geodesic flow.\nHyperbolicity implies the existence of a non-trivial expansive set $\\mathcal{E} \\subset SM$ consisting of flow lines\nwhich diverge from each other over time. More formally, if $d$ is the distance function induced by the Riemannian metric on $M$, a vector $v \\in SM$ is contained in $\\mathcal{E}$ if there is some\n $\\epsilon >0$ such that for all $w \\in SM $ with $d(c_v(t), c_w(\\mathbb{R})) < \\epsilon$ for all $t \\in \\mathbb{R}$,\nthe geodesics $c_v$, $c_w$ with initial conditions $v$ and $w$ agree up to a time shift. Furthermore, the expansive set\nis invariant under the geodesic flow.\n\nGeodesic flows are called expansive if $SM =\\mathcal{E}$.\nIn case that $(M,g)$ has no conjugate points it was proved by Ruggiero \\cite{rR94} that the expansiveness of the geodesic\nflow \nimplies that the fundamental group $\\pi_1(M)$ is Gromov hyperbolic. Furthermore the divergence property hold, i.e. for each pair of geodesic rays $c_1, c_2: [0, \\infty) \\to \\widetilde{M} $ on the universal cover we have \n$\\sup_{t \\ge 0} d(c_1(t),c_2(t)) = \\infty$. \nWhile negatively curved manifolds have expansive geodesic flows, this is not longer true for a typical manifold without conjugate points or even non-positive curvature. Nevertheless, in many cases the expansive set is not empty.\n\nIn the following we call an invariant Borel probability measure $\\nu \\in \\MMM_\\phi(SM)$ of the geodesic flow $\\phi^t: SM \\to SM$ non-expansive if its weight of the expansive set is vanishing, i.e. $\\nu(\\mathcal{E}) =0$. It is natural to assume that such measures have less complexity and are not of maximal entropy. Using this assumption we can prove the following.\n\nIn the following we call an invariant Borel probability measure $\\nu \\in \\MMM_\\phi(SM)$ of the geodesic flow $\\phi^t: SM \\to SM$ non-expansive if its weight of the expansive set is vanishing, i.e. $\\nu(\\EE) =0$. It is natural to assume that such measures have less complexity and are not of maximal entropy. Using this assumption we can prove the following.\n\nIn this section we prove first that $\\mu$ is the unique MME under the following conditions.\n\\begin{theorem}\\label{thm:uniqueMME-inj}\nLet $(M,g)$ be a closed Riemannian manifold without conjugate points with divergence property and with Gromov hyperbolic fundamental group. Let $\\delta $ the Gromov hyperbolicity constant of the universal cover $\\wM$ and assume that the injectivity radius $\\inj(M)$ of $M$ is larger than $16 \\delta$. Assume that at least one of the following two\nconditions is fulfilled:\n\\begin{enumerate}\n\\item \nThe entropy of non-expansive measures is strictly smaller than the topological entropy, i.e.\n$$ h_\\nu(\\phi)< \\htop(\\phi) \\; \\; \\text{for all} \\; \\; \\nu \\in \\MMM_\\phi(SM) \\; \\; with \\; \\;\\nu(\\EE)=0$$ \n\\item\nThe expansive set $\\EE \\subset SM$ has non-empty interior and the geodesic flow $\\phi^t: SM \\to SM$ is entropy expansive\nat some scale larger than $8 \\delta$. \n\\end{enumerate}\nThen the measure $\\mu$ is the unique measure of maximal entropy of the geodesic flow.\n\\end{theorem}\n\nNow we can prove Theorem \\ref{thm:main} and Theorem \\ref{thm:expansiveMME} using Theorem \\ref{thm:uniqueMME-inj}\nas follows.\n\\begin{proof}\nLet $(M,g)$ be a closed manifold Riemannian manifold without conjugate points and divergence property.\n Furthermore, assume that the fundamental group of $M$ is residually finite and Gromov hyperbolic fundamental group with Gromov hyperbolicity.\n By Proposition \\ref{prop:large-inj} there is a smooth Riemannian manifold $N$ and is for some \n$k \\in \\N$ a locally isometric $k$ to $1$ covering map $p\\colon N\\to M$ such that the injectivity radius of $N$ is larger \nthan $8 \\delta$. In particular $N$ has no conjugate, the divergence property\nand the geodesic $\\phi^t_{SM}: SM \\to SM$ is $k$ to $1$ factor of the \ngeodesic flow $\\phi^t_{SN} : SN \\to SN$ as well. If $\\mu$ is a invariant Borel probability measure on $SM$ \nwe denote by $\\tmu$ the canonical lift defined by\n$$\n\\tmu(A) = \\int_{SM} \\frac{1} {k}\\card \\{dp^{-1} (v) \\cap A \\}d\\mu(v),\n$$\nwhere $dp :SN \\to SM$ is the differential of $p$.\nObviously is $\\tmu$ is $\\phi^t_{SN}$ invariant Borel probability measure with $dp_*\\tmu =\\mu$ and\n$h_{\\tmu}(\\phi_{SN}) = h_{\\mu}(\\phi_{SM})$.\nSince the expansive set of $SN$ is the lift of the expansive set in $SM$ \nand the expansivity constants of $\\phi^t_{SM}: SM \\to SM$ and $\\phi^t_{SN} : SN \\to SN$ agree\nthe assumptions of Theorem \\ref{thm:main} or Theorem \\ref{thm:expansiveMME} imply that \n$\\phi^t_{SN} : SN \\to SN$ fullfils the assumptions\nof Theorem \\ref{thm:uniqueMME-inj}. Hence $\\phi^t_{SN} : SN \\to SN$ has a unique measure of maximal entropy.\n If $\\mu_1$ and $\\mu_2$ are measures of maximal entropy for $\\phi^t_{SM}$\nthen their lifts $\\tmu_1$ and $\\tmu_2$ are measures of maximal entropy for $\\phi^t_{SN}$ as well.\nIn particular $\\tmu_1 = \\tmu_2$ therefore $\\mu_1 = \\mu_2 = \\mu$.\n\nFor the construction of Green bundles we need to define stable and unstable Jacobi tensors. Since $(M.g)$ has no conjugate points\nfor each $r > 0$, there exists orthogonal Jacobi tensors along $c_v$ with\n\\begin{align*}\nS_{v,r}(0) = \\Id_{v^\\bot}, &\\quad S_{v,r}(r) = 0, \\\\\nU_{v,r}(0) = \\Id_{v^\\bot}, &\\quad U_{v,r}(-r) = 0.\n\\end{align*}\nNote that we have $U_{v,r}(t) = S_{-v,r}(-t)$. \nThe stable and unstable Jacobi tensor along $c_v$ are defined via the following initial conditions\n$$ S_v(0) = \\Id_{v^\\bot}, \\quad S_v'(0) = \\lim_{r \\to \\infty} S_{v,r}'(0), $$\nand similarly\n$$ U_v(0) = \\Id_{v^\\bot}, \\quad U_v'(0) = \\lim_{r \\to \\infty} U_{v,r}'(0). $$\nThe existence of $\\lim_{r \\to \\infty }S_{v,r}'(0)$ follows from the monotonicity $S'_{v,r}(0) 0$. Moreover, we introduce the symmetric endomorphisms\n$$ S(v) = S'_v(0) \\quad \\text{and} \\quad U(v) = U'_v(0). $$\nIn particular $U(v) -S(v) \\ge 0$. Since $U_{v,r}(t) = S_{-v,r}(-t)$ for all $t \\in \\R$ and $r>0$ we obtain $U_{v}(t) = S_{-v}(-t)$ and therefore\n$$ \nU(v) = U'_v(0) = - S'_{-v}(0) = -S(-v).\n$$\nFurthermore,\n\\begin{equation}\\label{eqn:invGbundles}\nU(\\phi^tv) = U'_v(t) U^{-1}_v(t) \\; \\text{and} \\;S(\\phi^tv) = S'_v(t) S^{-1}_v(t)\n\\end{equation}\n \\begin{definition}\\label{def:Green bundles}\n Let $(M,g)$ be a manifold without conjugate points. Then the subbundles $E^s$ and $E^$ of $TSM$ defined by\n $$\n E^s(v) =\\{(x, U(v) x) \\mid x \\in T_{\\pi (v)}M, x\\perp v \\}\n $$\n and\n $$\n E^u(v) =\\{(x, U(v) x) \\mid x \\in T_{\\pi (v)}M, x\\perp v \\}\n $$\n are called the stable and unstable Green bundles\n \\end{definition}\n \\begin{remark}\n From the equations \\ref{eqn:invGbundles} follows the invariance of the Green bundles under the geodesic flow $\\phi^t$, i.e.\n $$\n D\\phi^t(v)E^s(v) = E^s(\\phi^t(v)) \\; \\text{and} \\; D\\phi^t(v)E^u(v) = E^u(\\phi^t(v))\n $$\n for all $t \\in \\R$.\n \\end{remark}\n For the metric \nentropy of the geodesic flow, \nFreir\\'{e} and Ma\\~n\\'e \\cite{FM82}\n obtained, based on Ruelle's inequality \\cite{dR78} and Pesin's \\cite{jPe77} formula the following estimate. \n \\begin{theorem}\\label{thm: FM}\nLet $(M,g)$ be a compact manifold without conjugate points and $\\mu \\in \\MMM_\\phi (SM) $ be a\n$\\phi^t$-invariant\n probability measure for the geodesic flow.\nThen\n$$\nh_\\mu(\\phi^t) \\le \\int\\limits_{SM} \\tr U d \\mu.\n$$\nFurthermore equality holds if $\\mu$ \nis given by the Liouville measure.\n\\end{theorem}\n\\subsection{Positivity of the Liouville entropy}\nIn the \\cite{gK85} we showed that for manifolds without conjugate points and continuous asymptote positivity the metric entropy\nof the Liouville measure provided the topological entropy is positive as well.\nSince the paper is not easily accessible we provide the proof in this paper.\n\\begin{theorem}\\label{Kn-thesis}\nLet $(M,g)$ be a compact manifold without conjugate points and continuous Green bundles.\nIf the the geodesic flow $\\phi^t$ has positive topological entropy $\\htop(\\phi) $ then the metric entropy $h_\\lambda(\\phi)$ of the geodesic flow with respect to the Liouville measure $\\lambda$.\n\\end{theorem} \n\\begin{proof}\nAssume that $\\htop(\\phi) >0$. Then by the variational principle there exists a measure $\\mu \\in \\MMM_\\phi (SM) $ such that\n$$\nh_\\mu(\\phi) \\ge \\frac{1}{2} \\htop(\\phi) >0\n$$\nLet $F: SM \\to SM$ be the flip map given by $F(v) = -v$. Then the push forward measure $ \\mu_F$ given\nby $ \\mu_F(A) = \\mu (F(A))$ for all measurable sets $A \\subset SM$ is flow-invaraint as well since\n$$\n \\mu_F(\\phi^t A) = \\mu ( F(\\phi^t(A)) = \\mu ( \\phi^{-t} F(A)) = \\mu(F(A)) = \\mu_F(A)\n $$\n Consider the measure\n $\\nu = \\frac{1}{2}( \\mu + \\mu_F) $. The affine property of the metric entropy\n yields\n $$\n h_\\nu (\\phi) = \\frac{1}{2}h_\\mu (\\phi) + \\frac{1}{2}h_{\\mu_F} (\\phi) >0\n $$\n From theorem \\ref{thm: FM} we obtain $h_\\nu (\\phi_t) \\le \\int\\limits_{SM} \\mathrm{tr} U(v) d \\nu $.\n Since $\\tr U(v) = -\\tr S(-v)$ the flip invariance of $\\nu$ implies\n $$\n \\int\\limits_{SM} \\tr U(v) d \\nu = \\int\\limits_{SM} -\\tr S(v) d \\nu\n$$\nHence\n$$\n0 < h_\\nu (\\phi_t) \\le \\frac{1}{2}\\int\\limits_{SM} \\tr(U(v) - S(v)) d \\nu \n$$\nSince $\\tr(U(v) - S(v)) \\ge 0$ and by the continuity of the Green Bundles $v \\mapsto U(v)-S(v)$ is continuous the set\n$$\nQ:= \\{v \\in SM \\mid \\mathrm{tr} (U(v) - S(v) )> 0 \\}\n$$\nis open and non-empty by $ 0 < h_\\nu (\\phi_t)$.\nIn particular $\\lambda(Q) >0$ . Using that\n$ h_\\lambda (\\phi_t) = \\int\\limits_{SM} \\mathrm{tr} U(v) d \\lambda$ holds\nthe flip invariance of the Liouville measure $ \\lambda$ yields\n$$\nh_\\lambda (\\phi_t) = \\frac{1}{2}\\int\\limits_{SM} \\mathrm{tr} (U(v) - S(v) )d \\lambda >0.\n$$", + "post_theorem_intro_text_len": 7994, + "post_theorem_intro_text": "\\begin{remark}\n\\begin{itemize}\n\\item Closed Riemannian manifolds without conjugate points and Gromov hyperbolic fundamental group have positive topological\nentropy (see \\cite{CK02} and Remark \\ref{rem:conj-Grhyp}). \n\\item It follows from the work of Newhouse \\cite{sN89} that for smooth geodesic flows $\\phi^t$\nthere exists a measure $\\nu \\in \\MMM_\\phi(SM)$ of maximal entropy, i.e. $h_\\nu(\\phi)= \\htop(\\phi)$.\nHence, the assumption of our theorem forces $\\nu(\\mathcal{E})$ to be positive. In particular, the expansive set is not empty.\n\\item If $(M,g)$ is a closed and non-flat surface without conjugate points, then all assumptions of Theorem \\ref{thm:main}\nhold (see \\cite{CKW21}). Therefore such surfaces have a unique MME.\n\\item If $(M,g)$ is a closed smooth Riemannian manifold without conjugate points and expansive geodesic flow (i.e $\\mathcal{E} =SM$)\n the uniqueness of the MME has been obtained by Bosch\\'e in \\cite{aB18}.\nAs we mentioned above a closed manifold without conjugate points and expansive geodesic flow has the divergence\nproperty and Gromov hyperbolic fundamental group. Since for expansive geodesic flows non-expansive measures obviously do not exist, the result of Bosch\\'e follows from our theorem as a special case.\n\\item If $(M,g)$ is a closed rank 1 manifold of non-positive curvature the rank 1 (regular) set consists of orbits of the geodesic flow which do not have non-trivial parallel Jacobi-fields orthogonal to the geodesic (see e.g. \\cite{gK97} or \\cite{gK98}).\nIt is a consequence of the flat strip theorem that the regular set is contained in the expansive set.\nIn \\cite{gK98} we showed that for closed rank 1 manifolds the geodesic flow has a unique MME. Furthermore, the measure has full weight\non the regular, and hence, on the expansive set. A different proof of the uniqueness of the MME was later given in \\cite{BCFT}.\nSince there are examples of closed rank 1 manifolds whose fundamental group has $(\\mathbb{Z}^2,+)$ as a subgroup, the \nfundamental group is generally not Gromov hyperbolic (see \\cite{gK98}).\n\\end{itemize}\n\\end{remark}\nIn \\cite{CKW21} Climenhaga, War and the author proved Theorem \\ref{thm:main} under the more special condition of a background\nmetric of negative curvature and \nthe slightly stronger entropy gap assumption \n$$\n \\sup\\{h_\\nu(\\phi) : \\nu\\in \\MMM_\\phi(SM), \\nu(\\mathcal{E})=0\\} < \\htop(\\phi).\n $$\n for non-expansive measures.\n The entropy gap would follow from our condition that non-expansive measure do not have maximal\n entropy provided the expansive set is open. Namely, if $\\mathcal{E}$ is open and the entropy gap would not hold there would exist a sequence of measures \n $\\nu_n \\in \\MMM_\\phi(SM)$ with $\\nu_n(\\mathcal{E})=0 $ converging weakly to $\\nu$ and $\\lim_{n \\to \\infty}h_{\\nu_n}(\\phi) = \\htop(\\phi)$. Since $\\mathcal{E}$ is open, $\\nu(\\mathcal{E}) \\le \\liminf_{n \\to \\infty}\\nu_n(\\mathcal{E})= 0$ and by upper semi-continuity of entropy this would yield $h_{\\nu}(\\phi) = \\htop(\\phi)$. But this contradicts our assumption that $h_{\\nu}(\\phi) <\\htop(\\phi)$ for non-expansive measures.\n However, to our knowledge openness of the expansive set is not known in our setting.\n\nThe assumption of a background\nmetric of negative curvature implies that the fundamental group is Gromov hyperbolic (see subsection \\ref{subsec:hypgroups}). \nDue to the uniformization theorem for surfaces and the proof of the geometrization conjecture in dimension three\nGromov hyperbolicity of the fundamental group implies the existence of a metric with even constant negative curvature.\nHowever, for closed manifolds of dimension bigger than three, Gromov hyperbolicity of the fundamental group and existence of a metric without conjugate points might not be enough to yield a metric of negative curvature. In any case, to provide a solution to this question is a very difficult problem.\n\nIn \\cite{CKW21} the proof of the uniqueness of the measure of maximal entropy used the background metric of negative curvature\nto establish with the help of the Morse Lemma a coarse specification property \nfor the geodesic flow. Applying the work of Climenhaga and Thompson \\cite{CT16}, the specification property was used\nto prove the uniqueness of the MME.\n\n However, the proof of the above theorem does not require specification but relied on methods derived in a paper\n of the author \\cite{gK98} on the uniqueness of the measure of maximal entropy for geodesic flows on non-positively curved rank 1 manifolds.\\\\\n\n There is a interesting and quite flexible notion due to Bowen \\cite{rBo72}, called entropy expansiveness which hold for many dynamical systems for which expansiveness fails (see section \\ref{sec:e-expansiveness}).\n\n Given a closed Riemannian manifold $(M,g)$ without conjugate points and $(\\widetilde{M},g)$ \n be the universal cover with the lifted Riemannian metric denoted again by $g$. For $v \\in S\\widetilde{M}$ and $\\rho >0$ we define \n the set\n $$\n Z_{\\rho} (v) = \\{w \\in S\\widetilde{M} \\mid d(c_v(t), c_w(t)) \\le \\rho, t \\in \\mathbb{R} \\}.\n$$\n The geodesic flow $\\phi^t: SM \\to SM$ is called entropy expansive at scale $\\rho >0$\nif $\\htop(\\tilde \\phi, Z_\\rho(v)) = 0$, where $\\tilde \\phi^t$ is the geodesic flow lifted to $S \\widetilde{M}$.\nDue to the flat strip theorem geodesic flows on manifolds of non-positive curvature or more generally no focal points are entropy expansive at any scale (see \\cite[proposition 3.3]{gK98}).\nThis also holds for non-flat surfaces without conjugate points \\cite[Lemma 4.5]{GKOS14} even so the flat strip theorem fails in this case \\cite{kB92}. As far as we know, there is no example of a closed Riemannian manifold with metric without conjugate known for which the\ngeodesic flow is not entropy expansive for all or even some $\\rho >0$. Alternatively to Theorem \\ref{thm:main}, we obtain\nthe uniqueness of the MME under the following conditions.\n\\begin{MainThm}\\label{thm:expansiveMME}\n Let $(M,g)$ be a closed Riemannian manifold without conjugate points with divergence property and Gromov\n hyperbolic and residually finite fundamental group. Assume that the geodesic flow is entropy expansive at a scale larger than $8 \\delta$, where $\\delta$ is the Gromov hyperbolicity constant of the universal cover. If the expansive set has non-trivial interior then the geodesic flow\n has a unique measure of maximal entropy $\\mu$.\n\nFurthermore, $\\mu(\\mathcal{E}) =1$, $\\mu$ is mixing and fully supported on the unit tangent bundle.\n\\end{MainThm}\n\n Using the results of Climenhaga, War and the author proved in \\cite[Theorem 1.2]{CKW21} \n together with Theorem \\ref{thm:conj-Grhyp} and Remark \\ref{rem:conj-Grhyp},\n we can conclude that the measure of maximal entropy is given by the limiting distribution of closed orbits.\n\nFurthermore, as in \\cite[Theorem 1.2]{CKW22}, an estimate on the growth of pairwise non-free-homotopic closed geodesics,\nobtained by Margulis \\cite{gM69} in the case of negative curvature, follows. More precisely:\n\n \\begin{MainThm}\\label{thm:closed geodesics}\n\tLet $(M,g)$ be a closed Riemannian manifold such that the assumption in Theorem \\ref{thm:main} \n\tor Theorem \\ref{thm:expansiveMME} hold. Denote for $T>0$ by $\\mathcal{P}(T)$ be any maximal set of pairwise non-free-homotopic closed geodesics of minimal length in the free homotopy classes and $P(T) =\\mathrm{card} \\mathcal{P}(T) $ its cardinality. Consider the measures\n\\begin{equation*}\\label{eqn:nut}\n\\mu_T= \\frac {1}{P(T)} \\sum_{c\\in \\mathcal{P}(T)} \\frac{\\Leb_c}{T},\n\\end{equation*}\nwhere $\\Leb_c$ is Lebesgue measure (length) along the curve $\\dot{c}$ in the unit tangent bundle $SM$. Then\n\\begin{enumerate}\n\\item \\label{eqn:distr}\n\nThe measures $\\mu_T$ converge in the weak* topology as $T\\to \\infty$ to the measure of maximal entropy.\n\n\\item\nFurthermore, \n\\begin{equation*}\\label{eqn:margulis}\nP(T) \\sim \\frac{e^{hT}}{hT},\n\\end{equation*}\nwhich means that the ratio of $P(T)$ and $\\frac{e^{hT} }{hT} $ converges $1$ as $T \\to \\infty$.\n\\end{enumerate}\n\n\\end{MainThm}", + "sketch": "In \\cite{CKW21} the uniqueness of the MME in Theorem~\\ref{thm:main} was proved (under a background metric of negative curvature) by using the Morse Lemma to establish a \\emph{coarse specification property} for the geodesic flow, and then applying the work of Climenhaga--Thompson \\cite{CT16} to deduce uniqueness of the MME from specification.\n\nThe text also explains why their (slightly stronger) \\emph{entropy gap} assumption\n\\[\n\\sup\\{h_\\nu(\\phi): \\nu\\in\\MMM_\\phi(SM),\\,\\nu(\\mathcal E)=0\\}0$ such that for every $w\\in SM$ with $d(c_v(t),c_w(\\mathbb R))<\\varepsilon$ for all $t\\in\\mathbb R$, the geodesics $c_v$ and $c_w$ agree up to a time shift. Call a $\\phi^t$-invariant Borel probability measure $\\nu$ non-expansive if $\\nu(\\mathcal E)=0$. Suppose that every non-expansive measure $\\nu\\in\\mathcal M_\\phi(SM)$ satisfies\n$$\nh_\\nu(\\phi)0$, the following holds: \nLet $A_1,\\ldots,A_k\\subseteq \\RR$ be finite, each of size $n$. Then $$\n\\lvert f(A_1,\\ldots,A_k)\\rvert= \\Omega\\left( n^{\\frac{5r-4}{2r}-\\varepsilon}\\right), \n$$\nwhere the constant of proportionality depends on $\\deg(f)$, on $r$, and on $\\eps$.\n\\end{thm}", + "kspecial": "\\begin{align}\n f(x_1,\\ldots,x_k)&=h(p_1(x_1)+\\cdots +p_k(x_k))\\label{kspecial}\\\\\n f(x_1,\\ldots,x_k)&=h(p_1(x_1)\\cdot \\ldots \\cdot p_k(x_k))\\nonumber\n \\end{align}", + "specialbivariate": "\\begin{align}\nf(x,y)&=h(p(x)+q(y))\\quad\\text{or}\\nonumber\\\\\nf(x,y)&=h(p(x)q(y)),\\label{specialbivariate}\n\\end{align}", + "thm:kvarER": "\\begin{thm}[{\\bf \\cite{RazShaDeZee4D, RazShe}}]\\label{thm:kvarER}\n Let $k\\ge 3$ and let $f\\in \\RR[x_1,\\ldots, x_k]$. Then one of the following holds:\\\\\n (i) For every $A_1,\\ldots, A_k\\in \\RR$ each of size $n$ one has\n $$\n f(A_1,\\ldots,A_k)|=\\Omega\\left(n^{3/2}\\right)$$\\\\\n (ii) $f$ is of one of the forms:\n \\begin{align}\n f(x_1,\\ldots,x_k)&=h(p_1(x_1)+\\cdots +p_k(x_k))\\label{kspecial}\\\\\n f(x_1,\\ldots,x_k)&=h(p_1(x_1)\\cdot \\ldots \\cdot p_k(x_k))\\nonumber\n \\end{align}\n\\end{thm}", + "thm:app": "\\begin{thm}\\label{thm:app}\n Let $\\nu$ be the moment curve in $\\RR^d$ and let $P\\subset \\nu$ be any finite set of size $n$. Then, for every $\\eps>0$, \n $$\n|\\Delta(P)|=\\Omega\\left(n^{\\frac{5d-4}{2d}-\\eps}\\right),\n$$\n where the implicit constant depends only on $\\eps$ and on $d$.\n\\end{thm}", + "rank1char": "\\begin{thm}\\label{rank1char}\nLet $k\\ge3$ and let $f\\in\\RR\\left[x_1 \\ldots,x_k\\right]$. Assume that $f$ depends non-trivially on each of its variables and that $$\n{\\rm rank}(f) = 1.$$\nThen $f$ has one of the forms\n\\begin{align}\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)+\\cdots+p_{k}\\left(x_{k}\\right)\\right)\\ \\ \\text{ or}\\nonumber\\\\\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)\\cdot\\ldots\\cdot p_{k}\\left(x_{k}\\right)\\right),\\label{special1}\n\\end{align}\nfor some univariate real polynomials $h(x), p_1(x),\\ldots,p_k(x)$.\n\\end{thm}" + }, + "pre_theorem_intro_text_len": 1788, + "pre_theorem_intro_text": "In many cases in combinatorial geometry, counting questions involving distances, slopes, collinearity, etc., can be reformulated as analogous counting questions involving grid points lying on certain algebraic varieties. A unified study of such problems began with a question of Elekes~\\cite{Ele} about expansion of bivariate real polynomials $f(x,y)$. \nSpecifically, he asked: For a bivariate polynomial $f\\in \\mathbb{R}[x,y]$ and given finite sets $A,B\\subset \\mathbb{R}$, how small can be the image set\n$$\nf(A,B)=\\{f(a,b)\\mid a\\in A, b\\in B\\}.\n$$\n\nElekes conjectured that the image of $f$ on a $n\\times n$ Cartesian product must be of cardinality superlinear in $n$, unless $f$ has a very concrete {\\it special form}. This was confirmed in 2000 by Elekes and R\\'onyai~\\cite{EleRon00} who proved the following dichotomy:\nEither $f$ has one of the forms\n\\begin{align}\nf(x,y)&=h(p(x)+q(y))\\quad\\text{or}\\nonumber\\\\\nf(x,y)&=h(p(x)q(y)),\\label{specialbivariate}\n\\end{align} for some univariate real polynomials $p,q,h$, or, otherwise, for every finite $A,B\\subset \\mathbb{R}$, each of size $n$, we have\n\\begin{equation}\\label{eq:ER}\n|f(A,B)|=\\omega(n).\n\\end{equation}\n\nIn case $f$ is not one of the forms in \\eqref{specialbivariate}, the lower bound on $|f(A,B)|$ was improved in \\cite{RazShaDeZee} to be $\\Omega(n^{4/3})$, and further improved in \\cite{SolZha} to be $\\Omega(n^{3/2})$, which is currently the best known lower bound for bivariate polynomials that are not special. \n\nAn analogue of the Elekes--R\\'onyai problem can be formulated for polynomials in more than two variables. The trivariate case was studied by Raz, Sharir, and De Zeeuw~\\cite{RazShaDeZee4D}, and the general $k$-variate case was established by Raz and Shem Tov~\\cite{RazShe}. They obtain the following result.", + "context": "In many cases in combinatorial geometry, counting questions involving distances, slopes, collinearity, etc., can be reformulated as analogous counting questions involving grid points lying on certain algebraic varieties. A unified study of such problems began with a question of Elekes~\\cite{Ele} about expansion of bivariate real polynomials $f(x,y)$. \nSpecifically, he asked: For a bivariate polynomial $f\\in \\mathbb{R}[x,y]$ and given finite sets $A,B\\subset \\mathbb{R}$, how small can be the image set\n$$\nf(A,B)=\\{f(a,b)\\mid a\\in A, b\\in B\\}.\n$$\n\nElekes conjectured that the image of $f$ on a $n\\times n$ Cartesian product must be of cardinality superlinear in $n$, unless $f$ has a very concrete {\\it special form}. This was confirmed in 2000 by Elekes and R\\'onyai~\\cite{EleRon00} who proved the following dichotomy:\nEither $f$ has one of the forms\n\\begin{align}\nf(x,y)&=h(p(x)+q(y))\\quad\\text{or}\\nonumber\\\\\nf(x,y)&=h(p(x)q(y)),\\label{specialbivariate}\n\\end{align} for some univariate real polynomials $p,q,h$, or, otherwise, for every finite $A,B\\subset \\mathbb{R}$, each of size $n$, we have\n\\begin{equation}\\label{eq:ER}\n|f(A,B)|=\\omega(n).\n\\end{equation}\n\nIn case $f$ is not one of the forms in \\eqref{specialbivariate}, the lower bound on $|f(A,B)|$ was improved in \\cite{RazShaDeZee} to be $\\Omega(n^{4/3})$, and further improved in \\cite{SolZha} to be $\\Omega(n^{3/2})$, which is currently the best known lower bound for bivariate polynomials that are not special.\n\nAn analogue of the Elekes--R\\'onyai problem can be formulated for polynomials in more than two variables. The trivariate case was studied by Raz, Sharir, and De Zeeuw~\\cite{RazShaDeZee4D}, and the general $k$-variate case was established by Raz and Shem Tov~\\cite{RazShe}. They obtain the following result.", + "full_context": "In many cases in combinatorial geometry, counting questions involving distances, slopes, collinearity, etc., can be reformulated as analogous counting questions involving grid points lying on certain algebraic varieties. A unified study of such problems began with a question of Elekes~\\cite{Ele} about expansion of bivariate real polynomials $f(x,y)$. \nSpecifically, he asked: For a bivariate polynomial $f\\in \\mathbb{R}[x,y]$ and given finite sets $A,B\\subset \\mathbb{R}$, how small can be the image set\n$$\nf(A,B)=\\{f(a,b)\\mid a\\in A, b\\in B\\}.\n$$\n\nElekes conjectured that the image of $f$ on a $n\\times n$ Cartesian product must be of cardinality superlinear in $n$, unless $f$ has a very concrete {\\it special form}. This was confirmed in 2000 by Elekes and R\\'onyai~\\cite{EleRon00} who proved the following dichotomy:\nEither $f$ has one of the forms\n\\begin{align}\nf(x,y)&=h(p(x)+q(y))\\quad\\text{or}\\nonumber\\\\\nf(x,y)&=h(p(x)q(y)),\\label{specialbivariate}\n\\end{align} for some univariate real polynomials $p,q,h$, or, otherwise, for every finite $A,B\\subset \\mathbb{R}$, each of size $n$, we have\n\\begin{equation}\\label{eq:ER}\n|f(A,B)|=\\omega(n).\n\\end{equation}\n\nIn case $f$ is not one of the forms in \\eqref{specialbivariate}, the lower bound on $|f(A,B)|$ was improved in \\cite{RazShaDeZee} to be $\\Omega(n^{4/3})$, and further improved in \\cite{SolZha} to be $\\Omega(n^{3/2})$, which is currently the best known lower bound for bivariate polynomials that are not special.\n\nAn analogue of the Elekes--R\\'onyai problem can be formulated for polynomials in more than two variables. The trivariate case was studied by Raz, Sharir, and De Zeeuw~\\cite{RazShaDeZee4D}, and the general $k$-variate case was established by Raz and Shem Tov~\\cite{RazShe}. They obtain the following result.\n\n\\begin{abstract}\nLet $f\\in \\RR[x_1,\\ldots, x_k]$, for $k\\ge 2$. For any finite sets $A_1,\\ldots, A_k\\subset \\RR$, consider the set \n$$\nf(A_1,\\ldots, A_k):=\\{f(a_1,\\ldots, a_k)\\mid (a_1,\\cdots,a_k)\\in A_1\\times\\cdots \\times A_k\\},\n$$\nthat is, the image of $A_1\\times \\cdots\\times A_k$ under $f$.\nExtending a theorem of Elekes and R\\'onyai, which deals with the case $k=2$, and the result of Raz, Sharir, and De Zeeuw~\\cite{RazShaDeZee4D}, dealing with the case $k=3$, it is proved in Raz and Shem Tov~\\cite{RazShe}, that for every choice of finite $A_1,\\ldots, A_k\\subset \\RR$, each of size $n$, one has \n\\begin{equation}\\label{RSbound}\n|f(A_1,\\ldots,A_k)|=\\Omega(n^{3/2}),\n\\end{equation}\nunless $f$ has some degenerate special form.\n\nIn case $f$ is not one of the forms in \\eqref{specialbivariate}, the lower bound on $|f(A,B)|$ was improved in \\cite{RazShaDeZee} to be $\\Omega(n^{4/3})$, and further improved in \\cite{SolZha} to be $\\Omega(n^{3/2})$, which is currently the best known lower bound for bivariate polynomials that are not special.\n\nNote that the bound in Theorem~\\ref{thm:kvarER}, for non-special polynomials $f$, is independent of $k$, and in particular coincides with the bound for $k=3$.\nAt first glance this may appear to be merely a consequence of the proof. Indeed, the argument in \\cite{RazShe} reduces the $k$-variate case for $k \\ge 4$ to the trivariate case by fixing values for $k-3$ of the variables. They then show that if fixing any such subset of $k-3$ variables yields a special trivariate polynomial, then $f$ itself, as a $k$-variate polynomial, must be special in the sense of \\eqref{kspecial}.\n\nWe prove the following main result of the paper.\n\\begin{thm}\\label{mainthm}\nLet $k\\ge 3$ and let $f\\in\\RR[x_1\\ldots,x_k]$. \nAssume that ${\\rm rank}(f)=r\\ge 2$.\nThen, for every $\\varepsilon>0$, the following holds: \nLet $A_1,\\ldots,A_k\\subseteq \\RR$ be finite, each of size $n$. Then $$\n\\lvert f(A_1,\\ldots,A_k)\\rvert= \\Omega\\left( n^{\\frac{5r-4}{2r}-\\varepsilon}\\right), \n$$\nwhere the constant of proportionality depends on $\\deg(f)$, on $r$, and on $\\eps$.\n\\end{thm}\n\nWe observe that ${\\rm rank}(f)=1$ corresponds to the special forms from Theorem~\\ref{thm:kvarER}. Indeed, we have the following theorem. \n \\begin{thm}\\label{rank1char}\nLet $k\\ge3$ and let $f\\in\\RR\\left[x_1 \\ldots,x_k\\right]$. Assume that $f$ depends non-trivially on each of its variables and that $$\n{\\rm rank}(f) = 1.$$\nThen $f$ has one of the forms\n\\begin{align}\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)+\\cdots+p_{k}\\left(x_{k}\\right)\\right)\\ \\ \\text{ or}\\nonumber\\\\\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)\\cdot\\ldots\\cdot p_{k}\\left(x_{k}\\right)\\right),\\label{special1}\n\\end{align}\nfor some univariate real polynomials $h(x), p_1(x),\\ldots,p_k(x)$.\n\\end{thm}\n\n\\begin{prop}\\label{r_eq_k}\nLet $k\\ge 3$, $f\\in\\RR\\left[x_1\\ldots,x_k\\right]$, and assume that ${\\rm{rank}}(f)=k-1$. Then, for every $\\varepsilon>0$,\nthe following holds: Let $A_0,\\ldots,A_k\\subseteq \\RR$ be finite, each of size $n$. Then \n$$\n\\lvert f(A_1,\\ldots,A_k)\\rvert\n= \n\\Omega\\left(n^{\\frac{5(k-1)-4}{2(k-1)}-\\varepsilon}\\right), \n$$\nwhere the constant of proportionality depends on $\\deg(f)$, on $k$, and on $\\varepsilon$.\n\\end{prop}\n\nSo in order to complete the proof of Proposition~\\ref{r_eq_k}\nwe only need to prove \\eqref{upbndS}. Let $T=T_{f,x_1}$ be the coefficient map defined in the introduction. By assumption, ${\\rm rank}(J_T)=k-1$. Thus, there exist indices $\\left(i_1,\\ldots, i_{k-1}\\right) $ such that for \n$$\n\\hat{T}:(x_2\\ldots,x_k)\n\\mapsto(\\alpha_{i_1}(x_2\\ldots,x_k),\\ldots,\\alpha_{i_{k-1}}(x_2\\ldots,x_k)),\n$$ \nwe have \n$$\\det J_{\\hat{T}} \\not\\equiv 0.$$\nDefine \n\\begin{align*}\nS_0 &=\\left\\{\\left(a_{1},\\ldots,a_{k},b\\right)\\in S \\mid\\det J_{\\hat{T}}\\left(a_{2},\\ldots,a_{k}\\right)=0\\right\\}, \\\\\nS' & = S\\setminus S_0.\n\\end{align*} \nClearly $|S|= |S_0|+|S'|$.\nObserve that,\n$$\n|S_0|=|A_1|\\cdot\\left|\\left(A_2\\times\\cdots\\times A_k\\right) \\cap \\left\\{\\det J_{\\hat{T}}=0\\right\\}\\right|\\le n\\cdot \\deg(\\det J_{\\hat T})n^{k-2},\n$$\nwhere the inequality is due to the Schwartz--Zippel Lemma (see \\cite{Schw,Zipp}).\nThus, we get\n\\begin{equation}\n\\label{S0bound}\n\\left| S_0\\right|=O(n^{k-1}),\n\\end{equation}\nwhere the constant of proportionality depends on $\\deg(f)$ and on $k$.\n\nIn this section we prove Theorem~\\ref{rank1char}. For the proof we will use the following lemma from Raz and Shem Tov~\\cite{RazShe}.\n\\begin{lem}[{\\bf Raz--Shem Tov~\\cite[Lemma 2.3]{RazShe}}]\\label{RazShem}\nLet $f\\in\\RR[x_1,\\ldots,x_k]$. Assume\nthat\n\\begin{equation}\\label{rank1diff}\n\\frac{\\frac{\\partial f}{\\partial x_1}(x_1,\\ldots,x_k)\n}\n{\nr_1(x_1)\n}\n=\\cdots =\n\\frac{\\frac{\\partial f}{\\partial x_k}(x_1,\\ldots,x_k)\n}\n{\nr_k(x_k)\n},\n\\end{equation}\nfor some univariate real polynomials $r_1,\\ldots, r_k$. \nThen, $f$ is one of the forms\n\\begin{align*}\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)+\\cdots+p_{k}\\left(x_{k}\\right)\\right)\\ \\ \\text{ or}\\\\\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)\\cdot\\cdots\\cdot p_{k}\\left(x_{k}\\right)\\right),\n\\end{align*}\nfor some univariate real polynomials $h(x), p_1(x),\\ldots,p_k(x)$.\n\n\\begin{align}\n f(x_1,\\ldots,x_k)&=h(p_1(x_1)+\\cdots +p_k(x_k))\\label{kspecial}\\\\\n f(x_1,\\ldots,x_k)&=h(p_1(x_1)\\cdot \\ldots \\cdot p_k(x_k))\\nonumber\n \\end{align}\n\n\\begin{thm}\\label{rank1char}\nLet $k\\ge3$ and let $f\\in\\RR\\left[x_1 \\ldots,x_k\\right]$. Assume that $f$ depends non-trivially on each of its variables and that $$\n{\\rm rank}(f) = 1.$$\nThen $f$ has one of the forms\n\\begin{align}\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)+\\cdots+p_{k}\\left(x_{k}\\right)\\right)\\ \\ \\text{ or}\\nonumber\\\\\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)\\cdot\\ldots\\cdot p_{k}\\left(x_{k}\\right)\\right),\\label{special1}\n\\end{align}\nfor some univariate real polynomials $h(x), p_1(x),\\ldots,p_k(x)$.\n\\end{thm}\n\n\\begin{align}\nf(x,y)&=h(p(x)+q(y))\\quad\\text{or}\\nonumber\\\\\nf(x,y)&=h(p(x)q(y)),\\label{specialbivariate}\n\\end{align}\n\n\\begin{thm}[{\\bf \\cite{RazShaDeZee4D, RazShe}}]\\label{thm:kvarER}\n Let $k\\ge 3$ and let $f\\in \\RR[x_1,\\ldots, x_k]$. Then one of the following holds:\\\\\n (i) For every $A_1,\\ldots, A_k\\in \\RR$ each of size $n$ one has\n $$\n f(A_1,\\ldots,A_k)|=\\Omega\\left(n^{3/2}\\right)$$\\\\\n (ii) $f$ is of one of the forms:\n \\begin{align}\n f(x_1,\\ldots,x_k)&=h(p_1(x_1)+\\cdots +p_k(x_k))\\label{kspecial}\\\\\n f(x_1,\\ldots,x_k)&=h(p_1(x_1)\\cdot \\ldots \\cdot p_k(x_k))\\nonumber\n \\end{align}\n\\end{thm}", + "post_theorem_intro_text_len": 6334, + "post_theorem_intro_text": "Note that the bound in Theorem~\\ref{thm:kvarER}, for non-special polynomials $f$, is independent of $k$, and in particular coincides with the bound for $k=3$.\nAt first glance this may appear to be merely a consequence of the proof. Indeed, the argument in \\cite{RazShe} reduces the $k$-variate case for $k \\ge 4$ to the trivariate case by fixing values for $k-3$ of the variables. They then show that if fixing any such subset of $k-3$ variables yields a special trivariate polynomial, then $f$ itself, as a $k$-variate polynomial, must be special in the sense of \\eqref{kspecial}.\n\nIt is natural to expect that increasing the number of variables should force the image of $f$ to grow faster. However, certain polynomials in many variables can in effect behave like polynomials in fewer variables. For example, consider the $(k+2)$-variate polynomial\n$$\nf(x,y,z_1\\ldots,z_k)=xy+z_1+z_2+\\cdots +z_k.$$\nLet $A,B,C_1,\\ldots,C_k\\subset \\mathbb{R}$, where $A,B$ are arbitrary finite sets of size $n$ and $C_1=\\cdots=C_k=[n]$. Let \n$$\nC:=C_1+\\cdots +C_k=\\{k,k+1,\\ldots,kn\\}.\n$$ \nThen $|C|=\\Theta(n)$, and letting $g(x,y,z):=xy+z$, we have\n$$\nf(A,B,C_1,\\ldots,C_k)=g(A,B,C).$$\nIn this case, with the current techniques, it is unclear how to obtain a bound on the expansion of $f$ that improves upon the trivariate result for $g$.\n\n\\paragraph{Our results.}\nIn this paper, we recognize $k$-variate polynomials that are, in a precise sense, truly $k$-variate, and we improve the corresponding expansion bounds for them. More precisely, for a $k$-variate polynomial $f$, we introduce the notion of the {\\it rank} of $f$. If $f$ has rank $r$, then, in a rigorous sense, it is essentially $(r+1)$-variate, and the bound on the size of its image can be improved with an exponent that grows with $r$.\n\nWe now define the rank of a polynomial and then state our main result.\n\nLet $f\\in \\mathbb{R}\\left[x_1,\\ldots,x_k\\right]$ and let $d_{x_1}$ stand for the degree of $f$ with respect to the variable $x_1$. \nWrite $$\nf\\left(x_1,\\ldots,x_k\\right)=\\sum\\limits_{i=0}^{d_{x_1}}\\alpha_{i}\\left(x_2,\\ldots,x_k\\right)x_1^i.\n$$\nWe consider the {\\it coefficient map} $T=T_{f,x_1}:\\mathbb{R}^{k-1}\\to \\mathbb{R}^{d_{x_1}+1}$ given by $$\n\\left(x_2,\\ldots,x_k\\right)\\mapsto \\left(\\alpha_0\\left(x_2,\\ldots,x_k\\right),\\ldots,\\alpha_{d_{x_1}}\\left(x_2,\\ldots,x_k\\right)\\right).\n$$\nWe define the rank of $f$ with respect to the variable $x_1$ to be \n$$\n{\\rm rank}_{x_1}(f):={\\rm rank}(J_T),$$\nwhere $J_T$ stands for the Jacobian matrix of $T$.\nNote that \n$$\n0\\le {\\rm rank}_{x_1}(f)\\le k-1.\n$$ \nSimilarly, define ${\\rm rank}_{x_i}(f)$, for every $i=2,\\ldots, k$, where $x_i$ plays the role of $x_1$.\n\nFinally, define the {\\it rank} of the polynomial $f$ to be \n$$\n{\\rm rank}(f):=\\max_{1\\le i\\le k}{\\rm rank}_{x_i}(f).\n$$\n\n\\begin{exmp} \\label{example}\nLet $$\nf\\left(x_1,x_2,\\ldots,x_k\\right) = x_1x_k + x_2x_k^2\\dots + x_{k-1}x_k^{k-1}.$$\nThen $\\text{rank}(f)=\\text{rank}_{x_k}(f)=k-1$.\n\\end{exmp} \n\n\\begin{exmp}\nLet $$\nf(x_1,x_2,\\ldots,x_k) = p_1(x_1)x_k + p_2(x_1,x_2)x_k^2+\\dots + p_{k-1}(x_1,\\ldots,x_{k-1})x_k^{k-1},$$\nwhere $p_i$ is an $i$-variate polynomial that depends non-trivially on $x_i$.\nThen $\\text{rank}(f)=\\text{rank}_{x_k}(f)=k-1$. Indeed, in this case the matrix $J_{T_{f,x_k}}$ is upper-triangular.\n\\end{exmp}\n\nWe prove the following main result of the paper.\n\\begin{thm}\\label{mainthm}\nLet $k\\ge 3$ and let $f\\in\\mathbb{R}[x_1\\ldots,x_k]$. \nAssume that ${\\rm rank}(f)=r\\ge 2$.\nThen, for every $\\varepsilon>0$, the following holds: \nLet $A_1,\\ldots,A_k\\subseteq \\mathbb{R}$ be finite, each of size $n$. Then $$\n\\lvert f(A_1,\\ldots,A_k)\\rvert= \\Omega\\left( n^{\\frac{5r-4}{2r}-\\varepsilon}\\right), \n$$\nwhere the constant of proportionality depends on $\\deg(f)$, on $r$, and on $\\varepsilon$.\n\\end{thm}\n\n We observe that ${\\rm rank}(f)=1$ corresponds to the special forms from Theorem~\\ref{thm:kvarER}. Indeed, we have the following theorem. \n \\begin{thm}\\label{rank1char}\nLet $k\\ge3$ and let $f\\in\\mathbb{R}\\left[x_1 \\ldots,x_k\\right]$. Assume that $f$ depends non-trivially on each of its variables and that $$\n{\\rm rank}(f) = 1.$$\nThen $f$ has one of the forms\n\\begin{align}\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)+\\cdots+p_{k}\\left(x_{k}\\right)\\right)\\ \\ \\text{ or}\\nonumber\\\\\nf\\left(x_{1},\\ldots,x_{k}\\right) & =h\\left(p_{1}\\left(x_{1}\\right)\\cdot\\ldots\\cdot p_{k}\\left(x_{k}\\right)\\right),\\label{special1}\n\\end{align}\nfor some univariate real polynomials $h(x), p_1(x),\\ldots,p_k(x)$.\n\\end{thm}\n\n Finally, \n we present an application of our results to the following Erd\\H{o}s-type combinatorial geometric problem. \n Let $\\nu$ denote the moment curve in $\\mathbb{R}^d$, parameterized by\n $$\\nu(t)=(t,t^2,\\ldots,t^d),\\quad t\\in \\mathbb{R}.$$\n Let $P\\subset \\nu$ be a finite set of $n$ points. For any distinct $p_1,\\ldots, p_{d+1}\\in \\nu$, let $\\sigma=\\sigma(p_1,\\ldots,p_{d+1})$ denote the $d$-simplex which is the convex hull of $p_1,\\ldots,p_{d+1}$ in $\\mathbb{R}^d$, and let ${\\rm vol}(\\sigma)$ denote its $d$-dimensional volume.\n Define\n $$\n \\Delta(P)= \\left\\{{\\rm vol}(\\sigma(p_1,\\ldots,p_{d+1}))\\mid p_1,\\ldots,p_{d+1}\\in P\\right\\}.$$\n\n We have the following theorem.\n\\begin{thm}\\label{thm:app}\n Let $\\nu$ be the moment curve in $\\mathbb{R}^d$ and let $P\\subset \\nu$ be any finite set of size $n$. Then, for every $\\varepsilon>0$, \n $$\n|\\Delta(P)|=\\Omega\\left(n^{\\frac{5d-4}{2d}-\\varepsilon}\\right),\n$$\n where the implicit constant depends only on $\\varepsilon$ and on $d$.\n\\end{thm}\n\n Theorem~\\ref{thm:app} is obtained by identifying a $(d+1)$-variate polynomial $f$ whose expansion over a certain $n\\times \\cdots \\times n$ grid in $\\mathbb{R}^{d+1}$ corresponds to the number of distinct volumes of $d$-simplices spanned by $P$. We then show that $f$ has rank $d$ and apply our main Theorem~\\ref{mainthm}. \n\n\\paragraph{Organization of the paper.} The paper is organized as follows. In Section~\\ref{sec:pre} we recall an incidence bound that will serve as a key tool in our arguments. In Section~\\ref{sec:special}, we establish a special case of our main result, Theorem~\\ref{mainthm}, and in Section~\\ref{sec:proofmain}, we complete its proof. The proof of Theorem~\\ref{rank1char} is provided in Section~\\ref{sec:char}. Finally, Section~\\ref{sec:app} contains the proof of Theorem~\\ref{thm:app}.", + "sketch": "The post-theorem text explains that the proof in \\cite{RazShe} for Theorem~\\ref{thm:kvarER} “reduces the $k$-variate case for $k \\ge 4$ to the trivariate case by fixing values for $k-3$ of the variables.” It then argues that “if fixing any such subset of $k-3$ variables yields a special trivariate polynomial, then $f$ itself, as a $k$-variate polynomial, must be special in the sense of \\eqref{kspecial}.”", + "expanded_sketch": "The post-theorem text explains that the proof in Raz and Sheffer “reduces the $k$-variate case for $k \\ge 4$ to the trivariate case by fixing values for $k-3$ of the variables.” It then argues that “if fixing any such subset of $k-3$ variables yields a special trivariate polynomial, then $f$ itself, as a $k$-variate polynomial, must be special in the sense of\n\\begin{align}\n f(x_1,\\ldots,x_k)&=h(p_1(x_1)+\\cdots +p_k(x_k))\\label{kspecial}\\\\\n f(x_1,\\ldots,x_k)&=h(p_1(x_1)\\cdot \\ldots \\cdot p_k(x_k))\\nonumber\n \\end{align}\n.”", + "expanded_theorem": "[{\n\\bf \\cite{RazShaDeZee4D, RazShe}}]\\label{thm:kvarER}\n Let $k\\ge 3$ and let $f\\in \\mathbb{R}[x_1,\\ldots, x_k]$. Then one of the following holds:\\\\\n (i) For every $A_1,\\ldots, A_k\\in \\mathbb{R}$ each of size $n$ one has\n $$\n f(A_1,\\ldots,A_k)|=\\Omega\\left(n^{3/2}\\right)$$\\\\\n (ii) $f$ is of one of the forms:\n \\begin{align}\n f(x_1,\\ldots,x_k)&=h(p_1(x_1)+\\cdots +p_k(x_k))\\label{kspecial}\\\\\n f(x_1,\\ldots,x_k)&=h(p_1(x_1)\\cdot \\ldots \\cdot p_k(x_k))\\nonumber\n \\end{align}", + "theorem_type": [ + "Implication", + "Inequality or Bound" + ], + "mcq": { + "question": "Let $k\\ge 3$ and let $f\\in \\mathbb{R}[x_1,\\ldots,x_k]$ be a real polynomial in $k$ variables. For finite sets $A_1,\\ldots,A_k\\subset \\mathbb{R}$, define\n$$\nf(A_1,\\ldots,A_k):=\\{f(a_1,\\ldots,a_k):(a_1,\\ldots,a_k)\\in A_1\\times\\cdots\\times A_k\\}.\n$$\nWhich conclusion holds for such an $f$?", + "correct_choice": { + "label": "A", + "text": "Either for every choice of finite sets $A_1,\\ldots,A_k\\subset \\mathbb{R}$, each of size $n$, one has\n$$\n|f(A_1,\\ldots,A_k)|=\\Omega\\bigl(n^{3/2}\\bigr),\n$$\nor there exist univariate real polynomials $h,p_1,\\ldots,p_k$ such that $f$ has one of the two forms\n$$\nf(x_1,\\ldots,x_k)=h\\bigl(p_1(x_1)+\\cdots+p_k(x_k)\\bigr)\n$$\nor\n$$\nf(x_1,\\ldots,x_k)=h\\bigl(p_1(x_1)\\cdots p_k(x_k)\\bigr).\n$$" + }, + "choices": [ + { + "label": "B", + "text": "Either for every choice of finite sets $A_1,\\ldots,A_k\\subset \\mathbb{R}$, each of size $n$, one has\n$$\n|f(A_1,\\ldots,A_k)|=\\Omega\\bigl(n^{3/2}\\bigr),\n$$\nor there exist univariate real polynomials $h,p_1,\\ldots,p_k$ such that after fixing some $k-3$ variables to real constants, the resulting trivariate polynomial has one of the two forms\n$$\ng(x_i,x_j,x_\\ell)=h\\bigl(p_i(x_i)+p_j(x_j)+p_\\ell(x_\\ell)\\bigr)\n$$\nor\n$$\ng(x_i,x_j,x_\\ell)=h\\bigl(p_i(x_i)p_j(x_j)p_\\ell(x_\\ell)\\bigr).\n$$" + }, + { + "label": "C", + "text": "If $f$ is not of one of the two forms\n$$\nf(x_1,\\ldots,x_k)=h\\bigl(p_1(x_1)+\\cdots+p_k(x_k)\\bigr)\n$$\nor\n$$\nf(x_1,\\ldots,x_k)=h\\bigl(p_1(x_1)\\cdots p_k(x_k)\\bigr),\n$$\nthen for every choice of finite sets $A_1,\\ldots,A_k\\subset \\mathbb{R}$, each of size $n$, one has\n$$\n|f(A_1,\\ldots,A_k)|=\\Omega\\bigl(n^{3/2}\\bigr).\n$$" + }, + { + "label": "D", + "text": "Either for every choice of finite sets $A_1,\\ldots,A_k\\subset \\mathbb{R}$, each of size $n$, one has\n$$\n|f(A_1,\\ldots,A_k)|=\\Omega\\bigl(n^{2}\\bigr),\n$$\nor there exist univariate real polynomials $h,p_1,\\ldots,p_k$ such that $f$ has one of the two forms\n$$\nf(x_1,\\ldots,x_k)=h\\bigl(p_1(x_1)+\\cdots+p_k(x_k)\\bigr)\n$$\nor\n$$\nf(x_1,\\ldots,x_k)=h\\bigl(p_1(x_1)\\cdots p_k(x_k)\\bigr).\n$$" + }, + { + "label": "E", + "text": "Either there exist finite sets $A_1,\\ldots,A_k\\subset \\mathbb{R}$, each of size $n$, for which\n$$\n|f(A_1,\\ldots,A_k)|=\\Omega\\bigl(n^{3/2}\\bigr),\n$$\nor there exist univariate real polynomials $h,p_1,\\ldots,p_k$ such that $f$ has one of the two forms\n$$\nf(x_1,\\ldots,x_k)=h\\bigl(p_1(x_1)+\\cdots+p_k(x_k)\\bigr)\n$$\nor\n$$\nf(x_1,\\ldots,x_k)=h\\bigl(p_1(x_1)\\cdots p_k(x_k)\\bigr).\n$$" + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "B" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "case_split", + "tampered_component": "global-special-from-all-fixings replaced by existence of one special trivariate fixing", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "case_split", + "tampered_component": "dropped the converse direction in the dichotomy", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "case_split", + "tampered_component": "lower-bound exponent $3/2$ strengthened to $2$", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "case_split", + "tampered_component": "universal quantifier over all sets replaced by existential quantifier", + "template_used": "quantifier_dependence" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem only defines notation and asks for the valid conclusion; it does not reveal the structural dichotomy or the exponent, so there is no direct answer leakage." + }, + "TAS": { + "score": 1, + "justification": "This is largely a theorem-recognition item: the correct option is essentially the exact dichotomy theorem statement, though the presence of nearby variants prevents it from being a pure restatement." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to compare logical strength, quantifiers, and the exponent across choices, but the task is still mainly selecting the exact known theorem rather than deriving a conclusion from mathematical work." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong and plausible: one is a weaker true statement, one strengthens the bound incorrectly, one weakens the quantifier, and one introduces a subtle structural tampering. These reflect realistic mathematical failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with no answer leakage and high-quality distractors, but it mainly tests theorem recall/statement discrimination rather than deep generative reasoning." + } + }, + { + "id": "2511.02579v1", + "paper_link": "http://arxiv.org/abs/2511.02579v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "theorem", + "content": "\\label{TH-00}\nLet $u$ be a suitable weak solutions. Suppose that \nthe following two conditions hold:\n\\begin{equation}\n\\liminf_{r\\to 0}M(r):=\\liminf_{r\\to 0}\\int_{B_r}\\!\\! \\left( \\frac{|u|^2}{r^3} + \\frac{|\\nabla u|^2}{r} \\right) <\\infty, \\quad \\liminf_{r\\to 0}\\frac1{r^2}\\int_{B_r}\\!\\!(|u|^2+2P)u\\cdot\\frac x{|x|}>0.\n\\end{equation}\nThen for any $\\{r_k\\}_{k=1}^\\infty, r_k\\downarrow 0$ there is a subsequence $r_{k_m}$ such that \nthe scaled solutions $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$, converge to \na homogenous vector field of degree negative one, and hence \n$x=0$ is a regular point.", + "start_pos": 10011, + "end_pos": 10627, + "label": "TH-00" + }, + "ref_dict": { + "lem:positiveinf": "\\begin{lemma}\\label{lem:positiveinf}\nLet \n\\begin{equation}\nQ(r):= \\int_{B_r} \\Big\\{ \\frac{15}{4r^3}|u|^2 \n+ \n\\frac{1}{4r}|\\nabla u |^2 \n+ \n\\frac{3(r^2-|x|^2)}{4r^3} |\\nabla u|^2\\Big\\}\\, dx,\n\\end{equation}\nand \n\\begin{equation}\n\\text{\\Large$\\wp$}(r) =\\frac{1}{r^2} \\int_{B_r} \\Big\\{{|u|^2} + 2P \\Big\\} (u\\cdot \\frac{x}{|x|}) \\, dx .\n\\end{equation}\nIf $\\liminf_{r\\to 0} M(r)<\\infty$ and \n\\[\n\\liminf_{r\\to0^+}\\left[ Q(r)\n+\\text{\\Large$\\wp$}(r)\n\\right]> 0\n\\]\nthen $x=0$ is a regular point.\n\\end{lemma}", + "mH1": "\\begin{equation}\\label{mH1}\n\t\\mH(R):= \\left\\{ h\\in W^{1,2}(B_R) \\,\\bigg\\vert\\, \\exists \\zeta \\in W^{1,2}(\\mathbb{S}^{N-1}): \\ h(x)=\\frac{1}{|x|}\\zeta\\big( \\frac{x}{|x|} \\big) \\ \\text{for $x\\in B_R$ a.e.} \\right\\}.\n\\end{equation}", + "prop:dAdr": "\\begin{proposition}\\label{prop:dAdr}\n\tSuppose $N=5$ and $(u,P)$ is a suitable weak solution of \\eqref{st-NS-N}. For $r>0$, define\n\t\\begin{equation}\n\t\t\\left\\{\n\t\t\\begin{aligned}\n\t\t\t&D(r)\\vcentcolon= \\int_{B_r} \\Big\\{ \\frac{15}{4r^3}|u|^2 + \\frac{1}{4r}|\\nabla u |^2 + \\frac{3}{4r^3}\\left| \\nabla (|x|u) \\right|^2 + \\frac{3(r^2-|x|^2)}{4r^3} |\\nabla u|^2\\Big\\}\\, dx,\\\\\n\t\t\t&A(r) \\vcentcolon= \\frac{1}{r^3}\\int_{B_r}(x\\cdot \\nabla)\\frac{|u|^2}{2}\\, dx + \\frac{9}{4r^3}\\int_{ B_r}|u|^2 \\, dx - \\frac{1}{r^{2}} \\int_{B_r} \\big\\{ \\frac{|u|^2}{2} + P \\big\\} u\\cdot \\frac{x}{|x|} \\, dx .\n\t\t\\end{aligned}\n\t\t\\right. \n\t\\end{equation}\n\tThen the following differential equation holds for $r>0$,\n\t\\begin{equation*}\n\t\t\\frac{d A}{d r} \\ge \\frac{1}{r} D(r) + \\frac{2}{r^3} \\int_{B_r} \\Big\\{\\frac{|u|^2}{2} + P \\Big\\} (u\\cdot \\frac{x}{|x|}) \\, dx .\n\t\\end{equation*}\n\\end{proposition}", + "TH": "\\begin{theorem}\\label{TH}\nLet $u$ be a suitable weak solution of \\eqref{eq:problem}, $B_1\\subset \\Omega$, and\n\\[\nm:=\\liminf_{R\\to 0}M(R)<\\infty \\qquad \\text{where} \\quad M(R):=\\int_{B_R} \\Big( \\frac{|u|^2}{R^3} + \\frac{|\\nabla u|^2}{R} \\Big).\n\\] Let $\\mP_R[\\,\\cdot\\,]: \\mathcal W^{1,2}(B_R)\\to \\mH(R)$ be the projection operator for the space \\eqref{mH1}. There exists $\\ep(m)>0$ \nsuch that \nif \n\\begin{equation}\\label{eq:13-alt}\n\\frac{1}{R^3} \\int_{B_R} \\left| u - \\mP_R[u] \\right|^2 + \\frac{1}{R} \\int_{B_R} \\left| \\nabla u - \\nabla \\mP_R[u] \\right|^2 \\le \\ep(m) M(R)\n\\end{equation}\nholds\nfor all $R\\in (0, R_0)$ then $u$ is regular at $x=0$. \n\\end{theorem}", + "eq:problem-3d": "\\begin{equation}\\label{eq:problem-3d}\n\\left.\\begin{array}{rrr}\nu_t^i+u^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\mbox{in}\\ \\Omega\\times(0, T), \\Omega\\subset \\R^3.\n\\end{equation}", + "TH-00": "\\begin{theorem}\\label{TH-00}\nLet $u$ be a suitable weak solutions. Suppose that \nthe following two conditions hold:\n\\begin{equation}\n\\liminf_{r\\to 0}M(r):=\\liminf_{r\\to 0}\\int_{B_r}\\!\\! \\left( \\frac{|u|^2}{r^3} + \\frac{|\\nabla u|^2}{r} \\right) <\\infty, \\quad \\liminf_{r\\to 0}\\frac1{r^2}\\int_{B_r}\\!\\!(|u|^2+2P)u\\cdot\\frac x{|x|}>0.\n\\end{equation}\nThen for any $\\{r_k\\}_{k=1}^\\infty, r_k\\downarrow 0$ there is a subsequence $r_{k_m}$ such that \nthe scaled solutions $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$, converge to \na homogenous vector field of degree negative one, and hence \n$x=0$ is a regular point.\n\\end{theorem}", + "eq:problem": "\\begin{equation}\\label{eq:problem}\n\\left.\\begin{array}{rrr}\nu^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3, 4, 5,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\quad\\mbox{in}\\ \\Omega\\subset \\R^5\n\\end{equation}", + "prop:cubicEst": "\\begin{proposition}\\label{prop:cubicEst}\n\tFor fixed constants $\\delta_{1}, \\delta_{2}>0$, there exists $\\ep>0$ such that if $(u,p)$ is a Leray-Hopf solution to the Navier-Stokes equations satisfying\n\t\\begin{equation}\\label{h-A}\n\t\t\\frac{1}{R^3} \\int_{B_R} \\big|u- \\mP_R[u] \\big|^2 + \\frac{1}{R} \\int_{ B_R} \\big|\\nabla u - \\nabla \\mP_R[u] \\big|^2 \\le \\ep M[u](R)\n\t\\end{equation} \n\tfor some $R>0$, then \n\t\\begin{equation}\n\t\t\\bigg|\\frac{1}{R^2}\\int_{B_{R/2}} \\big( |u|^2 + 2 p \\big) \\, u \\cdot \\frac{x}{|x|} \\bigg| \\le \\delta_{1} + \\delta_{2} \\big(M[u](R)\\big)^{\\frac{3}{2}}. \n\t\\end{equation}\n\\end{proposition}" + }, + "pre_theorem_intro_text_len": 3941, + "pre_theorem_intro_text": "In this paper we study the local behavior of the weak solutions of the stationary incompressible \nNavier-Stokes equations in five space dimensions\n\\begin{equation}\\label{eq:problem}\n\\left.\\begin{array}{rrr}\nu^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3, 4, 5,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\quad\\mbox{in}\\ \\Omega\\subset \\mathbb{R}^5\n\\end{equation}\nwhere $\\Omega\\subset \\mathbb{R}^5$ is a domain . \n\nThe existence of weak solutions under various assumptions on the \nboundary data and $\\Omega$ has been established in \\cite{Galdi}, \\cite{FR-Pisa}, \\cite{Struwe-per}. \nMoreover, in \\cite{FR-Pisa}, \\cite{FR-arma}, \\cite{Struwe-per} the authors constructed \nsmooth solutions of \\eqref{eq:problem-3d}.\n\nThe problem \\eqref{eq:problem} has a number of similarities with the \ndynamic Navier-Stokes system in three space dimensions\n\\begin{equation}\\label{eq:problem-3d}\n\\left.\\begin{array}{rrr}\nu_t^i+u^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\mbox{in}\\ \\Omega\\times(0, T), \\Omega\\subset \\mathbb{R}^3.\n\\end{equation}\nFor instance, in both cases $u\\in L^{\\frac{10}3}_{loc}, P\\in L^{\\frac 53}_{loc}, \\nabla P\\in L^{\\frac54}_{loc}$, see \\cite{Seregin-book}. \nDue to this a number of mathematicians studied the stationary \nNavier-Stokes equations in higher dimensions in order to develop \nstronger analytical methods which may be applicable to the \ndynamic case \\eqref{eq:problem-3d}, see \\cite{Galdi}.\n\nIn this context, of particular interest is the problem of estimating the \ndimension of the singular points of suitable weak solutions $u$, i.e. the points where \n$u$ is not bounded. Scheffer \\cite{Scheffer} \nproved such results for \\eqref{eq:problem-3d} and later\nCaffarelli, Kohn and Nirenberg \\cite{CKN} improved upon it \nshowing that the Hausdorff dimension of the singular set in space-time is atmost one. \nWe note that the latter result can be established by a different method\nby looking at the \nsmall perturbations of the Stokes system \\cite{Lin}. \nFor \\eqref{eq:problem} the partial regularity is proved in \\cite{Struwe}.\n\nAt the possible singular point $(x_0, t_0)$ the scale invariance $u(x, t)\\mapsto ru(x_0+rx, t_0+r^2), r>0$ suggests that at the scale $r$, $u$ behaves like $1/r$ near $(x_0, t_0)$.\nA natural question that follows from this observation is whether \none can classify the scale invariant solutions.\nThis has been the main approach towards understanding the structure of possible singularities.\n\\v{S}ver\\'{a}k's classification for the self-similar solutions \\cite{Sverak} for Navier-Stokes equations \\eqref{eq:problem} shows that \na solution of the form $h(x)=\\frac{\\zeta(\\frac x{|x|})}{|x|}$, with some smooth vectorfield $\\zeta$, must be identically zero. \n\nAnother questions following from this result is whether the solutions sufficiently close to the self-similar one \nare in fact zero. It is easy to see that the self-similar vectorfields $h=\\frac{\\zeta(\\frac x{|x|})}{|x|}$ form a Hilbert subspace $\\mathcal{H}(R)$ of \nthe Sobolev space $\\mathcal W^{1, 2}(B_R)$ in $\\mathbb{R}^5$ with an appropriately scaled invariant norms,\n\\begin{equation}\\label{mH1}\n\t\\mathcal{H}(R):= \\left\\{ h\\in W^{1,2}(B_R) \\,\\bigg\\vert\\, \\exists \\zeta \\in W^{1,2}(\\mathbb{S}^{N-1}): \\ h(x)=\\frac{1}{|x|}\\zeta\\big( \\frac{x}{|x|} \\big) \\ \\text{for $x\\in B_R$ a.e.} \\right\\}.\n\\end{equation}\nThus the Hilbert projection theorem yields that $\\mathcal W^{1,2}(B_R) = \\mathcal{H}(R) \\oplus \\mathcal{H}(R)^{\\perp}$ and we can define the the corresponding projection operator as $\\mathcal{P}_R[\\,\\cdot\\,]: \\mathcal W^{1,2}(B_R)\\to \\mathcal{H}(R)$. Using this, we can measure the error $u-\\mathcal{P}_R[u]$ in terms of the $W^{1,2}$ norm of $u$. \n\nOur work is motivated by the following question. \n\n{\\textit{ If a suitable weak solution to the stationary Navier-Stokes system develops a singularity, \n can it asymptotically become self-similar?}}\n\nOur main result in this direction can be stated as follows:", + "context": "In this paper we study the local behavior of the weak solutions of the stationary incompressible \nNavier-Stokes equations in five space dimensions\n\\begin{equation}\\label{eq:problem}\n\\left.\\begin{array}{rrr}\nu^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3, 4, 5,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\quad\\mbox{in}\\ \\Omega\\subset \\mathbb{R}^5\n\\end{equation}\nwhere $\\Omega\\subset \\mathbb{R}^5$ is a domain .\n\nThe problem \\eqref{eq:problem} has a number of similarities with the \ndynamic Navier-Stokes system in three space dimensions\n\\begin{equation}\\label{eq:problem-3d}\n\\left.\\begin{array}{rrr}\nu_t^i+u^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\mbox{in}\\ \\Omega\\times(0, T), \\Omega\\subset \\mathbb{R}^3.\n\\end{equation}\nFor instance, in both cases $u\\in L^{\\frac{10}3}_{loc}, P\\in L^{\\frac 53}_{loc}, \\nabla P\\in L^{\\frac54}_{loc}$, see \\cite{Seregin-book}. \nDue to this a number of mathematicians studied the stationary \nNavier-Stokes equations in higher dimensions in order to develop \nstronger analytical methods which may be applicable to the \ndynamic case \\eqref{eq:problem-3d}, see \\cite{Galdi}.\n\nIn this context, of particular interest is the problem of estimating the \ndimension of the singular points of suitable weak solutions $u$, i.e. the points where \n$u$ is not bounded. Scheffer \\cite{Scheffer} \nproved such results for \\eqref{eq:problem-3d} and later\nCaffarelli, Kohn and Nirenberg \\cite{CKN} improved upon it \nshowing that the Hausdorff dimension of the singular set in space-time is atmost one. \nWe note that the latter result can be established by a different method\nby looking at the \nsmall perturbations of the Stokes system \\cite{Lin}. \nFor \\eqref{eq:problem} the partial regularity is proved in \\cite{Struwe}.\n\nAnother questions following from this result is whether the solutions sufficiently close to the self-similar one \nare in fact zero. It is easy to see that the self-similar vectorfields $h=\\frac{\\zeta(\\frac x{|x|})}{|x|}$ form a Hilbert subspace $\\mathcal{H}(R)$ of \nthe Sobolev space $\\mathcal W^{1, 2}(B_R)$ in $\\mathbb{R}^5$ with an appropriately scaled invariant norms,\n\\begin{equation}\\label{mH1}\n \\mathcal{H}(R):= \\left\\{ h\\in W^{1,2}(B_R) \\,\\bigg\\vert\\, \\exists \\zeta \\in W^{1,2}(\\mathbb{S}^{N-1}): \\ h(x)=\\frac{1}{|x|}\\zeta\\big( \\frac{x}{|x|} \\big) \\ \\text{for $x\\in B_R$ a.e.} \\right\\}.\n\\end{equation}\nThus the Hilbert projection theorem yields that $\\mathcal W^{1,2}(B_R) = \\mathcal{H}(R) \\oplus \\mathcal{H}(R)^{\\perp}$ and we can define the the corresponding projection operator as $\\mathcal{P}_R[\\,\\cdot\\,]: \\mathcal W^{1,2}(B_R)\\to \\mathcal{H}(R)$. Using this, we can measure the error $u-\\mathcal{P}_R[u]$ in terms of the $W^{1,2}$ norm of $u$.\n\n{\\textit{ If a suitable weak solution to the stationary Navier-Stokes system develops a singularity, \n can it asymptotically become self-similar?}}\n\nOur main result in this direction can be stated as follows:\n\n\\begin{equation}\\label{eq:problem}\n\\left.\\begin{array}{rrr}\nu^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3, 4, 5,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\quad\\mbox{in}\\ \\Omega\\subset \\R^5\n\\end{equation}\n\n\\begin{equation}\\label{eq:problem-3d}\n\\left.\\begin{array}{rrr}\nu_t^i+u^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\mbox{in}\\ \\Omega\\times(0, T), \\Omega\\subset \\R^3.\n\\end{equation}", + "full_context": "In this paper we study the local behavior of the weak solutions of the stationary incompressible \nNavier-Stokes equations in five space dimensions\n\\begin{equation}\\label{eq:problem}\n\\left.\\begin{array}{rrr}\nu^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3, 4, 5,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\quad\\mbox{in}\\ \\Omega\\subset \\mathbb{R}^5\n\\end{equation}\nwhere $\\Omega\\subset \\mathbb{R}^5$ is a domain .\n\nThe problem \\eqref{eq:problem} has a number of similarities with the \ndynamic Navier-Stokes system in three space dimensions\n\\begin{equation}\\label{eq:problem-3d}\n\\left.\\begin{array}{rrr}\nu_t^i+u^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\mbox{in}\\ \\Omega\\times(0, T), \\Omega\\subset \\mathbb{R}^3.\n\\end{equation}\nFor instance, in both cases $u\\in L^{\\frac{10}3}_{loc}, P\\in L^{\\frac 53}_{loc}, \\nabla P\\in L^{\\frac54}_{loc}$, see \\cite{Seregin-book}. \nDue to this a number of mathematicians studied the stationary \nNavier-Stokes equations in higher dimensions in order to develop \nstronger analytical methods which may be applicable to the \ndynamic case \\eqref{eq:problem-3d}, see \\cite{Galdi}.\n\nIn this context, of particular interest is the problem of estimating the \ndimension of the singular points of suitable weak solutions $u$, i.e. the points where \n$u$ is not bounded. Scheffer \\cite{Scheffer} \nproved such results for \\eqref{eq:problem-3d} and later\nCaffarelli, Kohn and Nirenberg \\cite{CKN} improved upon it \nshowing that the Hausdorff dimension of the singular set in space-time is atmost one. \nWe note that the latter result can be established by a different method\nby looking at the \nsmall perturbations of the Stokes system \\cite{Lin}. \nFor \\eqref{eq:problem} the partial regularity is proved in \\cite{Struwe}.\n\nAnother questions following from this result is whether the solutions sufficiently close to the self-similar one \nare in fact zero. It is easy to see that the self-similar vectorfields $h=\\frac{\\zeta(\\frac x{|x|})}{|x|}$ form a Hilbert subspace $\\mathcal{H}(R)$ of \nthe Sobolev space $\\mathcal W^{1, 2}(B_R)$ in $\\mathbb{R}^5$ with an appropriately scaled invariant norms,\n\\begin{equation}\\label{mH1}\n \\mathcal{H}(R):= \\left\\{ h\\in W^{1,2}(B_R) \\,\\bigg\\vert\\, \\exists \\zeta \\in W^{1,2}(\\mathbb{S}^{N-1}): \\ h(x)=\\frac{1}{|x|}\\zeta\\big( \\frac{x}{|x|} \\big) \\ \\text{for $x\\in B_R$ a.e.} \\right\\}.\n\\end{equation}\nThus the Hilbert projection theorem yields that $\\mathcal W^{1,2}(B_R) = \\mathcal{H}(R) \\oplus \\mathcal{H}(R)^{\\perp}$ and we can define the the corresponding projection operator as $\\mathcal{P}_R[\\,\\cdot\\,]: \\mathcal W^{1,2}(B_R)\\to \\mathcal{H}(R)$. Using this, we can measure the error $u-\\mathcal{P}_R[u]$ in terms of the $W^{1,2}$ norm of $u$.\n\n{\\textit{ If a suitable weak solution to the stationary Navier-Stokes system develops a singularity, \n can it asymptotically become self-similar?}}\n\nOur main result in this direction can be stated as follows:\n\n\\begin{equation}\\label{eq:problem}\n\\left.\\begin{array}{rrr}\nu^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3, 4, 5,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\quad\\mbox{in}\\ \\Omega\\subset \\R^5\n\\end{equation}\n\n\\begin{equation}\\label{eq:problem-3d}\n\\left.\\begin{array}{rrr}\nu_t^i+u^ju^i_j+P_i=\\Delta u^i, \\quad i=1, 2, 3,\\\\\n\\div u =0,\n\\end{array}\n\\right\\}\n\\mbox{in}\\ \\Omega\\times(0, T), \\Omega\\subset \\R^3.\n\\end{equation}\n\n{\\textit{ If a suitable weak solution to the stationary Navier-Stokes system develops a singularity, \n can it asymptotically become self-similar?}}\n\nIf $\\liminf_{r\\to 0}M(r)<\\infty$\nthen the singularity may occur only if the function \n\\begin{equation}\n\\text{\\Large$\\wp$}(r) =\\frac{1}{r^2} \\int_{B_r} \\Big\\{{|u|^2} + 2P \\Big\\} (u\\cdot \\frac{x}{|x|}) \\, dx .\n\\end{equation}\ntakes nonpositive values as $r\\to 0$. Moreover, \nif $u$ is of the form $\\frac{\\zeta(\\frac x{|x|})}{|x|}$ then one can check that \n$\\text{\\Large$\\wp$}(r) =0$. This observation motivates the formulation of \na condition in our next result that allows to control $\\text{\\Large$\\wp$}(r)$. \n\\begin{theorem}\\label{TH}\nLet $u$ be a suitable weak solution of \\eqref{eq:problem}, $B_1\\subset \\Omega$, and\n\\[\nm:=\\liminf_{R\\to 0}M(R)<\\infty \\qquad \\text{where} \\quad M(R):=\\int_{B_R} \\Big( \\frac{|u|^2}{R^3} + \\frac{|\\nabla u|^2}{R} \\Big).\n\\] Let $\\mP_R[\\,\\cdot\\,]: \\mathcal W^{1,2}(B_R)\\to \\mH(R)$ be the projection operator for the space \\eqref{mH1}. There exists $\\ep(m)>0$ \nsuch that \nif \n\\begin{equation}\\label{eq:13-alt}\n\\frac{1}{R^3} \\int_{B_R} \\left| u - \\mP_R[u] \\right|^2 + \\frac{1}{R} \\int_{B_R} \\left| \\nabla u - \\nabla \\mP_R[u] \\right|^2 \\le \\ep(m) M(R)\n\\end{equation}\nholds\nfor all $R\\in (0, R_0)$ then $u$ is regular at $x=0$. \n\\end{theorem}\n\n\\begin{proposition}\\label{prop:dAdr}\n Suppose $N=5$ and $(u,P)$ is a suitable weak solution of \\eqref{st-NS-N}. For $r>0$, define\n \\begin{equation}\n \\left\\{\n \\begin{aligned}\n &D(r)\\vcentcolon= \\int_{B_r} \\Big\\{ \\frac{15}{4r^3}|u|^2 + \\frac{1}{4r}|\\nabla u |^2 + \\frac{3}{4r^3}\\left| \\nabla (|x|u) \\right|^2 + \\frac{3(r^2-|x|^2)}{4r^3} |\\nabla u|^2\\Big\\}\\, dx,\\\\\n &A(r) \\vcentcolon= \\frac{1}{r^3}\\int_{B_r}(x\\cdot \\nabla)\\frac{|u|^2}{2}\\, dx + \\frac{9}{4r^3}\\int_{ B_r}|u|^2 \\, dx - \\frac{1}{r^{2}} \\int_{B_r} \\big\\{ \\frac{|u|^2}{2} + P \\big\\} u\\cdot \\frac{x}{|x|} \\, dx .\n \\end{aligned}\n \\right. \n \\end{equation}\n Then the following differential equation holds for $r>0$,\n \\begin{equation*}\n \\frac{d A}{d r} \\ge \\frac{1}{r} D(r) + \\frac{2}{r^3} \\int_{B_r} \\Big\\{\\frac{|u|^2}{2} + P \\Big\\} (u\\cdot \\frac{x}{|x|}) \\, dx .\n \\end{equation*}\n\\end{proposition}\n\\begin{proof}\n Let us consider the function \n \\begin{equation}\n \\phi(x)=\n \\left\\{\n \\begin{array}{lll}\n 1 \\quad & \\mbox{if} \\ |x|r+\\epsilon.\\\\\n \\end{array}\n \\right.\n \\end{equation}\n We then mollify this function and take $\\psi=\\phi*\\rho_\\delta$, \n where $\\rho$ is the mollification kernel. \n Note that $\\psi\\in \\mathcal C_c^\\infty(\\R^N)$.\n We use $\\psi$ as a test function in the local \n energy inequality to obtain \n \\begin{equation}\n \\int |\\nabla u|^2 \\psi\n \\le \n \\int \\left(-u \\nabla u +(|u|^2+2P)u\\right)\\cdot \\nabla \\psi.\n \\end{equation}\n\n\\begin{lemma}\\label{lemma:reg}\nSuppose $m:=\\liminf_{R\\to 0}M(R)<\\infty$. \n If there exists a sufficiently small $\\ep>0$, depending on $m$, such that $(u,P)$ is a suitable weak solution to the Navier-Stokes equations satisfying\n \\begin{equation}\\label{H-Assump}\n \\ep \\frac{1}{R^{3}}\\int_{B_R} |p| + \\frac{1}{R^3} \\int_{B_R} \\left| u - \\mP_R[u] \\right|^2 + \\frac{1}{R} \\int_{B_R} \\left| \\nabla u - \\nabla \\mP_R[u] \\right|^2 \\le \\ep M[u](R), \n \\end{equation}\n for all $R\\in(0,1]$ then $u$ is regular at $x=0$.\n\\end{lemma}\n\\begin{proof}\nIf $m\\le 8C_E$ then we can apply Proposition \\ref{prop:fM}, and \nhence the result follows.\n Now suppose $m\\in(8C_E, \\infty)$. In light of Lemma \\ref{lemma:MBound}, there exists $\\ep_1>0$ such that if $(u,P)$ satisfies \\eqref{H-Assump} with $\\ep\\in (0,\\ep_1)$ then $M(R)$ is uniformly bounded in $R\\in(0,1]$ and we set \n \\begin{equation}\\label{Mast}\n M_{\\ast} \\vcentcolon= \\sup\\limits_{0 \\frac12, \n\\quad \\int_{B_1} |\\bar u_i|^3+|\\bar p_i|^{\\frac32}\\le 2. \n\\end{equation}\nFrom the local energy inequality $u\\in \\mathcal W^{1, 2}_{loc}(B_1).$\nMoreover, the following equation is satisfied in distributional sense\n\\begin{equation}\n\\Delta \\bar p_i=-\\epsilon_i \\frac{\\partial^2 (\\bar u^k\\bar u^l)}{\\partial x_l\\partial x_k}, \n\\quad \\mbox {in}\\ B_1.\n\\end{equation}\nFrom the Poisson representation theorem we can write $\\bar p_i=h_i+g_i$, \nwhere $h_i$ is harmonic in $B_1$, and \n\\begin{equation}\n\\left\\{\n\\begin{array}{lll}\n\\Delta g_i=-\\epsilon_i \\frac{\\partial^2 (\\bar u^k\\bar u^l)}{\\partial x_l\\partial x_k}\n\\quad \\mbox {in}\\ B_{\\frac23},\\\\ \ng_i=0 \\quad \\mbox {on}\\ \\partial B_{\\frac23}.\n\\end{array}\n\\right.\n\\end{equation}\nFrom the Calder\\'on-Zygmund estimates \n$g_i$ is uniformly bounded in ${L^{5/3}(B_{2/3})}$.\nConsequently, $h_i\\in L^{3/2}(B_{2/3})$ uniformly, hence \nfrom the local estimates for the harmonic functions \n\\begin{align}\n\\int_{B_\\theta}|\\bar p_i-[\\bar p_i]_\\theta|^{\\frac32}\\le \n\\int_{B_\\theta}| h_i-[h_i]_\\theta|^{\\frac32}\n\\int_{B_\\theta}|g_i-[g_i]_\\theta|^{\\frac32}\\\\\n\\le C_0 \\theta^5\\theta^{3/2}+C_0\\epsilon_i\\int_{B_{2/3}}|\\bar u_i|^3.\n\\end{align}\nFor a suitable subsequence $\\bar u_i\\to \\bar u$ in $\\mathcal W^{1, 2}(B_{2/3})$\nand $\\bar p_i\\to \\bar p$ strongly in ${L^{3/2}(B_{2/3})}$.\nConsequently, for sufficiently large $i$, we have \n\\begin{equation}\\label{eq:Lin2}\n\\int_{B_\\theta}|\\bar p_i-[\\bar p_i]_\\theta|^{\\frac32} \\le C_0 \\theta^5\\theta^{3/2}\n\\end{equation}\nSince the limit $\\bar u$ solves the Stokes system, then it follows that \n$\\bar u$ is H\\\"older continuous with, say, exponent $2\\alpha_0$, and therefore \n$\\int_{B_\\theta}|\\bar u-[\\bar u]_\\theta|^{\\frac32}\\le \\frac14 \\theta^5\\theta^{\\alpha_0}$.\nFrom the strong convergence $\\bar u_i\\to \\bar u$ in $L^3(B_{2/3})$, we infer that \n\\begin{equation}\\label{eq:Lin3}\n\\int_{B_\\theta}|\\bar u-[\\bar u]_\\theta|^{\\frac32}\\le \\frac13 \\theta^5\\theta^{\\alpha_0}.\n\\end{equation}\nCombining \\eqref{eq:Lin2} and \\eqref{eq:Lin2} we get a contradiction with the \nsecond inequality in \\eqref{eq:Lin4}. \n\\end{proof}", + "post_theorem_intro_text_len": 4785, + "post_theorem_intro_text": "The proof of Theorem \\ref{TH-00} uses the monotonicity \nformula introduced in Proposition \\ref{prop:dAdr}, and \na scaling argument. See Lemma \\ref{lem:positiveinf} for the proof.\nNote that there are no smallness assumptions in the \nstatement of Theorem \\ref{TH-00}. \n\nIf $\\liminf_{r\\to 0}M(r)<\\infty$\nthen the singularity may occur only if the function \n\\begin{equation}\n\\text{\\Large$\\wp$}(r) =\\frac{1}{r^2} \\int_{B_r} \\Big\\{{|u|^2} + 2P \\Big\\} (u\\cdot \\frac{x}{|x|}) \\, dx .\n\\end{equation}\ntakes nonpositive values as $r\\to 0$. Moreover, \nif $u$ is of the form $\\frac{\\zeta(\\frac x{|x|})}{|x|}$ then one can check that \n$\\text{\\Large$\\wp$}(r) =0$. This observation motivates the formulation of \na condition in our next result that allows to control $\\text{\\Large$\\wp$}(r)$. \n\\begin{theorem}\\label{TH}\nLet $u$ be a suitable weak solution of \\eqref{eq:problem}, $B_1\\subset \\Omega$, and\n\\[\nm:=\\liminf_{R\\to 0}M(R)<\\infty \\qquad \\text{where} \\quad M(R):=\\int_{B_R} \\Big( \\frac{|u|^2}{R^3} + \\frac{|\\nabla u|^2}{R} \\Big).\n\\] Let $\\mP_R[\\,\\cdot\\,]: \\mathcal W^{1,2}(B_R)\\to \\mathcal{H}(R)$ be the projection operator for the space \\eqref{mH1}. There exists $\\varepsilon(m)>0$ \nsuch that \nif \n\\begin{equation}\\label{eq:13-alt}\n\\frac{1}{R^3} \\int_{B_R} \\left| u - \\mP_R[u] \\right|^2 + \\frac{1}{R} \\int_{B_R} \\left| \\nabla u - \\nabla \\mP_R[u] \\right|^2 \\le \\varepsilon(m) M(R)\n\\end{equation}\nholds\nfor all $R\\in (0, R_0)$ then $u$ is regular at $x=0$. \n\\end{theorem}\n\nIt is known that if $u\\in \\mathcal{H}$ then $u=0$ \\cite{Sverak}. In this context, Theorem \\ref{TH} \nstates that if $\\|u-\\mP_R[u]\\|_{\\mathcal{W}^{1,2}(R)}$ is small compared to $\\|u\\|_{\\mathcal{W}^{1,2}(R)}$ then \n$\\text{\\Large$\\wp$}(r)$ is smaller than $M^{\\frac32}(r)$, which \nafter application of Proposition \\ref{prop:dAdr} implies that \n$u=0$.\n\nAs opposed to the main result in \\cite{CKN}, we do not assume that $u$ is small in some scale invariant seminorm, reminiscent to \nthe ``local\" Reynolds number $\\frac1{r}\\int_{B_r}|\\nabla u|^2$. This leads us to the classification of the \nself-similar solution of the incompressible Euler equations in $\\mathbb{R}^5$. \nIn fact, we prove that for such solutions \nthe Bernoulli pressure is zero. This is the first key point in our proof of the main technical result, Proposition \\ref{prop:cubicEst}.\n\nThe second key point is the construction of a monotonicity formula for the suitable weak solutions, which follows from \nthe weak energy inequality. \n\nWe compare Theorem \\ref{TH} with the well-known \nregularity criteria for suitable weak solutions of \n\\eqref{eq:problem-3d}, which in its most general form, can be stated as follows:\nlet $Q(R)=B_R\\times(-R^2, 0)$ and define the local Reynolds numbers\n\\[\nE(R)=\\frac1R\\int_{Q(R)}|\\nabla u|^2, \\quad C(R)=\\frac1{R^2}\\int_{Q(R)}|u|^3.\n\\]\n\nThen the following statement holds: for every $M>0$ there is \n$\\varepsilon(M)>0$ such that \n$\\limsup_{R\\to 0} C(R)0$, define\n\t\\begin{equation}\n\t\t\\left\\{\n\t\t\\begin{aligned}\n\t\t\t&D(r)\\vcentcolon= \\int_{B_r} \\Big\\{ \\frac{15}{4r^3}|u|^2 + \\frac{1}{4r}|\\nabla u |^2 + \\frac{3}{4r^3}\\left| \\nabla (|x|u) \\right|^2 + \\frac{3(r^2-|x|^2)}{4r^3} |\\nabla u|^2\\Big\\}\\, dx,\\\\\n\t\t\t&A(r) \\vcentcolon= \\frac{1}{r^3}\\int_{B_r}(x\\cdot \\nabla)\\frac{|u|^2}{2}\\, dx + \\frac{9}{4r^3}\\int_{ B_r}|u|^2 \\, dx - \\frac{1}{r^{2}} \\int_{B_r} \\big\\{ \\frac{|u|^2}{2} + P \\big\\} u\\cdot \\frac{x}{|x|} \\, dx .\n\t\t\\end{aligned}\n\t\t\\right. \n\t\\end{equation}\n\tThen the following differential equation holds for $r>0$,\n\t\\begin{equation*}\n\t\t\\frac{d A}{d r} \\ge \\frac{1}{r} D(r) + \\frac{2}{r^3} \\int_{B_r} \\Big\\{\\frac{|u|^2}{2} + P \\Big\\} (u\\cdot \\frac{x}{|x|}) \\, dx .\n\t\\end{equation*}\n\\end{proposition}\n\nIt also uses a scaling argument. For the proof, see the following lemma.\n\n\\begin{lemma}\\label{lem:positiveinf}\nLet \n\\begin{equation}\nQ(r):= \\int_{B_r} \\Big\\{ \\frac{15}{4r^3}|u|^2 \n+ \n\\frac{1}{4r}|\\nabla u |^2 \n+ \n\\frac{3(r^2-|x|^2)}{4r^3} |\\nabla u|^2\\Big\\}\\, dx,\n\\end{equation}\nand \n\\begin{equation}\n\\text{\\Large$\\wp$}(r) =\\frac{1}{r^2} \\int_{B_r} \\Big\\{{|u|^2} + 2P \\Big\\} (u\\cdot \\frac{x}{|x|}) \\, dx .\n\\end{equation}\nIf $\\liminf_{r\\to 0} M(r)<\\infty$ and \n\\[\n\\liminf_{r\\to0^+}\\left[ Q(r)\n+\\text{\\Large$\\wp$}(r)\n\\right]> 0\n\\]\nthen $x=0$ is a regular point.\n\\end{lemma}”,", + "expanded_theorem": "\\label{TH-00}\nLet $u$ be a suitable weak solutions. Suppose that \nthe following two conditions hold:\n\\begin{equation}\n\\liminf_{r\\to 0}M(r):=\\liminf_{r\\to 0}\\int_{B_r}\\!\\! \\left( \\frac{|u|^2}{r^3} + \\frac{|\\nabla u|^2}{r} \\right) <\\infty, \\quad \\liminf_{r\\to 0}\\frac1{r^2}\\int_{B_r}\\!\\!(|u|^2+2P)u\\cdot\\frac x{|x|}>0.\n\\end{equation}\nTo prove the main theorem, we show that for any $\\{r_k\\}_{k=1}^\\infty, r_k\\downarrow 0$ there is a subsequence $r_{k_m}$ such that \nthe scaled solutions $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$, converge to \na homogenous vector field of degree negative one, and hence \n$x=0$ is a regular point.,", + "theorem_type": [ + "Implication", + "Universal–Existential" + ], + "mcq": { + "question": "Let \b$(u,P)$ be a suitable weak solution near $x=0$ of the stationary incompressible Navier\u0013Stokes system in five dimensions,\n\\[\nu^j\\partial_j u^i + \\partial_i P = \\Delta u^i, \\qquad \\operatorname{div}u=0,\n\\]\nin a domain $\\Omega\\subset\\mathbb{R}^5$. For $r>0$, write\n\\[\nM(r):=\\int_{B_r}\\left(\\frac{|u|^2}{r^3}+\\frac{|\\nabla u|^2}{r}\\right),\n\\]\nwhere $B_r$ is the ball of radius $r$ centered at $0$. Assume that\n\\[\n\\liminf_{r\\to 0} M(r)<\\infty\n\\]\nand also\n\\[\n\\liminf_{r\\to 0}\\frac1{r^2}\\int_{B_r} (|u|^2+2P)\\,u\\cdot \\frac{x}{|x|}>0.\n\\]\nFor a scale $r>0$, define the rescaled velocity field by\n\\[\nu_r(x)=r\\,u(rx).\n\\]\nWhich conclusion holds under these hypotheses?", + "correct_choice": { + "label": "A", + "text": "For every sequence $\\{r_k\\}_{k=1}^\\infty$ with $r_k\\downarrow 0$, there exists a subsequence $r_{k_m}$ such that the scaled fields $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$ converge to a vector field that is homogeneous of degree $-1$; consequently, $x=0$ is a regular point." + }, + "choices": [ + { + "label": "B", + "text": "There exists a sequence $r_k\\downarrow 0$ such that the scaled fields $u_{r_k}(x)=r_k u(r_kx)$ converge to a vector field that is homogeneous of degree $-1$; consequently, $x=0$ is a regular point." + }, + { + "label": "C", + "text": "There exists a sequence $r_k\\downarrow 0$ such that, after passing to a subsequence, the scaled fields $u_{r_k}(x)=r_k u(r_kx)$ converge to a vector field that is homogeneous of degree $-1$." + }, + { + "label": "D", + "text": "For every sequence $\\{r_k\\}_{k=1}^\\infty$ with $r_k\\downarrow 0$, the full sequence of scaled fields $u_{r_k}(x)=r_k u(r_kx)$ converges to a unique vector field that is homogeneous of degree $-1$; consequently, $x=0$ is a regular point." + }, + { + "label": "E", + "text": "For every sequence $\\{r_k\\}_{k=1}^\\infty$ with $r_k\\downarrow 0$, there exists a subsequence $r_{k_m}$ such that the scaled fields $u_{r_{k_m}}(x)=r_{k_m}u(r_{k_m}x)$ converge to a vector field that is homogeneous of degree $-1$; moreover, this implies that $x=0$ is a singular point." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "scaling", + "tampered_component": "quantifier order over blow-up sequences", + "template_used": "quantifier_dependence" + }, + { + "label": "C", + "sketch_hook_type": "scaling", + "tampered_component": "dropped regularity conclusion and universal quantifier over sequences", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "monotonicity", + "tampered_component": "subsequence compactness replaced by full-sequence unique limit", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "monotonicity", + "tampered_component": "sign of the monotonicity-based regularity consequence", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem states hypotheses and asks for the resulting conclusion, but it does not explicitly reveal the correct option. The correct answer must be distinguished from nearby variants involving different quantifiers, convergence strength, and regularity consequences." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question: the hypotheses are presented and the conclusion is selected. However, it is not a pure tautology because the choices differ in meaningful ways (every sequence vs. some sequence, subsequence vs. full sequence, regular vs. singular point)." + }, + "GPS": { + "score": 1, + "justification": "Moderate reasoning is required. A solver must track subtle logical structure and identify the strongest justified conclusion, but the problem mainly tests precise theorem comprehension rather than generating a new mathematical argument." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong and plausible: they reflect common mathematical errors involving quantifier order, unjustified uniqueness/full-sequence convergence, omission of the regularity consequence, and reversal of the conclusion about regularity." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with no answer leakage and high-quality distractors. Its main weakness is that it functions largely as a theorem-conclusion recognition item, so it tests precision with logical details more than deep generative reasoning." + } + }, + { + "id": "2511.02138v1", + "paper_link": "http://arxiv.org/abs/2511.02138v1", + "theorems_cnt": 7, + "theorem": { + "env_name": "theorem", + "content": "\\label{th:exp_growth}\n\tLet $\\Gamma$ be a countable finitely presented group.\n\tIf $\\Delta_1$ has a spectral gap on \t$\\ell_{0,c}^2(E)$ \tthen either $\\Gamma$ has exponential growth, \t or $\\Gamma$ is virtually infinite cyclic.", + "start_pos": 14022, + "end_pos": 14275, + "label": "th:exp_growth" + }, + "ref_dict": { + "lemma:embed": "\\begin{lemma}[on loop embedding]\n\\label{lemma:embed}\nSuppose that $\\Gamma$ has subexponential growth and just one end.\n\nLet $C>20$, $x\\ge 1$. There exist $\\mathscr L>2DCx$ and injective naturally parametrized $\\gamma\\colon \\mathbb T_{\\mathscr L}\\to \\Cay(\\Gamma)$ such that, for $t_1, t_2\\in\\mathbb T_{\\mathscr L}$ with $\\gamma(t_1), \\gamma(t_2)\\in \\Gamma$, \n\\emph{\n\\begin{equation}\n\t\\label{eq:x_bilip}\n\\mbox{if }\\dist_{\\Cay(\\Gamma)}(\\gamma(t_1), \\gamma(t_2)) \\le x \n\\mbox{ then }\n\\dist_{\\mathbb T_{\\mathscr L}}(t_1, t_2) \\le Cx.\n\\end{equation}\n}\n\\end{lemma}", + "eq:orth_decomp": "\\begin{equation}\n\t\t\\label{eq:orth_decomp}\n\t\t\\{f\\in\\ell^2(E)\\colon \\partial f=0\\}= \\ell_{0,c}^2(E)\\oplus_{\\ell^2(E)} \\{du\\mid u\\colon G\\to \\R, \\, \\Delta_0 u = 0, \\, du\\in \\ell^2(E)\\}.\n\t\\end{equation}", + "th:Kesten": "\\begin{theorem}[\\cite{K59}]\t\\label{th:Kesten}\n\tLet $\\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\\Cay(\\Gamma)$, $$0\\in \\spec\\left(-\\Delta_0|_{\\ell^2(\\Gamma)}\\right)$$ if and only if $\\Gamma$ is amenable.\n\\end{theorem}", + "eq:isoperimetr": "\\begin{equation}\n\\label{eq:isoperimetr}\n\\|du\\|_{\\ell^1(E)} \\gtrsim \\|u\\|_{\\ell^1(\\Gamma)}\n\\end{equation}", + "th:exp_growth": "\\begin{theorem}\n\t\\label{th:exp_growth}\n\tLet $\\Gamma$ be a countable finitely presented group.\n\tIf $\\Delta_1$ has a spectral gap on \t$\\ell_{0,c}^2(E)$ \tthen either $\\Gamma$ has exponential growth, \t or $\\Gamma$ is virtually infinite cyclic. \\end{theorem}", + "predl:general_invariance": "\\begin{predl}\n \t\\label{predl:general_invariance} \n \tLet $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\\subset E_2$. \tAssume the following:\n \t\\begin{enumerate}\n \t\t\\item $G_1$ is connected;\n\n \t\t\\item degrees of vertices in $G_2$ are bounded from the above;\n\n \t\t\\item $\\sup\\limits_{e\\in E_2\\setminus E_1} \\dist_{G_1}(\\beg e, \\ennd e) < +\\infty$ with the obvious notation. \t\t\n \t\\end{enumerate}\n\t\\emph{(}In other words, metrics $\\dist_{G_1}$ and $\\dist_{G_2}$ on $V$ are bilipshitz equivalent.\\emph{)} Then, $G_1$ has a coexact $1$-Laplacian spectral gap \\emph{(}with some $D$ implied at the construction of $G_1^{(2)}$\\emph{)}\n\tif and only if $G_2$ has such a spectral gap \\emph{(}with some $D'$ for $G_2^{(2)}$\\emph{)}.\n \\end{predl}" + }, + "pre_theorem_intro_text_len": 7255, + "pre_theorem_intro_text": "\\label{sec:Intro}\n\nLet $G=(V,E)$ be countable oriented graph with degrees of vertices bounded from the above. Let $\\tilde G$ be the non-oriented graph obtained from $G$ by forgetting the orientation of edges. Pick $D\\in\\mathbb N$ large enough. Consider all cycles in $\\tilde G$ having lengths $\\le D$. In $G$, glue each such cycle with a polygon. Choose any orientation of the latter polygons. We arrive to oriented $2$-dimensional complex, denote it by $G^{(2)}$ with implicit dependence on~$D$. Denote by $F$ the set of $2$-dimensional faces in $G^{(2)}$ which are polygons. Sometimes we write $F=FG$ and also $E=EG$ to indicate the dependence of these sets on $G$. Any of sets $V, E, F$ is endowed with counting measure which we denote by $\\card$. In graph $G$, we define graph metric $\\dist_G$ at $V\\cup E$ along edges in $E$ so that any edge has length $1$.\n\nIf $\\Gamma$ is a finitely generated group, $S$ is any of its generating sets (symmetrized or not) then we may consider $G=\\Cay(\\Gamma,S)$, Cayley graph of $\\Gamma$; then $V=\\Gamma$. If $\\Gamma$ is also finitely presented, that is, given by a finite number of relations then we assume that $D$ in the definition of $G^{(2)}=\\Cay^{(2)}(\\Gamma,S)$ is $\\ge$ than length of any of the defining relations. For general $G$, we assume that $D$ is such that \n\\begin{equation*}\\mbox{$G^{(2)}$ is simply connected}\n\\end{equation*}\n(and that such $D$ does exist).\n\nA \\emph{$k$-cochain}, $k$ is $0$, or $1$, or $2$, is a function from $V$, or $E$, or $F$, respectively, to ${\\mathbb R}$. We often understand cochains as chains.\nDiscrete differentials (coboundaries)\n$$\n\\{0\\mbox{-cochains}\\}\\xrightarrow{d}\\{1\\mbox{-cochains}\\}\\xrightarrow{d}\\{2\\mbox{-cochains}\\}\n$$\nand boundary operators \n$$\n\\{2\\mbox{-cochains}\\}\\xrightarrow{\\partial}\\{1\\mbox{-cochains}\\}\\xrightarrow{\\partial}\\{0\\mbox{-cochains}\\}\n$$\nare introduced in the standard way with respect to the orientation of edges and faces. Since valencies of vertices are bounded, all these operators are also bounded with respect to $\\ell^2$-norms on cochains. We have $(d|_{\\ell^2(V)})^*=\\partial|_{\\ell^2(E)}$, $(d|_{\\ell^2(E)})^*=\\partial|_{\\ell^2(F)}$. Indeed, discrete integration by parts is valid for finitely supported cochains and is proved for $\\ell^2$-cochains by $\\ell^2$-approximation. \nIf $\\gamma$ is an oriented path in $G$ then we may define $1$-(co)chain $f_\\gamma$: for $e\\in E$, let $f_\\gamma(e)$ be the number of passes of $\\gamma$ through $e$ in its direction minus number of passes of $\\gamma$ over $e$ in its reversed direction. Then we have $\\partial f_\\gamma=0$.\n\nOur space of interest is\n\\begin{equation*}\n\t\\ell_{0,c}^2(E) := \\clos_{\\ell^2(E)}\\{f\\colon E\\to {\\mathbb R}\\mid \\partial f=0, \\, \\supp f\n\t\\,\\mbox{is finite}\\}.\\end{equation*}\nAny of $1$-cochains at the right-hand side can be (convexly) decomposed into simple finite loops. Thus, $\\ell_{0,c}^2(E)$ is $\\ell^2$-closed linear span of (co)chains of the form $f_\\gamma$ with $\\gamma$ a finite loop in $G$.\n\nLaplace operator on $0$-cochains is \n$$\n-\\Delta_0=\\partial d\\colon \\left(\\mbox{functions on }V\\right)\\to \\left(\\mbox{functions on }V\\right).\n$$\nA discrete integration by parts leads to the following Hodge-type decomposition: \n\n\\begin{predl}\n\n\tWe have \n\t\\begin{equation}\n\t\t\\label{eq:orth_decomp}\n\t\t\\{f\\in\\ell^2(E)\\colon \\partial f=0\\}= \\ell_{0,c}^2(E)\\oplus_{\\ell^2(E)} \\{du\\mid u\\colon G\\to {\\mathbb R}, \\, \\Delta_0 u = 0, \\, du\\in \\ell^2(E)\\}.\n\t\\end{equation}\n\\end{predl}\n\nThe second summand in the right-hand side of the latter relation is \\emph{$\\ell^2$-cohomology} of $G$. It is known to be invariant with respect to change of generating system in a group: the factorspace nature of cohomology allows to implement \"discrete change of variables\" from one to another set of generators. Cohomology is invariant with respect to more general quasiisometries. \nNow, we pass to spectral estimates for $1$-cochains. Define non-negative Laplacian operator $\\Delta_1:=\\partial d+d\\partial \\colon \\ell^2(E)\\to \\ell^2(E)$. On $\\ell^2_{0,c}(E)$, our space of interest, this reduces to $\\partial d$. \n\nWe have one more Hodge-type decomposition:\n$$\n\\ell^2(E) = \\clos_{\\ell^2(E)}\\{du\\mid u \\colon V\\to {\\mathbb R}, \\, \\supp u \\mbox{ is finite}\\} \\oplus_{\\ell^2(E)} \\{f\\in\\ell^2(E)\\colon \\partial f=0\\}.$$\nSpectral questions for $\\Delta_1$ on the first summand are generally reduced to the same for $\\Delta_0$ on $\\ell^2(V)$. What concerns decomposition (\\ref{eq:orth_decomp}) for $\\{f\\in\\ell^2(E)\\colon \\partial f=0\\}$, operator $\\Delta_1$ vanishes at the second summand of its right-hand side, $\\ell^2$-cohomology. Also, by the definition of $\\ell^2_{0,c}(E)$ and by $\\ell^2$-approximation, we see that $\\Delta_1(\\ell^2_{0,c}(E))\\subset\\ell^2_{0,c}(E)$.\n\n\\begin{define}\n\tWe say that $\\Delta_1$ \\emph{has a spectral gap at $\\ell^2_{0,c}(E)$} \\emph{(}or just that graph $G$ has \\emph{{coexact $1$-Laplacian spectral gap}}\\emph{)} if $$\\spec\\left(\\Delta_1|_{\\ell^2_{0,c}(E)}\\right)\\cap [0,\\varepsilon)=\\varnothing$$ for some $\\varepsilon>0$ small enough.\n\\end{define}\n\nApplying discrete integration by parts, we conclude that this is equivalent to the estimate \n\\begin{equation}\n\t\\label{eq:rot_estim}\n\\langle f, f \\rangle_{\\ell^2(E)}\t\\le 1/\\varepsilon\\cdot \\langle df, df\\rangle_{\\ell^2(F)}\t\n\\end{equation}\nfor $f\\in \\ell^2_{0,c}(E)$. It is enough to check the latter only for finitely supported closed $1$-cochains $f$. Also, we conclude that if $1$-Laplacian has a coexact spectral gap then it will be so if we enlarge $D$ in the construction of $G^{(2)}$ or glue some extra faces to $G^{(2)}$ in a locally finite manner. \n We state quasiinvariance result as below, with possibility to add only edges. It seems feasible to preserve spectral gap under more general quasiisometric transformations of a graph, the ones with possibility to add or remove vertices in a locally finite way. \n \\begin{predl}\n \t\\label{predl:general_invariance} \n \tLet $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\\subset E_2$. \tAssume the following:\n \t\\begin{enumerate}\n \t\t\\item $G_1$ is connected;\n\n \t\t\\item degrees of vertices in $G_2$ are bounded from the above;\n\n \t\t\\item $\\sup\\limits_{e\\in E_2\\setminus E_1} \\dist_{G_1}(\\beg e, \\ennd e) < +\\infty$ with the obvious notation. \t\t\n \t\\end{enumerate}\n\t\\emph{(}In other words, metrics $\\dist_{G_1}$ and $\\dist_{G_2}$ on $V$ are bilipshitz equivalent.\\emph{)} Then, $G_1$ has a coexact $1$-Laplacian spectral gap \\emph{(}with some $D$ implied at the construction of $G_1^{(2)}$\\emph{)}\n\tif and only if $G_2$ has such a spectral gap \\emph{(}with some $D'$ for $G_2^{(2)}$\\emph{)}.\n \\end{predl}\n\n \\noindent Proof of this quasiinvariance is similar to the proof of quasiiinvariance of $\\ell^2$-cohomology; both are based on orthogonal projection. We give a detailed argument in the Appendix. Notice also that the proof is constructive: we may estimate $D'$ via $D$ and the supremum from the third assumption of Proposition \\ref{predl:general_invariance}, and vice versa.\n\n \\begin{sled}\n \tFor two Cayley graphs of the same finitely-presented group but with different generating sets, properties of existence of a coexact $1$-Laplacian spectral gap on them are equivalent.\n \\end{sled}\n\nOur main result is", + "context": "If $\\Gamma$ is a finitely generated group, $S$ is any of its generating sets (symmetrized or not) then we may consider $G=\\Cay(\\Gamma,S)$, Cayley graph of $\\Gamma$; then $V=\\Gamma$. If $\\Gamma$ is also finitely presented, that is, given by a finite number of relations then we assume that $D$ in the definition of $G^{(2)}=\\Cay^{(2)}(\\Gamma,S)$ is $\\ge$ than length of any of the defining relations. For general $G$, we assume that $D$ is such that \n\\begin{equation*}\\mbox{$G^{(2)}$ is simply connected}\n\\end{equation*}\n(and that such $D$ does exist).\n\nA \\emph{$k$-cochain}, $k$ is $0$, or $1$, or $2$, is a function from $V$, or $E$, or $F$, respectively, to ${\\mathbb R}$. We often understand cochains as chains.\nDiscrete differentials (coboundaries)\n$$\n\\{0\\mbox{-cochains}\\}\\xrightarrow{d}\\{1\\mbox{-cochains}\\}\\xrightarrow{d}\\{2\\mbox{-cochains}\\}\n$$\nand boundary operators \n$$\n\\{2\\mbox{-cochains}\\}\\xrightarrow{\\partial}\\{1\\mbox{-cochains}\\}\\xrightarrow{\\partial}\\{0\\mbox{-cochains}\\}\n$$\nare introduced in the standard way with respect to the orientation of edges and faces. Since valencies of vertices are bounded, all these operators are also bounded with respect to $\\ell^2$-norms on cochains. We have $(d|_{\\ell^2(V)})^*=\\partial|_{\\ell^2(E)}$, $(d|_{\\ell^2(E)})^*=\\partial|_{\\ell^2(F)}$. Indeed, discrete integration by parts is valid for finitely supported cochains and is proved for $\\ell^2$-cochains by $\\ell^2$-approximation. \nIf $\\gamma$ is an oriented path in $G$ then we may define $1$-(co)chain $f_\\gamma$: for $e\\in E$, let $f_\\gamma(e)$ be the number of passes of $\\gamma$ through $e$ in its direction minus number of passes of $\\gamma$ over $e$ in its reversed direction. Then we have $\\partial f_\\gamma=0$.\n\n\\begin{define}\n We say that $\\Delta_1$ \\emph{has a spectral gap at $\\ell^2_{0,c}(E)$} \\emph{(}or just that graph $G$ has \\emph{{coexact $1$-Laplacian spectral gap}}\\emph{)} if $$\\spec\\left(\\Delta_1|_{\\ell^2_{0,c}(E)}\\right)\\cap [0,\\varepsilon)=\\varnothing$$ for some $\\varepsilon>0$ small enough.\n\\end{define}\n\nApplying discrete integration by parts, we conclude that this is equivalent to the estimate \n\\begin{equation}\n \\label{eq:rot_estim}\n\\langle f, f \\rangle_{\\ell^2(E)} \\le 1/\\varepsilon\\cdot \\langle df, df\\rangle_{\\ell^2(F)} \n\\end{equation}\nfor $f\\in \\ell^2_{0,c}(E)$. It is enough to check the latter only for finitely supported closed $1$-cochains $f$. Also, we conclude that if $1$-Laplacian has a coexact spectral gap then it will be so if we enlarge $D$ in the construction of $G^{(2)}$ or glue some extra faces to $G^{(2)}$ in a locally finite manner. \n We state quasiinvariance result as below, with possibility to add only edges. It seems feasible to preserve spectral gap under more general quasiisometric transformations of a graph, the ones with possibility to add or remove vertices in a locally finite way. \n \\begin{predl}\n \\label{predl:general_invariance} \n Let $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\\subset E_2$. Assume the following:\n \\begin{enumerate}\n \\item $G_1$ is connected;\n\n\\begin{sled}\n For two Cayley graphs of the same finitely-presented group but with different generating sets, properties of existence of a coexact $1$-Laplacian spectral gap on them are equivalent.\n \\end{sled}\n\nOur main result is", + "full_context": "If $\\Gamma$ is a finitely generated group, $S$ is any of its generating sets (symmetrized or not) then we may consider $G=\\Cay(\\Gamma,S)$, Cayley graph of $\\Gamma$; then $V=\\Gamma$. If $\\Gamma$ is also finitely presented, that is, given by a finite number of relations then we assume that $D$ in the definition of $G^{(2)}=\\Cay^{(2)}(\\Gamma,S)$ is $\\ge$ than length of any of the defining relations. For general $G$, we assume that $D$ is such that \n\\begin{equation*}\\mbox{$G^{(2)}$ is simply connected}\n\\end{equation*}\n(and that such $D$ does exist).\n\nA \\emph{$k$-cochain}, $k$ is $0$, or $1$, or $2$, is a function from $V$, or $E$, or $F$, respectively, to ${\\mathbb R}$. We often understand cochains as chains.\nDiscrete differentials (coboundaries)\n$$\n\\{0\\mbox{-cochains}\\}\\xrightarrow{d}\\{1\\mbox{-cochains}\\}\\xrightarrow{d}\\{2\\mbox{-cochains}\\}\n$$\nand boundary operators \n$$\n\\{2\\mbox{-cochains}\\}\\xrightarrow{\\partial}\\{1\\mbox{-cochains}\\}\\xrightarrow{\\partial}\\{0\\mbox{-cochains}\\}\n$$\nare introduced in the standard way with respect to the orientation of edges and faces. Since valencies of vertices are bounded, all these operators are also bounded with respect to $\\ell^2$-norms on cochains. We have $(d|_{\\ell^2(V)})^*=\\partial|_{\\ell^2(E)}$, $(d|_{\\ell^2(E)})^*=\\partial|_{\\ell^2(F)}$. Indeed, discrete integration by parts is valid for finitely supported cochains and is proved for $\\ell^2$-cochains by $\\ell^2$-approximation. \nIf $\\gamma$ is an oriented path in $G$ then we may define $1$-(co)chain $f_\\gamma$: for $e\\in E$, let $f_\\gamma(e)$ be the number of passes of $\\gamma$ through $e$ in its direction minus number of passes of $\\gamma$ over $e$ in its reversed direction. Then we have $\\partial f_\\gamma=0$.\n\n\\begin{define}\n We say that $\\Delta_1$ \\emph{has a spectral gap at $\\ell^2_{0,c}(E)$} \\emph{(}or just that graph $G$ has \\emph{{coexact $1$-Laplacian spectral gap}}\\emph{)} if $$\\spec\\left(\\Delta_1|_{\\ell^2_{0,c}(E)}\\right)\\cap [0,\\varepsilon)=\\varnothing$$ for some $\\varepsilon>0$ small enough.\n\\end{define}\n\nApplying discrete integration by parts, we conclude that this is equivalent to the estimate \n\\begin{equation}\n \\label{eq:rot_estim}\n\\langle f, f \\rangle_{\\ell^2(E)} \\le 1/\\varepsilon\\cdot \\langle df, df\\rangle_{\\ell^2(F)} \n\\end{equation}\nfor $f\\in \\ell^2_{0,c}(E)$. It is enough to check the latter only for finitely supported closed $1$-cochains $f$. Also, we conclude that if $1$-Laplacian has a coexact spectral gap then it will be so if we enlarge $D$ in the construction of $G^{(2)}$ or glue some extra faces to $G^{(2)}$ in a locally finite manner. \n We state quasiinvariance result as below, with possibility to add only edges. It seems feasible to preserve spectral gap under more general quasiisometric transformations of a graph, the ones with possibility to add or remove vertices in a locally finite way. \n \\begin{predl}\n \\label{predl:general_invariance} \n Let $G_1=(V,E_1)$ and $G_2=(V,E_2)$ be graphs with the same vertex set $V$ and with $E_1\\subset E_2$. Assume the following:\n \\begin{enumerate}\n \\item $G_1$ is connected;\n\n\\begin{sled}\n For two Cayley graphs of the same finitely-presented group but with different generating sets, properties of existence of a coexact $1$-Laplacian spectral gap on them are equivalent.\n \\end{sled}\n\nOur main result is\n\n\\renewcommand{\\abstractname}{} \n\\begin{abstract}\nLet $\\Gamma$ be a discrete finitely presented group. Pick any system $S$ of generators in $\\Gamma$. In Cayley graph $\\Cay(\\Gamma)=\\Cay(\\Gamma, S)$ with edge set~$E$, glue with oriented polygons all the group relations translated to all the points of $\\Gamma$; denote the obtained simply connected complex by $\\Cay^{(2)}(\\Gamma)$. \nWe study non-negative \\emph{Hodge--Laplace operator $\\Delta_1$} on edge functions which is defined via complex $\\Cay^{(2)}(\\Gamma)$; $\\Delta_1$ acts on\n$$\n\\ell^2_{0,c}(E):= \\clos_{\\ell^2(E)}\\left\\{\\mbox{finitely supported closed $1$-(co)chains in }\\Cay^{}(\\Gamma)\\right\\}.\n$$\n\nWe prove the following implication in the spirit of Kesten Theorem: \\emph{if $\\Delta_1|_{\\ell_{0,c}^2(E)}$ has a spectral gap then $\\Gamma$ either has exponential growth or is virtually $\\mathbb Z$}. \\end{abstract}\n\nOur main result is\n\nLet us recall the well-known Kesten Theorem on Laplacian on vertices of a graph:\n\n\\begin{theorem}[\\cite{K59}] \\label{th:Kesten}\n Let $\\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\\Cay(\\Gamma)$, $$0\\in \\spec\\left(-\\Delta_0|_{\\ell^2(\\Gamma)}\\right)$$ if and only if $\\Gamma$ is amenable.\n\\end{theorem}\n\nIf we assume the contrary to Theorem \\ref{th:exp_growth}, then, first, $\\Gamma$ cannot have two \\emph{ends} since in this case $\\Gamma$ is virtually cyclic, see, e.g., \\cite{Me08}. Second, $\\Gamma$ cannot have infinitely many ends because then $\\Gamma$ has exponential growth; the latter follows from Stallings Theorem and from results of \\cite{HB00} but, of course, can be proved directly. So, by Freudenthal--Hopf Theorem we may assume that $\\Gamma$ has one end.\n\nNow, let us briefly recall the proof of Kesten Theorem \\ref{th:Kesten} to compare it to our argument. Non-amenability of $\\Gamma$ means that\n\\begin{equation}\n\\label{eq:isoperimetr}\n\\|du\\|_{\\ell^1(E)} \\gtrsim \\|u\\|_{\\ell^1(\\Gamma)}\n\\end{equation}\nfor $u=\\mathds 1_E$, $E$ ranges all finite subsets in $\\Gamma$. By discrete version of coarea formula, this is equivalent to the same for any finitely supported $u\\colon \\Gamma\\to\\R$. \nThe spectral gap condition $0\\notin \\spec\\left(-\\Delta_0|_{\\ell^2(\\Gamma)}\\right)$ means that \n$$\n\\langle u, u\\rangle_{\\ell^2(\\Gamma)} \\lesssim \\langle du, du\\rangle_{\\ell^2(E)}, ~~~ u\\colon\\Gamma\\to\\R \\mbox{ is finitely supported}.\n$$\nTo obtain this from (\\ref{eq:isoperimetr}), it remains to insert $u^2$ instead of $u$ to (\\ref{eq:isoperimetr}) and apply Cauchy--Bunyakovsky--Schwartz inequality.\n\nThis paper is organized as follows. In Section \\ref{sec:maueutic}, we explain our interest to the study of $1$-Laplacian spectra. This Section is not used in the proof of Theorem \\ref{th:exp_growth}. In Section~\\ref{sec:loop}, we prove Lemma \\ref{lemma:embed} by dropping lots of geodesic perpendiculars in a branching way. In Section \\ref{sec:spectra_homology}, we conclude the proof of Theorem~\\ref{th:exp_growth}. This is done by approximating the resolvent $\\Delta^{-1}_1$ by polynomials of $\\Delta_1$ provided that $0\\notin\\spec\\Delta_1|_{\\ell_{0,c}^2(E)}$. Next, we apply this to $1$-cochain given by curve $\\gamma$ from Lemma \\ref{lemma:embed}. We put some metric control on the approximating process as implemented in $\\Cay(\\Gamma)$ and also make use of homological nature of $\\Delta_1$: this operator is divisible by $\\partial\\colon \\ell^2(F)\\to\\ell^2(E)$ at $\\ell_{0,c}^2(E)$. Finally, in Section \\ref{sec:examples} we check the most natural examples of Cayley graphs.\n\n\\begin{lemma}[on loop embedding]\n\\label{lemma:embed}\nSuppose that $\\Gamma$ has subexponential growth and just one end.\n\nLet $C>20$, $x\\ge 1$. There exist $\\mathscr L>2DCx$ and injective naturally parametrized $\\gamma\\colon \\mathbb T_{\\mathscr L}\\to \\Cay(\\Gamma)$ such that, for $t_1, t_2\\in\\mathbb T_{\\mathscr L}$ with $\\gamma(t_1), \\gamma(t_2)\\in \\Gamma$, \n\\emph{\n\\begin{equation}\n\t\\label{eq:x_bilip}\n\\mbox{if }\\dist_{\\Cay(\\Gamma)}(\\gamma(t_1), \\gamma(t_2)) \\le x \n\\mbox{ then }\n\\dist_{\\mathbb T_{\\mathscr L}}(t_1, t_2) \\le Cx.\n\\end{equation}\n}\n\\end{lemma}\n\n\\begin{theorem}[\\cite{K59}]\t\\label{th:Kesten}\n\tLet $\\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\\Cay(\\Gamma)$, $$0\\in \\spec\\left(-\\Delta_0|_{\\ell^2(\\Gamma)}\\right)$$ if and only if $\\Gamma$ is amenable.\n\\end{theorem}\n\n\\begin{theorem}\n\t\\label{th:exp_growth}\n\tLet $\\Gamma$ be a countable finitely presented group.\n\tIf $\\Delta_1$ has a spectral gap on \t$\\ell_{0,c}^2(E)$ \tthen either $\\Gamma$ has exponential growth, \t or $\\Gamma$ is virtually infinite cyclic. \\end{theorem}", + "post_theorem_intro_text_len": 6791, + "post_theorem_intro_text": "Let us recall the well-known Kesten Theorem on Laplacian on vertices of a graph:\n\n\\begin{theorem}[\\cite{K59}]\t\\label{th:Kesten}\n\tLet $\\Gamma$ be a finitely generated group. Then, for $0$-Laplacian in $\\Cay(\\Gamma)$, $$0\\in \\spec\\left(-\\Delta_0|_{\\ell^2(\\Gamma)}\\right)$$ if and only if $\\Gamma$ is amenable.\n\\end{theorem}\n\nNon-amenability of a group, that is, the existence of a spectral gap for $-\\Delta_0$, easily implies exponential growth. The reverse is not true, in general. Thus, it is natural to ask, for example, whether Baumslag--Solitar groups $\\langle\\mathscr a, \\mathscr b\\mid \\mathscr b^{-1}\\mathscr a\\mathscr b=\\mathscr a^n\\rangle$, $n\\in\\mathbb N$, have a spectral gap for $1$-Laplacian. These groups are \\emph{non-elementary amenable} but have exponential growth. Such groups are not covered by Theorem \\ref{th:exp_growth}, and it is still unclear for the author whether $1$-Laplacian has a spectral gap on them.\n\nIf we assume the contrary to Theorem \\ref{th:exp_growth}, then, first, $\\Gamma$ cannot have two \\emph{ends} since in this case $\\Gamma$ is virtually cyclic, see, e.g., \\cite{Me08}. Second, $\\Gamma$ cannot have infinitely many ends because then $\\Gamma$ has exponential growth; the latter follows from Stallings Theorem and from results of \\cite{HB00} but, of course, can be proved directly. So, by Freudenthal--Hopf Theorem we may assume that $\\Gamma$ has one end.\n\nFor $\\mathscr L>0$, denote by $\\mathbb T_{\\mathscr{L}}$ a circle of length $\\mathscr L$. On $\\mathbb T_{\\mathscr L}$, one may measure distances along this loop. To prove Theorem \\ref{th:exp_growth}, we need the following\n\n\\begin{lemma}[on loop embedding]\n\\label{lemma:embed}\nSuppose that $\\Gamma$ has subexponential growth and just one end.\n\nLet $C>20$, $x\\ge 1$. There exist $\\mathscr L>2DCx$ and injective naturally parametrized $\\gamma\\colon \\mathbb T_{\\mathscr L}\\to \\Cay(\\Gamma)$ such that, for $t_1, t_2\\in\\mathbb T_{\\mathscr L}$ with $\\gamma(t_1), \\gamma(t_2)\\in \\Gamma$, \n\\emph{\n\\begin{equation}\n\t\\label{eq:x_bilip}\n\\mbox{if }\\dist_{\\Cay(\\Gamma)}(\\gamma(t_1), \\gamma(t_2)) \\le x \n\\mbox{ then }\n\\dist_{\\mathbb T_{\\mathscr L}}(t_1, t_2) \\le Cx.\n\\end{equation}\n}\n\\end{lemma}\n\nIn fact, we are able to make $\\mathscr L$ arbitrarily large with fixed $x$.\n\nNow, let us briefly recall the proof of Kesten Theorem \\ref{th:Kesten} to compare it to our argument. Non-amenability of $\\Gamma$ means that\n\\begin{equation}\n\\label{eq:isoperimetr}\n\\|du\\|_{\\ell^1(E)} \\gtrsim \\|u\\|_{\\ell^1(\\Gamma)}\n\\end{equation}\nfor $u=\\mathds 1_E$, $E$ ranges all finite subsets in $\\Gamma$. By discrete version of coarea formula, this is equivalent to the same for any finitely supported $u\\colon \\Gamma\\to{\\mathbb R}$. \nThe spectral gap condition $0\\notin \\spec\\left(-\\Delta_0|_{\\ell^2(\\Gamma)}\\right)$ means that \n$$\n\\langle u, u\\rangle_{\\ell^2(\\Gamma)} \\lesssim \\langle du, du\\rangle_{\\ell^2(E)}, ~~~ u\\colon\\Gamma\\to{\\mathbb R} \\mbox{ is finitely supported}.\n$$\nTo obtain this from (\\ref{eq:isoperimetr}), it remains to insert $u^2$ instead of $u$ to (\\ref{eq:isoperimetr}) and apply Cauchy--Bunyakovsky--Schwartz inequality.\n\nIn the first step of the latter argument, we assemble a function $u\\colon \\Gamma\\to{\\mathbb R}$, say, non-negative one, from its super-level sets $\\mathds 1_{\\{u\\ge \\mathscr t\\}}$, $\\mathscr t$ ranges $[0,+\\infty)$; we also assemble $du$ from $d\\mathds 1_{\\{u\\ge \\mathscr t\\}}$. (Both decompositions are $\\ell^1$-convex.) Thus, in Kesten Theorem, we deal with \"sets of codimensions $0$ and $1$\". At least, we will have such genuine codimensions in the case of a manifold instead of a group, the corresponding result linking spectra and isoperimetry is known as Cheeger--Yau inequality, see \\cite{Cheeger}, \\cite{Yau}.\n\nUnlike this, in our argument we work with dimension $1$ sets --- loops, in particular, as in Lemma \\ref{lemma:embed}. Also, in Section \\ref{sec:spectra_homology} we bound $1$-cycles with $2$-dimensional surfaces. \n\nNotice also that an analogue of Cheeger--Yau inequality for $1$-forms was obtained in \\cite{BC22} in the case of manifolds. Coexact $1$-Laplacian spectrum is indeed related to appropriate isoperimetric ratio, namely, to $\\sup_{\\gamma}\\inf_h|h|/\\length\\gamma$ with $\\gamma$ ranging homologicaly trivial loops at a manifold $\\mathscr M$ and $h$ be a $2$-dimensional chain in $\\mathscr M$ bounding $\\gamma$; here, $|h|$ is area of $h$. Some Poincar\\'e-type estimates for operator $d$ on coclosed $1$-forms are possible if isoperimetric ratios as above are bounded from the below. But, in \\cite{BC22}, authors impose the condition of finite diameter of $\\mathscr M$ which is not our case; also, \\cite{BC22} does not deal with effects of negative curvature.\n\nWhat concerns spaces with negative curvature, let us mention recent works \\cite{AAGLZ24}, \\cite{R23} devoted to $3$-dimensional hyperbolic manifolds. It turns out that, first, $1$-coexact spectral gap is related to exponential growth of torsion $1$-homology of the manifolds; second, there are relations between the spectral gap and isoperimetric ratios. The latter estimates from \\cite{R23} are also volume-dependent, as in \\cite{BC22}.\n\n\\medskip\n\nThis paper is organized as follows. In Section \\ref{sec:maueutic}, we explain our interest to the study of $1$-Laplacian spectra. This Section is not used in the proof of Theorem \\ref{th:exp_growth}. In Section~\\ref{sec:loop}, we prove Lemma \\ref{lemma:embed} by dropping lots of geodesic perpendiculars in a branching way. In Section \\ref{sec:spectra_homology}, we conclude the proof of Theorem~\\ref{th:exp_growth}. This is done by approximating the resolvent $\\Delta^{-1}_1$ by polynomials of $\\Delta_1$ provided that $0\\notin\\spec\\Delta_1|_{\\ell_{0,c}^2(E)}$. Next, we apply this to $1$-cochain given by curve $\\gamma$ from Lemma \\ref{lemma:embed}. We put some metric control on the approximating process as implemented in $\\Cay(\\Gamma)$ and also make use of homological nature of $\\Delta_1$: this operator is divisible by $\\partial\\colon \\ell^2(F)\\to\\ell^2(E)$ at $\\ell_{0,c}^2(E)$. Finally, in Section \\ref{sec:examples} we check the most natural examples of Cayley graphs. \n\n\\medskip\n\n\\noindent {\\bf Some notation.} For a set $A$ we denote by $\\card A$ the number of its elements. \nIf $v_1, v_2$ are vertices of some oriented graph then we denote by $\\edge(v_1, v_2)$ the oriented edge in the graph under consideration provided that the edge exists. If $e$ is an edge in a oriented graph or $\\gamma$ is an oriented path in a metric space then we write $\\beg e$, $\\beg\\gamma$ for their beginnings and $\\ennd e$, $\\ennd\\gamma$ for their endpoints, respectively. The notation $\\length\\gamma$ is obvious. \n\nWe write $\\mathcal B_X(x, \\rho)$ for the open ball in a metric space $X$ centered in a point $x\\in X$ and having radius $\\rho\\ge 0$.", + "sketch": "Assuming the contrary to Theorem~\\ref{th:exp_growth}, one first reduces the possibilities for the number of ends: $\\Gamma$ \"cannot have two \\emph{ends} since in this case $\\Gamma$ is virtually cyclic\"; it also \"cannot have infinitely many ends because then $\\Gamma$ has exponential growth\". Hence, \"by Freudenthal--Hopf Theorem we may assume that $\\Gamma$ has one end.\"\n\nUnder the hypothesis that $\\Gamma$ has subexponential growth and one end, one proves Lemma~\\ref{lemma:embed} (\"on loop embedding\"), producing for given $C>20$, $x\\ge1$ an injective loop $\\gamma\\colon\\mathbb T_{\\mathscr L}\\to\\Cay(\\Gamma)$ with $\\mathscr L>2DCx$ and the metric control (\\ref{eq:x_bilip}): if $\\dist_{\\Cay(\\Gamma)}(\\gamma(t_1),\\gamma(t_2))\\le x$ then $\\dist_{\\mathbb T_{\\mathscr L}}(t_1,t_2)\\le Cx$ (and \"we are able to make $\\mathscr L$ arbitrarily large with fixed $x$\"). The lemma is proved \"by dropping lots of geodesic perpendiculars in a branching way.\" \n\nThe proof of Theorem~\\ref{th:exp_growth} is then completed in Section~\\ref{sec:spectra_homology}: assuming a spectral gap, i.e. \"$0\\notin\\spec\\Delta_1|_{\\ell_{0,c}^2(E)}$\", one \"approximat[es] the resolvent $\\Delta^{-1}_1$ by polynomials of $\\Delta_1$\" and applies this to the \"$1$-cochain given by curve $\\gamma$ from Lemma~\\ref{lemma:embed}$.\" One also \"put[s] some metric control on the approximating process\" in $\\Cay(\\Gamma)$ and uses the \"homological nature of $\\Delta_1$: this operator is divisible by $\\partial\\colon \\ell^2(F)\\to\\ell^2(E)$ at $\\ell_{0,c}^2(E)$\", together with the fact that \"we bound $1$-cycles with $2$-dimensional surfaces.\"", + "expanded_sketch": "Assuming the contrary to the main theorem, one first reduces the possibilities for the number of ends: $\\Gamma$ \"cannot have two \\emph{ends} since in this case $\\Gamma$ is virtually cyclic\"; it also \"cannot have infinitely many ends because then $\\Gamma$ has exponential growth\". Hence, \"by Freudenthal--Hopf Theorem we may assume that $\\Gamma$ has one end.\"\n\nUnder the hypothesis that $\\Gamma$ has subexponential growth and one end, one proves the following lemma.\n\n\\begin{lemma}[on loop embedding]\n\\label{lemma:embed}\nSuppose that $\\Gamma$ has subexponential growth and just one end.\n\nLet $C>20$, $x\\ge 1$. There exist $\\mathscr L>2DCx$ and injective naturally parametrized $\\gamma\\colon \\mathbb T_{\\mathscr L}\\to \\Cay(\\Gamma)$ such that, for $t_1, t_2\\in\\mathbb T_{\\mathscr L}$ with $\\gamma(t_1), \\gamma(t_2)\\in \\Gamma$, \n\\emph{\n\\begin{equation}\n\t\\label{eq:x_bilip}\n\\mbox{if }\\dist_{\\Cay(\\Gamma)}(\\gamma(t_1), \\gamma(t_2)) \\le x \n\\mbox{ then }\n\\dist_{\\mathbb T_{\\mathscr L}}(t_1, t_2) \\le Cx.\n\\end{equation}\n}\n\\end{lemma}\n\nThis produces for given $C>20$, $x\\ge1$ an injective loop $\\gamma\\colon\\mathbb T_{\\mathscr L}\\to\\Cay(\\Gamma)$ with $\\mathscr L>2DCx$ and the metric control given by the displayed equation above (and \"we are able to make $\\mathscr L$ arbitrarily large with fixed $x$\"). The lemma is proved \"by dropping lots of geodesic perpendiculars in a branching way.\" \n\nIn completing the proof of the main theorem, assuming a spectral gap, i.e. \"$0\\notin\\spec\\Delta_1|_{\\ell_{0,c}^2(E)}$\", one \"approximat[es] the resolvent $\\Delta^{-1}_1$ by polynomials of $\\Delta_1$\" and applies this to the \"$1$-cochain given by curve $\\gamma$ from the lemma above.\" One also \"put[s] some metric control on the approximating process\" in $\\Cay(\\Gamma)$ and uses the \"homological nature of $\\Delta_1$: this operator is divisible by $\\partial\\colon \\ell^2(F)\\to\\ell^2(E)$ at $\\ell_{0,c}^2(E)$\", together with the fact that \"we bound $1$-cycles with $2$-dimensional surfaces.\"", + "expanded_theorem": "\\label{th:exp_growth}\n\tLet $\\Gamma$ be a countable finitely presented group.\n\tIf $\\Delta_1$ has a spectral gap on \t$\\ell_{0,c}^2(E)$ \tthen either $\\Gamma$ has exponential growth,\t or $\\Gamma$ is virtually infinite cyclic.,", + "theorem_type": [ + "Implication", + "Classification or Bijection" + ], + "mcq": { + "question": "Let \\(\\Gamma\\) be a countable finitely presented group. Choose a finite generating set \\(S\\), let \\(\\operatorname{Cay}(\\Gamma,S)\\) be its Cayley graph with edge set \\(E\\), and form the simply connected 2-complex \\(\\operatorname{Cay}^{(2)}(\\Gamma)\\) by gluing oriented polygons corresponding to all translates of the defining relations. Let \\(\\Delta_1\\) be the nonnegative Hodge--Laplace operator on edge functions determined by this 2-complex, and let\n\\[\n\\ell^2_{0,c}(E):=\\overline{\\{\\text{finitely supported closed 1-cochains on }\\operatorname{Cay}(\\Gamma,S)\\}}^{\\,\\ell^2(E)}.\n\\]\nAssume that \\(\\Delta_1\\) has a spectral gap on \\(\\ell^2_{0,c}(E)\\), meaning that for some \\(\\varepsilon>0\\),\n\\[\n\\operatorname{spec}\\bigl(\\Delta_1\\big|_{\\ell^2_{0,c}(E)}\\bigr)\\cap[0,\\varepsilon)=\\varnothing.\n\\]\nWhich conclusion about \\(\\Gamma\\) holds?", + "correct_choice": { + "label": "A", + "text": "Either \\(\\Gamma\\) has exponential growth, or \\(\\Gamma\\) is virtually infinite cyclic (that is, it has a finite-index subgroup isomorphic to \\(\\mathbb Z\\))." + }, + "choices": [ + { + "label": "B", + "text": "Either \\(\\Gamma\\) is nonamenable, or \\(\\Gamma\\) is virtually infinite cyclic." + }, + { + "label": "C", + "text": "\\(\\Gamma\\) is either of exponential growth or virtually cyclic." + }, + { + "label": "D", + "text": "If \\(\\Delta_1\\) has a spectral gap on \\(\\ell^2_{0,c}(E)\\), then \\(\\Gamma\\) has exponential growth unless it is finite." + }, + { + "label": "E", + "text": "Either \\(\\Gamma\\) has exponential growth, or \\(\\Gamma\\) has exactly one end." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "replacing growth conclusion by amenability-type alternative", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "replace virtually infinite cyclic by weaker virtually cyclic", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "finiteness", + "tampered_component": "exceptional virtually-Z case replaced by finite-case exception", + "template_used": "boundary_range" + }, + { + "label": "E", + "sketch_hook_type": "case_split", + "tampered_component": "one-end reduction under contradiction promoted to final alternative", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives only the hypothesis and setup; it does not explicitly state or strongly hint at the specific conclusion \"exponential growth or virtually infinite cyclic.\"" + }, + "TAS": { + "score": 1, + "justification": "This is essentially a theorem-recall/application item: given a highly specific hypothesis, the student is asked for the resulting conclusion. The options do create some contrast, but the item remains close to a direct theorem restatement." + }, + "GPS": { + "score": 1, + "justification": "There is some reasoning pressure because one must distinguish the strongest correct conclusion from nearby alternatives, especially the weaker true option \"virtually cyclic.\" However, for a prepared student the item is mostly recognition/recall rather than deep derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic confusions: swapping in amenability/nonamenability, weakening \"virtually infinite cyclic\" to \"virtually cyclic,\" mishandling the exceptional case, or replacing the conclusion with an ends statement." + }, + "total_score": 6, + "overall_assessment": "A solid but theorem-centric MCQ: it avoids answer leakage and has strong distractors, but it mainly tests recognition of a specific result rather than substantial generative reasoning." + } + }, + { + "id": "2511.19744v1", + "paper_link": "http://arxiv.org/abs/2511.19744v1", + "theorems_cnt": 5, + "theorem": { + "env_name": "thm", + "content": "\\label{thm:main}\n Assume \\cref{conj:main}. If $n$ is a positive integer, then $t(n)\\geq 1$. If $5\\nmid n$, then $t(n)\\geq 2$. If $3\\mid n$, then $t(n)\\geq 3$.", + "start_pos": 271680, + "end_pos": 271862, + "label": "thm:main" + }, + "ref_dict": { + "table:toda sets": "\\begin{table}[t]\n\\centering\n\\caption{Sets of Toda primes}\\label{table:toda sets}\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n1 & $3,5$\\\\\n2 & $3,5$\\\\\n3 & $5,7,13$\\\\\n4 & $3,5,17$\\\\\n5 & $3,11$\\\\\n6 & $5,7,13$\\\\\n7 & $3,5,29$\\\\\n8 & $3,5,17$\\\\\n9 & $5,7,13,19,37$\\\\\n10 & $3,11,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n11 & $3,5,23$\\\\\n12 & $5,7,13,17$\\\\\n13 & $3,5,53$\\\\\n14 & $3,5,29$\\\\\n15 & $7,11,13,31,61$\\\\\n16 & $3,5,17$\\\\\n17 & $3,5$\\\\\n18 & $5,7,13,19,37,73$\\\\\n19 & $3,5$\\\\\n20 & $3,11,17,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n21 & $5,13,29,43$\\\\\n22 & $3,5,23,89$\\\\\n23 & $3,5,47$\\\\\n24 & $5,7,13,17,97$\\\\\n25 & $3,11,101$\\\\\n26 & $3,5,53$\\\\\n27 & $5,7,13,19,37,109$\\\\\n28 & $3,5,17,29,113$\\\\\n29 & $3,5,59$\\\\\n30 & $7,11,13,31,41,61$\\\\\n\\hline\n\\end{tabular}\n\\end{table}", + "lem:t(n)=3": "\\begin{lem}\\label{lem:t(n)=3}\n Assume \\cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\\{5,7,13\\}$.\n\\end{lem}", + "ques:at least one": "\\begin{ques}[Mikhailov]\\label{ques:at least one}\n Does every positive integer have a Toda prime?\n\\end{ques}", + "fig:tn": "\\begin{figure}[p]\n \\includegraphics[width=.75\\linewidth]{tn.png}\n \\caption{$t(n)$ for $n\\leq 100000$}\\label{fig:tn}\n\\end{figure}", + "thm:main": "\\begin{thm}\\label{thm:main}\n Assume \\cref{conj:main}. If $n$ is a positive integer, then $t(n)\\geq 1$. If $5\\nmid n$, then $t(n)\\geq 2$. If $3\\mid n$, then $t(n)\\geq 3$.\n\\end{thm}", + "conj:main": "\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}" + }, + "pre_theorem_intro_text_len": 3502, + "pre_theorem_intro_text": "The fourth stable homotopy group of spheres is trivial, meaning that $\\pi_{n+4}(S^n)=0$ for all $n>5$. In contrast to this, it is a theorem that $S^m$ has no trivial higher homotopy groups when $m\\in\\{2,3,4,5\\}$, as we will briefly explain.\n\nCurtis proved that $\\pi_n(S^4)\\neq 0$ for all $n\\geq 4$ \\cite{Cur69}. Curtis also proved that $\\pi_n(S^2)\\neq 0$ for all $n\\not\\equiv 1\\mod 8$. These same results were obtained (via different methods) by Mimura, Mori, and Oda \\cite{MMO75}. The proof that $\\pi_n(S^5)\\neq 0$ for all $n\\geq 5$ was given by Mori \\cite{Mor75} and Mahowald \\cite{Mah75,Mah82}.\n\nSince $\\pi_n(S^2)\\cong\\pi_n(S^3)$ for all $n\\geq 3$, the remaining case was $\\pi_n(S^3)$ with $n\\equiv 1\\mod 8$. This last case was proved by Gray \\cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \\cite{IMW16} using different methods. In \\emph{op.~cit.}, the authors note that \\cite[Theorem 5.2(ii)]{Tod66} implies that\n\\[\\mathbb{Z}/p\\subseteq\\pi_{2(p-1)k+1}(S^3)\\]\nwhenever $\\gcd(p,k)=1$ \\cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\\gcd(p,k)=1$ and $4n=(p-1)k$, then $\\pi_n(S^3)\\neq 0$ for all $n\\equiv 1\\mod 8$. This leads one to the following definition.\n\n\\begin{defn}\n Let $n$ be an integer. A \\emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\\mid 4n$ and $\\gcd(p,\\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \\cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \\cref{fig:tn}).\n\\end{defn}\n\n\\begin{table}[t]\n\\centering\n\\caption{Sets of Toda primes}\\label{table:toda sets}\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n1 & $3,5$\\\\\n2 & $3,5$\\\\\n3 & $5,7,13$\\\\\n4 & $3,5,17$\\\\\n5 & $3,11$\\\\\n6 & $5,7,13$\\\\\n7 & $3,5,29$\\\\\n8 & $3,5,17$\\\\\n9 & $5,7,13,19,37$\\\\\n10 & $3,11,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n11 & $3,5,23$\\\\\n12 & $5,7,13,17$\\\\\n13 & $3,5,53$\\\\\n14 & $3,5,29$\\\\\n15 & $7,11,13,31,61$\\\\\n16 & $3,5,17$\\\\\n17 & $3,5$\\\\\n18 & $5,7,13,19,37,73$\\\\\n19 & $3,5$\\\\\n20 & $3,11,17,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n21 & $5,13,29,43$\\\\\n22 & $3,5,23,89$\\\\\n23 & $3,5,47$\\\\\n24 & $5,7,13,17,97$\\\\\n25 & $3,11,101$\\\\\n26 & $3,5,53$\\\\\n27 & $5,7,13,19,37,109$\\\\\n28 & $3,5,17,29,113$\\\\\n29 & $3,5,59$\\\\\n30 & $7,11,13,31,41,61$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nIf every positive integer has a Toda prime, then one can greatly simplify the proof that $\\pi_n(S^3)\\neq 0$ for $n\\geq 3$. This was asked on MathOverflow \\cite{MO} (and attributed to Roman Mikhailov) several years ago.\n\n\\begin{ques}[Mikhailov]\\label{ques:at least one}\n Does every positive integer have a Toda prime?\n\\end{ques}\n\nIn fact, it appears that every positive integer has at least two Toda primes.\n\n\\begin{conj}\\label{conj:at least two}\n If $n$ is a positive integer, then $t(n)\\geq 2$.\n\\end{conj}\n\nWe tried to answer \\cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\\cref{conj:main}). We will give some heuristic evidence for this conjecture in \\cref{sec:heuristic}.\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}", + "context": "Since $\\pi_n(S^2)\\cong\\pi_n(S^3)$ for all $n\\geq 3$, the remaining case was $\\pi_n(S^3)$ with $n\\equiv 1\\mod 8$. This last case was proved by Gray \\cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \\cite{IMW16} using different methods. In \\emph{op.~cit.}, the authors note that \\cite[Theorem 5.2(ii)]{Tod66} implies that\n\\[\\mathbb{Z}/p\\subseteq\\pi_{2(p-1)k+1}(S^3)\\]\nwhenever $\\gcd(p,k)=1$ \\cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\\gcd(p,k)=1$ and $4n=(p-1)k$, then $\\pi_n(S^3)\\neq 0$ for all $n\\equiv 1\\mod 8$. This leads one to the following definition.\n\n\\begin{defn}\n Let $n$ be an integer. A \\emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\\mid 4n$ and $\\gcd(p,\\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \\cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \\cref{fig:tn}).\n\\end{defn}\n\nIf every positive integer has a Toda prime, then one can greatly simplify the proof that $\\pi_n(S^3)\\neq 0$ for $n\\geq 3$. This was asked on MathOverflow \\cite{MO} (and attributed to Roman Mikhailov) several years ago.\n\n\\begin{conj}\\label{conj:at least two}\n If $n$ is a positive integer, then $t(n)\\geq 2$.\n\\end{conj}\n\nWe tried to answer \\cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\\cref{conj:main}). We will give some heuristic evidence for this conjecture in \\cref{sec:heuristic}.\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{figure}[p]\n \\includegraphics[width=.75\\linewidth]{tn.png}\n \\caption{$t(n)$ for $n\\leq 100000$}\\label{fig:tn}\n\\end{figure}\n\n\\begin{ques}[Mikhailov]\\label{ques:at least one}\n Does every positive integer have a Toda prime?\n\\end{ques}\n\n\\begin{table}[t]\n\\centering\n\\caption{Sets of Toda primes}\\label{table:toda sets}\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n1 & $3,5$\\\\\n2 & $3,5$\\\\\n3 & $5,7,13$\\\\\n4 & $3,5,17$\\\\\n5 & $3,11$\\\\\n6 & $5,7,13$\\\\\n7 & $3,5,29$\\\\\n8 & $3,5,17$\\\\\n9 & $5,7,13,19,37$\\\\\n10 & $3,11,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n11 & $3,5,23$\\\\\n12 & $5,7,13,17$\\\\\n13 & $3,5,53$\\\\\n14 & $3,5,29$\\\\\n15 & $7,11,13,31,61$\\\\\n16 & $3,5,17$\\\\\n17 & $3,5$\\\\\n18 & $5,7,13,19,37,73$\\\\\n19 & $3,5$\\\\\n20 & $3,11,17,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n21 & $5,13,29,43$\\\\\n22 & $3,5,23,89$\\\\\n23 & $3,5,47$\\\\\n24 & $5,7,13,17,97$\\\\\n25 & $3,11,101$\\\\\n26 & $3,5,53$\\\\\n27 & $5,7,13,19,37,109$\\\\\n28 & $3,5,17,29,113$\\\\\n29 & $3,5,59$\\\\\n30 & $7,11,13,31,41,61$\\\\\n\\hline\n\\end{tabular}\n\\end{table}", + "full_context": "Since $\\pi_n(S^2)\\cong\\pi_n(S^3)$ for all $n\\geq 3$, the remaining case was $\\pi_n(S^3)$ with $n\\equiv 1\\mod 8$. This last case was proved by Gray \\cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \\cite{IMW16} using different methods. In \\emph{op.~cit.}, the authors note that \\cite[Theorem 5.2(ii)]{Tod66} implies that\n\\[\\mathbb{Z}/p\\subseteq\\pi_{2(p-1)k+1}(S^3)\\]\nwhenever $\\gcd(p,k)=1$ \\cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\\gcd(p,k)=1$ and $4n=(p-1)k$, then $\\pi_n(S^3)\\neq 0$ for all $n\\equiv 1\\mod 8$. This leads one to the following definition.\n\n\\begin{defn}\n Let $n$ be an integer. A \\emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\\mid 4n$ and $\\gcd(p,\\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \\cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \\cref{fig:tn}).\n\\end{defn}\n\nIf every positive integer has a Toda prime, then one can greatly simplify the proof that $\\pi_n(S^3)\\neq 0$ for $n\\geq 3$. This was asked on MathOverflow \\cite{MO} (and attributed to Roman Mikhailov) several years ago.\n\n\\begin{conj}\\label{conj:at least two}\n If $n$ is a positive integer, then $t(n)\\geq 2$.\n\\end{conj}\n\nWe tried to answer \\cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\\cref{conj:main}). We will give some heuristic evidence for this conjecture in \\cref{sec:heuristic}.\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{figure}[p]\n \\includegraphics[width=.75\\linewidth]{tn.png}\n \\caption{$t(n)$ for $n\\leq 100000$}\\label{fig:tn}\n\\end{figure}\n\n\\begin{ques}[Mikhailov]\\label{ques:at least one}\n Does every positive integer have a Toda prime?\n\\end{ques}\n\n\\begin{table}[t]\n\\centering\n\\caption{Sets of Toda primes}\\label{table:toda sets}\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n1 & $3,5$\\\\\n2 & $3,5$\\\\\n3 & $5,7,13$\\\\\n4 & $3,5,17$\\\\\n5 & $3,11$\\\\\n6 & $5,7,13$\\\\\n7 & $3,5,29$\\\\\n8 & $3,5,17$\\\\\n9 & $5,7,13,19,37$\\\\\n10 & $3,11,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n11 & $3,5,23$\\\\\n12 & $5,7,13,17$\\\\\n13 & $3,5,53$\\\\\n14 & $3,5,29$\\\\\n15 & $7,11,13,31,61$\\\\\n16 & $3,5,17$\\\\\n17 & $3,5$\\\\\n18 & $5,7,13,19,37,73$\\\\\n19 & $3,5$\\\\\n20 & $3,11,17,41$\\\\\n\\hline\n\\end{tabular}\n\\quad\n\\begin{tabular}{|r|l|}\n\\hline\n$n$ & $T(n)$\\\\\n\\hline\n21 & $5,13,29,43$\\\\\n22 & $3,5,23,89$\\\\\n23 & $3,5,47$\\\\\n24 & $5,7,13,17,97$\\\\\n25 & $3,11,101$\\\\\n26 & $3,5,53$\\\\\n27 & $5,7,13,19,37,109$\\\\\n28 & $3,5,17,29,113$\\\\\n29 & $3,5,59$\\\\\n30 & $7,11,13,31,41,61$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\nIn \\cref{sec:lemmas}, we state and prove a few simple lemmas. We prove \\cref{thm:main} in \\cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \\cref{lem:t(n)=3}. We conclude with \\cref{sec:observations}, where we pose a couple questions that arose while working on this project.\n\n\\begin{lem}\\label{lem:not 5 or not 3 and 5}\n Let $p\\in\\{3,5\\}$. If $p\\nmid n$, then $p\\in T(n)$. In particular:\n \\begin{itemize}\n \\item If $5\\nmid n$, then $t(n)\\geq 1$.\n \\item If $3,5\\nmid n$, then $t(n)\\geq 2$.\n \\end{itemize}\n\\end{lem}\n\\begin{proof}\n Let $p\\in\\{3,5\\}$. Then $p-1\\mid 4n$, and $\\gcd(4n,p)=1$ by assumption. Thus $\\gcd(\\frac{4n}{p-1},p)=1$, so $p\\in T(n)$.\n\\end{proof}\n\n\\begin{lem}\\label{lem:t(n)=3}\n Assume \\cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\\{5,7,13\\}$.\n\\end{lem}\n\\begin{proof}\n We will induct on $\\omega(n)$. Our base cases will consist of $\\omega(n)\\leq 4$. Note that if $p\\nmid n$ for each $p\\in\\{5,7,13\\}$, then $T(n)\\supseteq\\{5,7,13\\}$. In particular, we may restrict our attention to multiples of these three primes. Moreover, $t(ap)\\geq t(a)$ for any prime $p\\not\\in T(a)$ by \\cref{cor:multiply by prime}, so we may assume that every prime factor $p\\mid n$ is a Toda prime of some divisor of $n$.\n \\begin{itemize}\n \\item The case of $\\omega(n)=1$ is just the calculation $T(3)=\\{5,7,13\\}$.\n \\item For $\\omega(n)=2$, we just need to compute $t(15)=t(39)=5$ and $t(21)=4$.\n \\item For $\\omega(n)=3$, we first compute $t(3\\cdot 5\\cdot 7)=9$ and $t(3\\cdot 5\\cdot 13)=t(3\\cdot 7\\cdot 13)=8$. It remains to compute, for each $p\\in\\{5,7,13\\}$, the Toda primes of $3pq$ for each $q\\in T(3p)$. Using the code provided in \\cref{sec:code}, we find that $t(3pq)\\geq 4$ for all such $p,q$.\n \\item For $\\omega(n)=4$, we first compute $t(3\\cdot 5\\cdot 7\\cdot 13)=16$. For the remaining computations in this case, we use the code in \\cref{sec:code}.\n \\begin{itemize}\n \\item If $\\{p,q\\}\\subseteq\\{5,7,13\\}$ and $r\\in T(3pq)$, then $t(3pqr)\\geq 9$.\n \\item If $p\\in\\{5,7,13\\}$ and $\\{q,r\\}\\subseteq T(3p)$, then $t(3pqr)\\geq 7$.\n \\item If $p\\in\\{5,7,13\\}$, $q\\in T(3p)$, and $r\\in T(3pq)$, then $t(3pqr)\\geq 5$.\n \\end{itemize}\n \\end{itemize}\n\n\\begin{cor}\\label{prop:divisible by 3}\n Assume \\cref{conj:main}. If $3\\mid n$, then $t(n)\\geq 3$.\n\\end{cor}\n\\begin{proof}\n By \\cref{cor:multiply by divisor}, we may assume that $n$ is square-free. By \\cref{cor:odd}, we may further assume that $n$ is odd. The result now follows from \\cref{lem:t(n)=3}.\n\\end{proof}\n\nFirstly, if there exists $q'\\in T(3p)-\\{5,7,13,q\\}$ such that $q'\\nmid n$, then $\\{5,7,13,q,q'\\}-\\{p\\}\\subseteq T(n)$, and we are done. In fact, the Toda primes of $n$ are precisely those primes among $\\{2d+1:d\\mid 2n\\}$ that are not factors of $n$. Thus if\n\\begin{equation}\\label{eq:heuristic}\n\\{2d+1\\text{ prime}:d\\mid 2n\\}-(\\Omega(n)\\cup\\{5,7,13,q\\})\n\\end{equation}\nis non-empty, then $t(n)\\geq 4$. Our heuristic for \\cref{conj:main} is that the set $\\{2d+1:d\\mid 2n\\}$ consists of $2^{\\omega(n)+1}$ elements, while $\\Omega(n)\\cup\\{5,7,13,q\\}$ consists of $\\omega(n)+3$ elements.\n\n\\begin{prop}\\label{prop:t(p)}\nAssume $p\\geq 7$ is a prime. Let $\\vphi$ denote the totient function. If $\\vphi(x)=4p$ for some integer $x$, then $T(p)=\\{3,5,2p+1\\}$ or $\\{3,5,4p+1\\}$. Otherwise, $T(p)=\\{3,5\\}$.\n\\end{prop}\n\\begin{proof}\n One can directly check that $3,5\\in T(p)$ for all primes greater than 5. Now by Euler's product formula, we have $\\vphi(x)=p_1^{e_1-1}(p_1-1)\\cdots p_m^{e_m-1}(p_m-1)$, where $x=\\prod_{i=1}^m p_i^{e_i}$ is the prime factorization of $x$. It follows that there exists $x$ such that $\\vphi(x)=4p$ if and only if one of the following cases holds:\n \\begin{enumerate}[(i)]\n \\item $x=2^2\\cdot q$, where $q$ is an odd prime such that $q-1=2p$. In this case, $q$ is a Toda prime of $p$ with $\\frac{4p}{q-1}=2$.\n \\item $x=2^r\\cdot 3\\cdot q$, where $r\\in\\{0,1\\}$ and $q$ is an odd prime such that $q-1=2p$. In this case, $q$ is a Toda prime of $p$ with $\\frac{4p}{q-1}=2$.\n \\item $x=2^r\\cdot q$, where $r\\in\\{0,1\\}$ and $q$ is an odd prime such that $q-1=4p$. In this case, $q$ is a Toda prime of $p$ with $\\frac{4p}{q-1}=1$.\n \\item $x=2^r\\cdot 5^2$, where $r\\in\\{0,1\\}$ (in which case $p=5$). This case is not relevant for this lemma, as we have assumed $p\\geq 7$.\n \\end{enumerate}\n It remains to show that no other primes can be the Toda prime of $p$. To this end, let $q>5$ be a Toda prime of $p$. Then $q-1\\mid 4p$, so we either have $q-1=4p$ or $q-1=2p$ (as $q-1$ is even and $p$ is odd). The existence of such a $q$ gives us a solution to $\\vphi(x)=4p$ as outlined in cases (i), (ii), and (iii).\n\n\\begin{lem}\\label{lem:strategy for denoms}\n Let $d$ be a Bernoulli denominator with $F(d)=4a$ for some integer $a$. If $\\{2pi+1:i\\mid 2a\\}$ contains a prime number for each $p\\in T(a)$, then \\cref{conj:general bernoulli} \\eqref{conj:toda for bernoulli} holds for this Bernoulli denominator.\n\\end{lem}\n\\begin{proof}\n We know that $p-1\\mid 4a$ with $\\gcd(\\frac{4a}{p-1},p)=1$ for all $p\\in T(a)$. Thus $p-1\\mid 4am$, and we have $\\gcd(\\frac{4am}{p-1},p)=1$ if and only if $p\\nmid m$. It therefore suffices to show that if $p\\mid m$ for some $p\\in T(a)$, then $D_{4am}>D_{4a}$.\n\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\n\n\\begin{lem}\\label{lem:t(n)=3}\n Assume \\cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\\{5,7,13\\}$.\n\\end{lem}\n\n\\begin{thm}\\label{thm:main}\n Assume \\cref{conj:main}. If $n$ is a positive integer, then $t(n)\\geq 1$. If $5\\nmid n$, then $t(n)\\geq 2$. If $3\\mid n$, then $t(n)\\geq 3$.\n\\end{thm}", + "post_theorem_intro_text_len": 541, + "post_theorem_intro_text": "In \\cref{sec:lemmas}, we state and prove a few simple lemmas. We prove \\cref{thm:main} in \\cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \\cref{lem:t(n)=3}. We conclude with \\cref{sec:observations}, where we pose a couple questions that arose while working on this project.\n\n\\subsection*{Acknowledgements}\nWe thank Nick Andersen, Pace Nielsen, and Kyle Pratt for helpful conversations. The author was partially supported by the NSF (DMS-2502365) and the Simons Foundation.", + "sketch": "We prove \\cref{thm:main} in \\cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \\cref{lem:t(n)=3}.", + "expanded_sketch": "No expanded sketch found.", + "expanded_theorem": "\\label{thm:main}\nAssume\n\\begin{conj}\\label{conj:main}\n Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\\in\\{5,7,13\\}$ such that $p\\mid n$, and that $r\\nmid n$ for $r\\in\\{5,7,13\\}-\\{p\\}$. Finally, assume there exists $q\\in T(3p)-\\{5,7,13\\}$ such that $q\\nmid n$. Then $t(n)\\geq 4$.\n\\end{conj}\nIf $n$ is a positive integer, then $t(n)\\geq 1$. If $5\\nmid n$, then $t(n)\\geq 2$. If $3\\mid n$, then $t(n)\\geq 3$.,\n", + "theorem_type": [ + "Implication", + "Inequality or Bound" + ], + "mcq": { + "question": "For a positive integer n, call an odd prime p a Toda prime of n if p-1 divides 4n and \\(\\gcd\\!\\left(p,\\frac{4n}{p-1}\\right)=1\\). Let \\(T(n)\\) be the set of Toda primes of n, and let \\(t(n)=|T(n)|\\). Assume the following hypothesis holds universally: whenever m is an odd, square-free multiple of 3, there exists \\(p\\in\\{5,7,13\\}\\) such that \\(p\\mid m\\) and \\(r\\nmid m\\) for every \\(r\\in\\{5,7,13\\}\\setminus\\{p\\}\\), and there exists \\(q\\in T(3p)\\setminus\\{5,7,13\\}\\) with \\(q\\nmid m\\), then \\(t(m)\\ge 4\\). Under this assumption, which quantitative estimate holds for Toda primes of a positive integer n?", + "correct_choice": { + "label": "A", + "text": "For every positive integer n, \\(t(n)\\ge 1\\); moreover, if \\(5\\nmid n\\), then \\(t(n)\\ge 2\\); and if \\(3\\mid n\\), then \\(t(n)\\ge 3\\)." + }, + "choices": [ + { + "label": "B", + "text": "For every positive integer \\(n\\), \\(t(n)\\ge 1\\); moreover, if \\(3\\nmid n\\), then \\(t(n)\\ge 2\\); and if \\(5\\mid n\\), then \\(t(n)\\ge 3\\)." + }, + { + "label": "C", + "text": "For every positive integer \\(n\\), \\(t(n)\\ge 1\\); moreover, if \\(5\\nmid n\\), then \\(t(n)\\ge 2\\)." + }, + { + "label": "D", + "text": "For every positive integer \\(n\\), \\(t(n)\\ge 2\\); moreover, if \\(5\\nmid n\\), then \\(t(n)\\ge 3\\); and if \\(3\\mid n\\), then \\(t(n)\\ge 4\\)." + }, + { + "label": "E", + "text": "For every positive integer \\(n\\), \\(t(n)\\ge 1\\); moreover, if \\(5\\nmid n\\), then \\(t(n)\\ge 2\\); and if \\(n\\) is an odd, square-free multiple of \\(3\\), then \\(t(n)\\ge 4\\)." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "B" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "special-role of the primes 3 and 5 in the baseline bounds", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped the final clause requiring \\(3\\mid n\\Rightarrow t(n)\\ge 3\\)", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "numerical lower bounds 1/2/3 replaced by stronger 2/3/4", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "scope of the \\(t(m)\\ge 4\\) hypothesis enlarged by dropping the auxiliary existence conditions on \\(p\\) and \\(q\\)", + "template_used": "quantifier_dependence" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly state the target estimate, and there is no direct textual giveaway of choice A. The hypothesis is technical, but it does not itself reveal the final 1/2/3 lower-bound pattern." + }, + "TAS": { + "score": 1, + "justification": "The item appears to ask for the theorem-level conclusion under a stated hypothesis, so it is close to theorem recall rather than a fully independent problem. Still, the answer choices differ by quantifiers, exceptional primes, and bound strength, so it is not a pure verbatim restatement." + }, + "GPS": { + "score": 1, + "justification": "There is some reasoning pressure in comparing nearby quantitative claims and tracking how the special roles of 3 and 5 affect the bounds. However, the stem is too specialized to support much derivation from first principles, so the task leans more toward recognizing the intended conclusion than generating it." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and well targeted: one swaps the roles of 3 and 5, one gives a weaker true-looking statement, one overstates the bounds, and one improperly broadens the hypothesis. These reflect realistic mathematical failure modes involving quantifiers and strength of conclusions." + }, + "total_score": 6, + "overall_assessment": "A reasonably strong MCQ with little answer leakage and high-quality distractors, but it is somewhat theorem-recall driven and only moderately tests genuine generative reasoning." + } + }, + { + "id": "2511.19681v1", + "paper_link": "http://arxiv.org/abs/2511.19681v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "theorem", + "content": "[$2\\pi^2$-Theorem, Marques-Neves~\\cite{MN14}]\\label{2pi2-theorem}\nLet $\\Sigma\\subset\\varmathbb{S}^3$ be a smooth embedded surface of genus at least $1$. \nThen its canonical five-parameter family \n\\[\n\\{\\Sigma_{(v,t)}\\}_{(v,t)\\in \\mathring{B}^4\\times[-\\pi,\\pi]}\n\\]\nsatisfies\n\\[\n\\max_{(v,t)\\in \\mathring{B}^4\\times[-\\pi,\\pi]}\n\\mathrm{Area}\\bigl(\\Sigma_{(v,t)}\\bigr)\\;\\geqslant\\; 2\\pi^2.\n\\]\nMoreover, equality holds if and only if $\\Sigma$ is the Clifford torus.", + "start_pos": 39790, + "end_pos": 40269, + "label": "2pi2-theorem" + }, + "ref_dict": { + "away from boundary of conformal group": "\\begin{proposition}\\label{away from boundary of conformal group}\nThere exist constants $\\eta_1>0$, $\\delta_0>0$, and a function \n$\\gamma=\\gamma(\\delta_0)>0$ with $\\gamma(\\delta_0)\\to 0$ as $\\delta_0\\to 0$,\nand a universal radius $r_0>0$, such that the following holds.\n\nLet $\\Sigma\\subset\\mathbb{S}^3$ be a smooth closed torus \nand assume that $\\Sigma$ is $(\\gamma,r_0)$-regular and $\\mathcal{W}(\\Sigma)\\leq 8\\pi$ \nConsider the canonical family \n\\[\n\\{\\Sigma_{(v,t)}\\}_{(v,t)\\in\\mathring{B}^4\\times[-\\pi,\\pi]},\n\\]\nand let $(v_0,t_0)$ be any maximizer of the area , i.e.\n\\[\n\\mathrm{Area}(\\Sigma_{(v_0,t_0)}) \n = \\max_{(v,t)\\in\\mathring{B}^4\\times[-\\pi,\\pi]}\n \\mathrm{Area}(\\Sigma_{(v,t)}).\n\\]\nThen\n\\[\n|v_0|\\le 1-\\eta_1.\n\\]\n\\end{proposition}", + "Thm: main": "\\begin{theorem}\\label{Thm: main}\nLet $V=\\underline{v}(\\Sigma,1)$ be an integral $2$-varifold in $\\mathbb{S}^3$\nwith unit density and mean curvature $H\\in L^2(d\\mu)$, and assume its support\n$\\Sigma$ has genus at least $1$. \nIf the Willmore energy\n\\[\n\\mathcal{W}(V)=\\int_\\Sigma\\Bigl(1+\\tfrac14 |H|^2\\Bigr)\\,d\\mu\n\\]\nsatisfies\n\\[\n\\mathcal{W}(V)\\le 2\\pi^2+\\delta^2,\n\\qquad \\delta>0 \\text{ sufficiently small},\n\\]\nthen, after a suitable conformal transformation of $\\mathbb{S}^3$, there exists\na homeomorphism\n\\[\nf:\\mathbb{S}^1\\times\\mathbb{S}^1 \\longrightarrow \\Sigma \\subset\\mathbb{S}^3\n\\]\nsuch that in the standard coordinates $(\\theta,\\varphi)$ on\n$\\mathbb{S}^1\\times\\mathbb{S}^1$ the pull-back metric takes the form\n\\[\nf^{*}g_{\\mathbb{S}^3}\n= e^{2u}\\,(a\\, d\\theta^2 + 2 b\\, d\\theta\\, d\\varphi + c\\, d\\varphi^2).\n\\]\n\nLet $f_0:\\mathbb{S}^1\\times\\mathbb{S}^1\\to\\mathbb{S}^3$ denote the Clifford torus\nembedding\n\\[\nf_0(\\theta,\\varphi)\n= \\frac{1}{\\sqrt{2}}(\\cos\\theta,\\;\\sin\\theta,\\;\\cos\\varphi,\\;\\sin\\varphi),\n\\]\nand write its image as $\\mathbb{T}^2:=f_0(\\mathbb{S}^1\\times\\mathbb{S}^1)$. \nThen\n\\[\nf_0^{*} g_{\\mathbb{S}^3}\n= \\tfrac12 (d\\theta^2+d\\varphi^2).\n\\]\n\nThe following quantitative rigidity estimates hold:\n\\begin{enumerate}\n\n\\item[\\rm(1)]\n\\[\n\\| f - f_0 \\|_{W^{2,2}(\\mathbb{S}^1\\times\\mathbb{S}^1)} \\le C\\,\\delta.\n\\]\n\n\\item[\\rm(2)]\n\\[\n\\|u\\|_{L^\\infty(\\mathbb{S}^1\\times\\mathbb{S}^1)} \\le C\\,\\delta.\n\\]\n\n\\item[\\rm(3)]\n\\[\n|a-\\tfrac12| + |b| + |c-\\tfrac12| \\le C\\,\\delta.\n\\]\n\n\\end{enumerate}\nHere $C$ is a universal constant independent of $\\Sigma$.\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 5710, + "pre_theorem_intro_text": "For an immersed surface $f:\\Sigma\\to \\varmathbb{S}^3$ in the round three-sphere, the Willmore energy is defined by\n\\[\n\\mathcal{W}(\\Sigma)\n=\\int_{\\Sigma}\\!\\left(1+\\tfrac{1}{4}|H|^2\\right)d\\mu_f,\n\\]\nwhere $H$ is the mean curvature and $d\\mu_f$ is the area element of $f^{*}g_{\\varmathbb{S}^3}$. \nA fundamental property of $\\mathcal{W}$ is its conformal invariance. \nBy the theorem of Marques and Neves resolving the Willmore conjecture~\\cite{MN14}, any closed oriented surface $\\Sigma\\subset\\varmathbb{S}^3$ with genus at least one satisfies\n\\[\n\\mathcal{W}(\\Sigma)\\geqslant 2\\pi^2,\n\\]\nwith equality if and only if $\\Sigma$ is, up to a conformal transformation, the Clifford torus\n\\[\n\\varmathbb{T}^2=\\varmathbb{S}^1\\!\\left(\\tfrac{1}{\\sqrt2}\\right)\\times\n\\varmathbb{S}^1\\!\\left(\\tfrac{1}{\\sqrt2}\\right)\\subset\\varmathbb{S}^3.\n\\]\n\nThe purpose of this paper is to study the quantitative rigidity of this theorem. \nGiven $\\delta<\\delta_0\\ll1$ and a surface $\\Sigma$ with\n\\[\n\\mathcal{W}(\\Sigma)\\leqslant 2\\pi^2+\\delta^2,\n\\]\nwe ask how close $\\Sigma$ must be to a conformal image of the Clifford torus, in a sense that simultaneously controls the $W^{2,2}$-parametrization, the $L^\\infty$-bound of the conformal factor, and the conformal structure. \nThis parallels the classical quantitative rigidity for the genus zero case, where De Lellis-M\\\"uller~\\cite{dLMu,dLMu2} proved optimal linear estimates for nearly umbilical surfaces, later extended to higher codimension by Lamm-Sch\\\"atzle~\\cite{LS-2014} and to varifolds by Bi-Zhou~\\cite{BZ25}.\n\nA first step toward our result is to understand the topology of surfaces whose Willmore energy is close to the Clifford value. \nUsing existing existence and energy asymptotic results, one obtains that if a surface $\\Sigma$ satisfies $g(\\Sigma)\\ge1$ and $\\mathcal{W}(\\Sigma)$ is sufficiently close to $2\\pi^2$, then in fact $g(\\Sigma)=1$. \nThis follows directly from the existence theory for Willmore minimizers (Simon for genus 1~\\cite{LS93} and Bauer-Kuwert for genus $\\ge2$~\\cite{BK03}), together with the large-genus limit of Kuwert-Li-Sch\\\"atzle~\\cite{KLS10} and the Willmore conjecture~\\cite{MN14}. \n\nOnce the genus has been identified, Leon Simon's results~\\cite{LS93} imply that any sequence of tori \n\\(\\Sigma_k \\subset \\varmathbb{S}^3\\) with \\(\\mathcal{W}(\\Sigma_k)\\to 2\\pi^2\\) admits, after suitable conformal reparametrization, \na subsequence converging as varifolds to a limit surface whose genus is at least one. \nBy lower semicontinuity of the Willmore energy, this limit has Willmore energy at most \\(2\\pi^2\\). \nIn view of the Willmore conjecture, the only possibility is that the limit is itself a torus of Willmore energy \\(2\\pi^2\\).\nIt is well known to experts (see for instance Kuwert-Li~\\cite{KL-2012}, Sch\\\"atzle~\\cite{Sch13} and Rivi\\'ere\\cite{R14}) that, after parametrizing by a normalized flat torus metric, such a sequence is then qualitatively close to the Clifford torus. Section~2 contains a detailed proof of this qualitative convergence. We in fact formulate and prove a slightly more general statement: the model\nsurface may be any fixed genus~$\\ge1$ minimal surface (not only the Clifford\ntorus), provided that no energy concentration occurs---in the precise sense\nthat the mean curvature has small $L^{2}$-norm and the area ratio is close to~$1$.\nOur approach proceeds by first establishing convergence of the surfaces\n\\emph{as subsets of $\\varmathbb{S}^{3}$} under these assumptions, and only\nafterwards upgrading the convergence to that of the associated conformal\nparametrizations. This is somewhat different in spirit from earlier\ntreatments, which work directly with compactness of $W^{2,2}$ conformal\nimmersions into Euclidean space.\n\nThe main difficulty is to promote this qualitative convergence to a quantitative, in particular linear, stability estimate. \n\nA natural idea would be to expand the Willmore functional around the Clifford torus: one considers the second variation together with the higher-order remainder, analyzes the spectral properties of the resulting symmetric quadratic form, and then derives quantitative stability from the coercivity of this linearized operator. However, in order for this approach to be effective, the remainder must be genuinely dominated by the quadratic term, which requires the surface to be $C^1$ or at least Lipschitz close to the Clifford torus. Establishing such a priori closeness is delicate in our weak $W^{2,2}$-conformal setting.\n\nFor this reason, following the strategy of Marques-Neves, we avoid a direct perturbative analysis of the Willmore functional. Instead, using the Heintze-Karcher-type inequality to relate the Willmore energy to the area of the canonical family $\\Sigma_{(v,t)}$, we reformulate the problem in Section~3 as a minimal-surface stability problem. \n\nTo describe our approach, we recall the canonical 5-parameter family of Marques-Neves~\\cite{MN14}. \nFor $v\\in\\mathring{B}^4$, let\n\\[\nF_v(x)=(1-|v|^2)\\frac{x-v}{|x-v|^2}-v\n\\]\nbe the corresponding conformal transformation. \nIf $\\varmathbb{S}^3\\setminus\\Sigma=A\\cup A^*$ is the decomposition into connected components, we denote $\\Sigma_v:=F_v(\\Sigma)$ and $\\varmathbb{S}^3\\setminus\\Sigma_v=A_v\\cup A_v^*$ with $A_v=F_v(A)$. \nLet $d_v$ be the signed distance to $\\Sigma_v$, and define\n\\[\n\\Sigma_{(v,t)}=\\partial\\{x\\mid d_v(x)0$, $\\delta_0>0$, and a function \n$\\gamma=\\gamma(\\delta_0)>0$ with $\\gamma(\\delta_0)\\to 0$ as $\\delta_0\\to 0$,\nand a universal radius $r_0>0$, such that the following holds.\n\nLet $\\Sigma\\subset\\mathbb{S}^3$ be a smooth closed torus \nand assume that $\\Sigma$ is $(\\gamma,r_0)$-regular and $\\mathcal{W}(\\Sigma)\\leq 8\\pi$ \nConsider the canonical family \n\\[\n\\{\\Sigma_{(v,t)}\\}_{(v,t)\\in\\mathring{B}^4\\times[-\\pi,\\pi]},\n\\]\nand let $(v_0,t_0)$ be any maximizer of the area , i.e.\n\\[\n\\mathrm{Area}(\\Sigma_{(v_0,t_0)}) \n = \\max_{(v,t)\\in\\mathring{B}^4\\times[-\\pi,\\pi]}\n \\mathrm{Area}(\\Sigma_{(v,t)}).\n\\]\nThen\n\\[\n|v_0|\\le 1-\\eta_1.\n\\]\n\\end{proposition}", + "post_theorem_intro_text_len": 7890, + "post_theorem_intro_text": "With the five-parameter family $\\{\\Sigma_{(v,t)}\\}$ of\nMarques-Neves recalled above, we now indicate how it enters our proof.\n\nThe key point is to show that if a genus--one surface $\\Sigma$ satisfies\n$\\mathcal{W}(\\Sigma)\\leqslant 2\\pi^2+\\delta^2$, then a \\emph{conformal}\ntransformation of $\\varmathbb{S}^3$ yields a surface $\\Sigma'$ with\n\\[\n\\mathrm{Area}(\\Sigma')\\;\\geqslant\\;2\\pi^2 - C\\delta^2.\n\\]\n\nThe $2\\pi^2$ theorem guarantees that the full $(v,t)$-family contains a slice\nwhose area is at least $2\\pi^2$. Since the variables $v$ encode conformal\ntransformations and $t$ is the signed distance parameter, our aim is to\nrealize such a high-area slice \\emph{within the conformal}\nfour-parameter subfamily. \nFor this, it is necessary to control the distance parameter $t$ in terms of\nthe energy gap~$\\delta$, ensuring that the maximizing slice must lie close to\nthe $t=0$ slice.\n\nA closer examination of the Heintze-Karcher inequality used by\nMarques-Neves shows that, \\emph{once the five-parameter family is built\nfrom the quantitatively normalized surface obtained in Section~2}, the\nproblem reduces to proving that the maximizing parameter $(v_0,t_0)$ stays a\ndefinite distance away from the boundary of the conformal group. \nEquivalently, there exists a universal constant $\\eta_1>0$ such that\n\\[\n|v_0|\\;\\leqslant\\;1-\\eta_1 .\n\\]\nThis reduction is carried out in Section~3, where the technical\nproof of Proposition~\\ref{away from boundary of conformal group} is deferred to the Appendix.\n\nTo clarify why such a uniform bound is true, we now outline the geometric\nintuition behind it. When the conformal parameter $v$ approaches the boundary of\n$\\mathring{B}^4$, the transformed surface $\\Sigma_{(v,t)}$ becomes\nquantitatively close to a geodesic sphere, which forces its\narea to stay strictly below $2\\pi^2$. \nThus such $v$ cannot maximize the canonical family.\n\nMore precisely, when $t=0$, the qualitative stability implies that $\\Sigma_v= \\Sigma_{(v,0)}$ is Hausdorff close to a geodesic sphere $S$, and the region enclosed by $\\Sigma_v$ is similarly close to that enclosed by~$S$. \nA direct computation then shows that\n\\[\n\\mathrm{Area}(\\Sigma_v)\n\\]\nis close to $\\mathrm{Area}(S)$, which is at most $4\\pi$. Since $\\Sigma_{(v,t)}$ is obtained from $\\Sigma_v$ by shifting along its normal directions, the area difference satisfies an estimate of the form\n\\[\n\\bigl|\\mathrm{Area}(\\Sigma_{(v,t)})-\\mathrm{Area}(\\Sigma_v)\\bigr|\n \\;\\lesssim\\; |t|\\,\\mathcal{W}(\\Sigma_v).\n\\]\nHence, for sufficiently small $|t|$, the area remains below~$5\\pi$.\n\nFor larger $|t|$, we distinguish two cases. \nIf $\\Sigma_v$ stays a definite distance from the poles, we may apply the\nEuclidean model: a region trapped between two nearly parallel half–spaces of\nseparation $\\varepsilon\\ll1$ has the property that, for any fixed\n$t_0\\sim1$, the boundaries of its $t$-parallel sets ($t\\geqslant t_0$) become Lipschitz graphs over one\nbounding plane with Lipschitz constant $\\to0$ as $\\varepsilon/t_0\\to0$. \nTransferring this to the spherical setting yields that $\\Sigma_{(v,t)}$ is a\nsmall Lipschitz graph over a geodesic sphere and hence has area $<5\\pi$.\n\nIf instead $\\Sigma_v$ lies very close to a pole, its diameter is already\ntiny, and a simple comparison–geometry argument shows that\n$\\Sigma_{(v,t)}$ remains of area $<5\\pi$.\n\nThus in all cases $\\mathrm{Area}(\\Sigma_{(v,t)})<5\\pi$. Since $5\\pi < 2\\pi^2$, no parameter $(v,t)$ with $|v|$ sufficiently close to $1$ can maximize the canonical area. \nThis yields the desired estimate\n\\[\n|v_0| \\leqslant 1 - \\eta_1.\n\\]\n\nWith this uniform control of the conformal parameter, the remaining task is to\nderive a quantitative stability estimate for minimal surfaces. \nAssembling the arguments developed so far leads to our main quantitative\nrigidity theorem:\n\n\\begin{theorem}\\label{Thm: main}\nLet $V=\\underline{v}(\\Sigma,1)$ be an integral $2$-varifold in $\\varmathbb{S}^3$\nwith unit density and mean curvature $H\\in L^2(d\\mu)$, and assume its support\n$\\Sigma$ has genus at least $1$. \nIf the Willmore energy\n\\[\n\\mathcal{W}(V)=\\int_\\Sigma\\Bigl(1+\\tfrac14 |H|^2\\Bigr)\\,d\\mu\n\\]\nsatisfies\n\\[\n\\mathcal{W}(V)\\leqslant 2\\pi^2+\\delta^2,\n\\qquad \\delta>0 \\text{ sufficiently small},\n\\]\nthen, after a suitable conformal transformation of $\\varmathbb{S}^3$, there exists\na homeomorphism\n\\[\nf:\\varmathbb{S}^1\\times\\varmathbb{S}^1 \\longrightarrow \\Sigma \\subset\\varmathbb{S}^3\n\\]\nsuch that in the standard coordinates $(\\theta,\\varphi)$ on\n$\\varmathbb{S}^1\\times\\varmathbb{S}^1$ the pull-back metric takes the form\n\\[\nf^{*}g_{\\varmathbb{S}^3}\n= e^{2u}\\,(a\\, d\\theta^2 + 2 b\\, d\\theta\\, d\\varphi + c\\, d\\varphi^2).\n\\]\n\nLet $f_0:\\varmathbb{S}^1\\times\\varmathbb{S}^1\\to\\varmathbb{S}^3$ denote the Clifford torus\nembedding\n\\[\nf_0(\\theta,\\varphi)\n= \\frac{1}{\\sqrt{2}}(\\cos\\theta,\\;\\sin\\theta,\\;\\cos\\varphi,\\;\\sin\\varphi),\n\\]\nand write its image as $\\varmathbb{T}^2:=f_0(\\varmathbb{S}^1\\times\\varmathbb{S}^1)$. \nThen\n\\[\nf_0^{*} g_{\\varmathbb{S}^3}\n= \\tfrac12 (d\\theta^2+d\\varphi^2).\n\\]\n\nThe following quantitative rigidity estimates hold:\n\\begin{enumerate}\n\n\\item[\\rm(1)]\n\\[\n\\| f - f_0 \\|_{W^{2,2}(\\varmathbb{S}^1\\times\\varmathbb{S}^1)} \\leqslant C\\,\\delta.\n\\]\n\n\\item[\\rm(2)]\n\\[\n\\|u\\|_{L^\\infty(\\varmathbb{S}^1\\times\\varmathbb{S}^1)} \\leqslant C\\,\\delta.\n\\]\n\n\\item[\\rm(3)]\n\\[\n|a-\\tfrac12| + |b| + |c-\\tfrac12| \\leqslant C\\,\\delta.\n\\]\n\n\\end{enumerate}\nHere $C$ is a universal constant independent of $\\Sigma$.\n\\end{theorem}\n\nRecent work of Rupp-Scharrer~\\cite{RS25}, making use of the regularity theorem\ndeveloped in~\\cite{BZ-2022b}, establishes density of smooth surfaces among\nintegral $2$-varifolds with square-integrable second fundamental form and\nWillmore energy below $8\\pi$. \nSince unit density together with $H\\in L^2$ already yields square-integrable\nsecond fundamental form by~\\cite{BZ-2022b}, their result is closely related to\nthe class of varifolds considered here.\n\nIn the appendix we prove a density statement suited to our setting:\ncompactly supported integral $2$-varifolds with unit density and\nsquare-integrable mean curvature can be approximated in $W^{2,2}$ by smooth\nembedded surfaces. \nOur construction provides global bilipschitz control between the limiting\nparametrization $F$ and its smoothings $F_\\eta$, which ensures\nembeddedness of the approximating surfaces without requiring any additional\nsmallness assumptions on the Willmore energy.\n\n\\medskip\n\\noindent\\textbf{Outline of the paper.}\nIn Section~2, we establish the topological stability and qualitative geometric stability for surfaces with Willmore energy sufficiently close to $2\\pi^2$. \nIn particular, we show that such a surface must have genus one and is, after suitable conformal normalization, qualitatively close to the Clifford torus.\n\nIn Section~3, we analyze the Marques-Neves canonical family and prove the key estimate that under the assumption $\\mathcal{W}(\\Sigma)\\leqslant 2\\pi^2+\\delta^2$, one can find a conformal transformation of $\\varmathbb{S}^3$ such that the transformed surface $\\Sigma'$ satisfies\n\\[\n\\mathrm{Area}(\\Sigma')\\geqslant 2\\pi^2 - C\\delta^2,\n\\]\nfor a universal constant $C>0$. \nThis reduces the Willmore stability problem to a quantitative stability statement for minimal surfaces.\n\nSection~4 contains the linearization and stability analysis. \nUsing the estimate from Section~3, we derive the quantitative bounds on the conformal factor, the conformal structure, and the $W^{2,2}$-distance to the Clifford torus, thereby proving Theorem~\\ref{Thm: main}.\n\nFinally, in the Appendix we provide the proof of Proposition~\\ref{away from boundary of conformal group} and we also establish the density and parametrization results for compactly\nsupported integral $2$-varifolds with unit density and $L^2$-mean curvature,\nshowing in particular that such varifolds admit global conformal\nparametrizations and can be approximated by smooth embedded surfaces.", + "sketch": "Using the Marques--Neves five-parameter family $\\{\\Sigma_{(v,t)}\\}$, the key point is to show that if a genus--one surface $\\Sigma$ satisfies $\\mathcal{W}(\\Sigma)\\leqslant 2\\pi^2+\\delta^2$, then after a \\emph{conformal} transformation one gets a surface $\\Sigma'$ with $\\mathrm{Area}(\\Sigma')\\geqslant 2\\pi^2-C\\delta^2$.\n\nThe $2\\pi^2$-Theorem~\\ref{2pi2-theorem} ensures the full $(v,t)$-family contains a slice of area at least $2\\pi^2$, and since $v$ encodes conformal transformations and $t$ is signed distance, the aim is to realize such a high-area slice \\emph{within the conformal} four-parameter subfamily; for this one must control $t$ in terms of the energy gap $\\delta$, so the maximizing slice lies close to $t=0$.\n\nA closer examination of the Heintze--Karcher inequality used by Marques--Neves shows that, once the family is built from the quantitatively normalized surface, the problem reduces to proving the maximizing parameter $(v_0,t_0)$ stays a definite distance away from the boundary of the conformal group, i.e. there exists $\\eta_1>0$ such that $|v_0|\\leqslant 1-\\eta_1$ (reduced in Section~3; technical proof deferred to the Appendix).\n\nGeometric intuition for this uniform bound: as $|v|\\to1$, $\\Sigma_{(v,t)}$ becomes quantitatively close to a geodesic sphere, forcing area strictly below $2\\pi^2$, so such $v$ cannot maximize. More precisely, for $t=0$, $\\Sigma_v=\\Sigma_{(v,0)}$ is Hausdorff close to a geodesic sphere $S$, so $\\mathrm{Area}(\\Sigma_v)$ is close to $\\mathrm{Area}(S)\\le 4\\pi$, and for small $|t|$ one uses\n\\[\n\\bigl|\\mathrm{Area}(\\Sigma_{(v,t)})-\\mathrm{Area}(\\Sigma_v)\\bigr|\\lesssim |t|\\,\\mathcal{W}(\\Sigma_v)\n\\]\nso the area stays $<5\\pi$. For larger $|t|$, either (i) if $\\Sigma_v$ stays away from the poles, an Euclidean-model argument shows $\\Sigma_{(v,t)}$ is a small Lipschitz graph over a geodesic sphere hence has area $<5\\pi$, or (ii) if $\\Sigma_v$ lies close to a pole, comparison geometry gives $\\mathrm{Area}(\\Sigma_{(v,t)})<5\\pi$. Thus for $|v|$ sufficiently close to $1$, $\\mathrm{Area}(\\Sigma_{(v,t)})<5\\pi<2\\pi^2$, proving $|v_0|\\le 1-\\eta_1$.\n\nWith this uniform control of $v$, the remaining task is to derive a quantitative stability estimate for minimal surfaces; the paper then uses the Section~3 area lower bound to reduce Willmore stability to minimal-surface stability, and in Section~4 performs linearization/stability analysis to obtain the quantitative bounds (conformal factor, conformal structure, and $W^{2,2}$-distance to the Clifford torus), thereby proving Theorem~\\ref{Thm: main}.", + "expanded_sketch": "Using the Marques--Neves five-parameter family $\\{\\Sigma_{(v,t)}\\}$, the key point is to show that if a genus--one surface $\\Sigma$ satisfies $\\mathcal{W}(\\Sigma)\\leqslant 2\\pi^2+\\delta^2$, then after a \\emph{conformal} transformation one gets a surface $\\Sigma'$ with $\\mathrm{Area}(\\Sigma')\\geqslant 2\\pi^2-C\\delta^2$.\n\nTo prove the main theorem, we use that the full $(v,t)$-family contains a slice of area at least $2\\pi^2$, and since $v$ encodes conformal transformations and $t$ is signed distance, the aim is to realize such a high-area slice \\emph{within the conformal} four-parameter subfamily; for this one must control $t$ in terms of the energy gap $\\delta$, so the maximizing slice lies close to $t=0$.\n\nA closer examination of the Heintze--Karcher inequality used by Marques--Neves shows that, once the family is built from the quantitatively normalized surface, the problem reduces to proving the maximizing parameter $(v_0,t_0)$ stays a definite distance away from the boundary of the conformal group, i.e. there exists $\\eta_1>0$ such that $|v_0|\\leqslant 1-\\eta_1$ (reduced later; technical proof deferred to an appendix).\n\nGeometric intuition for this uniform bound: as $|v|\\to1$, $\\Sigma_{(v,t)}$ becomes quantitatively close to a geodesic sphere, forcing area strictly below $2\\pi^2$, so such $v$ cannot maximize. More precisely, for $t=0$, $\\Sigma_v=\\Sigma_{(v,0)}$ is Hausdorff close to a geodesic sphere $S$, so $\\mathrm{Area}(\\Sigma_v)$ is close to $\\mathrm{Area}(S)\\le 4\\pi$, and for small $|t|$ one uses\n\\[\n\\bigl|\\mathrm{Area}(\\Sigma_{(v,t)})-\\mathrm{Area}(\\Sigma_v)\\bigr|\\lesssim |t|\\,\\mathcal{W}(\\Sigma_v)\n\\]\nso the area stays $<5\\pi$. For larger $|t|$, either (i) if $\\Sigma_v$ stays away from the poles, an Euclidean-model argument shows $\\Sigma_{(v,t)}$ is a small Lipschitz graph over a geodesic sphere hence has area $<5\\pi$, or (ii) if $\\Sigma_v$ lies close to a pole, comparison geometry gives $\\mathrm{Area}(\\Sigma_{(v,t)})<5\\pi$. Thus for $|v|$ sufficiently close to $1$, $\\mathrm{Area}(\\Sigma_{(v,t)})<5\\pi<2\\pi^2$, proving $|v_0|\\le 1-\\eta_1$.\n\nWith this uniform control of $v$, the remaining task is to derive a quantitative stability estimate for minimal surfaces; the paper then uses the earlier area lower bound to reduce Willmore stability to minimal-surface stability, and later performs linearization/stability analysis to obtain the quantitative bounds (conformal factor, conformal structure, and $W^{2,2}$-distance to the Clifford torus), thereby proving Theorem~\\ref{Thm: main}.", + "expanded_theorem": "[$2\\pi^2$-Theorem, Marques-Neves~\\cite{MN14}]\\label{2pi2-theorem}\nLet $\\Sigma\\subset\\varmathbb{S}^3$ be a smooth embedded surface of genus at least $1$. \nThen its canonical five-parameter family \n\\[\n\\{\\Sigma_{(v,t)}\\}_{(v,t)\\in \\mathring{B}^4\\times[-\\pi,\\pi]}\n\\]\nsatisfies\n\\[\n\\max_{(v,t)\\in \\mathring{B}^4\\times[-\\pi,\\pi]}\n\\mathrm{Area}\\bigl(\\Sigma_{(v,t)}\\bigr)\\;\\geqslant\\; 2\\pi^2.\n\\]\nMoreover, equality holds if and only if $\\Sigma$ is the Clifford torus.", + "theorem_type": [ + "Inequality or Bound", + "Biconditional or Equivalence" + ], + "mcq": { + "question": "Let \\(\\Sigma\\subset \\mathbb{S}^3\\) be a smooth embedded surface of genus at least \\(1\\). For \\(v\\in \\mathring{B}^4\\), define the conformal transformation\n\\[\nF_v(x)=(1-|v|^2)\\frac{x-v}{|x-v|^2}-v.\n\\]\nIf \\(\\mathbb{S}^3\\setminus \\Sigma=A\\cup A^*\\) is the decomposition into connected components, set \\(\\Sigma_v:=F_v(\\Sigma)\\), \\(A_v:=F_v(A)\\), let \\(d_v\\) be the signed distance to \\(\\Sigma_v\\), and define the canonical five-parameter family\n\\[\n\\Sigma_{(v,t)}=\\partial\\{x\\mid d_v(x)0.\\]\nChoose $R$ sufficiently large, so that $\\Omega\\subset B_R(0).$ Denote\n\\[\\bar{u}(x):=\\norm{g}_{L^{\\infty}(\\Omega)}\\frac{R^{2s}}{a}\\psi_R(x), ~\\mbox{with}~ \\psi_R(x):={\\psi(\\frac{x}{R})}.\\]\nThen it can be easily verified that $\\bar{u}$ and $-\\bar{u}$ are super-solution and sub-solution of \\eqref{eq1}, respectively. Therefore, \n\\begin{equation*}\n \\norm{u}_{L^{\\infty}(\\Omega)}\\le C\\norm{g}_{L^\\infty(\\Omega)},\n\\end{equation*}\nFrom the above construction, one can see that in order to obtain \\eqref{hd}, it is necessary that $\\Omega$ be bounded.\nWhen $\\Omega$ is an {\\em unbounded domain}, it is evident that the solution $u$ is required to be globally bounded. This requirement can not be fulfilled in the process of employing the blow-up and rescaling argument to obtain a priori estimates for solutions to a corresponding family of nonlinear fractional equations on {\\em unbounded domains with boundaries}.\n\nThis motivates us to establish a local version of the boundary regularity, in which, instesd of global one, only a local ${L^{\\infty}}$ norm of the solution is involved.\n\\begin{theorem}\\label{bdry C^s}\n Suppose $\\Omega$ is a unbounded domain with locally $C^{1,1}$ boundary, $00$ such that $B_\\varepsilon(x)\\subset \\Omega$. Then\n \\begin{align*}\n A_i(x,R) & = c_{n,s}\\int _{B_R\\backslash B_\\varepsilon(x)}\\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy \\\\\n &+c_{n,s}\\int _{B_\\varepsilon(x)}\\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy\n \\end{align*}\n By dominated convergence theorem, the first term converges to 0 as $i\\to\\infty$. For the second term,\n \\[\n \\lim _{i\\to\\infty} c_{n,s}\\int _{B_\\varepsilon(x)}\\frac{(u-u_i)(x)-(u-u_i)(y)}{|x-y|^{n+2s}}dy\\le c_{n,s}\\lim _{i\\to\\infty}[u-u_i]_{C^{2s+\\beta}(B_\\varepsilon(x))}\\varepsilon ^\\beta = 0.\n \\]\n Hence $\\lim _{i\\to\\infty}A_i(x,R)=0$. Therefore \n \\begin{equation}\n \\lim _{R\\to\\infty}\\lim _{i\\to\\infty}A_i(x,R)=0.\n \\end{equation}\n Then the same argument in \\cite{Du2023blowup} implies $\\lim _{i\\to\\infty}F_i(x,R)$ exists and\n \\[\n \\lim _{R\\to\\infty}\\lim _{i\\to\\infty}F_i(x,R) =\\lim _{R\\to\\infty}\\lim _{i\\to\\infty}F_i(0,R) = :b\\ge 0.\n \\]\n\\end{proof}", + "post_theorem_intro_text_len": 3811, + "post_theorem_intro_text": "The idea of the proof is that we divide a given solution into two parts: the potential part and the harmonic part. The regularity for the potential part is obtained by Proposition \\ref{otonserra}. For the harmonic part $h$, we rewrite it in terms of the Poisson representation formula in balls. Using this explicit expression, we first carry out a detailed analysis to derive an $\\alpha$ power order decay near $\\partial\\Omega$,\n\\begin{equation*}\n |h(x)|\\le C\\{\\lVertu\\rVert_{L^\\infty(\\Omega \\cap B_4)}+\\lVertu\\rVert_{\\cL_{2s}}\\}\\operatorname{dist}(x,\\partial \\Omega)^{\\alpha },\n \\end{equation*}\n where $\\alpha = \\min\\{s,1-s\\}$. Combining this decay estimate and the interior regularity result (Theorem \\ref{Holder thm}), we derive the $C^\\alpha$ boundary regularity. Then by an iteration process, we increase the power $\\alpha$ successively until it reaches the desired power $s$.\n\nAs an important application, we establish a priori estimate for the solutions to \\eqref{main2}. We assume that $\\Omega \\subset \\mathbb{R} ^n$ is an unbounded domain with uniformly $C^{1,1}$ boundary, and $f$ satisfies the following condition:\n\\begin{itemize}\n \\item $f(x,t):\\Omega \\times [0,\\infty)\\to\\mathbb{R}$ is uniformly H\\\"older continuous with respect to $x$ and continuous with respect to $t$.\n\\end{itemize}\n\\begin{theorem}\\label{1A1}\nAssume $1(1-\\alpha)/2$, the radius $\\rho_e(L_\\beta):= \\sup_{\\lambda\\in \\sigma_\\varepsilon(L_\\beta)}\\left\\lvert\\lambda\\right\\rvert$ of the essential spectrum satisfies\n \\[\\rho_e(L_\\beta)\\leq \\rho(\\mathcal{L}_{\\Re(\\beta) +\\alpha}):=\\sup_{\\lambda\\in \\sigma(\\mathcal{L}_{\\beta+\\alpha})}\\left\\lvert\\lambda\\right\\rvert .\\]\n Furthermore, the generalised eigenspaces belonging to eigenvalues $\\lambda$ with $\\left\\lvert\\lambda\\right\\rvert>\\rho_e(L_\\beta)$ consist of analytic functions which extend to generalised eigenfunctions of $\\mathcal{L}_\\beta$.", + "start_pos": 40259, + "end_pos": 40860, + "label": "theorem: radius of essential spectrum" + }, + "ref_dict": { + "equation: Mayer transfer operator": "\\begin{equation}\\label{equation: Mayer transfer operator}\n \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}f\\left(\\frac{1}{n+z}\\right),\n\\end{equation}", + "equation: Mayer transfer extended": "\\begin{equation}\\label{equation: Mayer transfer extended}\n \\mathcal{L}_{\\beta}(f)(z)=f(0)\\zeta_H(2\\beta,z+1)+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}", + "equation: Mayer transfer extended2": "\\begin{equation}\\label{equation: Mayer transfer extended2}\n \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=0}^k\\frac{d^nf}{dz^n}(0)\\frac{\\zeta_H(n+2\\beta,z+1)}{n!}+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}", + "theorem: radius of essential spectrum": "\\begin{theorem}\\label{theorem: radius of essential spectrum}\n On the half-plane $\\Re(\\beta) >(1-\\alpha)/2$, the radius $\\rho_e(L_\\beta):= \\sup_{\\lambda\\in \\sigma_\\varepsilon(L_\\beta)}\\abs{\\lambda}$ of the essential spectrum satisfies\n \\[\\rho_e(L_\\beta)\\leq \\rho(\\mathcal{L}_{\\Re(\\beta) +\\alpha}):=\\sup_{\\lambda\\in \\sigma(\\mathcal{L}_{\\beta+\\alpha})}\\abs{\\lambda} .\\]\n Furthermore, the generalised eigenspaces belonging to eigenvalues $\\lambda$ with $\\abs{\\lambda}>\\rho_e(L_\\beta)$ consist of analytic functions which extend to generalised eigenfunctions of $\\mathcal{L}_\\beta$. \n\\end{theorem}", + "remark: abstract": "\\begin{remark}\\label{remark: abstract}\n The zeroes of $Z(s)$ on the line $s=1/2+it$ are those for which there are eigenvalues of $\\Delta$ called \\enquote{Maass cusp forms} with eigenvalue $s(1-s)$. Furthermore the multiplicity of the zeroes and the eigenspaces of $\\Delta$ match. The other zeroes of $Z(s)$ are a simple zero at $s=1$ and at the values of $s$ for which $2s$ is a nontrivial zero of the Riemann zeta function $\\zeta$. Since $\\rho(\\mathcal{L}_a)<1$ when $a>1$, we see that the eigenspaces of $\\mathcal{L}_\\beta$ and $L_\\beta$ for the eigenvalues $\\lambda=\\pm 1$ are identical when $1>\\alpha>1-\\Re(\\beta)$. In particular, the eigenspaces associated to the Maass cusp forms are preserved when $\\alpha >1/2$, and the eigenspaces associated to the nontrivial zeroes of the Riemann zeta function\\footnote{In fact, the zeroes of $Z(s)$ associated with the nontrivial zeroes of $\\zeta$ always occur due to $1$ being an eigenvalue of $\\mathcal{L}_\\beta$.} to the right of the critical line are preserved when $\\alpha=3/4$.\n\\end{remark}" + }, + "pre_theorem_intro_text_len": 4494, + "pre_theorem_intro_text": "Let $G:(0,1)\\to [0,1)$ be the Gauss map \n\\begin{equation}\\label{equation: gauss map}\n G(x) = \\frac{1}{z}-\\left\\lfloor\\frac{1}{z}\\right\\rfloor.\n\\end{equation}\nWe associate to $G$ the following family of transfer operators which we formally define as acting on some space of functions $f:[0,1]\\to \\mathbb{C}$:\n\\begin{equation}\\label{equation: Mayer transfer operator}\n \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}f\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\nwhere $\\beta \\in \\{z\\in \\mathbb{C}: \\Re(\\beta)>1/2\\}$. When $\\beta=1$, this is the familiar Ruelle-Perron-Frobenius operator for $G$ and can formally be interpreted as the right-adjoint of the Koopman operator $f\\mapsto f\\circ G$. The operators in the case $\\beta \\neq 1$ were first introduced by Mayer \\cite{Mayer90}, who considered them acting on the Banach space of functions $A_\\infty(D)$ which are holomorphic on $D:=\\{z\\in\\mathbb{C}: \\left\\lvertz-1\\right\\rvert< 3/2\\}$ with continuous extension to $\\overline{D}$. \n\nFor this choice of Banach space, Mayer also showed that the operator-valued function $\\beta\\mapsto \\mathcal{L}_{\\beta}$ admits a meromorphic continuation on $\\mathbb{C}$, with simple poles at $\\beta=k/2$ for $k\\in\\{1,0,-1,-2,\\ldots\\}$. The continuation for $\\Re(\\beta) >0$ can be obtained by rewriting $f=f(0)+f^\\star$ with $f^\\star:=f-f(0)$ and noting that\n\\begin{equation}\\label{equation: Mayer transfer extended}\n \\mathcal{L}_{\\beta}(f)(z)=f(0)\\zeta_H(2\\beta,z+1)+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\nwhere $\\zeta_H(s,z)=\\sum_{n=0}^\\infty (n+z)^{-s}$ is the Hurwitz zeta function, which has an analytic continuation for all $s\\neq 1$ and a simple pole of residue $1$ at $s=1$. In fact, it can further be analytically continued to $\\beta \\geq -k/2$ for any $k\\in\\mathbb{N}$ by noticing that\n\\begin{equation}\\label{equation: Mayer transfer extended2}\n \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=0}^k\\frac{d^nf}{dz^n}(0)\\frac{\\zeta_H(n+2\\beta,z+1)}{n!}+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\nwhere $f^\\star(z)=f(z)- \\sum_{n=0}^k\\frac{d^nf}{d^nz}(0)z^n/n!$.\nFurthermore $\\mathcal{L}_{\\beta}$ is a nuclear operator on $A_\\infty(D)$ of order $0$ for all $\\beta$. \n\nThe Mayer transfer operator encodes geometric information about the modular surface, which is the quotient orbifold $\\mathbf{M}$ obtained from considering the Poincaré upper half plane $\\mathbb{H}:\\{x+iy:x,y\\in \\mathbb{R}, y>0\\}$ equipped with the usual Riemannian metric $ds^2=(dx^2+dy^2)/y^2$ and identifying points via equivalence relation $\\mathbb{H}\\ni z\\sim (az+b)/(cz+d)$ for all $a,b,c,d\\in\\mathbb{Z}$ with $ad-bc=1$. Since $\\mathcal{L}_{\\beta}$ is a nuclear operator, the Fredholm determinants $\\det(1-\\mathcal{L}_{\\beta})$ and $\\det(1+\\mathcal{L}_{\\beta})$ are well-defined. These relate to Selberg Zeta function $Z(s)$ by the formula \n\\begin{equation}\\label{equation: zeta functions}\n Z(s)= \\det(1-\\mathcal{L}_s)(1+\\mathcal{L}_s),\n\\end{equation}\nsee \\cite{Mayer91}. \nThe Selberg zeta function is closely connected with the spectral theory of the Laplace-Beltrami operator $\\Delta:=-y^2(d^2/dx^2+d^2/dy^2)$ \\cite{Iwaniec}. Equation \\eqref{equation: Mayer transfer extended} is thus a way of connecting the spectral theory of $\\Delta$ with that of $\\mathcal{L}_\\beta$. This connection has been elaborated upon by Lewis, Zagier, Mayer and Chang \\cite{Lewis97,ZagierLewis,MayerChang96,MayerChang98}.\n\nFrom a dynamical perspective, it is interesting to see to what extent the aforementioned spectral properties of $\\mathcal{L}_s$ survive when we consider e.g. Hölder continuous functions instead. For $\\alpha\\in (0,1)$, let $C^\\alpha([0,1])$ denote the space of $\\alpha$-Hölder functions on $[0,1]$. If $\\alpha = k+\\eta$ with $k\\in\\mathbb{N}$ and $\\eta\\in [0,1)$, let $C^\\alpha([0,1])$ be the set of functions for which the $k$-th derivative exists and is continuous or $\\eta$-Hölder if $\\eta=0$ or $\\eta\\in (0,1)$ respectively. Denote by $L_\\beta$ the operator \\eqref{equation: Mayer transfer extended2} acting on $C^\\alpha([0,1])$, which we shall show to be well-defined and continuous for large enough $\\alpha$. These operators are not compact, but we can decompose the spectrum $\\sigma(L_\\beta)$ into the essential spectrum $\\sigma_e(L_\\beta)$ and the discrete spectrum $\\sigma_{\\mathrm{disc}}(L_\\beta)$ which consists of isolated eigenvalues of finite algebraic multiplicity. We prove the following.", + "context": "Let $G:(0,1)\\to [0,1)$ be the Gauss map \n\\begin{equation}\\label{equation: gauss map}\n G(x) = \\frac{1}{z}-\\left\\lfloor\\frac{1}{z}\\right\\rfloor.\n\\end{equation}\nWe associate to $G$ the following family of transfer operators which we formally define as acting on some space of functions $f:[0,1]\\to \\mathbb{C}$:\n\\begin{equation}\\label{equation: Mayer transfer operator}\n \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}f\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\nwhere $\\beta \\in \\{z\\in \\mathbb{C}: \\Re(\\beta)>1/2\\}$. When $\\beta=1$, this is the familiar Ruelle-Perron-Frobenius operator for $G$ and can formally be interpreted as the right-adjoint of the Koopman operator $f\\mapsto f\\circ G$. The operators in the case $\\beta \\neq 1$ were first introduced by Mayer \\cite{Mayer90}, who considered them acting on the Banach space of functions $A_\\infty(D)$ which are holomorphic on $D:=\\{z\\in\\mathbb{C}: \\left\\lvertz-1\\right\\rvert< 3/2\\}$ with continuous extension to $\\overline{D}$.\n\nFor this choice of Banach space, Mayer also showed that the operator-valued function $\\beta\\mapsto \\mathcal{L}_{\\beta}$ admits a meromorphic continuation on $\\mathbb{C}$, with simple poles at $\\beta=k/2$ for $k\\in\\{1,0,-1,-2,\\ldots\\}$. The continuation for $\\Re(\\beta) >0$ can be obtained by rewriting $f=f(0)+f^\\star$ with $f^\\star:=f-f(0)$ and noting that\n\\begin{equation}\\label{equation: Mayer transfer extended}\n \\mathcal{L}_{\\beta}(f)(z)=f(0)\\zeta_H(2\\beta,z+1)+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\nwhere $\\zeta_H(s,z)=\\sum_{n=0}^\\infty (n+z)^{-s}$ is the Hurwitz zeta function, which has an analytic continuation for all $s\\neq 1$ and a simple pole of residue $1$ at $s=1$. In fact, it can further be analytically continued to $\\beta \\geq -k/2$ for any $k\\in\\mathbb{N}$ by noticing that\n\\begin{equation}\\label{equation: Mayer transfer extended2}\n \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=0}^k\\frac{d^nf}{dz^n}(0)\\frac{\\zeta_H(n+2\\beta,z+1)}{n!}+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\nwhere $f^\\star(z)=f(z)- \\sum_{n=0}^k\\frac{d^nf}{d^nz}(0)z^n/n!$.\nFurthermore $\\mathcal{L}_{\\beta}$ is a nuclear operator on $A_\\infty(D)$ of order $0$ for all $\\beta$.\n\nThe Mayer transfer operator encodes geometric information about the modular surface, which is the quotient orbifold $\\mathbf{M}$ obtained from considering the Poincaré upper half plane $\\mathbb{H}:\\{x+iy:x,y\\in \\mathbb{R}, y>0\\}$ equipped with the usual Riemannian metric $ds^2=(dx^2+dy^2)/y^2$ and identifying points via equivalence relation $\\mathbb{H}\\ni z\\sim (az+b)/(cz+d)$ for all $a,b,c,d\\in\\mathbb{Z}$ with $ad-bc=1$. Since $\\mathcal{L}_{\\beta}$ is a nuclear operator, the Fredholm determinants $\\det(1-\\mathcal{L}_{\\beta})$ and $\\det(1+\\mathcal{L}_{\\beta})$ are well-defined. These relate to Selberg Zeta function $Z(s)$ by the formula \n\\begin{equation}\\label{equation: zeta functions}\n Z(s)= \\det(1-\\mathcal{L}_s)(1+\\mathcal{L}_s),\n\\end{equation}\nsee \\cite{Mayer91}. \nThe Selberg zeta function is closely connected with the spectral theory of the Laplace-Beltrami operator $\\Delta:=-y^2(d^2/dx^2+d^2/dy^2)$ \\cite{Iwaniec}. Equation \\eqref{equation: Mayer transfer extended} is thus a way of connecting the spectral theory of $\\Delta$ with that of $\\mathcal{L}_\\beta$. This connection has been elaborated upon by Lewis, Zagier, Mayer and Chang \\cite{Lewis97,ZagierLewis,MayerChang96,MayerChang98}.\n\nFrom a dynamical perspective, it is interesting to see to what extent the aforementioned spectral properties of $\\mathcal{L}_s$ survive when we consider e.g. Hölder continuous functions instead. For $\\alpha\\in (0,1)$, let $C^\\alpha([0,1])$ denote the space of $\\alpha$-Hölder functions on $[0,1]$. If $\\alpha = k+\\eta$ with $k\\in\\mathbb{N}$ and $\\eta\\in [0,1)$, let $C^\\alpha([0,1])$ be the set of functions for which the $k$-th derivative exists and is continuous or $\\eta$-Hölder if $\\eta=0$ or $\\eta\\in (0,1)$ respectively. Denote by $L_\\beta$ the operator \\eqref{equation: Mayer transfer extended2} acting on $C^\\alpha([0,1])$, which we shall show to be well-defined and continuous for large enough $\\alpha$. These operators are not compact, but we can decompose the spectrum $\\sigma(L_\\beta)$ into the essential spectrum $\\sigma_e(L_\\beta)$ and the discrete spectrum $\\sigma_{\\mathrm{disc}}(L_\\beta)$ which consists of isolated eigenvalues of finite algebraic multiplicity. We prove the following.\n\n\\begin{equation}\\label{equation: Mayer transfer extended}\n \\mathcal{L}_{\\beta}(f)(z)=f(0)\\zeta_H(2\\beta,z+1)+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\n\n\\begin{equation}\\label{equation: Mayer transfer extended2}\n \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=0}^k\\frac{d^nf}{dz^n}(0)\\frac{\\zeta_H(n+2\\beta,z+1)}{n!}+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}", + "full_context": "Let $G:(0,1)\\to [0,1)$ be the Gauss map \n\\begin{equation}\\label{equation: gauss map}\n G(x) = \\frac{1}{z}-\\left\\lfloor\\frac{1}{z}\\right\\rfloor.\n\\end{equation}\nWe associate to $G$ the following family of transfer operators which we formally define as acting on some space of functions $f:[0,1]\\to \\mathbb{C}$:\n\\begin{equation}\\label{equation: Mayer transfer operator}\n \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}f\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\nwhere $\\beta \\in \\{z\\in \\mathbb{C}: \\Re(\\beta)>1/2\\}$. When $\\beta=1$, this is the familiar Ruelle-Perron-Frobenius operator for $G$ and can formally be interpreted as the right-adjoint of the Koopman operator $f\\mapsto f\\circ G$. The operators in the case $\\beta \\neq 1$ were first introduced by Mayer \\cite{Mayer90}, who considered them acting on the Banach space of functions $A_\\infty(D)$ which are holomorphic on $D:=\\{z\\in\\mathbb{C}: \\left\\lvertz-1\\right\\rvert< 3/2\\}$ with continuous extension to $\\overline{D}$.\n\nFor this choice of Banach space, Mayer also showed that the operator-valued function $\\beta\\mapsto \\mathcal{L}_{\\beta}$ admits a meromorphic continuation on $\\mathbb{C}$, with simple poles at $\\beta=k/2$ for $k\\in\\{1,0,-1,-2,\\ldots\\}$. The continuation for $\\Re(\\beta) >0$ can be obtained by rewriting $f=f(0)+f^\\star$ with $f^\\star:=f-f(0)$ and noting that\n\\begin{equation}\\label{equation: Mayer transfer extended}\n \\mathcal{L}_{\\beta}(f)(z)=f(0)\\zeta_H(2\\beta,z+1)+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\nwhere $\\zeta_H(s,z)=\\sum_{n=0}^\\infty (n+z)^{-s}$ is the Hurwitz zeta function, which has an analytic continuation for all $s\\neq 1$ and a simple pole of residue $1$ at $s=1$. In fact, it can further be analytically continued to $\\beta \\geq -k/2$ for any $k\\in\\mathbb{N}$ by noticing that\n\\begin{equation}\\label{equation: Mayer transfer extended2}\n \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=0}^k\\frac{d^nf}{dz^n}(0)\\frac{\\zeta_H(n+2\\beta,z+1)}{n!}+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\nwhere $f^\\star(z)=f(z)- \\sum_{n=0}^k\\frac{d^nf}{d^nz}(0)z^n/n!$.\nFurthermore $\\mathcal{L}_{\\beta}$ is a nuclear operator on $A_\\infty(D)$ of order $0$ for all $\\beta$.\n\nThe Mayer transfer operator encodes geometric information about the modular surface, which is the quotient orbifold $\\mathbf{M}$ obtained from considering the Poincaré upper half plane $\\mathbb{H}:\\{x+iy:x,y\\in \\mathbb{R}, y>0\\}$ equipped with the usual Riemannian metric $ds^2=(dx^2+dy^2)/y^2$ and identifying points via equivalence relation $\\mathbb{H}\\ni z\\sim (az+b)/(cz+d)$ for all $a,b,c,d\\in\\mathbb{Z}$ with $ad-bc=1$. Since $\\mathcal{L}_{\\beta}$ is a nuclear operator, the Fredholm determinants $\\det(1-\\mathcal{L}_{\\beta})$ and $\\det(1+\\mathcal{L}_{\\beta})$ are well-defined. These relate to Selberg Zeta function $Z(s)$ by the formula \n\\begin{equation}\\label{equation: zeta functions}\n Z(s)= \\det(1-\\mathcal{L}_s)(1+\\mathcal{L}_s),\n\\end{equation}\nsee \\cite{Mayer91}. \nThe Selberg zeta function is closely connected with the spectral theory of the Laplace-Beltrami operator $\\Delta:=-y^2(d^2/dx^2+d^2/dy^2)$ \\cite{Iwaniec}. Equation \\eqref{equation: Mayer transfer extended} is thus a way of connecting the spectral theory of $\\Delta$ with that of $\\mathcal{L}_\\beta$. This connection has been elaborated upon by Lewis, Zagier, Mayer and Chang \\cite{Lewis97,ZagierLewis,MayerChang96,MayerChang98}.\n\nFrom a dynamical perspective, it is interesting to see to what extent the aforementioned spectral properties of $\\mathcal{L}_s$ survive when we consider e.g. Hölder continuous functions instead. For $\\alpha\\in (0,1)$, let $C^\\alpha([0,1])$ denote the space of $\\alpha$-Hölder functions on $[0,1]$. If $\\alpha = k+\\eta$ with $k\\in\\mathbb{N}$ and $\\eta\\in [0,1)$, let $C^\\alpha([0,1])$ be the set of functions for which the $k$-th derivative exists and is continuous or $\\eta$-Hölder if $\\eta=0$ or $\\eta\\in (0,1)$ respectively. Denote by $L_\\beta$ the operator \\eqref{equation: Mayer transfer extended2} acting on $C^\\alpha([0,1])$, which we shall show to be well-defined and continuous for large enough $\\alpha$. These operators are not compact, but we can decompose the spectrum $\\sigma(L_\\beta)$ into the essential spectrum $\\sigma_e(L_\\beta)$ and the discrete spectrum $\\sigma_{\\mathrm{disc}}(L_\\beta)$ which consists of isolated eigenvalues of finite algebraic multiplicity. We prove the following.\n\n\\begin{equation}\\label{equation: Mayer transfer extended}\n \\mathcal{L}_{\\beta}(f)(z)=f(0)\\zeta_H(2\\beta,z+1)+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\n\n\\begin{equation}\\label{equation: Mayer transfer extended2}\n \\mathcal{L}_{\\beta}(f)(z)=\\sum_{n=0}^k\\frac{d^nf}{dz^n}(0)\\frac{\\zeta_H(n+2\\beta,z+1)}{n!}+\\sum_{n=1}^\\infty \\frac{1}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right),\n\\end{equation}\n\nMany of the usual theorems in complex analysis continue to hold for Banach or topological vector space-valued functions. In particular, holomorphic implies infinitely differentiable with a locally converging Taylor expansion. The unique extension theorem also continues to hold.\nFinally, note that it is straightforward to show that $\\beta\\to L_\\beta$ is a meromorphic function on $\\Re(\\beta)>(1-\\alpha)/2$ for any $\\alpha\\in\\mathbb{R}_{>0}$ with derivative \n\\[ \\left(\\frac{d L_\\beta}{d\\beta}\\right)(f)(z)= -\\sum_{n=0}^k(n+2\\beta)\\frac{d^nf}{dz^n}(0)\\frac{\\zeta_H(n+2\\beta+1,z+1)}{n!} -2 \\sum_{n=1}^\\infty \\frac{\\ln(n+z)}{(n+z)^{2\\beta}}(f^\\star)\\left(\\frac{1}{n+z}\\right).\\]\n\\section{Uniformly Bounded Essential Spectrum}\\label{section: uniformly bounded essential spectrum}\nBy \\eqref{equation: Nussbaum}, we can estimate the essential spectral radius by finding for each $l\\in \\mathbb{N}$ a compact operator $T'$ for which $\\|L_\\beta^l - T' \\|$ is small. A first naive approach would be to consider the operator $P$ which sends $f$ to a piecwise linear approximation that interpolates $f$ on the points $0,\\frac{1}{N},\\frac{2}{N},\\cdots,1$ and considering $\\|L_\\beta^l-L_\\beta^l\\circ P\\|$ as $N\\to \\infty$. For $\\Re(\\beta)>0$, we indeed have by Lemma \\ref{lemma: mayer operator inequalities} that for any $\\varepsilon>0$ there exists some $N$ for which $\\lvert (L_\\beta-L_\\beta\\circ P) (f)\\rvert_\\alpha(x)\\leq \\mathfrak{L}_{\\Re(\\beta)+\\alpha}((1+\\varepsilon)|f-P(f)|_\\alpha)(x)+\\varepsilon$. However, we do not have sufficient control over the $\\|\\cdot\\|_\\infty$-norm. We remedy this by considering a \\textit{countable} interpolation which ensures $(L_\\beta^l\\circ P)f$ remains a (nonlinear) interpolation of\n$L_\\beta^l f$.\n\nLet us now prove most of Theorem \\ref{theorem: radius of essential spectrum}. \n\\begin{theorem}\\label{theorem: spectral theorem}\nLet $0<\\alpha<1 $. Then for all $\\beta$ with $\\Re(\\beta)>(1-\\alpha)/2$ the transfer operator $L_\\beta$ satisfies \n\\[\\rho_e(L_\\beta) \\leq \\rho(\\mathfrak{L}_{\\Re(\\beta)+\\alpha}).\\]\n\\end{theorem}\n\\begin{proof}\n By Nussbaum's formula and Proposition \\ref{proposition: compact operator}, it suffices to prove that there is some constant $C''>0$ for which \\[\\limsup_{N\\to\\infty} \\|L_{\\beta}^l -L_{\\beta}^l \\circ P_{l,N} \\| \\leq C''\\left\\|\\mathfrak{L}_{\\Re(\\beta)+\\alpha}^l\\right\\| \\text{ for all } l.\\]\n\n\\end{proof}\n\\begin{remark}\\label{remark: spectral radius}\nBy using the spectral radius formula $\\rho(\\mathfrak{L}_{\\Re(\\beta)+\\alpha})=\\limsup_{l\\to\\infty}\\|\\mathfrak{L}_{\\Re(\\beta)+\\alpha}^l\\|^{1/l}$ and bounding every function $f\\in L^\\infty([0,1])$ by a constant, we obtain that $\\rho(\\mathfrak{L}_{\\Re(\\beta)+\\alpha})\\leq \\rho(\\mathcal{L}_{\\Re(\\beta)+\\alpha})$ and hence that \n\\[\\rho(\\mathfrak{L}_{\\Re(\\beta)+\\alpha}) = \\lambda_1(\\Re(\\beta)+\\alpha),\\]\nwhere $\\lambda_1(t)$ for $t>1/2$ is the eigenvalue of maximum modulus of $\\mathcal{L}_{t}$. The eigenvalue is guaranteed to be simple and positive by the Perron-Frobenius theorem. In fact $\\lambda_1(t)$ is a real-analytic and decreasing function of $t$ with $\\lambda(1)=1$, see \\cite{Mayer90,Vallee98}. \n\\end{remark}\nTheorem \\ref{theorem: radius of essential spectrum} for $\\alpha \\in (0,1)$ then follows from Remark \\ref{remark: spectral radius} and from the following corollary of Theorem \\ref{theorem: spectral theorem}.\n\\begin{corollary}\\label{Corollary: Ruelle}\n Let $\\lambda \\in \\mathbb{C}$ satisfy $\\abs{\\lambda}>\\rho(\\mathfrak{L}_{\\Re(\\beta)+\\alpha})$. Then $\\lambda$ is an eigenvalue of $L_\\beta$ if and only if it is an eigenvalue of $\\mathcal{L}_\\beta$. In this case the respective eigenspaces $E_\\lambda^\\alpha$ and $E_\\lambda^\\omega$ are equal. \n\\end{corollary}\nThis is an example of a more general principle for operators on Banach spaces, see e.g. \\cite{Grabiner}. For our purposes, we may directly adapt a proof due to Ruelle (Corollary 3.3 in \\cite{Ruelle89}), which we provide in the appendix.\n\n\\end{proof}\nWe see by Proposition $4.1$ that a generic solution of \\eqref{equation: three term functional} in $C^\\alpha((-1,\\infty))$ satisfies $f= O(z^{1-2\\Re(\\beta)})$ if $\\Re(\\beta)>(1-\\alpha)/2$. On the other hand, we see that the condition $Q=0$ in \\eqref{equation: Qinfinity} or in \\eqref{equation: Q0} is equivalent to the condition $L_\\beta f= \\lambda f$, which by Theorem \\ref{theorem: radius of essential spectrum} implies $f$ extends to a holomorphic function in $\\mathbb{C}\\setminus (-\\infty,-1]$ if $\\abs{\\lambda}\\leq \\rho(\\mathcal{L}_{\\Re(\\beta)+\\alpha})$. For example, we have the following corollary of Proposition \\ref{equation: proposition: }, which we can view as a stronger version of the Bootstrapping procedure by Lewis and Zagier in \\cite{ZagierLewis}.\n\\begin{corollary}\n Let $\\Re(\\beta)=1/2$. Suppose $f:(-1,\\infty)\\to \\mathbb{C}$ is some solution of \\eqref{equation: three term functional} for $\\abs{\\lambda} = 1$. Assume furthermore it satisfies the mild growth condition $f(z)= C_0x^{1-2\\beta}+o(1)$ as $z\\to \\infty$.\n Then if $f$ is $1/2+\\varepsilon$-Hölder for some $\\varepsilon>0$, it is automatically holomorphically extendable to the cut plane $\\mathbb{C}\\setminus (-\\infty,-1]$.\n\\end{corollary}\nThis corollary immediately follows from Theorem \\ref{theorem: radius of essential spectrum} and the fact that $\\rho(\\mathcal{L}_a)<1$ if $a>1$. We remark that when $\\Re(\\beta)=1/2$ and $\\lambda =1$ or $-1$, the space of eigenvalues of $\\mathcal{L}_{\\beta}$ is in bijection with the space of even, respectively uneven Maass cusp forms on the modular surface with eigenvalue $\\beta(1-\\beta)$ and we have that $C_0=0$, i.e. $f(z)= o(1)$ as $z\\to \\infty$, see \\cite{Lewis97, ZagierLewis}.\n\n\\begin{proposition}\\label{Proposition: Constructing Q}\n Let $\\alpha>0$, $\\lambda\\in\\mathbb{C}\\setminus\\{0\\}$ and let $Q\\in C^\\alpha(\\mathbb{R}/\\mathbb{Z})$.\n For all $\\beta$ for which $\\Re(\\beta)> (1-\\alpha)/2$ and \\[r(\\mathcal{L}_{\\Re(\\beta)+\\alpha})< \\abs{\\lambda},\\]\n the function\n \\[f=(1 - \\lambda^{-1}L_\\beta)^{-1}Q,\\]\n is a meromorphic expression over all $\\beta$ satisfying the conditions in this proposition. Furthermore $f$ satisfies \\eqref{equation: three term functional}, \\eqref{equation: Qinfinity} and \\eqref{equation: Q0}.\n\\end{proposition}\n\\begin{proof}\n We interpret $(\\lambda - L_\\beta)^{-1}$ in the sense of Remark \\ref{remark: extending L_beta}. More specifically, for $L_\\beta$ acting on $ C^r([a,(1-a)^{-1}])$, the expression $(1 - \\lambda^{-1}L_\\beta)^{-1}\\left(\\rstr{Q}{[a,(1-a)^{-1}]}\\right)$ is meromorphic and satisfies \n\\begin{equation}\\label{equation: expansion for resolvent}\n \\begin{split}\n &(1 - \\lambda^{-1}L_\\beta)^{-1}\\left(\\rstr{Q}{[a,(1-a)^{-1}]}\\right)(z)=\\sum_{n=0}^\\infty \\lambda^{-n}L_\\beta^n\\left(\\rstr{Q}{[a,(1-a)^{-1}]}\\right)(z).\\\\\n &= Q(z)+\\sum_{n=1}^\\infty \\lambda^{-n}\\sum_{a_1,\\ldots,a_n\\in\\mathbb{N}} Q([a_1,\\ldots,a_n+z])\\Pi_{k=1}^n[a_k,\\ldots,a_n+z]^{2\\beta}\n \\end{split}\n\\end{equation}\nif $\\Re(\\beta)$ is large enough. By Theorem \\ref{theorem: radius of essential spectrum}, we see that $(1 - \\lambda^{-1}L_\\beta)^{-1}$ is a meromorphic operator-valued function for all $a\\leq1$. Hence the functions $(1 - \\lambda^{-1}L_\\beta)^{-1}\\left(\\rstr{Q}{[a,(1-a)^{-1}]}\\right)$ are \nmeromorphic functions over the half-plane consisting of all $\\beta$ satisfying the conditions in the proposition. We see by the second line of \\eqref{equation: expansion for resolvent} and uniqueness of analytic continuation that these functions agree for different $a$ on the intersection of their domains. Hence we may define $(1 - \\lambda^{-1}L_\\beta)^{-1}Q(z)$ for any $z\\in (-1,\\infty)$ by choosing $a$ in \\eqref{equation: expansion for resolvent} small enough.", + "post_theorem_intro_text_len": 2204, + "post_theorem_intro_text": "The statement in the abstract concerning Maass cusp forms and nontrivial zeroes of the Riemann zeta function, which we make more precise in Remark \\ref{remark: abstract}, turns out to follow immediately from this theorem.\n\nComparing the behaviour of transfer operators for different levels of regularity is a somewhat common theme in dynamics. One typically expects for a transfer operator associated to a sufficiently \\enquote{nice} expanding map that the essential spectral radius vanishes for the operator acting on spaces of increasingly smooth functions, see e.g. \\cite{Butterley}. In a sense the above theorem is therefore not too suprising. Indeed, we shall show that if $\\alpha$ is a natural number, the proof of Theorem \\ref{theorem: radius of essential spectrum} is a mundane application of standard techniques. \n\n However, the proof of the case $1/2<\\alpha < 1$ is perhaps more interesting. In that case, we still obtain uniformly bounded estimates for $\\rho_e(L_\\beta)$ on vertical lines $\\Re(\\beta)=\\sigma$ even when $\\sigma < 1/2$, where the function $\\beta\\mapsto L_\\beta$ is unbounded. However, by \\eqref{equation: Mayer transfer operator}, the unboundedness in $\\beta$ is due to the contribution rank one operator $f\\mapsto f(0)\\zeta_H(2\\beta,z+1)$, which does not contribute to the essential spectral radius. Nevertheless, we do not obtain this result using a Doeblin-Fortet-Lasota-Yorke inequality, so our proof is substantially different from the standard approaches to these type of problems. \n\nIn Section \\ref{section: three-term functional equation}, we interpret this result in terms of Lewis' three-term functional equation. In particular, we can interpret this as an extension of the \\enquote{bootstrapping} result of Lewis and Zagier, who proved in \\cite{ZagierLewis} that real-analytic solutions of this equation on $(1,\\infty)$ satisfying certain growth conditions automatically extend to holomorphic functions on the cut plane $\\mathbb{C}\\backslash(-\\infty,-1]$. We show that for $\\Re(\\beta)>0$, we may even replace real-analyticity with a \\textit{Hölder condition} and obtain that it is still the restriction of a holomorphic function on $\\mathbb{C}\\backslash(-\\infty,-1]$.", + "sketch": "If $\\alpha$ is a natural number, “the proof of Theorem \\ref{theorem: radius of essential spectrum} is a mundane application of standard techniques.”\n\nFor $1/2<\\alpha<1$, the proof is “substantially different from the standard approaches”: one “still obtain[s] uniformly bounded estimates for $\\rho_e(L_\\beta)$ on vertical lines $\\Re(\\beta)=\\sigma$ even when $\\sigma<1/2$, where the function $\\beta\\mapsto L_\\beta$ is unbounded,” and this is because “the unboundedness in $\\beta$ is due to the contribution rank one operator $f\\mapsto f(0)\\zeta_H(2\\beta,z+1)$, which does not contribute to the essential spectral radius.” The result is “not obtain[ed]… using a Doeblin-Fortet-Lasota-Yorke inequality.”", + "expanded_sketch": "If $\\alpha$ is a natural number, the proof of the main theorem is a mundane application of standard techniques.\n\nFor $1/2<\\alpha<1$, the proof is substantially different from the standard approaches: one still obtains uniformly bounded estimates for $\\rho_e(L_\\beta)$ on vertical lines $\\Re(\\beta)=\\sigma$ even when $\\sigma<1/2$, where the function $\\beta\\mapsto L_\\beta$ is unbounded, and this is because the unboundedness in $\\beta$ is due to the contribution rank one operator $f\\mapsto f(0)\\zeta_H(2\\beta,z+1)$, which does not contribute to the essential spectral radius. The result is not obtained using a Doeblin-Fortet-Lasota-Yorke inequality.", + "expanded_theorem": "\\label{theorem: radius of essential spectrum}\n On the half-plane $\\Re(\\beta) >(1-\\alpha)/2$, the radius $\\rho_e(L_\\beta):= \\sup_{\\lambda\\in \\sigma_\\varepsilon(L_\\beta)}\\left\\lvert\\lambda\\right\\rvert$ of the essential spectrum satisfies\n \\[\\rho_e(L_\\beta)\\leq \\rho(\\mathcal{L}_{\\Re(\\beta) +\\alpha}):=\\sup_{\\lambda\\in \\sigma(\\mathcal{L}_{\\beta+\\alpha})}\\left\\lvert\\lambda\\right\\rvert .\\]\n Furthermore, the generalised eigenspaces belonging to eigenvalues $\\lambda$ with $\\left\\lvert\\lambda\\right\\rvert>\\rho_e(L_\\beta)$ consist of analytic functions which extend to generalised eigenfunctions of $\\mathcal{L}_\\beta$.", + "theorem_type": [ + "Inequality or Bound", + "Universal" + ], + "mcq": { + "question": "Let \u0007lpha>0, write \u0007lpha=k+\u0003b7 with k\\in\\mathbb N and \\eta\\in[0,1), and let C^\\alpha([0,1]) be the usual H\u0000f6lder space. For \\beta\\in\\mathbb C, define the analytically continued Mayer transfer operator L_\\beta on C^\\alpha([0,1]) by\n\\[\nL_{\\beta}(f)(z)=\\sum_{n=0}^k\\frac{f^{(n)}(0)}{n!}\\,\\zeta_H(n+2\\beta,z+1)+\\sum_{m=1}^\\infty \\frac{1}{(m+z)^{2\\beta}}\\,f^*\\!\\left(\\frac{1}{m+z}\\right),\n\\]\nwhere\n\\[\nf^*(z)=f(z)-\\sum_{n=0}^k\\frac{f^{(n)}(0)}{n!}z^n,\n\\]\nand \\zeta_H(s,z)=\\sum_{m=0}^\\infty (m+z)^{-s} is the Hurwitz zeta function. Let \\mathcal L_\\gamma denote the corresponding holomorphic Mayer transfer operator\n\\[\n\\mathcal L_\\gamma(g)(z)=\\sum_{m=1}^\\infty \\frac{1}{(m+z)^{2\\gamma}}\\,g\\!\\left(\\frac{1}{m+z}\\right).\n\\]\nIf \\sigma_e(L_\\beta) denotes the essential spectrum of L_\\beta and\n\\[\n\\rho_e(L_\\beta):=\\sup_{\\lambda\\in\\sigma_e(L_\\beta)}|\\lambda|,\n\\qquad\n\\rho(\\mathcal L_{\\Re(\\beta)+\\alpha}):=\\sup_{\\mu\\in\\sigma(\\mathcal L_{\\beta+\\alpha})}|\\mu|,\n\\]\nwhich statement holds for all \\beta with \\Re(\\beta)>(1-\\alpha)/2?", + "correct_choice": { + "label": "A", + "text": "For every \\beta with \\Re(\\beta)>(1-\\alpha)/2, the essential spectral radius satisfies\n\\[\n\\rho_e(L_\\beta)\\le \\rho(\\mathcal L_{\\Re(\\beta)+\\alpha}).\n\\]\nMoreover, for any eigenvalue \\lambda of L_\\beta with |\\lambda|>\\rho_e(L_\\beta), the corresponding generalized eigenspace consists of analytic functions, and those functions extend to generalized eigenfunctions of \\mathcal L_\\beta." + }, + "choices": [ + { + "label": "B", + "text": "For every \\beta with \\Re(\\beta)\\ge (1-\\alpha)/2, the essential spectral radius satisfies\n\\[\n\\rho_e(L_\\beta)\\le \\rho(\\mathcal L_{\\beta+\\alpha}).\n\\]\nMoreover, for any eigenvalue \\lambda of L_\\beta with |\\lambda|\\ge \\rho_e(L_\\beta), the corresponding generalized eigenspace consists of analytic functions, and those functions extend to generalized eigenfunctions of \\mathcal L_\\beta." + }, + { + "label": "C", + "text": "For every \\beta with \\Re(\\beta)>(1-\\alpha)/2, the essential spectral radius satisfies\n\\[\n\\rho_e(L_\\beta)\\le \\rho(\\mathcal L_{\\Re(\\beta)+\\alpha}).\n\\]" + }, + { + "label": "D", + "text": "For every \\beta with \\Re(\\beta)>(1-\\alpha)/2, the essential spectral radius satisfies\n\\[\n\\rho_e(L_\\beta)< \\rho(\\mathcal L_{\\Re(\\beta)+\\alpha}).\n\\]\nMoreover, for any eigenvalue \\lambda of L_\\beta with |\\lambda|>\\rho_e(L_\\beta), the corresponding generalized eigenspace consists of holomorphic functions on a neighborhood of [0,1], and every such generalized eigenfunction extends uniquely to an eigenfunction of \\mathcal L_\\beta with the same eigenvalue." + }, + { + "label": "E", + "text": "For every \\beta with \\Re(\\beta)>(1-\\alpha)/2, the essential spectral radius satisfies\n\\[\n\\rho_e(L_\\beta)\\le \\rho(\\mathcal L_{\\Re(\\beta)}).\n\\]\nMoreover, for any eigenvalue \\lambda of L_\\beta with |\\lambda|>\\rho_e(L_\\beta), the corresponding generalized eigenspace consists of analytic functions, and those functions extend to generalized eigenfunctions of \\mathcal L_{\\Re(\\beta)}." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "characteristic", + "tampered_component": "strict half-plane condition and dependence on \\Re(\\beta) rather than full \\beta", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "analyticity and extension of generalized eigenspaces above the essential spectral radius", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "generalized eigenfunctions vs eigenfunctions; non-strict radius inequality strengthened to strict and extension target sharpened", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "loss of the +\\alpha shift coming from Hölder regularity in the comparison operator", + "template_used": "quantifier_dependence" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives definitions and asks for the valid conclusion, but it does not explicitly state the theorem’s conclusion or otherwise reveal the correct option. The answer must be identified from the choices." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-restatement question: the correct option essentially reproduces the full theorem, while distractors are perturbed variants. It is not fully tautological because the choices encode competing conclusions, but it still mainly tests exact theorem recall." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish subtle mathematical differences such as strict vs. non-strict inequalities, dependence on Re(β) versus β, the +α shift, and generalized eigenspaces versus eigenfunctions. However, the task is still mostly recognition of the precise theorem statement rather than substantial derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong: they are mathematically close to the correct statement, vary in realistic ways, and reflect common failure modes such as boundary-condition weakening, unjustified strengthening, dropping the regularity shift, or omitting part of the conclusion." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-identification MCQ with good distractors and little answer leakage, but it leans more toward precise recall of a known statement than genuine generative mathematical reasoning." + } + }, + { + "id": "2511.03847v1", + "paper_link": "http://arxiv.org/abs/2511.03847v1", + "theorems_cnt": 4, + "theorem": { + "env_name": "thm", + "content": "\\label{thm:gawron-generalization}\n\tIf $z\\in \\mathcal S$, then $|z| > \\tfrac 14$.", + "start_pos": 108483, + "end_pos": 108579, + "label": "thm:gawron-generalization" + }, + "ref_dict": { + "thm:MO-ineq": "\\begin{thm}\\label{thm:MO-ineq}\n\tFor all real numbers $t$ and all positive integers $n$,\n\t\\begin{equation}\\label{eqn:tao-ineq}\n\t\t\\abs{\\left(1+\\frac{it}n\\right)^n - 1} \\geq \\big|e^{it} - 1\\big| = 2\\sin\\frac t2.\n\t\\end{equation}\n\\end{thm}", + "table:stern-16": "\\begin{table}[ht]\n\t\\input{table-stern-16}\n\t\\caption{\\small The first $16$ terms of the sequence $\\stern n\\lm$.}\n\t\\label{table:stern-16}\n\\end{table}", + "thm:roots-disk": "\\begin{thm}\\label{thm:roots-disk}\n\tAll roots of $\\stern n\\lm$ lie outside the disk $\\{\\abs{z-2}\\leq 1\\}\\subseteq\\C$.\n\\end{thm}", + "thm:gawron-generalization": "\\begin{thm}\\label{thm:gawron-generalization}\n\tIf $z\\in \\calS$, then $|z| > \\tfrac 14$.\n\\end{thm}", + "conj:location-roots": "\\begin{conj}\\label{conj:location-roots}\n\tAll elements of $\\calS$ lie in the half-plane $\\{\\Re w < 1\\}$. \n\\end{conj}", + "thm:stern-cont-frac": "\\begin{corr}[{\\cite[Theorem 1]{Schinzel2014}}]\\label{thm:stern-cont-frac}\n\tLet $a_1,\\ldots, a_t$ be positive integers, where $t\\geq 2$. Then \n\t\\begin{equation}\\label{eqn:stern-cont-frac}\n\t\t\\frac{\\stern{[[a_1,\\ldots, a_t]]}{\\lm}}{\\stern{[[a_2,\\ldots, a_t]]}{\\lm}}\n\t\t=\n\t\t(\\lm)_{a_1} + \\cfrac{\\lm^{a_1}}{(\\lm)_{a_2} + \\cfrac{\\lm^{a_2}}{\\cdots + \\cfrac{\\cdots}{(\\lm)_{a_{t-1}} + \\cfrac{\\lm^{a_{t-1}}}{(\\lm)_{a_t}}}}}\\,.\n\t\\end{equation}\n\\end{corr}", + "eqn:stronger-statement": "\\begin{equation}\\label{eqn:stronger-statement}\n\t\tb_{2n+1} > \\tfrac12\\max\\{b_n,b_{n+1}\\} > 0\n\t\\end{equation}", + "fig:stern-roots": "\\begin{figure}[ht]\n\t\\centering\n\t\\captionsetup{width=0.75\\linewidth}\n\t\\includegraphics[scale=0.75]{sternpoly-roots-deg-21-BW}\n\t\\caption{\\small The zeros of $\\stern n\\lm$ in the range $\\{a+bi:-4\\leq a\\leq 1, |b|\\leq 3\\}$, where $1\\leq n<2^{21}$ is odd.}\n\t\\label{fig:stern-roots}\n\\end{figure}", + "thm:prime-poly-irred": "\\begin{corr}\\label{thm:prime-poly-irred}\n\tFor each prime number $p$, the Stern polynomial $\\stern p \\lm$ is irreducible in $\\Q$.\n\\end{corr}", + "prop:basic-patterns": "\\begin{prop}\\label{prop:basic-patterns}\n\tLet $n$ be a nonnegative integer.\n\n\t\\vspace{-3pt}\n\n\t\\begin{enumerate}\n\t\t\\item $\\stern{n}{0} = 0$ when $n$ is even, and $\\stern n 0 = 1$ when $n$ is odd.\n\t\t\\item $\\stern{n}{1} = s_n$, where $s_n$ is the Stern sequence \\eqref{eqn:stern-seq}.\n\t\t\\item $\\stern{n}{2} = n$.\n\t\t\\item When $n = 2^r$ for a nonnegative integer $r$, $\\stern n \\lm = \\lm^r$.\n\t\t\\item When $n = 2^r - 1$ for a nonnegative integer $r$, $\\stern n\\lm = (\\lm)_r$.\n\t\\end{enumerate}\n\\end{prop}", + "thm:parabola": "\\begin{thm}[Parabola Theorem, {\\cite[Theorem 3.43]{Lorentzen2008}}]\\label{thm:parabola}\n\tFor fixed $|\\alpha| < \\tfrac\\pi 2$, let\n\t\\[\n\tV_\\alpha\\coloneqq -\\tfrac12 + e^{i\\alpha}\\overline{\\mathbb H} = \\{w\\in\\C: \\Re(we^{- i\\alpha})\\geq -\\tfrac12\\cos\\alpha\\}\n\t\\]\n\tand\n\t\\[\n\tE_\\alpha \\coloneqq \\{a\\in\\C: |a| - \\Re(ae^{-2i\\alpha})\\leq \\tfrac12\\cos^2\\alpha\\}.\n\t\\]\n\tThen $E_\\alpha$ is the element set for continued fractions $\\boldsymbol{K}(a_n | 1)$ corresponding to the value set $V_\\alpha$.\n\\end{thm}" + }, + "pre_theorem_intro_text_len": 2292, + "pre_theorem_intro_text": "The Stern sequence $(s_n)_{n\\geq 0}$ of positive integers, named after Moritz Abraham Stern, is given by $s_0 = 0$, $s_1 = 1$, and \n\\begin{equation}\\label{eqn:stern-seq}\n\ts_{2n} = s_n,\\quad s_{2n+1} = s_n + s_{n+1}.\n\\end{equation}\nThere is a vast body of literature (e.g. \\cite{Coons2014,Northshield2010,Reznick2008,Stanley2019} and the references therein) about the Stern sequence. Perhaps the most striking property of the Stern sequence is that every positive rational number appears exactly once in the sequence $(s_{n+1}/s_n)_{n\\geq 1}$, giving an explicit enumeration of $\\mathbb Q^+$. Stern proved this fact 15 years before Cantor introduced the notion of a countable set!\n\n\\par In \\cite{Klavzar2007}, Klav\\v{z}ar, Milutinovi\\'{c}, and Petr define a polynomial analogue $\\stern n\\lambda$ of the Stern sequence via $\\stern 0\\lambda = 0$, $\\stern 1\\lambda = 1$, and \n\\begin{equation}\\label{eqn:stern-def}\n\t\\stern{2n}\\lambda = \\lambda \\stern n\\lambda,\\quad \\stern{2n+1}\\lambda = \\stern n\\lambda + \\stern{n+1}\\lambda.\n\\end{equation}\nThe first $16$ terms of the sequence $\\stern n\\lambda$ are given in Table \\ref{table:stern-16}. This table suggests many patterns, several of which are recorded in Proposition \\ref{prop:basic-patterns}.\n\n\\begin{table}[ht]\n\t\\input{table-stern-16}\n\t\\caption{\\small The first $16$ terms of the sequence $\\stern n\\lambda$.}\n\t\\label{table:stern-16}\n\\end{table}\n\nWhile there are several papers which analyze the sequence $\\stern n\\lambda$ (see \\cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials. Because $\\stern{2n}\\lambda = \\lambda \\stern n \\lambda$ for all $n\\geq 1$, it suffices to examine the roots of $\\stern n\\lambda$ only when $n$ is odd. Define\n\\[\n\\mathcal S\\coloneqq \\{z\\in\\mathbb C: \\stern n z = 0\\text{ for some odd }n\\geq 1\\}.\n\\]\nIn \\cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\\tfrac12$, and $-\\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\\tfrac14,\\tfrac14]$ cannot be in $\\mathcal S$. His methods extend easily to show that any complex number $z$ with $|z| \\leq \\tfrac14$ cannot be in $\\mathcal S$. For completeness, we record the proof here.", + "context": "The Stern sequence $(s_n)_{n\\geq 0}$ of positive integers, named after Moritz Abraham Stern, is given by $s_0 = 0$, $s_1 = 1$, and \n\\begin{equation}\\label{eqn:stern-seq}\n s_{2n} = s_n,\\quad s_{2n+1} = s_n + s_{n+1}.\n\\end{equation}\nThere is a vast body of literature (e.g. \\cite{Coons2014,Northshield2010,Reznick2008,Stanley2019} and the references therein) about the Stern sequence. Perhaps the most striking property of the Stern sequence is that every positive rational number appears exactly once in the sequence $(s_{n+1}/s_n)_{n\\geq 1}$, giving an explicit enumeration of $\\mathbb Q^+$. Stern proved this fact 15 years before Cantor introduced the notion of a countable set!\n\n\\par In \\cite{Klavzar2007}, Klav\\v{z}ar, Milutinovi\\'{c}, and Petr define a polynomial analogue $\\stern n\\lambda$ of the Stern sequence via $\\stern 0\\lambda = 0$, $\\stern 1\\lambda = 1$, and \n\\begin{equation}\\label{eqn:stern-def}\n \\stern{2n}\\lambda = \\lambda \\stern n\\lambda,\\quad \\stern{2n+1}\\lambda = \\stern n\\lambda + \\stern{n+1}\\lambda.\n\\end{equation}\nThe first $16$ terms of the sequence $\\stern n\\lambda$ are given in Table \\ref{table:stern-16}. This table suggests many patterns, several of which are recorded in Proposition \\ref{prop:basic-patterns}.\n\n\\begin{table}[ht]\n \\input{table-stern-16}\n \\caption{\\small The first $16$ terms of the sequence $\\stern n\\lambda$.}\n \\label{table:stern-16}\n\\end{table}\n\nWhile there are several papers which analyze the sequence $\\stern n\\lambda$ (see \\cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials. Because $\\stern{2n}\\lambda = \\lambda \\stern n \\lambda$ for all $n\\geq 1$, it suffices to examine the roots of $\\stern n\\lambda$ only when $n$ is odd. Define\n\\[\n\\mathcal S\\coloneqq \\{z\\in\\mathbb C: \\stern n z = 0\\text{ for some odd }n\\geq 1\\}.\n\\]\nIn \\cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\\tfrac12$, and $-\\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\\tfrac14,\\tfrac14]$ cannot be in $\\mathcal S$. His methods extend easily to show that any complex number $z$ with $|z| \\leq \\tfrac14$ cannot be in $\\mathcal S$. For completeness, we record the proof here.\n\n\\begin{prop}\\label{prop:basic-patterns}\n\tLet $n$ be a nonnegative integer.\n\n\t\\vspace{-3pt}\n\n\t\\begin{enumerate}\n\t\t\\item $\\stern{n}{0} = 0$ when $n$ is even, and $\\stern n 0 = 1$ when $n$ is odd.\n\t\t\\item $\\stern{n}{1} = s_n$, where $s_n$ is the Stern sequence \\eqref{eqn:stern-seq}.\n\t\t\\item $\\stern{n}{2} = n$.\n\t\t\\item When $n = 2^r$ for a nonnegative integer $r$, $\\stern n \\lm = \\lm^r$.\n\t\t\\item When $n = 2^r - 1$ for a nonnegative integer $r$, $\\stern n\\lm = (\\lm)_r$.\n\t\\end{enumerate}\n\\end{prop}\n\n\\begin{table}[ht]\n\t\\input{table-stern-16}\n\t\\caption{\\small The first $16$ terms of the sequence $\\stern n\\lm$.}\n\t\\label{table:stern-16}\n\\end{table}", + "full_context": "The Stern sequence $(s_n)_{n\\geq 0}$ of positive integers, named after Moritz Abraham Stern, is given by $s_0 = 0$, $s_1 = 1$, and \n\\begin{equation}\\label{eqn:stern-seq}\n s_{2n} = s_n,\\quad s_{2n+1} = s_n + s_{n+1}.\n\\end{equation}\nThere is a vast body of literature (e.g. \\cite{Coons2014,Northshield2010,Reznick2008,Stanley2019} and the references therein) about the Stern sequence. Perhaps the most striking property of the Stern sequence is that every positive rational number appears exactly once in the sequence $(s_{n+1}/s_n)_{n\\geq 1}$, giving an explicit enumeration of $\\mathbb Q^+$. Stern proved this fact 15 years before Cantor introduced the notion of a countable set!\n\n\\par In \\cite{Klavzar2007}, Klav\\v{z}ar, Milutinovi\\'{c}, and Petr define a polynomial analogue $\\stern n\\lambda$ of the Stern sequence via $\\stern 0\\lambda = 0$, $\\stern 1\\lambda = 1$, and \n\\begin{equation}\\label{eqn:stern-def}\n \\stern{2n}\\lambda = \\lambda \\stern n\\lambda,\\quad \\stern{2n+1}\\lambda = \\stern n\\lambda + \\stern{n+1}\\lambda.\n\\end{equation}\nThe first $16$ terms of the sequence $\\stern n\\lambda$ are given in Table \\ref{table:stern-16}. This table suggests many patterns, several of which are recorded in Proposition \\ref{prop:basic-patterns}.\n\n\\begin{table}[ht]\n \\input{table-stern-16}\n \\caption{\\small The first $16$ terms of the sequence $\\stern n\\lambda$.}\n \\label{table:stern-16}\n\\end{table}\n\nWhile there are several papers which analyze the sequence $\\stern n\\lambda$ (see \\cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials. Because $\\stern{2n}\\lambda = \\lambda \\stern n \\lambda$ for all $n\\geq 1$, it suffices to examine the roots of $\\stern n\\lambda$ only when $n$ is odd. Define\n\\[\n\\mathcal S\\coloneqq \\{z\\in\\mathbb C: \\stern n z = 0\\text{ for some odd }n\\geq 1\\}.\n\\]\nIn \\cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\\tfrac12$, and $-\\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\\tfrac14,\\tfrac14]$ cannot be in $\\mathcal S$. His methods extend easily to show that any complex number $z$ with $|z| \\leq \\tfrac14$ cannot be in $\\mathcal S$. For completeness, we record the proof here.\n\n\\begin{prop}\\label{prop:basic-patterns}\n\tLet $n$ be a nonnegative integer.\n\n\t\\vspace{-3pt}\n\n\t\\begin{enumerate}\n\t\t\\item $\\stern{n}{0} = 0$ when $n$ is even, and $\\stern n 0 = 1$ when $n$ is odd.\n\t\t\\item $\\stern{n}{1} = s_n$, where $s_n$ is the Stern sequence \\eqref{eqn:stern-seq}.\n\t\t\\item $\\stern{n}{2} = n$.\n\t\t\\item When $n = 2^r$ for a nonnegative integer $r$, $\\stern n \\lm = \\lm^r$.\n\t\t\\item When $n = 2^r - 1$ for a nonnegative integer $r$, $\\stern n\\lm = (\\lm)_r$.\n\t\\end{enumerate}\n\\end{prop}\n\n\\begin{table}[ht]\n\t\\input{table-stern-16}\n\t\\caption{\\small The first $16$ terms of the sequence $\\stern n\\lm$.}\n\t\\label{table:stern-16}\n\\end{table}\n\nWhile there are several papers which analyze the sequence $\\stern n\\lm$ (see \\cite{Dilcher2017,Dilcher2018,Gawron2014,Ulas2011} and the references therein), not much is known about the zeros of Stern polynomials. Because $\\stern{2n}\\lm = \\lm \\stern n \\lm$ for all $n\\geq 1$, it suffices to examine the roots of $\\stern n\\lm$ only when $n$ is odd. Define\n\\[\n\\calS\\coloneqq \\{z\\in\\C: \\stern n z = 0\\text{ for some odd }n\\geq 1\\}.\n\\]\nIn \\cite[Theorem 2.1]{Gawron2014}, Gawron proves that $0$, ${-}1$, ${-}\\tfrac12$, and $-\\tfrac13$ are the only rational zeros of any Stern polynomial by showing, more generally, that any real number in the closed interval $[-\\tfrac14,\\tfrac14]$ cannot be in $\\calS$. His methods extend easily to show that any complex number $z$ with $|z| \\leq \\tfrac14$ cannot be in $\\calS$. For completeness, we record the proof here.\n\n\\begin{proof}\n Let $z$ be any complex number with $|z| \\leq \\tfrac 14$. Let $b_n\\coloneqq |\\stern n z|$ for each positive integer $n$. We show more strongly that\n \\begin{equation}\\label{eqn:stronger-statement}\n b_{2n+1} > \\tfrac12\\max\\{b_n,b_{n+1}\\} > 0\n \\end{equation}\n for all $n\\geq 1$.\n\n\\begin{thm}[Parabola Theorem, {\\cite[Theorem 3.43]{Lorentzen2008}}]\\label{thm:parabola}\n For fixed $|\\alpha| < \\tfrac\\pi 2$, let\n \\[\n V_\\alpha\\coloneqq -\\tfrac12 + e^{i\\alpha}\\overline{\\mathbb H} = \\{w\\in\\C: \\Re(we^{- i\\alpha})\\geq -\\tfrac12\\cos\\alpha\\}\n \\]\n and\n \\[\n E_\\alpha \\coloneqq \\{a\\in\\C: |a| - \\Re(ae^{-2i\\alpha})\\leq \\tfrac12\\cos^2\\alpha\\}.\n \\]\n Then $E_\\alpha$ is the element set for continued fractions $\\boldsymbol{K}(a_n | 1)$ corresponding to the value set $V_\\alpha$.\n\\end{thm}\nThe region $V_\\alpha$ is a half-plane whose boundary is a line intersecting the real axis at $z = -\\tfrac12$. In particular, because $|\\alpha| < \\tfrac\\pi 2$, $V_\\alpha$ contains the half-line $[-\\tfrac12,\\infty)$. The boundary of $E_\\alpha$ is a parabola with focus at the origin and vertex at $-\\tfrac14 e^{2i\\alpha}\\cos^2\\alpha$. This parabola intersects the real axis at $z = -\\tfrac14$.\n\nWe now tie the continued fraction theory back to Stern polynomials.\nRecall that Corollary \\ref{thm:stern-cont-frac} expresses the ratio\nof two Stern polynomials as a continued fraction with elements in $\\C$. \nThis form on its own is difficult to work with. We instead note (e.g. \\cite[Corollary 2.15]{Lorentzen2008}) that any continued fraction $\\boldsymbol{K}(a_n|b_n)$ with $b_j\\neq 0$ is equivalent to the continued fraction $\\boldsymbol{K}(c_n|1)$, where $c_j = \\tfrac{a_j}{b_jb_{j+1}}$. As an explicit example, \n\\begin{equation}\\label{eqn:cfrac-example}\n \\frac{\\stern{[[2,3,5]]}\\lm}{\\stern{[[3,5]]}\\lm} = (\\lm)_2 + \\cfrac{\\lm^2}{(\\lm)_3 + \\cfrac{\\lm^3}{(\\lm)_5}}\n = (\\lm)_2 \\left[1 + \\cfrac{\\dfrac{\\lm^2}{(\\lm)_2(\\lm)_3}}{1 + \\cfrac{\\lm^3}{(\\lm)_3(\\lm)_5}}\\right].\n\\end{equation}\n\\begin{remark}\n While continued fractions of the form $\\boldsymbol{K}(1|d_n)$ are more common, the coefficients $d_n$ are significantly messier than the coefficients $c_n$ (see \\cite[Corollary 2.15]{Lorentzen2008}). We opt to use the less-common form $\\boldsymbol{K}(c_n|1)$ to simplify our analysis.\n\\end{remark}\nEquation \\eqref{eqn:cfrac-example} suggests that the ratios $\\tfrac{z^a}{(z)_a(z)_b}$, where $a$ and $b$ are any positive integers, may play some importance.\nFor this reason, we define $z_{a,b}\\coloneqq \\tfrac{z^a}{(z)_a(z)_b}$ and \n\\[\n\\calA_z\\coloneqq \\left\\{z_{a,b}:(a,b)\\in\\N^2\\right\\}.\n\\]\nSuppose $\\calA_z\\subseteq E_\\alpha$ for some angle $\\alpha$ with $|\\alpha| < \\tfrac{\\pi}2$. \nThen $V_\\alpha$ is a value set for the continued fraction \\eqref{eqn:stern-cont-frac} at $\\lm = z$. In particular, $\\infty$ is not a possible value for this fraction, so $\\stern n z \\neq 0$ for any $z$.\n\n\\begin{itemize}\n \\item First suppose $x\\in[0,\\pi]$. Because both $\\sin x$ and $x - \\tfrac{x^2}{\\pi}$ are symmetric about the axis $x = \\tfrac\\pi 2$, it suffices to prove the inequality for $x\\in[0,\\tfrac\\pi 2]$. Within this smaller interval, cosine is concave down with $\\cos 0 = 1$ and $\\cos\\tfrac \\pi 2 = 0$, so\n \\begin{equation}\\label{eqn:cos-ineq}\n \\cos x \\geq 1 - \\frac{2x}{\\pi}.\n \\end{equation}\n Integrating both sides of \\eqref{eqn:cos-ineq} yields \\eqref{ineq-sinc} in this case.\n \\item Now suppose $x\\notin[0,\\pi]$. By symmetry, it suffices to prove the inequality for $x\\geq \\pi$. In this interval, $1 - \\tfrac{2x}{\\pi} \\leq -1 \\leq \\cos x$. It follows that \\eqref{eqn:cos-ineq}, and thus \\eqref{ineq-sinc}, holds as well.\n \\end{itemize}\n\\end{proof}\n\n\\begin{thm}\\label{thm:min-geom-series}\n Let $n$ be a positive integer, and suppose $z\\in\\calB$. Then $|(z)_n| \\geq \\min(n,\\tfrac{11}2)$.\n\\end{thm}\n\nNow let\n \\[\n M\\coloneqq 2\\pi\\left(1 - \\frac{5.5}{n}\\right) = \\pi\\left(2 - \\frac{11}n\\right).\n \\]\n There are two cases to consider. First, suppose $t\\leq M$. Then\n \\[\n n\\left(1 - \\frac{t}{2\\pi}\\right) \\geq n\\left(1 - \\frac{M}{2\\pi}\\right) = \\frac{11}2.\n \\]\n Now suppose $t\\geq M$. Because $n\\geq 15$, the coefficient $\\tfrac{1}{2n} - \\tfrac{11}{8n^2}$ is positive, so\n \\begin{equation}\\label{eqn:asymp-w-tn}\n t + t^3\\left(\\frac{1}{2n} - \\frac{11}{8n^2}\\right) \\geq \\pi\\left(2 - \\frac{11}n\\right) + \\pi^3\\left(2 - \\frac{11}n\\right)^3\\left(\\frac{1}{2n} - \\frac{11}{8n^2}\\right).\n \\end{equation}\n To estimate \\eqref{eqn:asymp-w-tn}, let $x = \\tfrac 1n$, so it suffices to analyze the polynomial\n \\[\n P(x) := \\pi(2-11x) + \\pi^3(2-11x)^3(\\tfrac 12x - \\tfrac{11}{8}x^2).\n \\]\n The polynomial $P$ has exactly one critical point in the interval $[0,\\tfrac 1{15}]$, occurring at $x = x_0 \\approx 0.024$. In particular, $P$ is increasing on $[0,x_0]$ and decreasing on $[x_0,\\tfrac1{15}]$. Compute\n \\[\n P(0) = 2\\pi\\quad\\text{and}\\quad P\\left(\\frac 1{15}\\right) = \\frac{19 \\pi }{15}+\\frac{336091 \\pi ^3}{6075000} \\approx 5.695 > \\frac{11}2.\n \\]\n It follows that $P(x) > \\tfrac{11}2$ for all $x\\in[0,\\tfrac{1}{15}]$, and thus $|w_{n,t}^n - 1| \\geq M|w_{n,t}-1|$ for all $t\\in[0,2\\pi]$ and $n\\geq 15$. \n\\end{proof}\n\n\\begin{prop}\\label{prop:silli}\n Let $z\\in\\calB^+$. Then $0\\leq \\arg z \\leq \\tfrac\\pi 6$ and\n \\begin{equation}\\label{eqn:re-ineq}\n |\\Re z^{-4}|\\leq |\\Re z^{-2}|.\n \\end{equation}\n\\end{prop}\n\n\\begin{corr}[{\\cite[Theorem 1]{Schinzel2014}}]\\label{thm:stern-cont-frac}\n\tLet $a_1,\\ldots, a_t$ be positive integers, where $t\\geq 2$. Then \n\t\\begin{equation}\\label{eqn:stern-cont-frac}\n\t\t\\frac{\\stern{[[a_1,\\ldots, a_t]]}{\\lm}}{\\stern{[[a_2,\\ldots, a_t]]}{\\lm}}\n\t\t=\n\t\t(\\lm)_{a_1} + \\cfrac{\\lm^{a_1}}{(\\lm)_{a_2} + \\cfrac{\\lm^{a_2}}{\\cdots + \\cfrac{\\cdots}{(\\lm)_{a_{t-1}} + \\cfrac{\\lm^{a_{t-1}}}{(\\lm)_{a_t}}}}}\\,.\n\t\\end{equation}\n\\end{corr}", + "post_theorem_intro_text_len": 4598, + "post_theorem_intro_text": "\\begin{proof}\n\tLet $z$ be any complex number with $|z| \\leq \\tfrac 14$. Let $b_n\\coloneqq |\\stern n z|$ for each positive integer $n$. We show more strongly that\n\t\\begin{equation}\\label{eqn:stronger-statement}\n\t\tb_{2n+1} > \\tfrac12\\max\\{b_n,b_{n+1}\\} > 0\n\t\\end{equation}\n\tfor all $n\\geq 1$.\n\n\t\\par The proof of \\eqref{eqn:stronger-statement} proceeds by induction on $n$. The base case, $n = 1$, follows because\n\t\\[\n\tb_3 = |z + 1| \\geq \\tfrac 34 > \\tfrac 12 \n\t= \\tfrac 12\\max\\{1, |z|\\} = \\tfrac 12\\max\\{b_1,b_2\\}.\n\t\\]\n\n\t\\par There are two cases to consider. First assume $n=2k$ is even. Then\n\t\\begin{align*}\n\t\tb_{4k+1} &= |\\stern{4k+1}z| = |\\stern{2k+1} z + \\stern{2k} z|\\\\\n\t\t&= |z \\stern kz + \\stern{2k+1} z| \\geq b_{2k+1} - \\tfrac14 b_k\\\\\n\t\t&\\geq b_{2k+1} - \\tfrac12 b_{2k+1} = \\tfrac12 b_{2k+1}.\n\t\\end{align*}\n\tMoreover, $b_{2k+1} \\geq \\frac12 b_k$, and thus $\\max\\{b_{2k},b_{2k+1}\\} = b_{2k+1} > 0$. In this case, our inequality is proved.\n\n\t\\par Now assume $n = 2k + 1$ is odd. Then\n\t\\begin{align*}\n\t\tb_{4k+3} &= |\\stern{4k+3}z| = |\\stern{2k+1} z + \\stern{2k+2} z|\\\\\n\t\t&= |\\stern{2k+1} z + z\\stern{k+1} z| \\geq b_{2k+1} - \\frac 14 b_{k+1}\\\\\n\t\t&\\geq b_{2k+1} - \\tfrac12 b_{2k+1} = \\tfrac12 b_{2k+1}.\n\t\\end{align*}\n\tMoreover, $b_{2k+1} \\geq \\frac12 b_{k+1}$, and so in this case we also have $\\max\\{b_{2k},b_{2k+1}\\} = b_{2k+1} > 0$. We have exhausted both cases, completing the proof of Theorem \\ref{thm:gawron-generalization}.\n\\end{proof}\n\nIn \\cite{Dilcher2017}, Dilcher et. al. focus more specifically on the complex roots of $\\stern n\\lambda$. Their paper makes the following conjecture.\n\n\\begin{conj}\\label{conj:location-roots}\n\tAll elements of $\\mathcal S$ lie in the half-plane $\\{\\operatorname{Re} w < 1\\}$. \n\\end{conj}\n\nBy generalizing the Enestrom-Kakeya theorem, they prove Conjecture \\ref{conj:location-roots} for several classes of positive integers $n$ taking the form $2^n\\pm k$, where $k$ is fixed and $2^n \\geq k$. These are the only two papers the author could find which discuss the complex zeros of $\\stern n\\lambda$.\n\n\\par Figure \\ref{fig:stern-roots} shows a snapshot of $\\mathcal S$. One striking feature of this figure is the contrasting behavior of these roots within the half-planes $\\{\\operatorname{Re} w\\geq 0\\}$ and $\\{\\operatorname{Re} w < 0\\}$. These differences present difficulties in fully characterizing the geometry of $\\mathcal S$.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\captionsetup{width=0.75\\linewidth}\n\t\\includegraphics[scale=0.75]{sternpoly-roots-deg-21-BW}\n\t\\caption{\\small The zeros of $\\stern n\\lambda$ in the range $\\{a+bi:-4\\leq a\\leq 1, |b|\\leq 3\\}$, where $1\\leq n<2^{21}$ is odd.}\n\t\\label{fig:stern-roots}\n\\end{figure}\n\nIn this paper, we partially resolve Conjecture \\ref{conj:location-roots} by establishing the following result.\n\n\\begin{thm}\\label{thm:roots-disk}\n\tAll roots of $\\stern n\\lambda$ lie outside the disk $\\{\\left|z-2\\right|\\leq 1\\}\\subseteq\\mathbb C$.\n\\end{thm}\n\nTo prove Theorem \\ref{thm:roots-disk}, we use a continued fraction representation for a ratio of Stern polynomials (Theorem \\ref{thm:stern-cont-frac}) independently discovered by Reznick \\cite{Reznick2008} and Schinzel \\cite{Schinzel2014}. This allows us to use the Parabola Theorem (Theorem \\ref{thm:parabola}) to show that, for certain values of $z\\in\\mathbb C$, the denominators of these continued fractions can never be zero. Along the way, we establish inequalities in $\\mathbb C$ relating to the sums $1 + z + \\cdots + z^{n-1}$ which may be of independent interest. The most notable of these inequalities is Theorems \\ref{thm:MO-ineq}, which proves a lower bound for this geometric series whenever $\\operatorname{Re} z \\geq 1$.\n\nAs a corollary, we obtain the following surprising fact, resolving a conjecture of Ulas and Ulas (\\cite{Ulas2011}).\n\n\\begin{corr}\\label{thm:prime-poly-irred}\n\tFor each prime number $p$, the Stern polynomial $\\stern p \\lambda$ is irreducible in $\\mathbb Q$.\n\\end{corr}\nUlas and Ulas had verified this conjecture computationally for the first million primes $p$. Additionally, Schinzel in \\cite{Schinzel2011} proved Corollary \\ref{thm:prime-poly-irred} for all primes $p < 2017$ by using finite differences to bound the leading coefficient of any proper divisor of $\\stern n\\lambda$. However, these previous attempts to prove the conjecture were algebraic in nature, whereas our proof depends on the analytic properties of $\\mathcal S$.\n\n\\paragraph{Acknowledgments.} This paper is adapted from the author's PhD dissertation \\cite{Altizio2025}. The author thanks his advisor, Dr. Bruce Reznick, for helpful correspondence.", + "sketch": "Let $z$ be any complex number with $|z|\\leq \\tfrac14$ and set $b_n\\coloneqq |\\stern n z|$. The proof shows the stronger claim\n\\begin{equation}\\label{eqn:stronger-statement}\n b_{2n+1} > \\tfrac12\\max\\{b_n,b_{n+1}\\} > 0\n\\end{equation}\nfor all $n\\ge1$, and proceeds by induction on $n$.\n\nBase case $n=1$: \n\\[\n b_3=|z+1|\\ge \\tfrac34>\\tfrac12=\\tfrac12\\max\\{1,|z|\\}=\\tfrac12\\max\\{b_1,b_2\\}.\n\\]\n\nInductive step splits into two cases.\n\n(i) If $n=2k$ is even, then\n\\[\n b_{4k+1}=|\\stern{4k+1}z|=|\\stern{2k+1}z+\\stern{2k}z|=|z\\,\\stern k z+\\stern{2k+1}z|\n \\ge b_{2k+1}-\\tfrac14 b_k\\ge b_{2k+1}-\\tfrac12 b_{2k+1}=\\tfrac12 b_{2k+1}.\n\\]\nMoreover, $b_{2k+1}\\ge \\tfrac12 b_k$, hence $\\max\\{b_{2k},b_{2k+1}\\}=b_{2k+1}>0$, giving \\eqref{eqn:stronger-statement}.\n\n(ii) If $n=2k+1$ is odd, then\n\\[\n b_{4k+3}=|\\stern{4k+3}z|=|\\stern{2k+1}z+\\stern{2k+2}z|=|\\stern{2k+1}z+z\\,\\stern{k+1}z|\n \\ge b_{2k+1}-\\tfrac14 b_{k+1}\\ge b_{2k+1}-\\tfrac12 b_{2k+1}=\\tfrac12 b_{2k+1}.\n\\]\nMoreover, $b_{2k+1}\\ge \\tfrac12 b_{k+1}$, so again $\\max\\{b_{2k},b_{2k+1}\\}=b_{2k+1}>0$. These two cases exhaust the induction, completing the proof of Theorem~\\ref{thm:gawron-generalization}.", + "expanded_sketch": "No expanded sketch found.", + "expanded_theorem": "\\label{thm:gawron-generalization}\n\tIf $z\\in \\mathcal S$, then $|z| > \\tfrac 14$.", + "theorem_type": [ + "Implication", + "Inequality or Bound" + ], + "mcq": { + "question": "For the Stern polynomials \\(\\operatorname{St}_n(\\lambda)\\) defined by \\(\\operatorname{St}_0(\\lambda)=0\\), \\(\\operatorname{St}_1(\\lambda)=1\\), and for all integers \\(n\\ge 0\\),\n\\[\n\\operatorname{St}_{2n}(\\lambda)=\\lambda\\,\\operatorname{St}_n(\\lambda),\\qquad\n\\operatorname{St}_{2n+1}(\\lambda)=\\operatorname{St}_n(\\lambda)+\\operatorname{St}_{n+1}(\\lambda),\n\\]\ndefine\n\\[\n\\mathcal S:=\\{z\\in\\mathbb C: \\operatorname{St}_n(z)=0\\text{ for some odd }n\\ge 1\\}.\n\\]\nWhich quantitative estimate holds for every \\(z\\in\\mathcal S\\)?", + "correct_choice": { + "label": "A", + "text": "Every \\(z\\in\\mathcal S\\) satisfies \\(|z|>\\tfrac14\\)." + }, + "choices": [ + { + "label": "B", + "text": "Every \\(z\\in\\mathcal S\\) satisfies \\(|z|\\ge \\tfrac14\\)." + }, + { + "label": "C", + "text": "Every \\(z\\in\\mathcal S\\) satisfies \\(|z|>0\\)." + }, + { + "label": "D", + "text": "Every \\(z\\in\\mathcal S\\) satisfies \\(\\Re(z)>-\\tfrac14\\)." + }, + { + "label": "E", + "text": "Every \\(z\\in\\mathcal S\\) satisfies \\(|z|>\\tfrac13\\)." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "characteristic", + "tampered_component": "strictness_at_radius_1_4", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "characteristic", + "tampered_component": "quantitative_radius_bound_dropped_to_nonzero_modulus", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "modulus_bound_replaced_by_real_part_bound", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "characteristic", + "tampered_component": "sharp_constant_1_4_strengthened_to_1_3", + "template_used": "stronger_trap" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the Stern polynomials and the set \\(\\mathcal S\\), but it does not explicitly state or strongly hint at the bound \\(|z|>\\tfrac14\\). The correct estimate is not leaked by wording alone." + }, + "TAS": { + "score": 1, + "justification": "This is close to a theorem-recall item: after defining \\(\\mathcal S\\), it asks directly which universal estimate holds. The multiple choices introduce some comparison, but the task is still largely a reformulation of the target conclusion rather than a genuinely new inference problem." + }, + "GPS": { + "score": 1, + "justification": "There is some reasoning pressure because the student must distinguish between a sharp strict bound, a weak true statement, and stronger or unrelated false variants. However, it mainly tests recognition of the known result rather than substantial derivation or synthesis." + }, + "DQS": { + "score": 2, + "justification": "The distractors are well designed: \\(\\ge \\tfrac14\\) probes strict vs. non-strict misunderstanding, \\(|z|>0\\) is a weaker true statement, \\(|z|>\\tfrac13\\) is an overstrengthening trap, and \\(\\Re(z)>-\\tfrac14\\) is a distinct but irrelevant-looking alternative. They are plausible and target common mathematical failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with strong distractors and no answer leakage, but it is somewhat theorem-recall based and only moderately tests generative reasoning." + } + }, + { + "id": "2511.22601v1", + "paper_link": "http://arxiv.org/abs/2511.22601v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:main theorem}\n Let $f \\colon S^2 \\to S^2$ be an expanding Thurston map and $d$ be a visual metric on $S^2$ for $f$.\n Let $\\beta \\in (0, 1]$ and $\\phi \\in C^{0, \\beta}(S^2, d)$ be an eventually positive real-valued H\\\"{o}lder continuous function satisfying the $\\beta$-strong non-integrability condition (with respect to $f$ and $d$). \n Then there exists a unique positive number $\\rootpressure > 0$ with topological pressure $P(f, -\\rootpressure \\phi) = 0$ and there exists $N_{f} \\in \\mathbb{N}$ depending only on $f$ such that for each $N \\in \\mathbb{N}$ with $N \\geqslant N_{f}$, the following statement holds for the iterate $F \\coloneqq f^{N}$ and the potential $\\Phi \\coloneqq \\sum_{i = 0}^{N - 1} \\phi \\circ f^{i}$:\n\n Denote $\\alpha \\coloneqq \\frac{\\mathrm{d}}{\\mathrm{d} t} P(F, t \\Phi) |_{t = -\\rootpressure}$ and $\\sigma \\coloneqq \\sqrt{ \\frac{\\mathrm{d}^2}{\\mathrm{d} t^2} P(F, t \\Phi) |_{t = -\\rootpressure } }$.\n Let $\\sequen{I_{n}}$ be a sequence of intervals contained in a compact set $K \\subseteq \\mathbb{R}$ with $\\sequen[\\big]{|I_{n}|^{-1}}$ having sub-exponential growth.\n Then\n \\[\n \\pi_{F, \\Phi}(n; \\alpha, I_n) \\sim \\frac{ \\int_{I_{n}} e^{\\rootpressure t} \\, \\mathrm{d}t }{ \\sqrt{2\\pi} \\, \\sigma } \\, \\frac{e^{\\rootpressure \\alpha n}}{n^{3/2}} \\qquad \\text{as } n \\to +\\infty.\n \\]", + "start_pos": 217609, + "end_pos": 218959, + "label": "thm:main theorem" + }, + "ref_dict": { + "def:strong non-integrability condition": "\\begin{definition}[Strong non-integrability condition] \\label{def:strong non-integrability condition}\n Let $f \\colon S^2 \\mapping S^2$ be an expanding Thurston map and $d$ be a visual metric on $S^2$ for $f$.\n Fix $\\holderexp \\in (0, 1]$.\n Let $\\potential \\in \\holderspacesphere$ be a real-valued \\holder continuous function with an exponent $\\holderexp$.\n \\begin{enumerate}[label=\\rm{(\\arabic*)}]\n \\smallskip\n \\item We say that $\\potential$ satisfies the \\emph{$(\\mathcal{C}, \\holderexp)$-strong non-integrability condition} (with respect to $f$ and $d$), for a Jordan curve $\\mathcal{C} \\subseteq S^2$ with $\\post{f} \\subseteq \\mathcal{C}$, if there exist\n \\begin{enumerate}[label=\\rm{(\\alph*)}]\n \\smallskip\n \\item numbers $\\juxtapose{N}{M} \\in \\n$, $\\varepsilon \\in (0, 1)$, \n \\smallskip\n \\item $M$-tiles $Y^{M}_{\\black} \\in \\cTile{M}{\\black}$, $Y^{M}_{\\white} \\in \\cTile{M}{\\white}$\n \\end{enumerate}\n such that for each $\\colour \\in \\colours$, each integer $m \\geqslant M$, and each $m$-tile $X \\in \\Tile{m}$ with $X \\subseteq Y^{M}_{\\colour}$, there exist two points $\\juxtapose{x_{1}}{x_{2}} \\in X$ with the following properties:\n \\begin{enumerate}\n \\smallskip\n \\item $\\min \\set[\\big]{ d \\parentheses[\\big]{ x_{1}, S^2 \\mysetminus X }, d \\parentheses[\\big]{ x_{2}, S^2 \\mysetminus X }, d(x_{1}, x_{2}) } \\geqslant \\diameter{d}{X}$, and\n \\smallskip\n \\item for each integer $n \\geqslant N$, there exist two $(n + M)$-tiles $\\juxtapose{X^{n + M}_{\\colour, 1}}{X^{n + M}_{\\colour, 2}} \\in \\Tile{n + M}$ such that $Y^{M}_{\\colour} = f^{n} \\parentheses[\\big]{ X^{n + M}_{\\colour, 1} } = f^{n} \\parentheses[\\big]{ X^{n + M}_{\\colour, 2} }$ and \n \\[\n \\abs{ S_{n}\\potential(\\varsigma_{1}(x_{1})) - S_{n}\\potential(\\varsigma_{2}(x_{1})) - S_{n}\\potential(\\varsigma_{1}(x_{2})) + S_{n}\\potential(\\varsigma_{2}(x_{2})) } \\geqslant \\varepsilon d( x_{1}, x_{2})^{\\holderexp},\n \\]\n where we write $\\varsigma_{i} \\define \\parentheses[\\big]{ f^{n}|_{X^{n + M}_{\\colour, i}} }^{-1}$ for each $i \\in \\set{1, 2}$.\n \\end{enumerate}\n \\smallskip\n\n \\item We say that $\\potential$ satisfies the \\emph{$\\holderexp$-strong non-integrability condition} (with respect to $f$ and $d$) if $\\potential$ satisfies the $(\\mathcal{C}, \\holderexp)$-strong non-integrability condition with respect to $f$ and $d$ for some Jordan curve $\\mathcal{C} \\subseteq S^2$ with $\\post{f} \\subseteq \\mathcal{C}$.\n\n \\smallskip\n\n \\item We say that $\\potential$ satisfies the \\emph{strong non-integrability condition} (with respect to $f$ and $d$) if $\\potential$ satisfies the $\\holderexp'$-strong non-integrability condition with respect to $f$ and $d$ for some $\\holderexp' \\in (0, \\holderexp]$.\n \\end{enumerate}\n\\end{definition}", + "def:eventually positive functions": "\\begin{definition}[Eventually positive function] \\label{def:eventually positive functions}\n Let $g \\colon X \\mapping X$ be a map on a set $X$, and $\\varphi \\colon X \\mapping \\real$ be a real-valued function on $X$.\n Then $\\varphi$ is \\emph{eventually positive} if there exists $N \\in \\n$ such that $S_n \\varphi(x) > 0$ for each $x \\in X$ and each $n \\in \\n$ with $n \\geqslant N$. \n\\end{definition}", + "sub:Thurston_maps": "\\begin{equation} \\label{eq:Variational Principle for entropy}\n h_{\\operatorname{top}}(g) = \\sup\\{h_{\\mu}(g) \\describe\\mu \\in \\mathcal{M}(X, g)\\}.\n\\end{equation}\nA measure $\\mu$ that attains the supremum in \\eqref{eq:Variational Principle for pressure} is called an \\emph{equilibrium state} for the map $g$ and the potential $\\psi$. A measure $\\mu$ that attains the supremum in \\eqref{eq:Variational Principle for entropy} is called a \\emph{measure of maximal entropy} of $g$.\n\nLet $\\widetilde{X}$ be another compact metric space. If $\\mu$ is a measure on $X$ and the map $\\pi \\colon X \\mapping \\widetilde{X}$ is continuous, then the \\emph{push-forward} $\\pi_{*} \\mu$ of $\\mu$ by $\\pi$ is the measure given by $\\pi_{*}\\mu(A) \\define \\mu \\parentheses[\\big]{ \\pi^{-1}(A) }$ for all Borel sets $A \\subseteq \\widetilde{X}$. \n\n\\subsection{Thurston maps}\\label{sub:Thurston_maps}\n\nIn this subsection, we go over some key concepts and results on Thurston maps, and expanding Thurston maps in particular. \nFor a more thorough treatment of the subject, we refer to \\cite{bonk2017expanding,li2017ergodic}.\n\nLet $S^2$ denote an oriented topological $2$-sphere and $f \\colon S^2 \\mapping S^2$ be a branched covering map. \nWe denote by $\\deg_f(x)$ the local degree of $f$ at $x \\in S^2$.\nThe \\emph{degree} of $f$ is $\\deg{f} = \\sum_{x\\in f^{-1}(y)} \\deg_{f}(x)$ for $y\\in S^2$ and is independent of $y$. \n\nA point $x\\in S^2$ is a \\emph{critical point} of $f$ if $\\deg_f(x) \\geqslant 2$. \nThe set of critical points of $f$ is denoted by $\\crit{f}$. A point $y\\in S^2$ is a \\emph{postcritical point} of $f$ if $y = f^n(x)$ for some $x \\in \\crit{f}$ and $n\\in \\n$. \nThe set of postcritical points of $f$ is denoted by $\\post{f}$. \nWe observe that $\\post{f} = \\post{f^{n}}$ for all $n \\in \\n$.\n\n\\begin{definition}[Thurston maps]\n A Thurston map is a branched covering map $f \\colon S^2 \\mapping S^2$ on $S^2$ with $\\deg f \\geqslant 2$ and $\\card{ \\post{f} } < +\\infty$.\n\\end{definition}\n\nWe now recall the notation for cell decompositions of $S^2$ as used in \\cite{bonk2017expanding} and \\cite{li2017ergodic}. A \\emph{cell of dimension $n$} in $S^2$, $n \\in \\{1, \\, 2\\}$, is a subset $c \\subseteq S^2$ that is homeomorphic to the closed unit ball $\\overline{\\mathbb{B}^n}$ in $\\real^n$, where $\\mathbb{B}^{n}$ is the open unit ball in $\\real^{n}$. We define the \\emph{boundary of $c$}, denoted by $\\partial c$, to be the set of points corresponding to $\\partial \\mathbb{B}^n$ under such a homeomorphism between $c$ and $\\overline{\\mathbb{B}^n}$. The \\emph{interior of $c$} is defined to be $\\inte{c} = c \\mysetminus \\partial c$. For each point $x\\in S^2$, the set $\\{x\\}$ is considered as a \\emph{cell of dimension $0$} in $S^2$. For a cell $c$ of dimension $0$, we adopt the convention that $\\partial c = \\emptyset$ and $\\inte{c} = c$. \n\nLet $f \\colon S^2 \\mapping S^2$ be a Thurston map, and $\\mathcal{C}\\subseteq S^2$ be a Jordan curve containing $\\post{f}$. \nThen the pair $f$ and $\\mathcal{C}$ induces natural cell decompositions $\\mathbf{D}^n(f,\\mathcal{C})$ of $S^2$, for each $n \\in \\n_0$, in the following way:\n\nBy the Jordan curve theorem, the set $S^2 \\mysetminus \\mathcal{C}$ has two connected components. We call the closure of one of them the \\emph{white $0$-tile} for $(f,\\mathcal{C})$, denoted by $X^0_{\\white}$, and the closure of the other one the \\emph{black $0$-tile} for $(f,\\mathcal{C})$, denoted be $X^0_{\\black}$. \nThe set of $0$-\\emph{tiles} is $\\mathbf{X}^0(f, \\mathcal{C}) \\define \\set[\\big]{ X^0_{\\black}, \\, X^0_{\\white} }$. \nThe set of $0$-\\emph{vertices} is $\\mathbf{V}^0(f, \\mathcal{C}) \\define \\post{f}$. \nWe set $\\overline{\\mathbf{V}}^0(f, \\mathcal{C}) \\define \\set[\\big]{ \\{x\\} \\describe x\\in \\mathbf{V}^0(f,\\mathcal{C}) }$. \nThe set of $0$-\\emph{edges} $\\mathbf{E}^0(f,\\mathcal{C})$ consists of the closures of the connected components of $\\mathcal{C} \\mysetminus \\post{f}$. \nThen we get a cell decomposition\\[\n \\mathbf{D}^0(f,\\mathcal{C}) \\define \\mathbf{X}^0(f, \\mathcal{C}) \\cup \\mathbf{E}^0(f,\\mathcal{C}) \\cup \\overline{\\mathbf{V}}^0(f,\\mathcal{C})\n\\]\nof $S^2$ consisting of \\emph{cells of level }$0$, or $0$-\\emph{cells}.\n\nWe can recursively define the unique cell decomposition $\\mathbf{D}^n(f,\\mathcal{C})$ for $n\\in \\n$, consisting of $n$-\\emph{cells}, such that $f$ is cellular for $\\parentheses[\\big]{ \\mathbf{D}^{n + 1}(f,\\mathcal{C}), \\mathbf{D}^n(f,\\mathcal{C}) }$. \nSee \\cite[Lemma~5.12]{bonk2017expanding} for details. \nWe denote by $\\mathbf{X}^n(f,\\mathcal{C})$ the set of $n$-cells of dimension 2, called $n$-\\emph{tiles}; by $\\mathbf{E}^n(f,\\mathcal{C})$ the set of $n$-cells of dimension $1$, called $n$-\\emph{edges}; by $\\overline{\\mathbf{V}}^n(f,\\mathcal{C})$ the set of $n$-cells of dimension $0$; and by $\\mathbf{V}^n(f,\\mathcal{C})$ the set $\\set[\\big]{x \\describe \\{x\\} \\in \\overline{\\mathbf{V}}^n(f,\\mathcal{C}) }$, called the set of $n$-\\emph{vertices}. \n\n\\smallskip\n\nFor $n\\in \\n_0$, we define the \\emph{set of black $n$-tiles} as\\[\n \\mathbf{X}^n_{\\black}(f,\\mathcal{C}) \\define \\left\\{ X \\in \\mathbf{X}^n (f,\\mathcal{C}) \\describe f^n(X) = X^0_{\\black} \\right\\},\n\\]\nand the \\emph{set of white $n$-tiles} as\\[\n \\mathbf{X}^n_{\\white}(f,\\mathcal{C}) \\define \\left\\{X\\in \\mathbf{X}^n(f,\\mathcal{C}) \\describe f^n(X) = X^0_{\\white}\\right\\}.\n\\]\n\nFrom now on, if the map $f$ and the Jordan curve $\\mathcal{C}$ are clear from the context, we will sometimes omit $(f,\\mathcal{C})$ in the notation above.\n\nWe can now give a definition of expanding Thurston maps.\n\n\\begin{definition}[Expansion] \\label{def:expanding_Thurston_maps}\n A Thurston map $f \\colon S^2 \\mapping S^2$ is called \\emph{expanding} if there exists a metric $d$ on $S^2$ that induces the standard topology on $S^2$ and a Jordan curve $\\mathcal{C} \\subseteq S^2$ containing $\\post{f}$ such that\n \\begin{equation} \\label{eq:definition of expansion}\n \\lim_{n \\to +\\infty} \\max\\{ \\diameter{d}{X} \\describe X \\in \\mathbf{X}^n(f,\\mathcal{C}) \\} = 0.\n \\end{equation}", + "thm:main theorem": "\\begin{theorem} \\label{thm:main theorem}\n Let $f \\colon S^2 \\to S^2$ be an expanding Thurston map and $d$ be a visual metric on $S^2$ for $f$.\n Let $\\beta \\in (0, 1]$ and $\\phi \\in C^{0, \\beta}(S^2, d)$ be an eventually positive real-valued \\holder continuous function satisfying the $\\beta$-strong non-integrability condition (with respect to $f$ and $d$). \n Then there exists a unique positive number $\\rootpressure > 0$ with topological pressure $P(f, -\\rootpressure \\phi) = 0$ and there exists $N_{f} \\in \\n$ depending only on $f$ such that for each $N \\in \\n$ with $N \\geqslant N_{f}$, the following statement holds for the iterate $F \\coloneqq f^{N}$ and the potential $\\Phi \\coloneqq \\sum_{i = 0}^{N - 1} \\phi \\circ f^{i}$:\n\n Denote $\\alpha \\coloneqq \\frac{\\mathrm{d}}{\\mathrm{d} t} P(F, t \\Phi) |_{t = -\\rootpressure}$ and $\\sigma \\coloneqq \\sqrt{ \\frac{\\mathrm{d}^2}{\\mathrm{d} t^2} P(F, t \\Phi) |_{t = -\\rootpressure } }$.\n Let $\\sequen{I_{n}}$ be a sequence of intervals contained in a compact set $K \\subseteq \\real$ with $\\sequen[\\big]{|I_{n}|^{-1}}$ having sub-exponential growth.\n Then\n \\[\n \\pi_{F, \\Phi}(n; \\alpha, I_n) \\sim \\frac{ \\int_{I_{n}} e^{\\rootpressure t} \\, \\mathrm{d}t }{ \\sqrt{2\\pi} \\, \\sigma } \\, \\frac{e^{\\rootpressure \\alpha n}}{n^{3/2}} \\qquad \\text{as } n \\to +\\infty.\n \\]\n\\end{theorem}", + "rem:chordal metric visual metric qs equiv": "\\begin{remark}\\label{rem:chordal metric visual metric qs equiv}\n If $f \\colon \\ccx \\mapping \\ccx$ is a rational expanding Thurston map, then a visual metric is quasisymmetrically equivalent to the chordal metric on the Riemann sphere $\\ccx$ (see \\cite[Theorem~18.1~(ii)]{bonk2017expanding}). \n Here the chordal metric $\\sigma$ on $\\ccx$ is given by $\\sigma (z, w) \\define \\frac{2\\abs{z - w}}{\\sqrt{1 + \\abs{z}^2} \\sqrt{1 + \\abs{w}^2}}$ for all $\\juxtapose{z}{w} \\in \\cx$, and $\\sigma(\\infty, z) = \\sigma(z, \\infty) \\define \\frac{2}{\\sqrt{1 + \\abs{z}^2}}$ for all $z \\in \\cx$. \n Quasisymmetric embeddings of bounded connected metric spaces are \\holder continuous (see \\cite[Section~11.1 and Corollary~11.5]{heinonen2001lectures}). \n Accordingly, the classes of \\holder continuous functions on $\\ccx$ equipped with the chordal metric and on $S^2 = \\ccx$ equipped with any visual metric for $f$ are the same (up to a change of the \\holder exponent).\n\\end{remark}" + }, + "pre_theorem_intro_text_len": 4167, + "pre_theorem_intro_text": "\\label{sec:Introduction}\n\nPeriodic orbits serve as the skeleton of chaotic dynamics, encoding essential information about the system's long-term behavior.\nA fundamental objective in this field is counting these orbits, a problem analogous to the Prime Number Theorem in number theory.\nRecently, the dynamical counterpart---the Prime Orbit Theorem---was successfully established for expanding Thurston maps \\cite{li2024prime:dirichlet, li2024prime:split}, which serve as topological models for postcritically-finite rational maps.\nWhile this answers the question of ``how many'' orbits exist, it leaves open the more subtle question of ``how they are distributed'' with respect to statistical observables.\nUnderstanding this fine-scale distribution is crucial for characterizing the fluctuations inherent in the system.\n\nIn this paper, we investigate the asymptotic distribution of primitive periodic orbits restricted to shrinking intervals for expanding Thurston maps.\nWe count orbits whose Birkhoff sums for a given potential lie within a prescribed family of shrinking intervals.\nwe obtain a precise asymptotic formula for the number of these constrained orbits, which resemble a local central limit theorem.\nUnlike the central limit theorem, which describes the distribution of Birkhoff sums on the scale of $\\sqrt{n}$, the local central limit theorem can probe the density at a given point.\nOur main result refines the coarse counting of the Prime Orbit Theorem \\cite{li2024prime:dirichlet} for expanding Thurston maps.\n\nThe problem of counting orbits in shrinking intervals has been successfully addressed for uniformly expanding systems.\nFor instance, Petkov and Stoyanov \\cite{petkovDistributionPeriodsClosed2012} investigated the distribution of closed orbits for hyperbolic flows, and Sharp and Stylianou \\cite{sharpStatisticsMultipliersHyperbolic2022} studied the multipliers and holonomies for hyperbolic rational maps.\nHowever, these results rely heavily on the hyperbolicity and smoothness of the underlying systems.\nThe context of non-uniformly expanding dynamics, particularly for branched covering maps, remains largely unexplored.\nTo the best of our knowledge, our work is the first to address this problem in such a setting.\n\n\\subsection{Main results}\\label{sub:Main results}\n\nLet $f \\colon S^2 \\to S^2$ be an expanding Thurston map and $\\phi \\colon S^2 \\to \\mathbb{R}$ be a real-valued H\\\"{o}lder continuous function. \nThe topological $2$-sphere $S^2$ is equipped with a \\emph{visual metric} $d$ (see Section~\\ref{sub:Thurston_maps} for details). \nA periodic orbit $\\tau = \\set[\\big]{x, f(x), \\dots, f^{n-1}(x)}$ (where $f^n(x) = x$) is called \\emph{primitive} if $f^m(x) \\ne x$ for each integer $m$ with $1 \\leqslant m < n$. \nWe denote the set of primitive periodic orbits of $f$ by $\\mathfrak{P}(f)$. \nFor each $\\tau \\in \\mathfrak{P}(f)$, we write $\\phi(\\tau) \\coloneqq \\sum_{x \\in \\tau} \\phi(x)$.\n\nIn this article, we investigate the asymptotic distribution of primitive periodic orbits subject to constraints on their Birkhoff sums. \nSpecifically, for a given number $\\alpha \\in \\mathbb{R}$ and a sequence $\\sequen{I_{n}}$ of intervals contained in a compact set $K \\subseteq \\mathbb{R}$, we study the asymptotic behavior of \n\\[\n \\pi_{f, \\phi}(n; \\alpha, I_n) \\coloneqq \\operatorname{card} \\set[\\big]{ \\tau \\in \\mathfrak{P}_n(f) : \\phi(\\tau) - n \\alpha \\in I_{n}}\n\\]\nas $n \\to +\\infty$, where $\\mathfrak{P}_n(f) \\coloneqq \\set[\\big]{\\tau \\in \\mathfrak{P}(f) : |\\tau| = n}$. \n\nTo obtain precise estimates, we impose specific conditions on the potential $\\phi$ and the sequence $\\sequen{I_{n}}$.\nWe assume that $\\phi$ is \\emph{eventually positive} and satisfies the \\emph{strong non-integrability condition} (see Definitions~\\ref{def:eventually positive functions} and~\\ref{def:strong non-integrability condition}). \nFurthermore, denoting the length of $I_{n}$ by $|I_n|$, we assume that the sequence $\\sequen[\\big]{|I_{n}|^{-1}}$ exhibits sub-exponential growth, i.e., $\\limsup_{n \\to +\\infty} \\frac{1}{n} \\log \\parentheses[\\big]{ |I_{n}|^{-1} } = 0$.\n\nWe write $A(n) \\sim B(n)$ as $n \\to +\\infty$ if $\\lim_{n \\to +\\infty} A(n) / B(n) = 1$.", + "context": "Periodic orbits serve as the skeleton of chaotic dynamics, encoding essential information about the system's long-term behavior.\nA fundamental objective in this field is counting these orbits, a problem analogous to the Prime Number Theorem in number theory.\nRecently, the dynamical counterpart---the Prime Orbit Theorem---was successfully established for expanding Thurston maps \\cite{li2024prime:dirichlet, li2024prime:split}, which serve as topological models for postcritically-finite rational maps.\nWhile this answers the question of ``how many'' orbits exist, it leaves open the more subtle question of ``how they are distributed'' with respect to statistical observables.\nUnderstanding this fine-scale distribution is crucial for characterizing the fluctuations inherent in the system.\n\nIn this paper, we investigate the asymptotic distribution of primitive periodic orbits restricted to shrinking intervals for expanding Thurston maps.\nWe count orbits whose Birkhoff sums for a given potential lie within a prescribed family of shrinking intervals.\nwe obtain a precise asymptotic formula for the number of these constrained orbits, which resemble a local central limit theorem.\nUnlike the central limit theorem, which describes the distribution of Birkhoff sums on the scale of $\\sqrt{n}$, the local central limit theorem can probe the density at a given point.\nOur main result refines the coarse counting of the Prime Orbit Theorem \\cite{li2024prime:dirichlet} for expanding Thurston maps.\n\nLet $f \\colon S^2 \\to S^2$ be an expanding Thurston map and $\\phi \\colon S^2 \\to \\mathbb{R}$ be a real-valued H\\\"{o}lder continuous function. \nThe topological $2$-sphere $S^2$ is equipped with a \\emph{visual metric} $d$ (see Section~\\ref{sub:Thurston_maps} for details). \nA periodic orbit $\\tau = \\set[\\big]{x, f(x), \\dots, f^{n-1}(x)}$ (where $f^n(x) = x$) is called \\emph{primitive} if $f^m(x) \\ne x$ for each integer $m$ with $1 \\leqslant m < n$. \nWe denote the set of primitive periodic orbits of $f$ by $\\mathfrak{P}(f)$. \nFor each $\\tau \\in \\mathfrak{P}(f)$, we write $\\phi(\\tau) \\coloneqq \\sum_{x \\in \\tau} \\phi(x)$.\n\nIn this article, we investigate the asymptotic distribution of primitive periodic orbits subject to constraints on their Birkhoff sums. \nSpecifically, for a given number $\\alpha \\in \\mathbb{R}$ and a sequence $\\sequen{I_{n}}$ of intervals contained in a compact set $K \\subseteq \\mathbb{R}$, we study the asymptotic behavior of \n\\[\n \\pi_{f, \\phi}(n; \\alpha, I_n) \\coloneqq \\operatorname{card} \\set[\\big]{ \\tau \\in \\mathfrak{P}_n(f) : \\phi(\\tau) - n \\alpha \\in I_{n}}\n\\]\nas $n \\to +\\infty$, where $\\mathfrak{P}_n(f) \\coloneqq \\set[\\big]{\\tau \\in \\mathfrak{P}(f) : |\\tau| = n}$.\n\nTo obtain precise estimates, we impose specific conditions on the potential $\\phi$ and the sequence $\\sequen{I_{n}}$.\nWe assume that $\\phi$ is \\emph{eventually positive} and satisfies the \\emph{strong non-integrability condition} (see Definitions~\\ref{def:eventually positive functions} and~\\ref{def:strong non-integrability condition}). \nFurthermore, denoting the length of $I_{n}$ by $|I_n|$, we assume that the sequence $\\sequen[\\big]{|I_{n}|^{-1}}$ exhibits sub-exponential growth, i.e., $\\limsup_{n \\to +\\infty} \\frac{1}{n} \\log \\parentheses[\\big]{ |I_{n}|^{-1} } = 0$.\n\nWe write $A(n) \\sim B(n)$ as $n \\to +\\infty$ if $\\lim_{n \\to +\\infty} A(n) / B(n) = 1$.", + "full_context": "Periodic orbits serve as the skeleton of chaotic dynamics, encoding essential information about the system's long-term behavior.\nA fundamental objective in this field is counting these orbits, a problem analogous to the Prime Number Theorem in number theory.\nRecently, the dynamical counterpart---the Prime Orbit Theorem---was successfully established for expanding Thurston maps \\cite{li2024prime:dirichlet, li2024prime:split}, which serve as topological models for postcritically-finite rational maps.\nWhile this answers the question of ``how many'' orbits exist, it leaves open the more subtle question of ``how they are distributed'' with respect to statistical observables.\nUnderstanding this fine-scale distribution is crucial for characterizing the fluctuations inherent in the system.\n\nIn this paper, we investigate the asymptotic distribution of primitive periodic orbits restricted to shrinking intervals for expanding Thurston maps.\nWe count orbits whose Birkhoff sums for a given potential lie within a prescribed family of shrinking intervals.\nwe obtain a precise asymptotic formula for the number of these constrained orbits, which resemble a local central limit theorem.\nUnlike the central limit theorem, which describes the distribution of Birkhoff sums on the scale of $\\sqrt{n}$, the local central limit theorem can probe the density at a given point.\nOur main result refines the coarse counting of the Prime Orbit Theorem \\cite{li2024prime:dirichlet} for expanding Thurston maps.\n\nLet $f \\colon S^2 \\to S^2$ be an expanding Thurston map and $\\phi \\colon S^2 \\to \\mathbb{R}$ be a real-valued H\\\"{o}lder continuous function. \nThe topological $2$-sphere $S^2$ is equipped with a \\emph{visual metric} $d$ (see Section~\\ref{sub:Thurston_maps} for details). \nA periodic orbit $\\tau = \\set[\\big]{x, f(x), \\dots, f^{n-1}(x)}$ (where $f^n(x) = x$) is called \\emph{primitive} if $f^m(x) \\ne x$ for each integer $m$ with $1 \\leqslant m < n$. \nWe denote the set of primitive periodic orbits of $f$ by $\\mathfrak{P}(f)$. \nFor each $\\tau \\in \\mathfrak{P}(f)$, we write $\\phi(\\tau) \\coloneqq \\sum_{x \\in \\tau} \\phi(x)$.\n\nIn this article, we investigate the asymptotic distribution of primitive periodic orbits subject to constraints on their Birkhoff sums. \nSpecifically, for a given number $\\alpha \\in \\mathbb{R}$ and a sequence $\\sequen{I_{n}}$ of intervals contained in a compact set $K \\subseteq \\mathbb{R}$, we study the asymptotic behavior of \n\\[\n \\pi_{f, \\phi}(n; \\alpha, I_n) \\coloneqq \\operatorname{card} \\set[\\big]{ \\tau \\in \\mathfrak{P}_n(f) : \\phi(\\tau) - n \\alpha \\in I_{n}}\n\\]\nas $n \\to +\\infty$, where $\\mathfrak{P}_n(f) \\coloneqq \\set[\\big]{\\tau \\in \\mathfrak{P}(f) : |\\tau| = n}$.\n\nTo obtain precise estimates, we impose specific conditions on the potential $\\phi$ and the sequence $\\sequen{I_{n}}$.\nWe assume that $\\phi$ is \\emph{eventually positive} and satisfies the \\emph{strong non-integrability condition} (see Definitions~\\ref{def:eventually positive functions} and~\\ref{def:strong non-integrability condition}). \nFurthermore, denoting the length of $I_{n}$ by $|I_n|$, we assume that the sequence $\\sequen[\\big]{|I_{n}|^{-1}}$ exhibits sub-exponential growth, i.e., $\\limsup_{n \\to +\\infty} \\frac{1}{n} \\log \\parentheses[\\big]{ |I_{n}|^{-1} } = 0$.\n\nWe write $A(n) \\sim B(n)$ as $n \\to +\\infty$ if $\\lim_{n \\to +\\infty} A(n) / B(n) = 1$.\n\nWe write $A(n) \\sim B(n)$ as $n \\to +\\infty$ if $\\lim_{n \\to +\\infty} A(n) / B(n) = 1$.\n\nRecall that a postcritically-finite rational map is expanding if and only if it has no periodic critical points (see \\cite[Proposition~2.3]{bonk2017expanding}). \nTherefore, when we restrict our attention to rational maps, we obtain the following corollary of Theorem~\\ref{thm:main theorem} and Remark~\\ref{rem:chordal metric visual metric qs equiv}.\n\n\\begin{corollary}\\label{coro:main theorem for postcritically-finite rational maps}\n Let $f \\colon \\ccx \\mapping \\ccx$ be a postcritically-finite rational map without periodic critical points.\n Let $\\sigma$ be the chordal metric or the spherical metric on the Riemann sphere $\\ccx$, and $\\phi \\in C^{0, \\holderexp} \\parentheses[\\big]{ \\ccx, \\sigma }$ be an eventually positive real-valued \\holder continuous function with exponent $\\holderexp \\in (0, 1]$ satisfying the $\\holderexp$-strong non-integrability condition (with respect to $f$ and a visual metric).\n Then there exists a unique positive number $\\rootpressure > 0$ with topological pressure $P(f, -\\rootpressure \\phi) = 0$ and there exists $N_f \\in \\n$ depending only on $f$ such that for each $N \\in \\n$ with $N \\geqslant N_f$, the following statements hold for $F \\define f^N$ and $\\Phi \\define \\sum_{i=0}^{N-1} \\phi \\circ f^i$:\n\nDenote $\\alpha \\coloneqq \\frac{\\mathrm{d}}{\\mathrm{d} t} P(F, t \\Phi) |_{t = -\\rootpressure}$ and $\\sigma \\coloneqq \\sqrt{ \\frac{\\mathrm{d}^2}{\\mathrm{d} t^2} P(F, t \\Phi) |_{t = -\\rootpressure } }$.\n Let $\\sequen{I_{n}}$ be a sequence of intervals contained in a compact set $K \\subseteq \\real$ with $\\sequen[\\big]{|I_{n}|^{-1}}$ having sub-exponential growth.\n Then\n \\[\n \\pi_{F, \\Phi}(n; \\alpha, I_n) \\sim \\frac{ \\int_{I_{n}} e^{\\rootpressure t} \\, \\mathrm{d}t }{ \\sqrt{2\\pi} \\, \\sigma } \\, \\frac{e^{\\rootpressure \\alpha n}}{n^{3/2}} \\qquad \\text{as } n \\to +\\infty.\n \\]\n\\end{corollary}\n\nWe now consider the following three quantities:\n \\begin{align*}\n A_1(n) &\\define \\int_{ \\abs{t} < \\varepsilon \\sigma \\sqrt{n} } \\, \\abs[\\bigg]{ \\frac{\\ell_{n}^{-1}}{e^{P(f, -\\rootpressure \\normpotential)n}} \\widehat{\\psi}_{n}\\parentheses[\\Big]{ \\frac{t}{2 \\pi \\sigma \\sqrt{n}} } \\normpartifun[\\Big]{-\\rootpressure + \\frac{\\imaginary t}{\\sigma \\sqrt{n}}} \\, - \\, \\ell_{n}^{-1} e^{-\\frac{t^{2}}{2}} \\! \\int_{\\real} \\! \\psi_{n}(x) \\,\\mathrm{d}x } \\,\\mathrm{d}t, \\\\\n A_2(n) &\\define \\int_{ \\abs{t} \\geqslant \\varepsilon \\sigma \\sqrt{n} } \\, \\abs[\\bigg]{ \\frac{\\ell_{n}^{-1}}{e^{P(f, -\\rootpressure \\normpotential)n}} \\widehat{\\psi}_{n}\\parentheses[\\Big]{ \\frac{t}{2 \\pi \\sigma \\sqrt{n}} } \\normpartifun[\\Big]{-\\rootpressure + \\frac{\\imaginary t}{\\sigma \\sqrt{n}}} } \\,\\mathrm{d}t, \\\\\n A_3(n) &\\define \\int_{ \\abs{t} \\geqslant \\varepsilon \\sigma \\sqrt{n} } \\, \\abs[\\bigg]{ \\ell_{n}^{-1} e^{-\\frac{t^{2}}{2}} \\! \\int_{\\real} \\! \\psi_{n}(x) \\,\\mathrm{d}x } \\,\\mathrm{d}t.\n \\end{align*}\n Here $\\varepsilon \\in (0, 1)$ is chosen to be smaller than $\\min\\set[\\Big]{ \\delta , \\frac{\\sigma^{2}}{4 C_{\\delta}}, t_0 }$, where the constants $\\delta$ and $C_{\\delta}$ are given by Lemma~\\ref{lem:Taylor extension of pressure function in imaginary part}, and the constant $t_0$ is given by Lemma~\\ref{lem:partition function estimate:expanding Thurston map:bounded imaginary bounded cases}~\\ref{item:lem:partition function estimate:expanding Thurston map:bounded imaginary bounded cases:local part}.\n It follows from Claim~1 that \n \\[\n A(n) \\leqslant \\frac{1}{\\sqrt{2\\pi}} \\parentheses[\\big]{ A_1(n) + A_2(n) + A_3(n)} \\qquad \\text{for each } n \\in \\n.\n \\]\n Thus it suffices to show that $\\lim_{n \\to +\\infty} A_{i}(n) = 0$ for each $i \\in \\{1, 2, 3\\}$.\n\nLet $T > 1$ be the constant given by Proposition~\\ref{prop:partition function estimate:expanding Thurston map:unbounded imaginary}.\n By Lemma~\\ref{lem:partition function estimate:expanding Thurston map:bounded imaginary bounded cases}~\\ref{item:lem:partition function estimate:expanding Thurston map:bounded imaginary bounded cases:intermediate part}, there exists $\\vartheta \\in (0, 1)$ such that\n \\begin{equation} \\label{eq:temp:prop:estimate of partition function with periodic orbits:Claim3:medium case}\n \\int_{ \\varepsilon \\leqslant \\frac{ \\abs{t}}{\\sigma \\sqrt{n} } \\leqslant T } \\abs[\\bigg]{ \\frac{\\ell_{n}^{-1}}{e^{P(f, -\\rootpressure \\normpotential)n}} \\widehat{\\psi}_{n}\\parentheses[\\Big]{ \\frac{t}{2 \\pi \\sigma \\sqrt{n}} } \\normpartifun[\\Big]{-\\rootpressure + \\frac{\\imaginary t}{\\sigma \\sqrt{n}}} } \\,\\mathrm{d}t\n = \\mathcal{O}\\parentheses[\\big]{ \\sigma \\sqrt{n} \\, \\ell_{n}^{-1} \\uniformnorm{\\widehat{\\psi}_{n}} \\vartheta^{n} }\n \\end{equation}\n as $n \\to +\\infty$, where $\\ell_{n}^{-1} \\uniformnorm{\\widehat{\\psi}_{n}} \\leqslant \\ell_{n}^{-1} \\int_{\\real} \\! \\psi_{n}(x) \\,\\mathrm{d}x = \\int_{\\real} \\! \\psi(y) e^{\\rootpressure \\ell_{n} y} \\,\\mathrm{d}y$ is uniformly bounded for $n \\in \\n$ since $\\psi$ has compact support. \n Since $\\psi_n \\in C^{4}(\\real, \\real)$, we have that for all $n \\in \\n$ and $t \\in \\real$,\n \\[\n (2 \\pi \\imaginary t)^{4} \\widehat{\\psi}_{n}(t) = \\widehat{\\psi_{n}^{(4)}}(t).\n \\] \n Moreover, since $\\psi$ has compact support, we have that \n \\[\n \\begin{split}\n \\ell_{n}^{3} \\, \\uniformnorm[\\Big]{ \\widehat{\\psi_{n}^{(4)}} } \n \\leqslant \\ell_{n}^{3} \\! \\int_{\\real} \\abs[\\big]{ \\psi_{n}^{(4)}(x) } \\,\\mathrm{d}x \n &= \\int_{\\real} \\, \\abs[\\bigg]{ \\sum_{k=0}^{4} \\binom{4}{k} \\ell_{n}^{k} \\, \\psi^{(4 - k)}(y) (\\rootpressure )^{k} e^{\\rootpressure \\ell_{n} y} } \\,\\mathrm{d}y \\leqslant C'\n \\end{split}\n \\]\n for some constant $C' \\geqslant 0$, which is independent of $n$. \n Then, by Proposition~\\ref{prop:partition function estimate:expanding Thurston map:unbounded imaginary}, there exist $C > 0$ and $\\rho \\in (0, 1)$ such that for each integer $n \\geqslant 2$,\n \\begin{align*}\n &\\int_{ \\abs{t} > T \\sigma \\sqrt{n} } \\, \\abs[\\bigg]{ \\frac{\\ell_{n}^{-1}}{e^{P(f, -\\rootpressure \\normpotential)n}} \\widehat{\\psi}_{n}\\parentheses[\\Big]{ \\frac{t}{2 \\pi \\sigma \\sqrt{n}} } \\normpartifun[\\Big]{-\\rootpressure + \\frac{\\imaginary t}{\\sigma \\sqrt{n}}} } \\,\\mathrm{d}t \\\\\n &\\qquad \\leqslant \\int_{ \\abs{t} > T \\sigma \\sqrt{n} } \\, \\abs[\\bigg]{ \\ell_{n}^{-1} \\widehat{\\psi}_{n}\\parentheses[\\Big]{ \\frac{t}{2 \\pi \\sigma \\sqrt{n}} } C \\abs[\\Big]{ \\frac{t}{\\sigma\\sqrt{n}} }^{2 + \\varepsilon} \\rho^{n} } \\,\\mathrm{d}t\\\\\n &\\qquad = \\int_{ \\abs{t} > T } C \\sigma \\sqrt{n} \\, \\ell_{n}^{-1} \\rho^{n} \\abs{t}^{2 + \\varepsilon} \\abs[\\Big]{ \\widehat{\\psi}_{n}\\parentheses[\\Big]{ \\frac{t}{2 \\pi} } } \\,\\mathrm{d}t \\\\\n &\\qquad = C \\sigma \\sqrt{n} \\, \\ell_{n}^{-4} \\rho^{n} \\int_{ \\abs{t} > T } \\abs{t}^{-2 + \\varepsilon} \\, \\ell_{n}^{3} \\abs[\\Big]{ \\widehat{\\psi^{(4)}_{n}} \\parentheses[\\Big]{ \\frac{t}{2 \\pi} } } \\,\\mathrm{d}t \\\\\n &\\qquad \\leqslant C C' \\sigma \\sqrt{n} \\, \\ell_{n}^{-4} \\rho^{n} \\int_{ \\abs{t} > T } \\abs{t}^{-2 + \\varepsilon} \\,\\mathrm{d}t.\n \\end{align*}\n Combining this with \\eqref{eq:temp:prop:estimate of partition function with periodic orbits:Claim3:medium case} and recalling that the sequences $\\sequen{\\ell_{n}^{-1}}$ are of sub-exponential growth, we establish Claim~3.", + "post_theorem_intro_text_len": 5653, + "post_theorem_intro_text": "Recall that a postcritically-finite rational map is expanding if and only if it has no periodic critical points (see \\cite[Proposition~2.3]{bonk2017expanding}). \nTherefore, when we restrict our attention to rational maps, we obtain the following corollary of Theorem~\\ref{thm:main theorem} and Remark~\\ref{rem:chordal metric visual metric qs equiv}.\n\n\\begin{corollary}\\label{coro:main theorem for postcritically-finite rational maps}\n Let $f \\colon \\ccx \\rightarrow \\ccx$ be a postcritically-finite rational map without periodic critical points.\n Let $\\sigma$ be the chordal metric or the spherical metric on the Riemann sphere $\\ccx$, and $\\phi \\in C^{0, \\beta} \\parentheses[\\big]{ \\ccx, \\sigma }$ be an eventually positive real-valued H\\\"{o}lder continuous function with exponent $\\beta \\in (0, 1]$ satisfying the $\\beta$-strong non-integrability condition (with respect to $f$ and a visual metric).\n Then there exists a unique positive number $\\rootpressure > 0$ with topological pressure $P(f, -\\rootpressure \\phi) = 0$ and there exists $N_f \\in \\mathbb{N}$ depending only on $f$ such that for each $N \\in \\mathbb{N}$ with $N \\geqslant N_f$, the following statements hold for $F \\coloneqq f^N$ and $\\Phi \\coloneqq \\sum_{i=0}^{N-1} \\phi \\circ f^i$:\n\n Denote $\\alpha \\coloneqq \\frac{\\mathrm{d}}{\\mathrm{d} t} P(F, t \\Phi) |_{t = -\\rootpressure}$ and $\\sigma \\coloneqq \\sqrt{ \\frac{\\mathrm{d}^2}{\\mathrm{d} t^2} P(F, t \\Phi) |_{t = -\\rootpressure } }$.\n Let $\\sequen{I_{n}}$ be a sequence of intervals contained in a compact set $K \\subseteq \\mathbb{R}$ with $\\sequen[\\big]{|I_{n}|^{-1}}$ having sub-exponential growth.\n Then\n \\[\n \\pi_{F, \\Phi}(n; \\alpha, I_n) \\sim \\frac{ \\int_{I_{n}} e^{\\rootpressure t} \\, \\mathrm{d}t }{ \\sqrt{2\\pi} \\, \\sigma } \\, \\frac{e^{\\rootpressure \\alpha n}}{n^{3/2}} \\qquad \\text{as } n \\to +\\infty.\n \\]\n\\end{corollary}\n\n\\subsection{Strategy and organization}\nOur approach relies on a combination of thermodynamic formalism and operator theory, specifically adapted to the branched covering setting.\n\nThe main technical obstacle in studying Thurston maps is the presence of critical points, which disrupts the functional analytic properties of the standard Ruelle transfer operator.\nTo overcome this, we employ the \\emph{split Ruelle operators} introduced in \\cite{li2024prime:split}.\nThe idea is to decompose the sphere into ``black'' and ``white'' tiles (based on a checkerboard coloring induced by an invariant Jordan curve) and define a pair of operators acting on functions supported on these tiles.\nThis construction effectively ``unfolds'' the singularities, allowing us to recover good spectral properties.\n\nTo obtain the precise asymptotics required for the local central limit theorem, we need to control the decay of the characteristic function of the Birkhoff sums.\nIn terms of operator theory, this translates to bounding the spectral radius of the twisted transfer operator $\\mathbb{L}_{s\\phi}$ as the complex parameter $s$ moves along the imaginary axis.\nThe detailed estimates are separated into three parts: the unbounded part, the bounded part, and the local part.\nFor the unbounded part, we employ Dolgopyat-type estimates for the split Ruelle operators established in \\cite{li2024prime:split}.\nThis requires checking a strong non-integrability condition (Definition~\\ref{def:strong non-integrability condition}), which in particular implies that the potential is not cohomologous to a constant.\nFor the bounded part, we employ Ruelle's estimate (see Appendix~\\ref{sec:Appendix:Ruelle lemma}).\nFor the local part, we employ the complex Ruelle--Perron--Frobenius theorem \\cite[Theorem~2]{pollicott1984complex} and arguments in \\cite{pollicottZetaFunctionsPeriodic1990}.\n\nFinally, to count orbits in intervals $I_n$, we approximate the indicator function $\\mathbbm{1}_{I_n}$ by smooth test functions.\nWe then apply Fourier transforms to relate the smoothed count to partition functions, which allows us to apply the established decay estimates.\n\n\\smallskip\n\nThe paper is organized as follows.\nIn Section~\\ref{sec:Preliminaries}, we fix our notation, review fundamental concepts from thermodynamic formalism, and recall key results from the theory of expanding Thurston maps.\nIn Section~\\ref{sec:The Assumptions}, we collect the main assumptions used throughout the paper.\nThe technical core of the paper is Section~\\ref{sec:Pressure function and partition function estimates}, where we employ previous results and derive crucial decay estimates for associated partition functions.\nThese estimates are then used in Section~\\ref{sec:Proof of the main theorem} to prove Theorem~\\ref{thm:main theorem}.\n\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000section/Proof.tex\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u00000000664\u00000000000\u00000000000\u000000000072144\u000015112046574\u0000013046\u0000 0\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000ustar \u0000root\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000root\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000\u0000", + "sketch": "To prove Theorem~\\ref{thm:main theorem}, the approach uses “a combination of thermodynamic formalism and operator theory, specifically adapted to the branched covering setting.” The “main technical obstacle” is that “critical points … disrupt the functional analytic properties of the standard Ruelle transfer operator,” so the argument “employ[s] the \\emph{split Ruelle operators}” by “decompos[ing] the sphere into ‘black’ and ‘white’ tiles … and defin[ing] a pair of operators acting on functions supported on these tiles,” which “unfolds the singularities” and restores “good spectral properties.”\n\nFor the local central limit theorem asymptotics, one “control[s] the decay of the characteristic function of the Birkhoff sums,” i.e. “bound[s] the spectral radius of the twisted transfer operator $\\mathbb{L}_{s\\phi}$ as the complex parameter $s$ moves along the imaginary axis.” The needed estimates are split into “three parts: the unbounded part, the bounded part, and the local part”: for the unbounded part use “Dolgopyat-type estimates for the split Ruelle operators,” which “requires checking a strong non-integrability condition … (in particular [implying] that the potential is not cohomologous to a constant)”; for the bounded part use “Ruelle’s estimate”; and for the local part use “the complex Ruelle--Perron--Frobenius theorem … and arguments in \\cite{pollicottZetaFunctionsPeriodic1990}.”\n\nFinally, “to count orbits in intervals $I_n$,” one “approximate[s] the indicator function $\\mathbbm{1}_{I_n}$ by smooth test functions,” then “apply Fourier transforms to relate the smoothed count to partition functions,” so that the previously established decay estimates can be applied.", + "expanded_sketch": "To prove the main theorem, the approach uses “a combination of thermodynamic formalism and operator theory, specifically adapted to the branched covering setting.” The “main technical obstacle” is that “critical points … disrupt the functional analytic properties of the standard Ruelle transfer operator,” so the argument “employ[s] the \\emph{split Ruelle operators}” by “decompos[ing] the sphere into ‘black’ and ‘white’ tiles … and defin[ing] a pair of operators acting on functions supported on these tiles,” which “unfolds the singularities” and restores “good spectral properties.”\n\nFor the local central limit theorem asymptotics, one “control[s] the decay of the characteristic function of the Birkhoff sums,” i.e. “bound[s] the spectral radius of the twisted transfer operator $\\mathbb{L}_{s\\phi}$ as the complex parameter $s$ moves along the imaginary axis.” The needed estimates are split into “three parts: the unbounded part, the bounded part, and the local part”: for the unbounded part use “Dolgopyat-type estimates for the split Ruelle operators,” which “requires checking a strong non-integrability condition … (in particular [implying] that the potential is not cohomologous to a constant)”; for the bounded part use “Ruelle’s estimate”; and for the local part use “the complex Ruelle--Perron--Frobenius theorem … and arguments in Mark Pollicott, \\emph{Zeta functions, periodic points and oscillations of the prime orbit theorem} (1990).”\n\nFinally, “to count orbits in intervals $I_n$,” one “approximate[s] the indicator function $\\mathbbm{1}_{I_n}$ by smooth test functions,” then “apply Fourier transforms to relate the smoothed count to partition functions,” so that the previously established decay estimates can be applied.", + "expanded_theorem": "\\label{thm:main theorem}\n Let $f \\colon S^2 \\to S^2$ be an expanding Thurston map and $d$ be a visual metric on $S^2$ for $f$.\n Let $\\beta \\in (0, 1]$ and $\\phi \\in C^{0, \\beta}(S^2, d)$ be an eventually positive real-valued H\\\"{o}lder continuous function satisfying the $\\beta$-strong non-integrability condition (with respect to $f$ and $d$). \n Then there exists a unique positive number $\\rootpressure > 0$ with topological pressure $P(f, -\\rootpressure \\phi) = 0$ and there exists $N_{f} \\in \\mathbb{N}$ depending only on $f$ such that for each $N \\in \\mathbb{N}$ with $N \\geqslant N_{f}$, the following statement holds for the iterate $F \\coloneqq f^{N}$ and the potential $\\Phi \\coloneqq \\sum_{i = 0}^{N - 1} \\phi \\circ f^{i}$:\n\n Denote $\\alpha \\coloneqq \\frac{\\mathrm{d}}{\\mathrm{d} t} P(F, t \\Phi) |_{t = -\\rootpressure}$ and $\\sigma \\coloneqq \\sqrt{ \\frac{\\mathrm{d}^2}{\\mathrm{d} t^2} P(F, t \\Phi) |_{t = -\\rootpressure } }$.\n Let $\\sequen{I_{n}}$ be a sequence of intervals contained in a compact set $K \\subseteq \\mathbb{R}$ with $\\sequen[\\big]{|I_{n}|^{-1}}$ having sub-exponential growth.\n Then\n \\[\n \\pi_{F, \\Phi}(n; \\alpha, I_n) \\sim \\frac{ \\int_{I_{n}} e^{\\rootpressure t} \\, \\mathrm{d}t }{ \\sqrt{2\\pi} \\, \\sigma } \\, \\frac{e^{\\rootpressure \\alpha n}}{n^{3/2}} \\qquad \\text{as } n \\to +\\infty.\n \\],", + "theorem_type": [ + "Asymptotic or Limit", + "Uniqueness" + ], + "mcq": { + "question": "Let $f\\colon S^2\\to S^2$ be an expanding Thurston map, let $d$ be a visual metric on $S^2$ for $f$, let $\\beta\\in(0,1]$, and let $\\phi\\in C^{0,\\beta}(S^2,d)$ be an eventually positive real-valued H\\\"older continuous function satisfying the $\\beta$-strong non-integrability condition with respect to $f$ and $d$. For $N\\in\\mathbb N$, set $F:=f^N$ and $\\Phi:=\\sum_{i=0}^{N-1}\\phi\\circ f^i$. If $\\tau=\\{x,F(x),\\dots,F^{n-1}(x)\\}$ is a primitive periodic orbit of $F$ (meaning $F^n(x)=x$ and $F^m(x)\\neq x$ for $1\\le m0$ such that $P(f,-s_0\\phi)=0$, and there exists $N_f\\in\\mathbb N$, depending only on $f$, such that for every $N\\ge N_f$, if $F=f^N$ and $\\Phi=\\sum_{i=0}^{N-1}\\phi\\circ f^i$, then with\n\\[\n\\alpha:=\\left.\\frac{d}{dt}P(F,t\\Phi)\\right|_{t=-s_0},\\qquad\n\\sigma:=\\sqrt{\\left.\\frac{d^2}{dt^2}P(F,t\\Phi)\\right|_{t=-s_0}},\n\\]\nfor every sequence of intervals $(I_n)$ contained in a compact set $K\\subseteq\\mathbb R$ such that $(|I_n|^{-1})$ has sub-exponential growth, one has\n\\[\n\\pi_{F,\\Phi}(n;\\alpha,I_n)\\sim \\frac{\\int_{I_n} e^{s_0 t}\\,dt}{\\sqrt{2\\pi}\\,\\sigma}\\,\\frac{e^{s_0\\alpha n}}{n^{3/2}}\n\\qquad\\text{as } n\\to+\\infty.\n\\]\"" + }, + "choices": [ + { + "label": "B", + "text": "There exists a unique positive number $s_0>0$ such that $P(f,-s_0\\phi)=0$, and for every $N\\in\\mathbb N$, if $F=f^N$ and $\\Phi=\\sum_{i=0}^{N-1}\\phi\\circ f^i$, then with\n\\[\n\\alpha:=\\left.\\frac{d}{dt}P(F,t\\Phi)\\right|_{t=-s_0},\\qquad\n\\sigma:=\\sqrt{\\left.\\frac{d^2}{dt^2}P(F,t\\Phi)\\right|_{t=-s_0}},\n\\]\nfor every sequence of intervals $(I_n)$ contained in a compact set $K\\subseteq\\mathbb R$ such that $(|I_n|^{-1})$ has sub-exponential growth, one has\n\\[\n\\pi_{F,\\Phi}(n;\\alpha,I_n)\\sim \\frac{\\int_{I_n} e^{s_0 t}\\,dt}{\\sqrt{2\\pi}\\,\\sigma}\\,\\frac{e^{s_0\\alpha n}}{n^{3/2}}\n\\qquad\\text{as } n\\to+\\infty.\n\\]" + }, + { + "label": "C", + "text": "There exists a unique positive number $s_0>0$ such that $P(f,-s_0\\phi)=0$, and there exists $N_f\\in\\mathbb N$, depending only on $f$, such that for every $N\\ge N_f$, if $F=f^N$ and $\\Phi=\\sum_{i=0}^{N-1}\\phi\\circ f^i$, then with\n\\[\n\\alpha:=\\left.\\frac{d}{dt}P(F,t\\Phi)\\right|_{t=-s_0},\n\\]\nfor every sequence of intervals $(I_n)$ contained in a compact set $K\\subseteq\\mathbb R$ such that $(|I_n|^{-1})$ has sub-exponential growth, one has\n\\[\n\\pi_{F,\\Phi}(n;\\alpha,I_n)=\\mathcal O\\!\\left(\\frac{e^{s_0\\alpha n}}{n^{3/2}}\\right)\n\\qquad\\text{as } n\\to+\\infty.\n\\]" + }, + { + "label": "D", + "text": "There exists a unique positive number $s_0>0$ such that $P(f,-s_0\\phi)=0$, and there exists $N_f\\in\\mathbb N$, depending only on $f$, such that for every $N\\ge N_f$, if $F=f^N$ and $\\Phi=\\sum_{i=0}^{N-1}\\phi\\circ f^i$, then with\n\\[\n\\alpha:=\\left.\\frac{d}{dt}P(F,t\\Phi)\\right|_{t=-s_0},\\qquad\n\\sigma:=\\sqrt{\\left.\\frac{d^2}{dt^2}P(F,t\\Phi)\\right|_{t=-s_0}},\n\\]\nfor every sequence of intervals $(I_n)$ contained in a compact set $K\\subseteq\\mathbb R$, one has\n\\[\n\\pi_{F,\\Phi}(n;\\alpha,I_n)\\sim \\frac{|I_n|}{\\sqrt{2\\pi}\\,\\sigma}\\,\\frac{e^{s_0\\alpha n}}{n^{3/2}}\n\\qquad\\text{as } n\\to+\\infty.\n\\]" + }, + { + "label": "E", + "text": "There exists a unique positive number $s_0>0$ such that $P(f,-s_0\\phi)=0$, and there exists $N_f\\in\\mathbb N$, depending only on $f$, such that for every $N\\ge N_f$, if $F=f^N$ and $\\Phi=\\sum_{i=0}^{N-1}\\phi\\circ f^i$, then with\n\\[\n\\alpha:=\\left.\\frac{d}{dt}P(F,t\\Phi)\\right|_{t=-s_0},\\qquad\n\\sigma:=\\sqrt{\\left.\\frac{d^2}{dt^2}P(F,t\\Phi)\\right|_{t=-s_0}},\n\\]\nfor every sequence of intervals $(I_n)$ contained in a compact set $K\\subseteq\\mathbb R$ such that $(|I_n|^{-1})$ has sub-exponential growth, one has\n\\[\n\\pi_{F,\\Phi}(n;\\alpha,I_n)\\sim \\frac{\\int_{I_n} e^{s_0 t}\\,dt}{\\sqrt{2\\pi}\\,\\sigma}\\,\\frac{e^{s_0\\alpha n}}{n^{1/2}}\n\\qquad\\text{as } n\\to+\\infty.\n\\]" + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "characteristic", + "tampered_component": "threshold iterate N\\ge N_f replaced by all N", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped the sharp asymptotic constant and equivalence, retaining only a coarse upper bound of the same order", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "smooth approximation/Fourier-weighted interval term replaced by bare length and sub-exponential hypothesis removed", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "local_part", + "tampered_component": "prime-orbit scale n^{-3/2} replaced by CLT-like n^{-1/2}", + "template_used": "stronger_trap" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem itself does not state the conclusion or uniquely reveal the correct option; it only gives the hypotheses and asks which resulting theorem statement is valid." + }, + "TAS": { + "score": 1, + "justification": "The item is essentially a theorem-recognition question: under a full list of hypotheses, the student must pick the exact conclusion. It is not a verbatim restatement in the stem, but it remains close to selecting the correct theorem statement." + }, + "GPS": { + "score": 1, + "justification": "There is some reasoning pressure because the options differ in subtle but meaningful ways (quantifier on N, sharp asymptotic vs bound, interval factor, power of n). However, it mainly tests precise recall/discrimination rather than constructing a conclusion from first principles." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic failure modes: overgeneralizing to all N, weakening an asymptotic to an O-bound, dropping the weighted interval term/sub-exponential condition, and confusing n^{-3/2} with n^{-1/2}." + }, + "total_score": 6, + "overall_assessment": "A solid high-level theorem-discrimination MCQ with little stem leakage and strong distractors, but it primarily assesses recognition of a precise result rather than genuinely generative mathematical reasoning." + } + }, + { + "id": "2511.15021v1", + "paper_link": "http://arxiv.org/abs/2511.15021v1", + "theorems_cnt": 3, + "theorem": { + "env_name": "Theorem", + "content": "\\label{thm:uniqueness within periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ be a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$ satisfying \\eqref{eq:A compatible}. Then there exists a unique periodic solution (up to addition of constants) to \\eqref{eq:periodic equation ma} with quadratic component $P$.", + "start_pos": 14162, + "end_pos": 14620, + "label": "thm:uniqueness within periodic" + }, + "ref_dict": { + "def:phi homogenization": "\\begin{Definition}\\label{def:phi homogenization}\nFix $\\lambda = \\lambda(n) > 0$ sufficiently small such that $\\sigma = C_1(n)\\lambda$ satisfies $4^{n+1}\\sigma < c(n)$, where $C_1(n)$ is the one in Lemma \\ref{lem:covering Lemma} and $c(n)$ is the one in Proposition \\ref{prop:basic}. For $\\Lambda\\ge \\max\\{\\Lambda_0,\\Lambda_1,\\Lambda_2\\}$, and any nonnegative functions $v \\in C(\\Omega)$, we define its $\\phi$-homogenization $M_{\\phi,\\Lambda}v(x):=M_{\\phi,\\lambda,\\Lambda}v(x)$ in $\\frac{7}{8}\\Omega$ through the following maximal averaging type operator:\n\\begin{equation}\\label{eq:phi homogenization}\n\\begin{split}\nM_{\\phi,\\Lambda}v(x)= \\sup\\left\\{ t\\in\\R:\\; \\exists \\ \\check{h}(x) t\\} \\cap S_h (x) )}{\\mu(S_h (x))} > \\lambda \\right\\}.\n\\end{split}\n\\end{equation} \n\\end{Definition}", + "eq:periodic equation ma": "\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\R^n,\n\\end{equation}", + "eq:A compatible": "\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation}", + "eq:periodic solution": "\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}", + "thm:solutions are periodic": "\\begin{Theorem}\\label{thm:solutions are periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $u$ be a convex solution of \\eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \\ref{def:periodic solution}.\n\\end{Theorem}", + "thm:uniqueness within periodic": "\\begin{Theorem}\\label{thm:uniqueness within periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ be a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$ satisfying \\eqref{eq:A compatible}. Then there exists a unique periodic solution (up to addition of constants) to \\eqref{eq:periodic equation ma} with quadratic component $P$. \n\\end{Theorem}", + "thm:harnack": "\\begin{Theorem}\\label{thm:harnack}\nLet $\\Omega$ and $\\widetilde{\\Omega}$ be open convex subsets of $\\mathbb{R}^n$ satisfying $B_1\\subset \\Omega \\subset B_n, B_1\\subset \\widetilde{\\Omega} \\subset B_n$, $n\\geq 2$, and let $\\phi\\in C^2(\\overline{\\Omega})$ and $\\tilde{\\phi}\\in C^2(\\overline{\\widetilde{\\Omega}})$ be convex functions satisfying, \n\\[\n\\mu_\\phi =\\mu \\quad \\text{in } \\Omega, \\quad \\phi=0 \\quad \\text{on } \\partial \\Omega,\n\\]\n\\[ \n\\mu_{\\tilde{\\phi}} =\\mu \\quad \\text{in } \\widetilde{\\Omega}, \\quad \\tilde{\\phi}=0 \\quad \\text{on } \\partial \\widetilde{\\Omega},\n\\] \nwhere $\\mu$ is a periodic Borel measure defined on $\\R^n$ satisfying \\eqref{eq:period gamma}, \\eqref{eq:normalized assumption}, and \\eqref{eq:small period assumption}. Assume that $v\\in C^2(\\Omega) \\cap C^2(\\widetilde{\\Omega})$ with $v\\geq 0$ satisfies\n\\[\nL_{\\phi}v \\leq 0 \\quad \\text{in } \\Omega, \\quad L_{\\tilde{\\phi}}v \\geq 0 \\quad \\text{in }\\widetilde{\\Omega}.\n\\] \n\nThen there exist positive constants $\\beta(n)$, $\\Lambda_9(n)$, and $C(n)$ such that if $\\Lambda>\\Lambda_9(n)$, then either \n\\[\n\\sup_{B_{1/4}}v\\leq C\\inf_{B_{1/4}}v,\n\\]\nor for $\\kappa =\\kappa(B_1)\\geq \\Lambda^2$ that\n\\[\n\\sup_{B_{1/2}} v \\geq e^{\\beta \\kappa^{\\frac{1}{6}}}\\sup_{B_{1/4}}v.\n\\] \nConsequently, we have\n\\[\n\\sup_{B_{1/4}}v \\leq C\\inf_{B_{1/4}}v+e^{-\\beta \\kappa^{\\frac{1}{6}}}\\sup_{B_{1/2}} v.\n\\]\n\\end{Theorem}", + "eq:periodic f": "\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}", + "def:periodic solution": "\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic with respect to $\\mathbb{Z}^n$ and $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\R^n$ and $c\\in\\R$. \n\\end{Definition}", + "thm:harnack super": "\\begin{Theorem}\\label{thm:harnack super} \nAssume that $v \\in C^2(\\Omega)$ is a nonnegative supersolution of \\eqref{eq:linearized ma phi} in $\\Omega$.\nThen there exist positive constants $\\Lambda_6(n)$, $\\epsilon(n) $ and $C(n) $ such that if \n\\[\n\\inf_{B_{1/4}} v \\leq 1,\n\\]\nthen for $\\Lambda\\ge \\Lambda_6(n)$, we have\n\\begin{equation}\\label{eq:harnack super} \n\\mu(\\{M_{\\phi,\\Lambda}v > t\\} \\cap B_{1/2})< Ct^{-\\epsilon} .\n\\end{equation} \n\\end{Theorem}", + "prop:semiconcave": "\\begin{Proposition}\\label{prop:semiconcave}\nLet $u$ be a convex solution to \\eqref{eq:periodic equation ma} with $\\mu$ being a $\\mathbb{Z}^n/8n$-periodic measure (where $\\mathbb{Z}^n := \\Gamma_{\\{e_1,\\cdots,e_n\\}}$) satisfying $\\mu(\\mathbb{Z}^n) = 1$. \nUnder the nondegeneracy condition\n\\begin{equation}\\label{eq:nondegenerate}\n\\Delta^2_{i} u(0):=\\Delta^2_{e_i} u(0) > 0 \\quad \\text{for all } 1 \\leq i \\leq n,\n\\end{equation}\nthere exists a unique compatible matrix $A$ satisfying:\n\\[\n\\sup_{ \\R^n}\\Delta_z^2 u=z^{\\top}Az ,\\quad \\forall z \\in \\mathbb{Z}^n.\n\\]\n\\end{Proposition}", + "thm:harnack sub": "\\begin{Theorem}\\label{thm:harnack sub} \nAssume that $v \\in C^2(\\Omega)$ with $v \\geq 0$ is a subsolution of \\eqref{eq:linearized ma phi} in $\\Omega$.\nThen, there exist positive constants $\\Lambda_7(n)$ and $\\beta(n)$ such that for any $\\epsilon > 0$, there is a constant $C_2(n, \\epsilon)$ with the following property: If $\\Lambda\\ge \\Lambda_7(n)$ and\n\\[\n\\min\\{\\mu(\\{v > t\\} \\cap B_{1/2}), \\mu(\\{M_{\\phi,\\Lambda}v > t\\} \\cap B_{1/2}) \\} \\leq t^{-\\epsilon} \\quad \\forall t>0, \n\\]\nthen either\n\\[\n\\sup_{B_{1/4} } v \\leq C_2(n,\\epsilon), \n\\] \nor for $\\kappa =\\kappa(B_1)\\geq \\Lambda^2$ that\n\\[\n\\sup_{B_{1/2} } v \\geq e^{\\beta \\kappa^{\\frac{1}{3}} } \\sup_{B_{1/4} }v.\n\\] \n\\end{Theorem}" + }, + "pre_theorem_intro_text_len": 1825, + "pre_theorem_intro_text": "In this paper, we study convex solutions to the Monge-Amp\\`ere equation\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\mathbb{R}^n,\n\\end{equation}\nwhere $\\mu\\not\\equiv 0$ is a nonnegative locally finite Borel measure on $\\mathbb{R}^n$ and is periodic in $n$ linearly independent directions. By exploiting the affine invariance of the equation, we may, without loss of generality, restrict our analysis to the case where $\\mu $ is periodic with respect to the integer lattice $\\mathbb{Z}^n := \\{(k_1,\\dots,k_n) \\mid k_i \\in \\mathbb{Z}\\} \\subset \\mathbb{R}^n$, i.e., for any Borel set $E \\subset \\mathbb{R}^n$, \n\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}\n\nLet $\\mathcal{S}_+^{n\\times n}$ denote the set of positive definite symmetric $n\\times n$ matrices.\n\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic with respect to $\\mathbb{Z}^n$ and $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\mathbb{R}^n$ and $c\\in\\mathbb{R}$. \n\\end{Definition}\n\nNote that if $u$ is a periodic solution to \\eqref{eq:periodic equation ma} and is expressed as \\eqref{eq:periodic solution}, then the quadratic component $P(x)$ always satisfies the compatibility condition\n\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation} \nwhere $\\mathbb{T}^n=\\mathbb{R}^n/\\mathbb{Z}^n$.\n\nOur first result is the uniqueness of periodic solutions.", + "context": "In this paper, we study convex solutions to the Monge-Amp\\`ere equation\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\mathbb{R}^n,\n\\end{equation}\nwhere $\\mu\\not\\equiv 0$ is a nonnegative locally finite Borel measure on $\\mathbb{R}^n$ and is periodic in $n$ linearly independent directions. By exploiting the affine invariance of the equation, we may, without loss of generality, restrict our analysis to the case where $\\mu $ is periodic with respect to the integer lattice $\\mathbb{Z}^n := \\{(k_1,\\dots,k_n) \\mid k_i \\in \\mathbb{Z}\\} \\subset \\mathbb{R}^n$, i.e., for any Borel set $E \\subset \\mathbb{R}^n$, \n\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}\n\nLet $\\mathcal{S}_+^{n\\times n}$ denote the set of positive definite symmetric $n\\times n$ matrices.\n\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic with respect to $\\mathbb{Z}^n$ and $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\mathbb{R}^n$ and $c\\in\\mathbb{R}$. \n\\end{Definition}\n\nNote that if $u$ is a periodic solution to \\eqref{eq:periodic equation ma} and is expressed as \\eqref{eq:periodic solution}, then the quadratic component $P(x)$ always satisfies the compatibility condition\n\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation} \nwhere $\\mathbb{T}^n=\\mathbb{R}^n/\\mathbb{Z}^n$.\n\nOur first result is the uniqueness of periodic solutions.\n\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\R^n,\n\\end{equation}", + "full_context": "In this paper, we study convex solutions to the Monge-Amp\\`ere equation\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\mathbb{R}^n,\n\\end{equation}\nwhere $\\mu\\not\\equiv 0$ is a nonnegative locally finite Borel measure on $\\mathbb{R}^n$ and is periodic in $n$ linearly independent directions. By exploiting the affine invariance of the equation, we may, without loss of generality, restrict our analysis to the case where $\\mu $ is periodic with respect to the integer lattice $\\mathbb{Z}^n := \\{(k_1,\\dots,k_n) \\mid k_i \\in \\mathbb{Z}\\} \\subset \\mathbb{R}^n$, i.e., for any Borel set $E \\subset \\mathbb{R}^n$, \n\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}\n\nLet $\\mathcal{S}_+^{n\\times n}$ denote the set of positive definite symmetric $n\\times n$ matrices.\n\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic with respect to $\\mathbb{Z}^n$ and $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\mathbb{R}^n$ and $c\\in\\mathbb{R}$. \n\\end{Definition}\n\nNote that if $u$ is a periodic solution to \\eqref{eq:periodic equation ma} and is expressed as \\eqref{eq:periodic solution}, then the quadratic component $P(x)$ always satisfies the compatibility condition\n\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation} \nwhere $\\mathbb{T}^n=\\mathbb{R}^n/\\mathbb{Z}^n$.\n\nOur first result is the uniqueness of periodic solutions.\n\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\R^n,\n\\end{equation}\n\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite periodic Borel measure on $\\R^n$. We show that any convex solution to the Monge-Amp\\`ere equation\n\\[ \n\\det D^2 u = \\mu \\quad \\text{in } \\R^n\n\\]\nadmits a unique decomposition (up to addition of constants) as the sum of a quadratic polynomial and a periodic function. This result extends,\nin full generality, the earlier works for the case $\\mu=f(x)\\,\\ud x$: when $\\log f \\in C^\\alpha$, it was established by Caffarelli and Li; and\nwhen $\\log f$ is merely bounded, it was proved by Li and Lu. Our result thus answers a question raised by Li and Lu. A key ingredient in the proof is a new dichotomous Harnack-type inequality for linearized Monge-Amp\\`ere equations with nonnegative periodic measures.\n\nLet $\\mathcal{S}_+^{n\\times n}$ denote the set of positive definite symmetric $n\\times n$ matrices.\n\\begin{Definition}[Periodic solution]\\label{def:periodic solution}\nA \\emph{periodic solution} to \\eqref{eq:periodic equation ma} is a convex function $u \\in C(\\mathbb{R}^n)$ satisfying \\eqref{eq:periodic equation ma} in the Alexandrov sense, and can be decomposed as\n\\begin{equation}\\label{eq:periodic solution}\nu(x) = v(x) +P(x),\n\\end{equation}\nwhere $v$ is periodic with respect to $\\mathbb{Z}^n$ and $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ is a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$, $b\\in\\R^n$ and $c\\in\\R$. \n\\end{Definition}\n\nOur first result is the uniqueness of periodic solutions.\n\nThe existence of periodic solutions follows easily from Theorem 2.1 in Li \\cite{li1990existence} and its proof. For the uniqueness, when $\\mu=f(x)\\,\\ud x$ is a positive measure, the case $\\log f \\in C^\\alpha$ was proved by Li \\cite{li1990existence}, and the case $\\log f\\in L^\\infty$ was shown by Li-Lu \\cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.\n\n\\begin{Theorem}\\label{thm:solutions are periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $u$ be a convex solution of \\eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \\ref{def:periodic solution}.\n\\end{Theorem}\n\n\\begin{Proposition}\\label{prop:semiconcave}\nLet $u$ be a convex solution to \\eqref{eq:periodic equation ma} with $\\mu$ being a $\\mathbb{Z}^n/8n$-periodic measure (where $\\mathbb{Z}^n := \\Gamma_{\\{e_1,\\cdots,e_n\\}}$) satisfying $\\mu(\\mathbb{Z}^n) = 1$. \nUnder the nondegeneracy condition\n\\begin{equation}\\label{eq:nondegenerate}\n\\Delta^2_{i} u(0):=\\Delta^2_{e_i} u(0) > 0 \\quad \\text{for all } 1 \\leq i \\leq n,\n\\end{equation}\nthere exists a unique compatible matrix $A$ satisfying:\n\\[\n\\sup_{ \\R^n}\\Delta_z^2 u=z^{\\top}Az ,\\quad \\forall z \\in \\mathbb{Z}^n.\n\\]\n\\end{Proposition} \n\\begin{proof}\nWithout loss of generality, we may assume $u(0) = 0$ and $u \\geq 0$. \nLet us define $S_h := \\{x \\in \\mathbb{R}^n : u(x) \\leq h\\}$, and select $\\Lambda_u>0$ such that the section $S_{\\Lambda_u}$ satisfies $\\kappa_{\\mathbb{Z}^n}(S_{\\Lambda_u}) \\geq \\Lambda_0^2$. \nLet us now normalize $S_h$ such that $B_1(0) \\subset T_h^{-1} S_h \\subset B_n(0)$ and consider the normalized function\n\\[\nu_{h}(x) =\\frac{u(T_h x)}{(\\det T_h)^{\\frac{2}{n}}} \\quad \\text{with } \\det D^2 u_h= \\mu_h=\\mu \\circ T_h .\n\\] \nDue to the periodicity of $\\mu$ and large $h$, we know that \n\\[\nc(n)h^{\\frac{n}{2}} \\leq \\det T_h \\leq C(n)h^{\\frac{n}{2}},\n\\] \nand thus, $c(n)h^{\\frac{n}{2}} \\leq |S_h| \\leq C(n)h^{\\frac{n}{2}}$. Applying Proposition \\ref{prop:basic} to $\\phi=u_h-h(\\det T_h)^{-\\frac{2}{n}}$, we obtain that for sufficiently large $h$, \n\\[\nc(n) (h/\\Lambda_u)^{\\frac{1}{3}} S_{\\Lambda_u} \\subset S_h \\subset C(n) (h/\\Lambda_u)^{\\frac{2}{3}} S_{\\Lambda_u},\n\\] \nand consequently, $B_{c(n) h^{\\frac{1}{4}}}\\subset S_h\\subset B_{C(n) h^{\\frac{3}{4}}}$ for all large $h$. That is, for sufficiently large $h$, we now have\n\\[\n\\quad c(n)h^{\\frac{1}{4}} I\\leq T_h \\leq C(n)h^{\\frac{3}{4}} I.\n\\]\nThe transformation law \\eqref{eq:transformation law} then yields the following estimates at corresponding points \n\\begin{equation}\\label{eq:transformh}\nc(n)h^{-\\frac{1}{2}}\\Delta^2_{e_i} u \\leq \\Delta^2_{T_h^{-1} e_i} u_h \\leq C(n)h^{\\frac{1}{2}}\\Delta^2_{e_i} u.\n\\end{equation}\n\n\\textbf{Step 4.} Suppose for some sequence $h \\to \\infty$, the rescaled functions $\\tilde{u}_h$ locally converge to a quadratic form $P_A(x) = \\frac{1}{2}x^{\\top}A x$ for $A \\in \\mathcal{S}_+^{n\\times n}$ with $\\det A = 1$. \nFor each fixed direction $i$, we denote\n\\[\n\\alpha=\\sup_{\\R^n}\\Delta^2_{e_i} u,\\quad \\beta= e_i^{\\top} A e_i.\n\\] \nNote that the estimate \\eqref{eq:w21 estimate} continues to hold for $\\tilde{u}_{h}$, from which we derive $\\alpha \\geq \\beta$.\nWe claim\n\\begin{equation}\\label{eq:w2infty sup}\n\\alpha=\\beta.\n\\end{equation}\nAssume by contradiction that $\\alpha = \\beta + 4s$ for some $s > 0$.\nThrough rescaling from infinity, we may assume without loss of generality that there exist points $x_h \\in B_{1/8}$ satisfying\n\\[\n\\alpha- \\Delta^2_{h^{\\frac{1}{2}}e_i} \\tilde{u}_{h}(x_h)= \\inf_{B_{1/8}} (\\alpha- \\Delta^2_{h^{\\frac{1}{2}} e_i} \\tilde{u}_{h} ) \\leq a_h,\n\\]\nwhere $a_h \\to 0 $ as $h \\to \\infty$. Define the normalized function $v = (\\alpha - \\Delta^2_{h^{\\frac{1}{2}}e_i} \\tilde{u}_{h})/a_h$ and set $\\delta = s/[C(\\beta + 2s)]$. Applying Lemma \\ref{lem:Theorem 7.3.1 modify} yields with $\\phi=\\tilde{u}_h-1$ and $\\Omega=S_{1,h}=\\{\\tilde{u}_h \\leq 1\\} $, we obtain that\n\\[\n\\tilde{\\mu}_h ( \\{v\\leq M_1(n,\\delta) \\} \\cap S_{1,h} ) \\geq (1-\\delta) \\tilde{\\mu}_h(S_{1,h} ),\n\\]\nwhere $\\tilde{\\mu}_h=\\tilde{f}_{h}\\,\\ud x$. For sufficiently small $a_h$, we consequently obtain\n\\[\n\\tilde{\\mu}_h\\left(\\left\\{\\Delta^2_{h^{\\frac{1}{2}} e_i} \\tilde{u}_h>\\beta +2 s\\right\\} \\cap S_{1,h}\\right)= \\tilde{\\mu}_h\\left(\\left\\{v <\\frac{2s}{a_h}\\right\\} \\cap S_{1,h}\\right)\\geq (1-\\delta) \\tilde{\\mu}_h (S_{1,h}).\n\\]\nThis leads to the integral lower bound\n\\[\n\\int_{S_{1,h}} \\Delta^2_{h^{\\frac{1}{2}} e_i} \\tilde{u}_h \\ud \\tilde{\\mu}_h \\geq (1-\\delta) (\\beta+2s) \\tilde{\\mu}_h (S_{1,h})\\geq \n(\\beta+s) \\tilde{\\mu}_h (S_{1,h}).\n\\]\nThis contradicts the uniform estimate \\eqref{eq:w21 estimate} after performing a suitable rescaling.\n\n\\begin{proof}[Proof of Theorems \\ref{thm:uniqueness within periodic} and \\ref{thm:solutions are periodic}] \nLet $u$ be a global solution of \\eqref{eq:periodic equation ma}. \nClearly, $u$ cannot be linear on any ray segment. Indeed, if it were linear on a ray, after subtracting its supporting hyperplane on this ray and by using the convexity of $u$, $u$ would be bounded within any cylinder having that ray as its axis. This would imply that the associated measure $\\mu$ is finite on any strictly smaller cylinder, contradicting the assumptions that $\\mu$ is periodic and $\\mu \\not\\equiv 0$. Therefore, there exists a sufficiently large integer $R$ such that $S_1(0) \\subset B_R(0)$.\nWe now rescale $u$ by defining $u(8nRx)/64n^2R^2$. For simplicity, we continue to denote this rescaled function as $u$. Under this scaling, one can verify that condition \\eqref{eq:nondegenerate} in Proposition \\ref{prop:semiconcave} is satisfied. Hence, by Proposition \\ref{prop:semiconcave}, there exists a compatible matrix $A \\in \\mathcal{S}_+^{n \\times n}$ such that \n\\[\n\\sup_{ \\R^n}\\Delta_z^2 u=z^{\\top}Az,\\quad \\forall z \\in \\mathbb{Z}^n.\n\\]\nLet $w$ be a periodic solution to \\eqref{eq:periodic equation ma} in the sense of Definition \\ref{def:periodic solution} with quadratic part $\\frac{1}{2}z^{\\top} A z $. Then $\\Delta_z^2 w=z^{\\top}Az$. Let \n\\[\nv(x)=u(x)-w(x).\n\\]\nBy subtracting a suitable linear function, we now assume that $v(0)=0$ and \n\\[\nv(e_i)=v(-e_i),\\quad 1\\leq i\\leq n.\n\\]\nNoting that $\\sup_{\\R^n} \\Delta_z^2 v = \\sup_{\\R^n} \\Delta_z^2 (u - w) = 0$, we then find that\n\\[\nv(\\pm ke_i)\\leq 0 \\quad \\text{for all } k\\in \\mathbb{Z},\\ 1\\leq i\\leq n.\n\\]", + "post_theorem_intro_text_len": 5293, + "post_theorem_intro_text": "The existence of periodic solutions follows easily from Theorem 2.1 in Li \\cite{li1990existence} and its proof. For the uniqueness, when $\\mu=f(x)\\,\\mathrm{d} x$ is a positive measure, the case $\\log f \\in C^\\alpha$ was proved by Li \\cite{li1990existence}, and the case $\\log f\\in L^\\infty$ was shown by Li-Lu \\cite{li2019monge}. Our theorem extends these results to general periodic Borel measures.\n\nOur second result is the classification below, which extends the main results of Caffarelli-Li \\cite{caffarelli2004liouville} and Li-Lu \\cite{li2019monge} to the full generality. In particular, it provides a complete answer to Question 1.1 in \\cite{li2019monge}. \n\n\\begin{Theorem}\\label{thm:solutions are periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying \\eqref{eq:periodic f}, and let $u$ be a convex solution of \\eqref{eq:periodic equation ma}. Then $u$ is a periodic solution in the sense of Definition \\ref{def:periodic solution}.\n\\end{Theorem}\n\nCaffarelli-Li \\cite{caffarelli2004liouville} first proved Theorem \\ref{thm:solutions are periodic} for positive measures $\\mu=f(x)\\,\\mathrm{d} x$ with $\\log f \\in C^\\alpha$, and conjectured the extension to merely bounded $\\log f$. This conjecture was resolved by Li-Lu \\cite{li2019monge}, who further asked whether the result holds for the nonnegative case $f \\geq 0$ with $f \\not\\equiv 0$. Theorem \\ref{thm:solutions are periodic} affirmatively answers this question -- and indeed establishes it in full generality.\n\n\\begin{Remark} \nWhen $\\mu \\equiv 0$, equation \\eqref{eq:periodic equation ma} reduces to the homogeneous Monge-Amp\\`ere equation $\\det D^2u = 0$. The solutions are precisely those functions that are linear along certain directions (see, e.g., Caffarelli-Nirenberg-Spruck \\cite{caffarelli1986dirichlet}). \nIn particular, writing $x = (x',x_n) \\in \\mathbb{R}^{n-1} \\times \\mathbb{R}$, any convex function of the form $u(x) = w(x')$ is a solution, and such solutions are not necessarily periodic.\n\\end{Remark}\n\nSince $\\mu$ is assumed to be nonnegative, equation \\eqref{eq:periodic equation ma} represents a possibly degenerate Monge-Amp\\`ere equation, for which many arguments in \\cite{caffarelli2004liouville,li2019monge} do not apply. The principal challenge stems from the behavior of $u$ on small sections: the associated measure $\\mu$ may not satisfy the doubling condition. This failure implies that sections lose the engulfing property, consequently invalidating the Vitali or Besicovitch covering lemma which is essential for the Calder\\'on-Zygmund decomposition on sections. \nTherefore, we cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\\'errez \\cite{caffarelli1997properties} for subsolutions or supersolutions of linearized Monge-Amp\\`ere equations. \n\nNonetheless, by exploiting the equation's periodic structure, we derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator (see Definition \\ref{def:phi homogenization}). This constitutes one of the key innovations of the present work. For supersolutions, we address the challenge by excluding contributions from small sections. Theorem \\ref{thm:harnack super} establishes decay estimates for homogenized level-sets of supersolutions within our framework. For subsolutions, for which standard $L^{\\infty}$ bounds fail, Theorem \\ref{thm:harnack sub} establishes an alternative dichotomy regarding their growth behavior: either the $L^{\\infty}$ bound holds or the $L^{\\infty}$ norm grows exponentially. Finally, we obtain in Theorem \\ref{thm:harnack} a Harnack inequality with a quantitatively controlled error term.\n\nAfter overcoming this critical difficulty arising from the degeneracy of equation \\eqref{eq:periodic equation ma}, the rest proof focuses on analyzing the second-order difference quotient $\\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart. As observed in \\cite{caffarelli2004liouville}, these quantities naturally arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\\`ere equations. This is another point where the periodicity of the measure plays a crucial role. We then establish the quadratic behavior of solutions near infinity, extending the arguments in \\cite{caffarelli2004liouville,li2019monge} for positive measures $\\mu=f(x)\\mathrm{d} x$ to the degenerate case considered here. In this step, we must also overcome additional difficulties due to the degeneracy of\n \\eqref{eq:periodic equation ma}. Consequently, when Theorem \\ref{thm:harnack} is applied to the difference of two solutions, exponential growth is precluded. Hence, the classical Harnack inequality holds for this difference, and our theorems would follow. \n\nThis paper is organized as follows. In Section \\ref{sec:harnack}, we study convex functions whose Monge-Amp\\`ere measure has small period, establishing a dichotomous Harnack type inequality for linearized Monge-Amp\\`ere equations. Section \\ref{sec:semiconvavity} investigates the uniqueness of compatible quadratic component of each solution in the sense of Proposition \\ref{prop:semiconcave}. Finally, in Section \\ref{sec:main theorem}, we prove Theorems \\ref{thm:uniqueness within periodic} and \\ref{thm:solutions are periodic}.", + "sketch": "The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \\cite{li1990existence} and its proof,” while Theorem~\\ref{thm:uniqueness within periodic} “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including Theorem~\\ref{thm:uniqueness within periodic}) in the nonnegative/degenerate setting: “the associated measure $\\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\\'errez.”\n\nTo overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” For supersolutions they do so “by excluding contributions from small sections,” obtaining “decay estimates for homogenized level-sets” (Theorem~\\ref{thm:harnack super}); for subsolutions (where “standard $L^{\\infty}$ bounds fail”), they prove a “dichotomy” (Theorem~\\ref{thm:harnack sub}): “either the $L^{\\infty}$ bound holds or the $L^{\\infty}$ norm grows exponentially”; combining these yields “a Harnack inequality with a quantitatively controlled error term” (Theorem~\\ref{thm:harnack}).\n\n“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \\cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying Theorem~\\ref{thm:harnack} “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and “our theorems would follow.”", + "expanded_sketch": "The post-theorem introduction says: existence of periodic solutions “follows easily from Theorem 2.1 in Li \\cite{li1990existence} and its proof,” while in establishing the main theorem it “extends these results to general periodic Borel measures.” It then explains the main difficulty in proving the results (including the main theorem) in the nonnegative/degenerate setting: “the associated measure $\\mu$ may not satisfy the doubling condition,” so “sections lose the engulfing property,” which “invalidat[es] the Vitali or Besicovitch covering lemma … essential for the Calder\\'on-Zygmund decomposition on sections,” and therefore one “cannot employ directly the weak Harnack inequalities of Caffarelli-Guti\\'errez.”\n\nTo overcome this, “by exploiting the equation's periodic structure,” the authors “derive truncated weak Harnack inequalities for both supersolutions and subsolutions via a maximal-type operator.” We first use the following theorem for supersolutions:\n\\begin{Theorem}\\label{thm:harnack super} \nAssume that $v \\in C^2(\\Omega)$ is a nonnegative supersolution of \\eqref{eq:linearized ma phi} in $\\Omega$.\nThen there exist positive constants $\\Lambda_6(n)$, $\\epsilon(n) $ and $C(n) $ such that if \n\\[\n\\inf_{B_{1/4}} v \\leq 1,\n\\]\nthen for $\\Lambda\\ge \\Lambda_6(n)$, we have\n\\begin{equation}\\label{eq:harnack super} \n\\mu(\\{M_{\\phi,\\Lambda}v > t\\} \\cap B_{1/2})< Ct^{-\\epsilon} .\n\\end{equation} \n\\end{Theorem}\nFor subsolutions (where “standard $L^{\\infty}$ bounds fail”), they prove the following “dichotomy”:\n\\begin{Theorem}\\label{thm:harnack sub} \nAssume that $v \\in C^2(\\Omega)$ with $v \\geq 0$ is a subsolution of \\eqref{eq:linearized ma phi} in $\\Omega$.\nThen, there exist positive constants $\\Lambda_7(n)$ and $\\beta(n)$ such that for any $\\epsilon > 0$, there is a constant $C_2(n, \\epsilon)$ with the following property: If $\\Lambda\\ge \\Lambda_7(n)$ and\n\\[\n\\min\\{\\mu(\\{v > t\\} \\cap B_{1/2}), \\mu(\\{M_{\\phi,\\Lambda}v > t\\} \\cap B_{1/2}) \\} \\leq t^{-\\epsilon} \\quad \\forall t>0, \n\\]\nthen either\n\\[\n\\sup_{B_{1/4} } v \\leq C_2(n,\\epsilon), \n\\] \nor for $\\kappa =\\kappa(B_1)\\geq \\Lambda^2$ that\n\\[\n\\sup_{B_{1/2} } v \\geq e^{\\beta \\kappa^{\\frac{1}{3}} } \\sup_{B_{1/4} }v.\n\\] \n\\end{Theorem}\nCombining these yields “a Harnack inequality with a quantitatively controlled error term,” namely the following theorem:\n\\begin{Theorem}\\label{thm:harnack}\nLet $\\Omega$ and $\\widetilde{\\Omega}$ be open convex subsets of $\\mathbb{R}^n$ satisfying $B_1\\subset \\Omega \\subset B_n, B_1\\subset \\widetilde{\\Omega} \\subset B_n$, $n\\geq 2$, and let $\\phi\\in C^2(\\overline{\\Omega})$ and $\\tilde{\\phi}\\in C^2(\\overline{\\widetilde{\\Omega}})$ be convex functions satisfying, \n\\[\n\\mu_\\phi =\\mu \\quad \\text{in } \\Omega, \\quad \\phi=0 \\quad \\text{on } \\partial \\Omega,\n\\]\n\\[ \n\\mu_{\\tilde{\\phi}} =\\mu \\quad \\text{in } \\widetilde{\\Omega}, \\quad \\tilde{\\phi}=0 \\quad \\text{on } \\partial \\widetilde{\\Omega},\n\\] \nwhere $\\mu$ is a periodic Borel measure defined on $\\R^n$ satisfying \\eqref{eq:period gamma}, \\eqref{eq:normalized assumption}, and \\eqref{eq:small period assumption}. Assume that $v\\in C^2(\\Omega) \\cap C^2(\\widetilde{\\Omega})$ with $v\\geq 0$ satisfies\n\\[\nL_{\\phi}v \\leq 0 \\quad \\text{in } \\Omega, \\quad L_{\\tilde{\\phi}}v \\geq 0 \\quad \\text{in }\\widetilde{\\Omega}.\n\\] \n\nThen there exist positive constants $\\beta(n)$, $\\Lambda_9(n)$, and $C(n)$ such that if $\\Lambda>\\Lambda_9(n)$, then either \n\\[\n\\sup_{B_{1/4}}v\\leq C\\inf_{B_{1/4}}v,\n\\]\nor for $\\kappa =\\kappa(B_1)\\geq \\Lambda^2$ that\n\\[\n\\sup_{B_{1/2}} v \\geq e^{\\beta \\kappa^{\\frac{1}{6}}}\\sup_{B_{1/4}}v.\n\\] \nConsequently, we have\n\\[\n\\sup_{B_{1/4}}v \\leq C\\inf_{B_{1/4}}v+e^{-\\beta \\kappa^{\\frac{1}{6}}}\\sup_{B_{1/2}} v.\n\\]\n\\end{Theorem}\n\n“After overcoming this critical difficulty,” the remaining proof “focuses on analyzing the second-order difference quotient $\\Delta^2_e u$ and the deviation between $u$ and its periodic counterpart,” which “arise as subsolutions and supersolutions to carefully selected linearized Monge-Amp\\`ere equations,” with periodicity “play[ing] a crucial role.” They then “establish the quadratic behavior of solutions near infinity,” extending arguments from \\cite{caffarelli2004liouville,li2019monge} to the degenerate case. Finally, applying the preceding theorem (the Harnack inequality with controlled error term) “to the difference of two solutions,” “exponential growth is precluded,” so “the classical Harnack inequality holds for this difference,” and this completes the proof strategy for the main theorem.", + "expanded_theorem": "\\label{thm:uniqueness within periodic}\nLet $\\mu \\not\\equiv 0$ be a nonnegative locally finite Borel measure satisfying\n\\begin{equation}\\label{eq:periodic f}\n\\mu (E+z)=\\mu (E),\\quad \\forall \\ z \\in \\mathbb{Z}^n.\n\\end{equation}\nand let $P(x)=\\frac{1}{2}x^{\\top}Ax+b\\cdot x+c$ be a quadratic function with $A\\in \\mathcal{S}_+^{n\\times n}$ satisfying\n\\begin{equation}\\label{eq:A compatible}\n \\det A =\\mu (\\mathbb{T}^n),\n\\end{equation}\nThen, to prove the main theorem, we show that there exists a unique periodic solution (up to addition of constants) to\n\\begin{equation}\\label{eq:periodic equation ma}\n\\det D^2 u = \\mu \\quad \\text{in } \\R^n,\n\\end{equation}\nwith quadratic component $P$.", + "theorem_type": [ + "Uniqueness", + "Existence" + ], + "mcq": { + "question": "Let \\(\\mu \\not\\equiv 0\\) be a nonnegative locally finite Borel measure on \\(\\mathbb{R}^n\\) that is periodic with respect to \\(\\mathbb{Z}^n\\), meaning that \\(\\mu(E+z)=\\mu(E)\\) for every Borel set \\(E\\subset \\mathbb{R}^n\\) and every \\(z\\in \\mathbb{Z}^n\\). Let \\(P(x)=\\frac12 x^{\\top}Ax+b\\cdot x+c\\), where \\(A\\in \\mathcal{S}_+^{n\\times n}\\) is positive definite symmetric, \\(b\\in\\mathbb{R}^n\\), \\(c\\in\\mathbb{R}\\), and \\(\\det A=\\mu(\\mathbb{T}^n)\\) with \\(\\mathbb{T}^n=\\mathbb{R}^n/\\mathbb{Z}^n\\). A periodic solution with quadratic component \\(P\\) means a convex function \\(u\\in C(\\mathbb{R}^n)\\) satisfying \\(\\det D^2u=\\mu\\) in \\(\\mathbb{R}^n\\) in the Alexandrov sense and admitting a decomposition \\(u(x)=v(x)+P(x)\\), where \\(v(x+z)=v(x)\\) for all \\(z\\in\\mathbb{Z}^n\\). Which statement holds?", + "correct_choice": { + "label": "A", + "text": "There exists a periodic solution with quadratic component \\(P\\), and it is unique up to addition of constants. Equivalently, there exists a convex \\(u\\in C(\\mathbb{R}^n)\\) solving \\(\\det D^2u=\\mu\\) in the Alexandrov sense such that \\(u(x)=v(x)+P(x)\\) for some \\(\\mathbb{Z}^n\\)-periodic function \\(v\\), and whenever \\(u_1=u_1^{\\mathrm{per}}+P\\) and \\(u_2=u_2^{\\mathrm{per}}+P\\) are two such periodic solutions, \\(u_1-u_2\\) is constant on \\(\\mathbb{R}^n\\)." + }, + "choices": [ + { + "label": "B", + "text": "There exists a periodic solution with quadratic component \\(P\\), and it is unique in the strict sense. Equivalently, there exists a convex \\(u\\in C(\\mathbb{R}^n)\\) solving \\(\\det D^2u=\\mu\\) in the Alexandrov sense such that \\(u(x)=v(x)+P(x)\\) for some \\(\\mathbb{Z}^n\\)-periodic function \\(v\\), and whenever \\(u_1=u_1^{\\mathrm{per}}+P\\) and \\(u_2=u_2^{\\mathrm{per}}+P\\) are two such periodic solutions, one has \\(u_1=u_2\\) on \\(\\mathbb{R}^n\\)." + }, + { + "label": "C", + "text": "There exists at most one periodic solution with quadratic component \\(P\\) up to addition of constants. Equivalently, whenever \\(u_1=u_1^{\\mathrm{per}}+P\\) and \\(u_2=u_2^{\\mathrm{per}}+P\\) are two such periodic solutions of \\(\\det D^2u=\\mu\\) in the Alexandrov sense, \\(u_1-u_2\\) is constant on \\(\\mathbb{R}^n\\)." + }, + { + "label": "D", + "text": "For every quadratic function \\(P(x)=\\frac12 x^{\\top}Ax+b\\cdot x+c\\) with \\(A\\in \\mathcal{S}_+^{n\\times n}\\), there exists a periodic solution with quadratic component \\(P\\), and it is unique up to addition of constants; in particular, the compatibility condition \\(\\det A=\\mu(\\mathbb{T}^n)\\) is not needed for existence or uniqueness." + }, + { + "label": "E", + "text": "There exists a periodic solution with quadratic component \\(P\\), and any two such periodic solutions differ by an affine function on \\(\\mathbb{R}^n\\). Equivalently, there exists a convex \\(u\\in C(\\mathbb{R}^n)\\) solving \\(\\det D^2u=\\mu\\) in the Alexandrov sense such that \\(u(x)=v(x)+P(x)\\) for some \\(\\mathbb{Z}^n\\)-periodic function \\(v\\), and whenever \\(u_1=u_1^{\\mathrm{per}}+P\\) and \\(u_2=u_2^{\\mathrm{per}}+P\\) are two such periodic solutions, \\(u_1-u_2=\\ell\\) for some affine function \\(\\ell\\)." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "uniqueness_mod_constants", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped_existence", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "compatibility_detA_equals_mass", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "difference_of_two_solutions_constant_not_affine", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives hypotheses and definitions but does not explicitly state the conclusion. It does not directly reveal that existence plus uniqueness up to constants is the correct option." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question: under the stated assumptions, the student is asked to identify the correct conclusion. However, the alternatives are meaningfully varied, so it is not a pure verbatim restatement." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish strict uniqueness, uniqueness up to constants, mere uniqueness without existence, and incorrect affine ambiguity. But the task is still mainly recognition of the theorem rather than substantial derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong and mathematically plausible: they test common overstatements and confusions about compatibility conditions and uniqueness classes. They are distinct and well-targeted." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-identification MCQ with strong distractors and little answer leakage, though it leans more toward recall of a known result than deep generative reasoning." + } + }, + { + "id": "2511.20502v2", + "paper_link": "http://arxiv.org/abs/2511.20502v2", + "theorems_cnt": 1, + "theorem": { + "env_name": "teo", + "content": "\\label{A}\n Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$. Let $\\alpha = f'(p) \\in (0,1)$. Then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.", + "start_pos": 9131, + "end_pos": 9553, + "label": "A" + }, + "ref_dict": { + "A": "\\begin{teo}\\label{A}\n Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$. Let $\\alpha = f'(p) \\in (0,1)$. Then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.\n\\end{teo}" + }, + "pre_theorem_intro_text_len": 2630, + "pre_theorem_intro_text": "Let $\\mathbb{D}$ be the unit disk, let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be \na holomorphic self map of $\\mathbb{D}$, and consider the dynamical system given by its iterates $\\{ f^n \\}_{n\\in \\mathbb{N}}$.\nThe Denjoy-Wolff Theorem states that if $f$ is not conjugate to a rotation, then there exists a point $p \\in \\overline{\\mathbb{D}}$ such that the orbit of every point in $\\mathbb{D}$ converges to $p$, that is, $f^n(z) \\to p$ as $n \\to \\infty$ for every $z \\in \\mathbb{D}$. We say that $p$ is the {\\em Denjoy-Wolff point} of $f$.\n\nIn the case where $f$ is an {\\em inner function}, that is, for $\\lambda$-almost every $\\zeta\\in\\partial \\mathbb{D}$ the radial limit \\[ f^*(\\zeta)=\\lim_{r\\to1} f(r\\zeta)\\] belongs to $\\partial\\mathbb{D}$ (where $\\lambda$ denotes the normalized Lebesgue measure on $\\partial\\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension\n$$\nf^{*} \\colon \\partial\\mathbb{D} \\longrightarrow \\partial\\mathbb{D}.\n$$\nAssuming that the Denjoy-Wolff point $p$ lies in $\\partial\\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.\nIn the seminal work of Aaronson, Doreing and Mañé \\cite{Aaronson78, mane_dynamics_1991}, this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\\lambda$-almost every point on $\\partial\\mathbb{D}$ converges to the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ if and only if \\[\\sum_{n\\geq 0} 1-|f^n(0|<\\infty.\\]\n\nIt is well-known that {\\em hyperbolic} inner functions (i.e. for which the angular derivative $f'(p)\\in (0,1)$, see Section \\ref{sect-hyperbolic-inner}) always satisfy the condition above. Going one step further, one may ask at which\nrate do these orbits approach the Denjoy–Wolff point.\n\nIf the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that $f$ behaves like the map $z \\mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\\partial \\mathbb{D}$ in a neighborhood of $p$. \n\nThe case when the Denjoy–Wolff point is a singularity is much more complicated, due to the lack of normal forms. In this paper, we establish an explicit rate of convergence for hyperbolic inner functions, covering the case when the Denjoy–Wolff point is a singularity. Namely, if we denote $A(p;a,b) \\coloneqq \\{ z \\in \\mathbb{C}\\colon a <|z-p| < b\\}$, we prove the following. \\\\", + "context": "Let $\\mathbb{D}$ be the unit disk, let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be \na holomorphic self map of $\\mathbb{D}$, and consider the dynamical system given by its iterates $\\{ f^n \\}_{n\\in \\mathbb{N}}$.\nThe Denjoy-Wolff Theorem states that if $f$ is not conjugate to a rotation, then there exists a point $p \\in \\overline{\\mathbb{D}}$ such that the orbit of every point in $\\mathbb{D}$ converges to $p$, that is, $f^n(z) \\to p$ as $n \\to \\infty$ for every $z \\in \\mathbb{D}$. We say that $p$ is the {\\em Denjoy-Wolff point} of $f$.\n\nIn the case where $f$ is an {\\em inner function}, that is, for $\\lambda$-almost every $\\zeta\\in\\partial \\mathbb{D}$ the radial limit \\[ f^*(\\zeta)=\\lim_{r\\to1} f(r\\zeta)\\] belongs to $\\partial\\mathbb{D}$ (where $\\lambda$ denotes the normalized Lebesgue measure on $\\partial\\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension\n$$\nf^{*} \\colon \\partial\\mathbb{D} \\longrightarrow \\partial\\mathbb{D}.\n$$\nAssuming that the Denjoy-Wolff point $p$ lies in $\\partial\\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.\nIn the seminal work of Aaronson, Doreing and Mañé \\cite{Aaronson78, mane_dynamics_1991}, this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\\lambda$-almost every point on $\\partial\\mathbb{D}$ converges to the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ if and only if \\[\\sum_{n\\geq 0} 1-|f^n(0|<\\infty.\\]\n\nIt is well-known that {\\em hyperbolic} inner functions (i.e. for which the angular derivative $f'(p)\\in (0,1)$, see Section \\ref{sect-hyperbolic-inner}) always satisfy the condition above. Going one step further, one may ask at which\nrate do these orbits approach the Denjoy–Wolff point.\n\nIf the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that $f$ behaves like the map $z \\mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\\partial \\mathbb{D}$ in a neighborhood of $p$.\n\nThe case when the Denjoy–Wolff point is a singularity is much more complicated, due to the lack of normal forms. In this paper, we establish an explicit rate of convergence for hyperbolic inner functions, covering the case when the Denjoy–Wolff point is a singularity. Namely, if we denote $A(p;a,b) \\coloneqq \\{ z \\in \\mathbb{C}\\colon a <|z-p| < b\\}$, we prove the following. \\\\", + "full_context": "Let $\\mathbb{D}$ be the unit disk, let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be \na holomorphic self map of $\\mathbb{D}$, and consider the dynamical system given by its iterates $\\{ f^n \\}_{n\\in \\mathbb{N}}$.\nThe Denjoy-Wolff Theorem states that if $f$ is not conjugate to a rotation, then there exists a point $p \\in \\overline{\\mathbb{D}}$ such that the orbit of every point in $\\mathbb{D}$ converges to $p$, that is, $f^n(z) \\to p$ as $n \\to \\infty$ for every $z \\in \\mathbb{D}$. We say that $p$ is the {\\em Denjoy-Wolff point} of $f$.\n\nIn the case where $f$ is an {\\em inner function}, that is, for $\\lambda$-almost every $\\zeta\\in\\partial \\mathbb{D}$ the radial limit \\[ f^*(\\zeta)=\\lim_{r\\to1} f(r\\zeta)\\] belongs to $\\partial\\mathbb{D}$ (where $\\lambda$ denotes the normalized Lebesgue measure on $\\partial\\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension\n$$\nf^{*} \\colon \\partial\\mathbb{D} \\longrightarrow \\partial\\mathbb{D}.\n$$\nAssuming that the Denjoy-Wolff point $p$ lies in $\\partial\\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.\nIn the seminal work of Aaronson, Doreing and Mañé \\cite{Aaronson78, mane_dynamics_1991}, this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\\lambda$-almost every point on $\\partial\\mathbb{D}$ converges to the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ if and only if \\[\\sum_{n\\geq 0} 1-|f^n(0|<\\infty.\\]\n\nIt is well-known that {\\em hyperbolic} inner functions (i.e. for which the angular derivative $f'(p)\\in (0,1)$, see Section \\ref{sect-hyperbolic-inner}) always satisfy the condition above. Going one step further, one may ask at which\nrate do these orbits approach the Denjoy–Wolff point.\n\nIf the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that $f$ behaves like the map $z \\mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\\partial \\mathbb{D}$ in a neighborhood of $p$.\n\nThe case when the Denjoy–Wolff point is a singularity is much more complicated, due to the lack of normal forms. In this paper, we establish an explicit rate of convergence for hyperbolic inner functions, covering the case when the Denjoy–Wolff point is a singularity. Namely, if we denote $A(p;a,b) \\coloneqq \\{ z \\in \\mathbb{C}\\colon a <|z-p| < b\\}$, we prove the following. \\\\\n\nIn the case where $f$ is an {\\em inner function}, that is, for $\\lambda$-almost every $\\zeta\\in\\partial \\mathbb{D}$ the radial limit \\[ f^*(\\zeta)=\\lim_{r\\to1} f(r\\zeta)\\] belongs to $\\partial\\mathbb{D}$ (where $\\lambda$ denotes the normalized Lebesgue measure on $\\partial\\mathbb{D}$), one can consider the dynamical system defined in the unit circle given by the radial extension\n$$\nf^{*} \\colon \\partial\\mathbb{D} \\longrightarrow \\partial\\mathbb{D}.\n$$\nAssuming that the Denjoy-Wolff point $p$ lies in $\\partial\\mathbb{D}$, a natural question to ask is whether points in the unit circle also converge to the Denjoy-Wolff point under the iteration of $f^*$.\nIn the seminal work of Aaronson, Doreing and Mañé \\cite{Aaronson78, mane_dynamics_1991}, this question is answered by means of a complete characterization in terms of infinite sums. More precisely, $\\lambda$-almost every point on $\\partial\\mathbb{D}$ converges to the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ if and only if \\[\\sum_{n\\geq 0} 1-|f^n(0|<\\infty.\\]\n\nIf the Denjoy-Wolff point $p\\in\\partial \\mathbb{D}$ is not a singularity (i.e. $f$ extends as a holomorphic map around $p$), then $p$ is an attracting fixed point for $f$. Using Koenigs' linearizing coordinates, we see that $f$ behaves like the map $z \\mapsto f'(p) z$ near $p$, showing geometric convergence to the fixed point for points on $\\partial \\mathbb{D}$ in a neighborhood of $p$.\n\nWithout being precise, we say that a shrinking target is a collection of arcs of $\\partial\\mathbb{D}$ shrinking in length, and the problem is to determine whether orbits hit these arcs almost surely. Indeed, the arcs $A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n})$ in Theorem \\ref{A} form a shrinking target, and we want to conclude that it is hit almost surely. Via Möbius transformations, we convert our autonomous system into a non-autonomous one fixing $0$, to be able to apply the criteria in \\cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely. We make use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.\n\\\n\nIn this section, we prove Theorem \\ref{A}, which states that if $f \\colon \\mathbb{D} \\to \\mathbb{D}$ is a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$, $\\alpha = f'(p) \\in (0,1)$, then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.\n\n\\begin{lemma}\\label{cotas+}\n Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic self map of $\\mathbb{D}$ and let $p \\in \\partial\\mathbb{D}$ be its Denjoy-Wolff point. Let $\\alpha = f'(p)$. Then, for every $\\delta > 0$, there exists a real constant $C \\geq 1$ such that $$\\frac{1}{C} (\\alpha - \\delta)^n \\leq |f^n(0) - p| \\leq C \\alpha^n,$$ for $n$ large enough.\n\nWe now prove that given $0 < \\varepsilon < 1$, $$(f^*)^n(\\zeta) \\in D(p, \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$, where $\\alpha = f'(p) \\in (0,1)$.\n\nConsider the arc $J_n(\\varepsilon) = D(p,\\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$, $n \\in \\mathbb{N}$, and define \n \\begin{align*}\n E(\\varepsilon) &\\coloneqq \\{ \\zeta \\in \\partial\\mathbb{D}\\colon f^n(\\zeta) \\in J_n(\\varepsilon) \\text{\\ for all $n$ large enough}\\}\\\\\n &= \\{ \\zeta \\in \\partial\\mathbb{D} \\colon (f^n(\\zeta)) \\text{\\ fails to hit\\ } (J_n(\\varepsilon)^c)\\},\n \\end{align*}\n where $J_n(\\varepsilon)^c = \\partial\\mathbb{D}\\setminus J_n(\\varepsilon)$. We will prove that $E(\\varepsilon)$ has full measure.\n\nWe finally prove that given $0 < \\varepsilon < 1$, $$(f^*)^n(\\zeta) \\not \\in D(p, \\alpha^{(1+\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.\n\n\\begin{teo}\\label{A}\n Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$. Let $\\alpha = f'(p) \\in (0,1)$. Then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.\n\\end{teo}", + "post_theorem_intro_text_len": 1298, + "post_theorem_intro_text": "\\\nThe proof of the theorem is based on the concept of \\textit{shrinking targets} of $\\partial\\mathbb{D}$ and their hitting properties (see Section \\ref{sec:shrinking}), developed in \\cite{benini_shrinking_2024} to analyze the recurrent behavior of compositions of inner functions fixing 0. \n\nWithout being precise, we say that a shrinking target is a collection of arcs of $\\partial\\mathbb{D}$ shrinking in length, and the problem is to determine whether orbits hit these arcs almost surely. Indeed, the arcs $A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n})$ in Theorem \\ref{A} form a shrinking target, and we want to conclude that it is hit almost surely. Via Möbius transformations, we convert our autonomous system into a non-autonomous one fixing $0$, to be able to apply the criteria in \\cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely. We make use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.\n\\\n\n{\\bf Acknowledgements. } The authors gratefully acknowledge the Barcelona Introduction to Mathematical Research (BIMR) program at the Centre de Recerca Matemàtica (CRM) for providing an excellent research environment and support during the development of this work.", + "sketch": "The proof of Theorem~\\ref{A} is \"based on the concept of \\textit{shrinking targets} of $\\partial\\mathbb{D}$ and their hitting properties\" (Section~\\ref{sec:shrinking}), as developed in \\cite{benini_shrinking_2024}. The arcs $A(p;\\alpha^{(1+\\varepsilon)n},\\alpha^{(1-\\varepsilon)n})$ appearing in Theorem~\\ref{A} \"form a shrinking target\", and the goal is to \"conclude that it is hit almost surely\" by boundary orbits. \"Via Möbius transformations\", the authors \"convert\" the autonomous system into \"a non-autonomous one fixing $0$\" in order to \"apply the criteria in \\cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely\". The argument also \"make[s] use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.\"", + "expanded_sketch": "The proof of Theorem~\\ref{A} is \"based on the concept of \\textit{shrinking targets} of $\\partial\\mathbb{D}$ and their hitting properties\" (Section~\\ref{sec:shrinking}), as developed in \\cite{benini_shrinking_2024}. The arcs $A(p;\\alpha^{(1+\\varepsilon)n},\\alpha^{(1-\\varepsilon)n})$ appearing in the main theorem \"form a shrinking target\", and the goal is to \"conclude that it is hit almost surely\" by boundary orbits. \"Via Möbius transformations\", the authors \"convert\" the autonomous system into \"a non-autonomous one fixing $0$\" in order to \"apply the criteria in \\cite{benini_shrinking_2024} to determine that the shrinking target is hit almost surely\". The argument also \"make[s] use of some facts about the rate of convergence of the orbit of $0$ to the Denjoy-Wolff point for hyperbolic inner functions.\"", + "expanded_theorem": "\\label{A}\n Let $f \\colon \\mathbb{D} \\to \\mathbb{D}$ be a hyperbolic inner function with Denjoy-Wolff point $p \\in \\partial{\\mathbb{D}}$. Let $\\alpha = f'(p) \\in (0,1)$. Then for all $0<\\varepsilon<1$, we have $$(f^*)^n(\\zeta) \\in A(p;\\alpha^{(1+\\varepsilon)n} , \\alpha^{(1-\\varepsilon)n}) \\cap \\partial\\mathbb{D}$$ for $n$ large enough and $\\lambda$-almost every $\\zeta \\in \\partial\\mathbb{D}$.,", + "theorem_type": [ + "Universal", + "Asymptotic or Limit" + ], + "mcq": { + "question": "Let \\(\\mathbb D\\) be the unit disk, let \\(\\partial\\mathbb D\\) be its boundary, and let \\(\\lambda\\) denote the normalized Lebesgue measure on \\(\\partial\\mathbb D\\). Suppose \\(f:\\mathbb D\\to\\mathbb D\\) is a holomorphic self-map that is an inner function, meaning that for \\(\\lambda\\)-almost every \\(\\zeta\\in\\partial\\mathbb D\\) the radial limit \\(f^*(\\zeta)=\\lim_{r\\to 1^-}f(r\\zeta)\\) exists and lies in \\(\\partial\\mathbb D\\). Assume that \\(f\\) is hyperbolic with Denjoy-Wolff point \\(p\\in\\partial\\mathbb D\\), so that \\(f^n(z)\\to p\\) for every \\(z\\in\\mathbb D\\), and let \\(\\alpha=f'(p)\\in(0,1)\\). For \\(a,b>0\\), define \\(A(p;a,b)=\\{z\\in\\mathbb C: a<|z-p|0$ and $k_0\\in\\mathbb{N}$ such that for every $k\\ge k_0$ every global maximum of $\\rho_k$ lies within a neighborhood of radius\n\\[\nC\\exp\\!\\Big(\\tfrac{k}{4}(l_1^2-l_2^2)\\Big)\n\\]\nof some point $p$ satisfying\n\\[\n\\mathrm{Hol}_{kL}(\\gamma_{p,v})=1\\quad\\text{for all }v\\in S_1.\n\\]\n\n\\item[(c)] Let $\\{v_1,\\dots,v_m\\}$ be a maximally linearly independent subset of $S_1$ and suppose $\\#S_1=2m$ (so $S_1=\\{\\pm v_1,\\dots,\\pm v_m\\}$). Then the statement of (b) holds. Moreover, there exists $C'>0$ and $k_0'$ such that for all $k\\ge k_0'$ every global minimum of $\\rho_k$ lies within a neighborhood of radius\n\\[\nC'\\exp\\!\\Big(\\tfrac{k}{4}(l_1^2-l_2^2)\\Big)\n\\]\nof some point $p$ with\n\\[\n\\mathrm{Hol}_{kL}(\\gamma_{p,v})=-1\\quad\\text{for all }v\\in S_1.\n\\]\n\\end{enumerate}\n\\end{theorem}", + "prop-surj": "\\begin{proposition}\\label{prop-surj}\n\t $\\Phi_\\vcal$ is surjective.\n\\end{proposition}" + }, + "pre_theorem_intro_text_len": 3500, + "pre_theorem_intro_text": "Let $(M,\\omega)$ be a K\\\"{a}hler manifold of complex dimension $n$, and let $L\\to M$ be a holomorphic line bundle equipped with a Hermitian metric $h$ whose Chern curvature equals $-i\\omega$. The Bergman space $\\hcal_k$ consists of holomorphic sections of $L^k$ that are square integrable:\n\\[\n\\int_M |s|_{h}^2\\,\\frac{\\omega^n}{n!}<\\infty.\n\\]\nWith the $L^2$--inner product induced by $h$ and $\\omega$, $\\hcal_k$ is a closed subspace of $L^2(M,L^k)$. The Bergman kernel $K_k(x,y)$ is the integral kernel of the orthogonal projection $L^2(M,L^k)\\to\\hcal_k$, characterized by\n\\[\ns(y)=\\int_M \\langle s(x),K_k(x,y)\\rangle_{h^k}\\,\\frac{\\omega^n(x)}{n!}\\qquad(\\forall s\\in\\hcal_k).\n\\]\nThe diagonal density (or Bergman kernel function) is the pointwise trace of $K_k$,\n\\[\n\\rho_k(x)=|K_k(x,x)|_{h^k}.\n\\]\n\nFor trivial bundles over domains in $\\mathbb C^n$ one uses the analogous definition for holomorphic $L^2$--functions.\n\nThe Bergman kernel has been intensively studied for domains in $\\mathbb C^n$ and for K\\\"{a}hler manifolds (see, e.g., \\cite{Fefferman1974,boutet1975singularite,kerzman1978cauchy,donaldson2001,Sun2011Expected,Donaldson2014Gromov,berman2011fekete,Donaldson15, dsz1,dsz2,dsz3,Shiffman2008NV,sz1,bsz0,bsz1}). Except in very symmetric cases (for instance complex space forms), explicit formulas are rare. A fundamental substitute is the Tian-Catlin-Zelditch expansion (\\cite{Tian1990On, Zelditch2000Szego, Lu2000On, Catlin}): when $M$ is compact,\n\\[\n\\rho_k(x)\\sim\\Big(\\frac{k}{2\\pi}\\Big)^n\\Big(1+\\frac{b_1(x)}{k}+\\frac{b_2(x)}{k^2}+\\cdots\\Big),\n\\]\nwhere each $b_j(x)$ is a universal polynomial in the curvature of the Riemannian metric corresponding to $\\omega$ and its covariant derivatives. There are extensions of this expansion to symplectic manifolds and orbifolds (see \\cite{MM,dailiuma}).\n\nIn \\cite{sun-rem-hyp}, the author made a progress by proving an explicit formula for the Bergman kernel function for positive line bundles on hyperbolic Riemann surfaces. The formula involves summation over all geodesic loops based at the point where the Bergman kernel is evaluated.\n In this paper, inspired by this work, we continue to study the Bergman kernel of abelian varieties.\n\n\\\n\nLet $X=\\mathbb C^n/\\Lambda$ be an abelian variety over $\\mathbb C$. Here, $\\Lambda$ is a lattice in $\\mathbb C^n$. Let $L\\to X$ be a positive line bundle over $X$, whose first Chern class is determined by a positive definite Hermitian form $H=(H_{ij})$ on $\\mathbb C^n$ such that $\\Im H(\\Lambda,\\Lambda)\\subset \\mathbb Z$. For each vector $v\\in \\mathbb C^n$, we denote by $|v|_H$ the norm of $v$ defined by $H$.\nLet $\\omega=\\pi \\sqrt{-1} \\sum H_{ij} dz^i\\wedge d\\bar{z}^j \\in 2\\pi c_1(L)$ be a flat K\\\"ahler metric on $X$. There exist a Hermitian metric $h$ on $L$ with curvature $- i \\omega$.\n\nFor each $p\\in X$, $\\exists \\tilde{p}\\in\\mathbb C^n$ such that $p=\\pi(\\tilde{p})$. Such a $\\tilde{p}$ is called a lift of $p$.\nFor each $v\\in \\Lambda$, let $\\gamma_{p,v}\\subset X$ be the smooth geodesic loop defined by $v$ starting at $p$ with orientation defined by $\\frac{\\partial}{\\partial a}$, where $a$ is the parameter of the line segment $\\tilde{p}+av$, $a\\in[0,1]$. It is easy to see that $\\gamma_{p,v}$ does not depend on the choice of $\\tilde{p}$, and that the length of $\\gamma_{p,v}$ is $|v|_H$. Let $\\nabla$ be the Chern connection of $(L,h)$.\n Let $e^{2\\pi i\\alpha_v(p)}=\\text{Hol}_{L}(\\gamma_{p,v})$ be the holonomy of $\\nabla$ along $\\gamma_{p,v}$. \nOur first result is the following formula:", + "context": "Let $(M,\\omega)$ be a K\\\"{a}hler manifold of complex dimension $n$, and let $L\\to M$ be a holomorphic line bundle equipped with a Hermitian metric $h$ whose Chern curvature equals $-i\\omega$. The Bergman space $\\hcal_k$ consists of holomorphic sections of $L^k$ that are square integrable:\n\\[\n\\int_M |s|_{h}^2\\,\\frac{\\omega^n}{n!}<\\infty.\n\\]\nWith the $L^2$--inner product induced by $h$ and $\\omega$, $\\hcal_k$ is a closed subspace of $L^2(M,L^k)$. The Bergman kernel $K_k(x,y)$ is the integral kernel of the orthogonal projection $L^2(M,L^k)\\to\\hcal_k$, characterized by\n\\[\ns(y)=\\int_M \\langle s(x),K_k(x,y)\\rangle_{h^k}\\,\\frac{\\omega^n(x)}{n!}\\qquad(\\forall s\\in\\hcal_k).\n\\]\nThe diagonal density (or Bergman kernel function) is the pointwise trace of $K_k$,\n\\[\n\\rho_k(x)=|K_k(x,x)|_{h^k}.\n\\]\n\nFor trivial bundles over domains in $\\mathbb C^n$ one uses the analogous definition for holomorphic $L^2$--functions.\n\nThe Bergman kernel has been intensively studied for domains in $\\mathbb C^n$ and for K\\\"{a}hler manifolds (see, e.g., \\cite{Fefferman1974,boutet1975singularite,kerzman1978cauchy,donaldson2001,Sun2011Expected,Donaldson2014Gromov,berman2011fekete,Donaldson15, dsz1,dsz2,dsz3,Shiffman2008NV,sz1,bsz0,bsz1}). Except in very symmetric cases (for instance complex space forms), explicit formulas are rare. A fundamental substitute is the Tian-Catlin-Zelditch expansion (\\cite{Tian1990On, Zelditch2000Szego, Lu2000On, Catlin}): when $M$ is compact,\n\\[\n\\rho_k(x)\\sim\\Big(\\frac{k}{2\\pi}\\Big)^n\\Big(1+\\frac{b_1(x)}{k}+\\frac{b_2(x)}{k^2}+\\cdots\\Big),\n\\]\nwhere each $b_j(x)$ is a universal polynomial in the curvature of the Riemannian metric corresponding to $\\omega$ and its covariant derivatives. There are extensions of this expansion to symplectic manifolds and orbifolds (see \\cite{MM,dailiuma}).\n\nIn \\cite{sun-rem-hyp}, the author made a progress by proving an explicit formula for the Bergman kernel function for positive line bundles on hyperbolic Riemann surfaces. The formula involves summation over all geodesic loops based at the point where the Bergman kernel is evaluated.\n In this paper, inspired by this work, we continue to study the Bergman kernel of abelian varieties.\n\nLet $X=\\mathbb C^n/\\Lambda$ be an abelian variety over $\\mathbb C$. Here, $\\Lambda$ is a lattice in $\\mathbb C^n$. Let $L\\to X$ be a positive line bundle over $X$, whose first Chern class is determined by a positive definite Hermitian form $H=(H_{ij})$ on $\\mathbb C^n$ such that $\\Im H(\\Lambda,\\Lambda)\\subset \\mathbb Z$. For each vector $v\\in \\mathbb C^n$, we denote by $|v|_H$ the norm of $v$ defined by $H$.\nLet $\\omega=\\pi \\sqrt{-1} \\sum H_{ij} dz^i\\wedge d\\bar{z}^j \\in 2\\pi c_1(L)$ be a flat K\\\"ahler metric on $X$. There exist a Hermitian metric $h$ on $L$ with curvature $- i \\omega$.\n\nFor each $p\\in X$, $\\exists \\tilde{p}\\in\\mathbb C^n$ such that $p=\\pi(\\tilde{p})$. Such a $\\tilde{p}$ is called a lift of $p$.\nFor each $v\\in \\Lambda$, let $\\gamma_{p,v}\\subset X$ be the smooth geodesic loop defined by $v$ starting at $p$ with orientation defined by $\\frac{\\partial}{\\partial a}$, where $a$ is the parameter of the line segment $\\tilde{p}+av$, $a\\in[0,1]$. It is easy to see that $\\gamma_{p,v}$ does not depend on the choice of $\\tilde{p}$, and that the length of $\\gamma_{p,v}$ is $|v|_H$. Let $\\nabla$ be the Chern connection of $(L,h)$.\n Let $e^{2\\pi i\\alpha_v(p)}=\\text{Hol}_{L}(\\gamma_{p,v})$ be the holonomy of $\\nabla$ along $\\gamma_{p,v}$. \nOur first result is the following formula:", + "full_context": "Let $(M,\\omega)$ be a K\\\"{a}hler manifold of complex dimension $n$, and let $L\\to M$ be a holomorphic line bundle equipped with a Hermitian metric $h$ whose Chern curvature equals $-i\\omega$. The Bergman space $\\hcal_k$ consists of holomorphic sections of $L^k$ that are square integrable:\n\\[\n\\int_M |s|_{h}^2\\,\\frac{\\omega^n}{n!}<\\infty.\n\\]\nWith the $L^2$--inner product induced by $h$ and $\\omega$, $\\hcal_k$ is a closed subspace of $L^2(M,L^k)$. The Bergman kernel $K_k(x,y)$ is the integral kernel of the orthogonal projection $L^2(M,L^k)\\to\\hcal_k$, characterized by\n\\[\ns(y)=\\int_M \\langle s(x),K_k(x,y)\\rangle_{h^k}\\,\\frac{\\omega^n(x)}{n!}\\qquad(\\forall s\\in\\hcal_k).\n\\]\nThe diagonal density (or Bergman kernel function) is the pointwise trace of $K_k$,\n\\[\n\\rho_k(x)=|K_k(x,x)|_{h^k}.\n\\]\n\nFor trivial bundles over domains in $\\mathbb C^n$ one uses the analogous definition for holomorphic $L^2$--functions.\n\nThe Bergman kernel has been intensively studied for domains in $\\mathbb C^n$ and for K\\\"{a}hler manifolds (see, e.g., \\cite{Fefferman1974,boutet1975singularite,kerzman1978cauchy,donaldson2001,Sun2011Expected,Donaldson2014Gromov,berman2011fekete,Donaldson15, dsz1,dsz2,dsz3,Shiffman2008NV,sz1,bsz0,bsz1}). Except in very symmetric cases (for instance complex space forms), explicit formulas are rare. A fundamental substitute is the Tian-Catlin-Zelditch expansion (\\cite{Tian1990On, Zelditch2000Szego, Lu2000On, Catlin}): when $M$ is compact,\n\\[\n\\rho_k(x)\\sim\\Big(\\frac{k}{2\\pi}\\Big)^n\\Big(1+\\frac{b_1(x)}{k}+\\frac{b_2(x)}{k^2}+\\cdots\\Big),\n\\]\nwhere each $b_j(x)$ is a universal polynomial in the curvature of the Riemannian metric corresponding to $\\omega$ and its covariant derivatives. There are extensions of this expansion to symplectic manifolds and orbifolds (see \\cite{MM,dailiuma}).\n\nIn \\cite{sun-rem-hyp}, the author made a progress by proving an explicit formula for the Bergman kernel function for positive line bundles on hyperbolic Riemann surfaces. The formula involves summation over all geodesic loops based at the point where the Bergman kernel is evaluated.\n In this paper, inspired by this work, we continue to study the Bergman kernel of abelian varieties.\n\nLet $X=\\mathbb C^n/\\Lambda$ be an abelian variety over $\\mathbb C$. Here, $\\Lambda$ is a lattice in $\\mathbb C^n$. Let $L\\to X$ be a positive line bundle over $X$, whose first Chern class is determined by a positive definite Hermitian form $H=(H_{ij})$ on $\\mathbb C^n$ such that $\\Im H(\\Lambda,\\Lambda)\\subset \\mathbb Z$. For each vector $v\\in \\mathbb C^n$, we denote by $|v|_H$ the norm of $v$ defined by $H$.\nLet $\\omega=\\pi \\sqrt{-1} \\sum H_{ij} dz^i\\wedge d\\bar{z}^j \\in 2\\pi c_1(L)$ be a flat K\\\"ahler metric on $X$. There exist a Hermitian metric $h$ on $L$ with curvature $- i \\omega$.\n\nFor each $p\\in X$, $\\exists \\tilde{p}\\in\\mathbb C^n$ such that $p=\\pi(\\tilde{p})$. Such a $\\tilde{p}$ is called a lift of $p$.\nFor each $v\\in \\Lambda$, let $\\gamma_{p,v}\\subset X$ be the smooth geodesic loop defined by $v$ starting at $p$ with orientation defined by $\\frac{\\partial}{\\partial a}$, where $a$ is the parameter of the line segment $\\tilde{p}+av$, $a\\in[0,1]$. It is easy to see that $\\gamma_{p,v}$ does not depend on the choice of $\\tilde{p}$, and that the length of $\\gamma_{p,v}$ is $|v|_H$. Let $\\nabla$ be the Chern connection of $(L,h)$.\n Let $e^{2\\pi i\\alpha_v(p)}=\\text{Hol}_{L}(\\gamma_{p,v})$ be the holonomy of $\\nabla$ along $\\gamma_{p,v}$. \nOur first result is the following formula:\n\nLastly, we state an off-diagonal decay estimate for the Bergman kernel. For $x,y\\in X$ let $\\mathfrak{G}_{x,y}$ denote the set of geodesic segments joining $x$ to $y$ (parameterized by arc length).\n\\begin{theorem}\\label{thm-off-diag}\n Let $(X,\\omega,L,h)$ be a polarized abelian variety as in Theorem \\ref{thm-main}. For every $k\\ge 1$ and every $x\\neq y$,\n \\[\n |K_k(x,y)|_{h^k}\\;\\le\\;\\Big(\\frac{k}{2\\pi}\\Big)^{\\!n}\\sum_{\\gamma\\in\\mathfrak{G}_{x,y}} e^{-\\frac{k}{4}\\,\\ell(\\gamma)^2},\n \\]\n where $\\ell(\\gamma)$ is the length of $\\gamma$.\n\\end{theorem}\n\nLet $K_k(z,w)$ be the Bergman kernel of $\\hcal_{\\C^n,k}$. Then we can write \\[K_k(z,w)=\\sum_{i=1}^{\\infty}f_i(z)\\otimes \\bar{f}_i(w),\\] where $\\{f_i\\}_{i=1}^\\infty$ is an orthonormal basis of $\\hcal_{\\C^n,k}$.\nFor each $v\\in \\Lambda$, we define \\[K^v_k(z,w)=\\sum_{i=1}^{\\infty}A_v^*f_i(z)\\otimes \\bar{f}_i(w).\\]\n\\begin{proposition}\n $K^v_k(z,w)=\\sum_{i=1}^{\\infty}A_v^*f_i(z)\\otimes \\bar{f}_i(w)$ is independent of the choice of orthonormal basis $\\{f_i\\}_{i=1}^\\infty$ of $\\hcal_{\\C^n,k}$.\n\\end{proposition}\n\\begin{proof}\n For any $f\\in \\hcal_{\\C^n,k}$, we have $f=\\sum_{i=1}^{\\infty}c_i A_v^*f_i$, since $\\{A_v^*f_i\\}$ is also an orthonormal basis of $\\hcal_{\\C^n,k}$. Then for any fixed $w$, \\[\\int_{\\C^n} (f,\\sum_{i=1}^{\\infty}A_v^*f_i(z)\\otimes \\bar{f}_i(w))_{\\pi^*h}\\frac{w_0^n}{n!}=\\sum_{i=1}^{\\infty}c_i f_i(w)=(A_v^{-1})^*f(w), \\]\n namely, $K^v_k(z,w)$ is the representation element for the functional $\\val_w\\circ A_{-v}^*$, hence independent of the choice of the orthonormal basis.\n\\end{proof}\n\\begin{defandthm}\\label{def-and-thm-k-gamma}\n We define\n\\[K_{k}^\\Lambda(z,w)\\eqd \\sum_{v\\in \\Lambda}K^v_k(z,w). \\]\nThen for $k>0$, and for each fixed $w$, it converges locally uniformly and absolutely. Moreover, we have that for $k>0$, $\\sum_{v\\in \\Lambda}K^v_k(z,w)\\in W_k$ for each fixed $w$.\n\\end{defandthm}\n\\begin{proof}\n Since $K^v_k(z,w)$ is independent of the choice of orthonormal basis, for each fixed $w$, we can choose the orthonormal basis $\\{f_i\\}$ such that $f_1=s_w$, a peak section at $w$. So $f_i(w)=0$ for $i>1$. So we have \\[K^v_k(z,w)=A_v^*s_w(z)\\otimes \\bar{s}_w(w).\\]\n So by Lemma \\ref{lem-sum-g-f-0}, we have $\\sum_{v\\in \\Lambda}K^v_k(z,w)$ converges locally uniformly and absolutely. And by Lemma \\ref{lem-sum-g-f-0-in-v-k}, $\\sum_{v\\in \\Lambda}K^v_k(z,w)\\in W_k$ for each fixed $w$.\n\\end{proof}\n\\begin{lemma}\\label{lem-int-s-sum-g-s-p}\n For any $s\\in W_k$, we have \n \\[\\int_{F_\\Lambda}(s,\\sum_{v\\in \\Lambda}A_v^*s_p)_{\\pi^*h}\\frac{\\omega_0^n}{n!}=(\\frac{2\\pi}{k})^{n/2}\\frac{s(p)}{s_p(p)}.\\]\n\\end{lemma}\n\\begin{proof}\n It suffices to show this for $p=0$. By Lemmas \\ref{lem-sum-g-f-0} and \\ref{lem-sum-g-f-0-in-v-k}, \n \\[ \\int_{F_\\Lambda}(s,\\sum_{v\\in \\Lambda}A_v^*s_0)_{\\pi^*h}\\frac{\\omega_0^n}{n!}=\\sum_{v\\in \\Lambda}\\int_{F_\\Lambda}(s,A_v^*s_0)_{\\pi^*h}\\frac{\\omega_0^n}{n!}.\\]\n On the other hand, since $s_0=(\\frac{k}{2\\pi})^{n/2}\\bm{e}_k$ is $L_1$ integrable on $\\C^n$, and $s$ is bounded on $\\C^n$, we have $(s,s_0)_{\\pi^*h}\\in L^1(\\C^n,\\frac{\\omega_0^n}{n!})$. So \\[\\sum_{v\\in \\Lambda}\\int_{F_\\Lambda}(s,A_v^*s_0)_{\\pi^*h}\\frac{\\omega_0^n}{n!}=\\int_{\\C^n} (s,s_0)_{\\pi^*h}\\frac{\\omega_0^n}{n!}.\\]\n Then by taking the Taylor expansion of $\\frac{s}{\\bm{e}_k}$, we get the right hand side equals $(\\frac{2\\pi}{k})^{n/2}\\frac{s(0)}{\\bm{e}_k(0)}=\\frac{s(0)}{s_0(0)}$.\n\\end{proof}\n\\begin{theorem}\\label{thm-k-gamma-bergman}\n For $k>0$, $K_{k}^\\Lambda(z,w)$ is the Bergman kernel of $W_k$.\n\\end{theorem}\n\\begin{proof}\n For fixed $w$, we have \\[K_{k}^\\Lambda(z,w)=(\\sum_{v\\in \\Lambda}A_v^*s_w(z))\\otimes \\bar{s}_w(w).\\]\n So for each $S\\in \\hcal_{X,k}$, we have \\[\\int_{F_\\Lambda} (S,K_{k}^\\Lambda(z,w))_{\\pi^*h}\\frac{\\omega_0^n}{n!}=\\frac{S(w)}{s_w(w)}s_w(w)=S(w).\\]\n We have proved the theorem.\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{thm-main}]\n For a point \\(p\\in X\\) choose a lift \\(\\tilde p\\in\\C^n\\); without loss of generality we may take \\(\\tilde p=0\\). By Theorem \\ref{thm-k-gamma-bergman} we have the identity\n \\[\n \\rho_{X,k}(0)=K_{k}^\\Lambda(0,0)\n = (\\frac{k}{2\\pi})^n \\;+\\; \\sum_{v\\in\\Lambda\\setminus\\{0\\}} K^v_k(0,0).\n \\]\n Recall that an element $v\\in \\Lambda$ is primitive if it can not be written as $v=mu$ for some $u\\in\\Lambda$ and $m>1$.\n Let \\(\\mathcal P\\) denote the set of primitive elements of \\(\\Lambda\\). Writing \\(\\langle v\\rangle=\\{mv:\\,m\\in\\Z \\}\\) for the cyclic subgroup generated by \\(v\\in\\mathcal P\\), we may reorganise the sum as\n \\[\n \\sum_{v\\in\\Lambda\\setminus\\{0\\}} K^v_k(0,0)\n = \\frac12\\sum_{v\\in\\mathcal P}\\sum_{u\\in\\langle v\\rangle\\setminus\\{0\\}} K^{u}_k(0,0).\n \\]\n\nApplying Theorem \\ref{thm-cylinder-main} to the quotient \\(\\C^n/\\langle v\\rangle\\) yields, for each \\(v\\in\\mathcal P\\),\n \\[\n \\sum_{u\\in\\langle v\\rangle\\setminus\\{1\\}} K^{u}_k(0,0)\n = (\\frac{k}{2\\pi})^n\\sum_{u\\in\\langle u\\rangle\\setminus\\{0\\}}\n e^{-k(\\frac{|u|}{2})^2}\\cos\\!\\big(2\\pi k\\alpha_{u}(p)\\big).\n \\]\n Summing over all primitive elements \\(g\\in\\mathcal P\\) gives the formula in Theorem \\ref{thm-main}. \n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{thm-off-diag}]\n By Theorem \\ref{thm-k-gamma-bergman}, we have \n \\[\\pi^*K_{X,k}(z,w)=\\sum_{v\\in \\Lambda}(A_v^*s_w(z))\\otimes \\bar{s}_w(w).\\]\n We can assume that $w=0$. Then since $|A_v^*s_0(z)|=(\\frac{k}{2\\pi})^{n/2}e^{-\\frac{k}{4}d^2(-v,z)}$. We have \\[|\\pi^*K_{X,k}(z,0)|_h\\leq (\\frac{k}{2\\pi})^{n}\\sum_{v\\in \\Lambda}e^{-\\frac{k}{4}d^2(-v,z)}.\\]\n Since $d(-v,z)$ can be identified with the length of the corresponding geodesic segment in $X$ joining $\\pi(z)$ and $\\pi(0)$, the theorem follows.\n\\end{proof}\n\\subsection{Proof of Theorem \\ref{thm-second}}\nFor a fixed line bundle $L$ on $X$ with $\\omega\\in 2\\pi c_1(L)$, let $h$ be a Hermitian metric on $L$ such that $\\Theta_h=-i\\omega$. Let $F$ denote the underlying $C^\\infty$ complex line bundle of $L$. Let $\\ucal$ be the space of unitary connections on $(F,h)$ whose curvatures satisfy the condition that the $(0,2)$-part is $0$. Let $\\nabla_0\\in \\ucal$ be the Chern connection of $(L,h)$.\n\nFor $L'$, we let $\\text{Hol}_{L'}(\\gamma_{p,v})=e^{2\\pi i \\alpha'_v(p)}$ and $\\phi'_p=\\alpha'_{v_1}(p)$. Then we also have \\[(P_*(\\frac{(2\\pi)^n}{k^n}\\rho_{L',k}-1))(x)=2\\nu \\sum_{m\\geq 1} e^{-\\frac{k}{4}|m|^2|v_1|_H^2}\\cos(2\\pi m (\\lambda t+\\phi'_p)). \\]\nTherefore, if $\\rho_{L',k}=\\rho_{L,k}$, we get \\[\\sum_{m\\geq 1} e^{-\\frac{k}{2}|m|^2|v_1|_H^2}\\cos(2\\pi m (\\lambda t+\\phi'_p))= \\sum_{m\\geq 1} e^{-\\frac{k}{2}|m|^2\\|v_1\\|^2}\\cos(2\\pi m (\\lambda t+\\phi_p)), \\] for $t\\in [0,1]$. So we get $\\phi_p'=\\phi_p, \\mod 1$, namely $\\text{Hol}_{L'}(\\gamma_{p,v_1})=\\text{Hol}_{L'}(\\gamma_{p,v_1})$. So we have proved the theorem.\n\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{thm-max-min}]\n Part (a) follows directly from Theorem \\ref{thm-main}. For part (b) and (c), we have that $\\exists C_1>0$, $k_0>0$ such that for $k\\geq k_0$, \\[\\sum_{v\\in \\Lambda, |v|_H>l_1}e^{-\\frac{k}{4}|v|_H^2}\\leq C_1 e^{-\\frac{k}{4}l_2^2}.\\] \n Let $p$ be a point satisfying the condition that $\\text{Hol}_{L}(\\gamma_{p,v})=1$ for all $v\\in S_1$. Suppose that $k\\geq k_0$ and $\\rho_k(q)>\\rho_k(p)$, then we have \\[\\sum_{v\\in S_1}e^{-\\frac{k}{4}l_1^2}(1-\\cos(2\\pi k\\alpha_v(q)))\\leq 2C_1 e^{-\\frac{k}{4}l_2^2}, \\]namely \\[\\sum_{v\\in S_1}(1-\\cos(2\\pi k\\alpha_v(q)))\\leq 2C_1 e^{-\\frac{k}{4}(l_2^2-l_1^2)}.\\] \n If we require that $\\alpha_v(q)\\in (-\\pi,\\pi]$ for all $v$, then this implies that $|\\alpha_v(q)|$ is small for $v\\in S_1$. \nLet $\\{v_1, v_2,\\cdots, v_m\\}$ be a maximally linearly independent subset of $S_1$.\n From the proof of Proposition \\ref{prop-surj}, it is easy to see that $\\exists C_2>0$, independent of $k$, such that when $\\epsilon$ is small enough, for any $q'\\in X$, if $1-\\cos(2\\pi k\\alpha_{v_j}(q'))<\\epsilon$, $1\\leq j\\leq m$, then $\\exists p'\\in X$ such that $d(p',q')l_1}} |v|_{H},\n\\]\nand write $S_1=\\{v\\in\\Lambda:\\;|v|_H=l_1\\}$.\n\n\\begin{theorem}\\label{thm-max-min}\n\\begin{enumerate}\n\\item[(a)] Assume $\\Im H(u,v)\\in 2\\mathbb{Z}$ for all $u,v\\in\\Lambda$. Then for every $k\\ge1$ the Bergman density $\\rho_k$ attains its maximum precisely at those points $p\\in X$ for which\n\\[\n\\mathrm{Hol}_L(\\gamma_{p,v})=1\\quad\\text{for all }v\\in\\Lambda.\n\\]\n\n\\item[(b)] Assume $\\Im H(u,v)\\in 2\\mathbb{Z}$ for all $u,v\\in S_1$. There exist constants $C>0$ and $k_0\\in\\mathbb{N}$ such that for every $k\\ge k_0$ every global maximum of $\\rho_k$ lies within a neighborhood of radius\n\\[\nC\\exp\\!\\Big(\\tfrac{k}{4}(l_1^2-l_2^2)\\Big)\n\\]\nof some point $p$ satisfying\n\\[\n\\mathrm{Hol}_{kL}(\\gamma_{p,v})=1\\quad\\text{for all }v\\in S_1.\n\\]\n\n\\item[(c)] Let $\\{v_1,\\dots,v_m\\}$ be a maximally linearly independent subset of $S_1$ and suppose $\\#S_1=2m$ (so $S_1=\\{\\pm v_1,\\dots,\\pm v_m\\}$). Then the statement of (b) holds. Moreover, there exists $C'>0$ and $k_0'$ such that for all $k\\ge k_0'$ every global minimum of $\\rho_k$ lies within a neighborhood of radius\n\\[\nC'\\exp\\!\\Big(\\tfrac{k}{4}(l_1^2-l_2^2)\\Big)\n\\]\nof some point $p$ with\n\\[\n\\mathrm{Hol}_{kL}(\\gamma_{p,v})=-1\\quad\\text{for all }v\\in S_1.\n\\]\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark}\n\\begin{itemize}\n\\item If $(v_1,\\dots,v_{2n})$ is any basis of $\\Lambda$, Proposition \\ref{prop-surj} implies there exist points $p$ with $\\mathrm{Hol}_L(\\gamma_{p,v_j})=1$ for all $j$. Hence the hypothesis $\\Im H(u,v)\\in2\\mathbb{Z}$ on $\\Lambda\\times\\Lambda$ ensures existence of points where $\\mathrm{Hol}_L(\\gamma_{p,v})=1$ for every $v\\in\\Lambda$.\n\\item In part (b) it is enough to verify the holonomy condition $\\mathrm{Hol}_{kL}(\\gamma_{p,v_j})=1$ for a maximally independent subset $\\{v_j\\}_{j=1}^m\\subset S_1$.\n\\end{itemize}\n\\end{remark}\nSimilar to Conjecture 1.6 in \\cite{sun-rem-hyp}, we propose the following rigidity conjecture for polarized abelian varieties:\n\n\\begin{conjecture}\n\tLet $(X,L)$ and $(X',L')$ be polarized abelian varieties of dimension $n$. Suppose there is a diffeomorphism $\\Phi:X\\to X'$ and an integer $k\\geq 1$ such that\n\t\\[\n\t\\rho_{L,k}=\\Phi^*\\rho_{L',k}.\n\t\\]\n\tThen $\\Phi$ is either a biholomorphism or an anti-biholomorphism, and moreover\n\t\\[\n\t\\Phi^*((L')^{\\!k})\\cong L^{\\!k}\\qquad\\text{or}\\qquad \\Phi^*((L')^{\\!k})\\cong\\overline{L^{\\!k}},\n\t\\]\n\trespectively. Here $\\overline{L}$ denotes the complex-conjugate line bundle of $L$ (transition functions given by complex conjugates).\n\\end{conjecture}\n\nTheorem \\ref{thm-second} gives partial evidence for this conjecture.\n\nLastly, we state an off-diagonal decay estimate for the Bergman kernel. For $x,y\\in X$ let $\\mathfrak{G}_{x,y}$ denote the set of geodesic segments joining $x$ to $y$ (parameterized by arc length).\n\\begin{theorem}\\label{thm-off-diag}\n\tLet $(X,\\omega,L,h)$ be a polarized abelian variety as in Theorem \\ref{thm-main}. For every $k\\ge 1$ and every $x\\neq y$,\n\t\\[\n\t|K_k(x,y)|_{h^k}\\;\\le\\;\\Big(\\frac{k}{2\\pi}\\Big)^{\\!n}\\sum_{\\gamma\\in\\mathfrak{G}_{x,y}} e^{-\\frac{k}{4}\\,\\ell(\\gamma)^2},\n\t\\]\n\twhere $\\ell(\\gamma)$ is the length of $\\gamma$.\n\\end{theorem}\n\nThe corresponding result in \\cite{sun-rem-hyp} was the first to relate the global off-diagonal decay to the geodesic distance (compared to many local results, see the references therein); the theorem above provides a second instance, now in the flat setting.\n\n\\medskip\nWe briefly outline the proofs. The argument for Theorem \\ref{thm-main} follows the same two-step strategy as in \\cite{sun-rem-hyp}: first establish the formula for the cylinder (quotients of $\\mathbb C^n$ by a cyclic subgroup), and then lift the global Bergman kernel to the universal cover and express it as a summation over the lattice $\\Lambda$. This strategy parallels the construction behind the Selberg trace formula. \n\nTheorem \\ref{thm-second} is obtained by integrating the kernel along complementary subtori to isolate the contributions of a chosen lattice direction; comparison of these averaged formulas recovers the holonomy data. Theorems \\ref{thm-max-min} and \\ref{thm-off-diag} are direct consequences of Theorem \\ref{thm-main}, while the off-diagonal decay is obtained by estimating the contribution of peak sections and summing over geodesic segments joining two points.\n\n\\medskip\n\n\\noindent\\textbf{Organization of the paper.}\nIn Section~2 we collect background material and derive the one-dimensional cylinder formula. Section~\\ref{section-cylinder} treats the higher-dimensional cylinder and proves the cylinder case of Theorem~\\ref{thm-main}. Section~\\ref{sec-proof-thm-main} carries out the lattice summation on the universal cover and completes the proof of Theorem~\\ref{thm-main}; the remaining theorems are then proved as applications.\n\n\\medskip\n\n\\noindent\\textbf{Acknowledgements.} The author would like to thank Professor Song Sun for many very helpful discussions.", + "sketch": "We briefly outline the proofs. The argument for Theorem \\ref{thm-main} follows the same two-step strategy as in \\cite{sun-rem-hyp}: first establish the formula for the cylinder (quotients of $\\mathbb C^n$ by a cyclic subgroup), and then lift the global Bergman kernel to the universal cover and express it as a summation over the lattice $\\Lambda$. This strategy parallels the construction behind the Selberg trace formula.", + "expanded_sketch": "We briefly outline the proofs. The argument for the main theorem follows the same two-step strategy as in Sun, “Removable singularities in hyperbolic geometry”: first establish the formula for the cylinder (quotients of $\\mathbb C^n$ by a cyclic subgroup), and then lift the global Bergman kernel to the universal cover and express it as a summation over the lattice $\\Lambda$. This strategy parallels the construction behind the Selberg trace formula.,", + "expanded_theorem": "\\label{thm-main}\n\t For $k\\geq 1$, the Bergman kernel $\\rho_k$ of $H^0(X,L^k)$ at each $p\\in X$ satisfies \n\t \\[\\rho_k(p)=(\\frac{k}{2\\pi})^n\\left(1+\\sum_{v\\in \\Lambda\\setminus \\{0\\}}e^{-\\frac{k}{4}(|v|_{H})^2}\\cos\\big(2\\pi k\\alpha_v(p)\\big)\\right). \\]", + "theorem_type": [ + "Universal", + "Equality or Bound" + ], + "mcq": { + "question": "Let \\(X=\\mathbb C^n/\\Lambda\\) be an abelian variety, where \\(\\Lambda\\subset \\mathbb C^n\\) is a lattice. Let \\(L\\to X\\) be a positive holomorphic line bundle whose first Chern class is determined by a positive definite Hermitian form \\(H\\) on \\(\\mathbb C^n\\) with \\(\\operatorname{Im} H(\\Lambda,\\Lambda)\\subset \\mathbb Z\\). Write \\(|v|_H\\) for the norm induced by \\(H\\), and let \\(\\omega=\\pi i\\sum H_{ij}\\,dz^i\\wedge d\\bar z^j\\in 2\\pi c_1(L)\\). Choose a Hermitian metric \\(h\\) on \\(L\\) with Chern curvature \\(-i\\omega\\). For each \\(p\\in X\\) and \\(v\\in\\Lambda\\), let \\(\\gamma_{p,v}\\) be the geodesic loop based at \\(p\\) determined by \\(v\\), and define \\(\\alpha_v(p)\\in \\mathbb R/\\mathbb Z\\) by \\(e^{2\\pi i\\alpha_v(p)}=\\operatorname{Hol}_L(\\gamma_{p,v})\\). For each integer \\(k\\ge 1\\), let \\(\\rho_k(p)\\) denote the diagonal Bergman kernel of \\(H^0(X,L^k)\\) at \\(p\\). Which statement holds for every such integer \\(k\\ge 1\\) and every point \\(p\\in X\\)?", + "correct_choice": { + "label": "A", + "text": "\\[\\rho_k(p)=\\left(\\frac{k}{2\\pi}\\right)^n\\left(1+\\sum_{v\\in \\Lambda\\setminus\\{0\\}} e^{-\\frac{k}{4}|v|_H^2}\\cos\\big(2\\pi k\\alpha_v(p)\\big)\\right).\\]" + }, + "choices": [ + { + "label": "B", + "text": "\\[\\rho_k(p)=\\left(\\frac{k}{2\\pi}\\right)^n\\left(1+\\sum_{v\\in \\Lambda} e^{-\\frac{k}{4}|v|_H^2}\\cos\\big(2\\pi k\\alpha_v(p)\\big)\\right).\\]" + }, + { + "label": "C", + "text": "\\[\\rho_k(p)=\\left(\\frac{k}{2\\pi}\\right)^n\\left(1+\\sum_{v\\in \\Lambda\\setminus\\{0\\}} e^{-\\frac{k}{4}|v|_H^2}c_v(p)\\right),\\qquad |c_v(p)|\\le 1\\ \\text{for all }v\\in\\Lambda\\setminus\\{0\\}.\\]" + }, + { + "label": "D", + "text": "\\[\\rho_k(p)=\\left(\\frac{k}{2\\pi}\\right)^n\\left(1+\\sum_{v\\in \\Lambda\\setminus\\{0\\}} e^{-\\frac{k}{2}|v|_H^2}\\cos\\big(2\\pi k\\alpha_v(p)\\big)\\right).\\]" + }, + { + "label": "E", + "text": "\\[\\rho_k(p)=\\left(\\frac{k}{2\\pi}\\right)^n\\left(1+\\sum_{v\\in \\Lambda\\setminus\\{0\\}} e^{-\\frac{k}{4}|v|_H^2}\\,e^{2\\pi i k\\alpha_v(p)}\\right).\\]" + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "trace_identity", + "tampered_component": "exclusion_of_zero_lattice_vector", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "trace_identity", + "tampered_component": "explicit_identification_of_each_coefficient_as_\\cos(2\\pi k\\alpha_v(p))", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "geometric_construction", + "tampered_component": "gaussian_weight_constant_k_over_4", + "template_used": "boundary_range" + }, + { + "label": "E", + "sketch_hook_type": "trace_identity", + "tampered_component": "real_trace_pairing_of_lattice_terms_into_cosine", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives the geometric setup and definitions but does not state or strongly hint at the exact Bergman-kernel formula. The correct answer is not leaked explicitly or trivially." + }, + "TAS": { + "score": 1, + "justification": "This is essentially a theorem-identification question: under the full hypotheses, the student must recognize the precise conclusion. It is not a verbatim restatement, since the options differ in subtle but meaningful ways, but it is still close to recall of a known formula." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is required to distinguish the exact identity from plausible near-misses: excluding the zero vector, pairing terms into a cosine, and getting the Gaussian factor right. However, the task is still primarily recognition of the exact theorem statement rather than substantial derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong: one includes the zero lattice vector, one weakens the conclusion, one uses the wrong Gaussian constant, and one forgets the real pairing into cosine. These are plausible, distinct, and mathematically aligned with common failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid high-level MCQ with no answer leakage and strong distractors, but it is still fairly close to theorem recall rather than a deeply generative reasoning task." + } + }, + { + "id": "2511.16782v1", + "paper_link": "http://arxiv.org/abs/2511.16782v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "thm", + "content": "\\label{mainthm}\n Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.", + "start_pos": 35776, + "end_pos": 35919, + "label": "mainthm" + }, + "ref_dict": { + "mainthm": "\\begin{thm}\\label{mainthm}\n Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.\n\\end{thm}", + "discuss": "\\begin{thm}\\label{mainthm}\n Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.\n\\end{thm}\n\n\\par In slightly more detail, \ngiven a boundaryless surface with negative Euler characteristic\nand a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\\colon S \\to S$, one can construct the veering triangulation of the mapping torus of $f$, whose\nflow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.\n\n\\subsection{Connections to literature}\\label{discuss}\n\n\\par Work of Penner-Papadopoulos \\cite[Theorem 4.1]{PP} is sometimes cited as producing $f$-invariant generic train tracks with irreducible transition matrix for a pseudo-Anosov $f$, however it was pointed out to us by Chris Leininger that this construction seems to only produce train tracks which are $f^n$-invariant with irreducible transition matrix for some $n$ possibly greater than one.\n\\par In the setting of \\cite[Theorem 4.1]{PP},\nthe support map $\\sigma$ produced for the $f$-invariant train track $\\tau$ is not required to map switches to switches, rather the transition matrix is defined by choosing test points in the interiors of branches and counting the number of times the image of each branch hits each test point. Since switches don't have to map to switches, a branch $b$ of $\\tau$ could e.g. be mapped by $\\sigma \\circ f$ over itself and a small piece of an adjacent branch. Then the transition matrix for $f$ will record that $b$ only maps over itself. However, for some $n$, $(\\sigma \\circ f)^n(b)$ will be mapped over all of the adjacent branch, so the transition matrix for $(\\sigma \\circ f)^n$ will record that $b$ maps over itself and the adjacent branch. Hence, it's possible that $f^n(\\tau) \\prec \\tau$ has irreducible transition matrix while $f(\\tau) \\prec \\tau$ does not.\n\n\\par Returning to discussion of the present work, we note that much of the theory of train tracks assumes that the tracks considered are \\textit{\\textbf{generic}}, i.e. trivalent, since this simplification can often be made without loss of generality. In particular, the train tracks associated to veering triangulations are always generic. However, if the flow graph for a pseudo-Anosov $f$ is not irreducible, \\Cref{mainthm} necessarily produces train tracks which are not generic. \nThis motivates the following question:\n\n\\begin{qn}\n For all $f\\colon S \\to S$ pseudo-Anosov, does there exist a \\textit{generic} invariant train track for $f$ with irreducible transition matrix?\n\\end{qn}\n\n\\par \\textbf{Acknowledgements.} I would like to thank Sam Taylor for his generous guidance and feedback. I would also like to thank Chi Cheuk Tsang for helpful conversations and comments on a draft of this paper.\n\n\\section{Background}\\label{background}\n\n\\par A \\textit{\\textbf{strongly connected component}} of a directed graph $G$ is a maximal subset $C$ of the vertices of $G$ such that for any two vertices in $C$, there is a path in $G$ from the first vertex to the second.\nIf $G$ has only one strongly connected component, it is called \\textit{\\textbf{strongly connected}}.\nA nonnegative square matrix $A$ is called \\textit{\\textbf{irreducible}} if for every pair of indices $i$, $j$ there is an $n>0$ such that the $(i,j)$-th entry of $A^n$ is positive. $A$ is irreducible if and only if it is the adjacency matrix of a strongly connected graph.\n\\par\n Given an orientable finite-type surface $S$ without boundary and negative Euler characteristic, a homeomorphism $f\\colon S \\to S$ is called \\textit{\\textbf{pseudo-Anosov}} if there is a pair of transverse measured singular foliations $(\\Lambda^u,\\mu_u)$ and $(\\Lambda^s,\\mu_s)$, called the \\textit{\\textbf{unstable}} and \\textit{\\textbf{stable}} foliations, and a constant $\\lambda > 1$, called the \\textit{\\textbf{stretch factor}}, such that $f(\\Lambda^u)=\\Lambda^u$, $f(\\Lambda^s)=\\Lambda^s$ and $f(\\mu^u) = \\lambda \\mu^u$, $f(\\mu^s) = \\lambda^{-1} \\mu^s$. No nontrivial power of a pseudo-Anosov map fixes the homotopy class of any essential closed curve in $S$. A pseudo-Anosov map is a diffeomorphism except at the singular points of $\\Lambda^u$.\n\n \\par The \\textit{\\textbf{mapping torus of $f$}} is the 3-manifold\n $$ M_f = \\faktor{(S \\times [0,1])}{ (f(p),0) \\sim (p,1)} $$ which fibers over the circle with fiber $S$ and monodromy $f$.\n\n \\par If all of the singularities of the foliations are at the punctures of $S$, we say $f$ is \\textit{\\textbf{fully punctured}}. We will study general pseudo-Anosovs by deleting the singularities of the foliations to obtain a new surface $S^\\circ$ on which the restriction of $f$ is fully punctured.\nFor full background on pseudo-Anosov homeomorphisms, see Fathi--Laudenbach--Poénaru \\cite{FLP}.\n\n\\subsection{Train tracks}\\label{tts}\n\\par A \\textit{\\textbf{train track}} $\\tau$ is a closed 1-complex embedded in a surface $S$ with a ``smoothing\" at each vertex so that $\\tau$ has a well-defined tangent space at each vertex, and $S \\setminus \\tau$ contains no nullgons, unpunctured monogons or unpunctured bigons. See Penner-Harer \\cite{PH}. The vertices of $\\tau$ are called \\textit{\\textbf{switches}} and the edges are called \\textit{\\textbf{branches}}. If the switches of $\\tau$ all have degree three, $\\tau$ is called \\textit{\\textbf{generic}}.\n\n\\par Given two train tracks $\\tau_1$ and $\\tau_2$ on $S$, $\\tau_2$ is said to \\textit{\\textbf{carry}} $\\tau_1$ (or $\\tau_1$ \\textit{\\textbf{is carried by}} $\\tau_2$), denoted $\\tau_1 \\prec \\tau_2$, if there is a $C^1$ map $\\sigma\\colon S \\to S$ called the \\textit{\\textbf{support map}} \nsuch that\n \\begin{enumerate}\n \\item $\\sigma$ is homotopic to the identity,\n \\item $\\sigma(\\tau_1) \\subseteq \\tau_2$,\n \\item for all points $x$ in $\\tau_1$,\n $d_x \\sigma_{|\\tau_1}\\colon T_x \\tau_1 \\to T_{h(x)}\\tau_2$ is an isomorphism of tangent spaces,\n \\end{enumerate} \nNote that carrying is transitive: if $\\tau_1 \\prec \\tau_2$ and $\\tau_2 \\prec \\tau_3$, then\n$\\tau_1 \\prec \\tau_3$. \nWe also write $\\tau_1 \\prec_\\sigma \\tau_2$ to indicate that $\\sigma$ is the support map for $\\tau_1 \\prec \\tau_2$. A carrying map $\\sigma$ for $\\tau_1 \\prec \\tau_2$ is called \\textit{\\textbf{combinatorial}} if for every switch $v$ in $\\tau_1$, $\\sigma(v)$ is a switch in $\\tau_2$.\n \\par\n Given $\\tau_1$ with branches $\\{b^1_j\\}_{j=1\\ldots n}$ and $\\tau_2$ with branches $\\{b^2_i\\}_{i=1\\ldots m}$ such that $\\tau_1$ is carried by $\\tau_2$ with combinatorial support map $\\sigma$,\n the \\textit{\\textbf{transition matrix}} is the $m \\times n$ matrix $T = [t_{i,j}]$ where \n $t_{i,j}$ is the number of times the image of $b^1_j$ under $\\sigma$ passes over $b^2_i$.\n\\par\n A \\textit{\\textbf{train path}} is a $C^1$ immersion $p\\colon [0,1] \\to \\tau$ so that the endpoints $\\{0,1\\} = \\partial[0,1]$ are mapped to switches. We will sometimes conflate $p$ with its image in $\\tau$. For $t \\in (0,1)$ such that $p(t)$ is a switch, the sub-paths $p_0 = p_{|[0,t]}\\colon [0,t] \\to \\tau$, $p_1 = p_{|[t,1]}\\colon [t,1] \\to \\tau$ are also train paths (after reparametrizing). Similarly, if $p_0$ and $p_1$ are train paths such that $p_0(1)=p_1(0)$ and the differentials $d_1 p_0(\\partial_t)$ and $d_0 p_1(\\partial_t)$ are either both positive or both negative. The concatenation $(p_0 p_1)\\colon [0,2] \\to \\tau$ is a train path after reparametrizing, hence any train path can be written as $(p_0\\ldots p_{n-1})$, where $p_i\\colon\\left[\\frac{i}{n},\\frac{i+1}{n}\\right] \\to \\tau$ has image consisting of a single branch. If $p$ is injective we say the train path is \\textit{\\textbf{embedded}}.\n\\par\n A \\textit{\\textbf{fold}} is a particular combinatorial support map $\\phi$ for $\\tau_1 \\prec \\tau_2$ which induces a bijection between the branches of $\\tau_1$ and $\\tau_2$\n except on a particular connected subgraph where it is defined according to either of the two pictures in \\Cref{fig:fold}. The three branches of $\\tau_1$ which are mapped over the small branch $e'$ in the domain of \\Cref{fig:fold} are said to \\textit{\\textbf{fold to}} $e'$. \n If there is a $\\tau_1 \\prec_\\phi \\tau_2$\n we say $\\tau_1$ \\textit{\\textbf{folds to}} $\\tau_2$, denoted $\\tau_1 \\leftharpoonup \\tau_2$. If $\\tau_1$ folds to $\\tau_2$, $\\tau_1$ is carried by $\\tau_2$. If $\\tau_1 \\prec \\tau_2$ with support map given by a sequence of folds we also say $\\tau_1$ folds to $\\tau_2$.\n \\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.66\\textwidth]{foldgood.pdf}\n \\caption{The two possible pictures of a fold. The fold on the left is a left fold and the one on the right is a right fold. Colors indicate the image of each branch on the top after folding.}\n \\label{fig:fold}\n\\end{figure}\n \\par\n A train track $\\tau$ is an \\textit{\\textbf{invariant}} track for a map $f: S \\to S$ with support map $\\sigma$ if $f(\\tau)$ is carried by $\\tau$ with support map $\\sigma$.\n If the transition matrix for $f(\\tau) \\prec_\\sigma \\tau$ is irreducible, we also say $\\tau$ is \\textit{\\textbf{irreducible}}.\nBy smoothly isotoping $\\tau$ if necessary, we may assume it is disjoint from the singularities of $\\Lambda^u$. This will be convenient later when we remove the singularities of $\\Lambda^u$. The following theorem due to Agol is the starting point of the construction of the veering triangulation of the mapping torus of a pseudo-Anosov $f$:\n\n\\begin{thm}[\\protect{\\cite[Theorem 3.5]{A}}]\\label{ttfs} Given $f\\colon S \\to S$ pseudo-Anosov, there exists a generic invariant train track $\\tau$ for $f$ such that $f(\\tau)$ folds to $\\tau$, i.e. there exist a finite sequence of train tracks \n$$f(\\tau) = \\tau_0 \\leftharpoonup \\tau_1 \\leftharpoonup \\ldots \\leftharpoonup \\tau_n = \\tau,$$ \nwhere $\\tau_i$ is carried by $\\tau_{i+1}$ with support map given by a single fold.\n\\end{thm}" + }, + "pre_theorem_intro_text_len": 2101, + "pre_theorem_intro_text": "\\label{intro}\nGiven a pseudo-Anosov homeomorphism $f$, an invariant train track provides a combinatorial way of recording how curves and other objects behave under iteration of $f$.\nThe existence of invariant train tracks with irreducible transition matrix\nallows the application of Perron-Frobenius theory to the study of the properties of these maps. \nFor example, when an invariant train track is irreducible it can be used to build a Markov partition for the map. \nThe existence of irreducible invariant train tracks is often taken for granted \n(See e.g. \\cite{FR}, \\cite{F}, \\cite{J}, \\cite{EF}, \\cite{M}, \\cite{Pe}) \nbut to our knowledge no complete proof is currently in the literature. See \\Cref{discuss} for more details.\n\n\\par The veering triangulation associated to $f$ provides a unified way to study both the topology of its mapping torus and the dynamics of its invariant train tracks.\nOne important datum attached to a veering triangulation is its \\textit{flow graph}, which encodes a Markov partition of a pseudo-Anosov map or flow. Flow graphs have been used by Landry--Minsky--Taylor used to count periodic points of pseudo-Anosov maps, and to define polynomial invariants of pseudo-Anosov flows without perfect fits \\cite{LMT1}.\nWhen the flow graph associated to the veering triangulation of a pseudo-Anosov mapping torus is strongly connected, the triangulation can be used to obtain invariant train tracks with \nirreducible transition matrices. \n\\par For general veering triangulations, Agol--Tsang have characterized the strongly connected components of the flow graph, which consist of one ``reduced component\" from which there are paths to every vertex, and some number of ``infinitesimal cycles\" which do not have any paths back to the rest of the graph \\cite{AT}. \nBy studying the interaction between these infinitesimal components and the invariant train tracks arising from the triangulation, we are able to produce new train tracks where the obstructions to irreducibility coming from the infinitesimal components have been bypassed. This allows us to prove the following:", + "context": "\\label{intro}\nGiven a pseudo-Anosov homeomorphism $f$, an invariant train track provides a combinatorial way of recording how curves and other objects behave under iteration of $f$.\nThe existence of invariant train tracks with irreducible transition matrix\nallows the application of Perron-Frobenius theory to the study of the properties of these maps. \nFor example, when an invariant train track is irreducible it can be used to build a Markov partition for the map. \nThe existence of irreducible invariant train tracks is often taken for granted \n(See e.g. \\cite{FR}, \\cite{F}, \\cite{J}, \\cite{EF}, \\cite{M}, \\cite{Pe}) \nbut to our knowledge no complete proof is currently in the literature. See \\Cref{discuss} for more details.\n\n\\par The veering triangulation associated to $f$ provides a unified way to study both the topology of its mapping torus and the dynamics of its invariant train tracks.\nOne important datum attached to a veering triangulation is its \\textit{flow graph}, which encodes a Markov partition of a pseudo-Anosov map or flow. Flow graphs have been used by Landry--Minsky--Taylor used to count periodic points of pseudo-Anosov maps, and to define polynomial invariants of pseudo-Anosov flows without perfect fits \\cite{LMT1}.\nWhen the flow graph associated to the veering triangulation of a pseudo-Anosov mapping torus is strongly connected, the triangulation can be used to obtain invariant train tracks with \nirreducible transition matrices. \n\\par For general veering triangulations, Agol--Tsang have characterized the strongly connected components of the flow graph, which consist of one ``reduced component\" from which there are paths to every vertex, and some number of ``infinitesimal cycles\" which do not have any paths back to the rest of the graph \\cite{AT}. \nBy studying the interaction between these infinitesimal components and the invariant train tracks arising from the triangulation, we are able to produce new train tracks where the obstructions to irreducibility coming from the infinitesimal components have been bypassed. This allows us to prove the following:", + "full_context": "\\label{intro}\nGiven a pseudo-Anosov homeomorphism $f$, an invariant train track provides a combinatorial way of recording how curves and other objects behave under iteration of $f$.\nThe existence of invariant train tracks with irreducible transition matrix\nallows the application of Perron-Frobenius theory to the study of the properties of these maps. \nFor example, when an invariant train track is irreducible it can be used to build a Markov partition for the map. \nThe existence of irreducible invariant train tracks is often taken for granted \n(See e.g. \\cite{FR}, \\cite{F}, \\cite{J}, \\cite{EF}, \\cite{M}, \\cite{Pe}) \nbut to our knowledge no complete proof is currently in the literature. See \\Cref{discuss} for more details.\n\n\\par The veering triangulation associated to $f$ provides a unified way to study both the topology of its mapping torus and the dynamics of its invariant train tracks.\nOne important datum attached to a veering triangulation is its \\textit{flow graph}, which encodes a Markov partition of a pseudo-Anosov map or flow. Flow graphs have been used by Landry--Minsky--Taylor used to count periodic points of pseudo-Anosov maps, and to define polynomial invariants of pseudo-Anosov flows without perfect fits \\cite{LMT1}.\nWhen the flow graph associated to the veering triangulation of a pseudo-Anosov mapping torus is strongly connected, the triangulation can be used to obtain invariant train tracks with \nirreducible transition matrices. \n\\par For general veering triangulations, Agol--Tsang have characterized the strongly connected components of the flow graph, which consist of one ``reduced component\" from which there are paths to every vertex, and some number of ``infinitesimal cycles\" which do not have any paths back to the rest of the graph \\cite{AT}. \nBy studying the interaction between these infinitesimal components and the invariant train tracks arising from the triangulation, we are able to produce new train tracks where the obstructions to irreducibility coming from the infinitesimal components have been bypassed. This allows us to prove the following:\n\n\\par In slightly more detail, \ngiven a boundaryless surface with negative Euler characteristic\nand a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\\colon S \\to S$, one can construct the veering triangulation of the mapping torus of $f$, whose\nflow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.\n\n\\begin{qn}\n For all $f\\colon S \\to S$ pseudo-Anosov, does there exist a \\textit{generic} invariant train track for $f$ with irreducible transition matrix?\n\\end{qn}\n\n\\par Given two train tracks $\\tau_1$ and $\\tau_2$ on $S$, $\\tau_2$ is said to \\textit{\\textbf{carry}} $\\tau_1$ (or $\\tau_1$ \\textit{\\textbf{is carried by}} $\\tau_2$), denoted $\\tau_1 \\prec \\tau_2$, if there is a $C^1$ map $\\sigma\\colon S \\to S$ called the \\textit{\\textbf{support map}} \nsuch that\n \\begin{enumerate}\n \\item $\\sigma$ is homotopic to the identity,\n \\item $\\sigma(\\tau_1) \\subseteq \\tau_2$,\n \\item for all points $x$ in $\\tau_1$,\n $d_x \\sigma_{|\\tau_1}\\colon T_x \\tau_1 \\to T_{h(x)}\\tau_2$ is an isomorphism of tangent spaces,\n \\end{enumerate} \nNote that carrying is transitive: if $\\tau_1 \\prec \\tau_2$ and $\\tau_2 \\prec \\tau_3$, then\n$\\tau_1 \\prec \\tau_3$. \nWe also write $\\tau_1 \\prec_\\sigma \\tau_2$ to indicate that $\\sigma$ is the support map for $\\tau_1 \\prec \\tau_2$. A carrying map $\\sigma$ for $\\tau_1 \\prec \\tau_2$ is called \\textit{\\textbf{combinatorial}} if for every switch $v$ in $\\tau_1$, $\\sigma(v)$ is a switch in $\\tau_2$.\n \\par\n Given $\\tau_1$ with branches $\\{b^1_j\\}_{j=1\\ldots n}$ and $\\tau_2$ with branches $\\{b^2_i\\}_{i=1\\ldots m}$ such that $\\tau_1$ is carried by $\\tau_2$ with combinatorial support map $\\sigma$,\n the \\textit{\\textbf{transition matrix}} is the $m \\times n$ matrix $T = [t_{i,j}]$ where \n $t_{i,j}$ is the number of times the image of $b^1_j$ under $\\sigma$ passes over $b^2_i$.\n\\par\n A \\textit{\\textbf{train path}} is a $C^1$ immersion $p\\colon [0,1] \\to \\tau$ so that the endpoints $\\{0,1\\} = \\partial[0,1]$ are mapped to switches. We will sometimes conflate $p$ with its image in $\\tau$. For $t \\in (0,1)$ such that $p(t)$ is a switch, the sub-paths $p_0 = p_{|[0,t]}\\colon [0,t] \\to \\tau$, $p_1 = p_{|[t,1]}\\colon [t,1] \\to \\tau$ are also train paths (after reparametrizing). Similarly, if $p_0$ and $p_1$ are train paths such that $p_0(1)=p_1(0)$ and the differentials $d_1 p_0(\\partial_t)$ and $d_0 p_1(\\partial_t)$ are either both positive or both negative. The concatenation $(p_0 p_1)\\colon [0,2] \\to \\tau$ is a train path after reparametrizing, hence any train path can be written as $(p_0\\ldots p_{n-1})$, where $p_i\\colon\\left[\\frac{i}{n},\\frac{i+1}{n}\\right] \\to \\tau$ has image consisting of a single branch. If $p$ is injective we say the train path is \\textit{\\textbf{embedded}}.\n\\par\n A \\textit{\\textbf{fold}} is a particular combinatorial support map $\\phi$ for $\\tau_1 \\prec \\tau_2$ which induces a bijection between the branches of $\\tau_1$ and $\\tau_2$\n except on a particular connected subgraph where it is defined according to either of the two pictures in \\Cref{fig:fold}. The three branches of $\\tau_1$ which are mapped over the small branch $e'$ in the domain of \\Cref{fig:fold} are said to \\textit{\\textbf{fold to}} $e'$. \n If there is a $\\tau_1 \\prec_\\phi \\tau_2$\n we say $\\tau_1$ \\textit{\\textbf{folds to}} $\\tau_2$, denoted $\\tau_1 \\leftharpoonup \\tau_2$. If $\\tau_1$ folds to $\\tau_2$, $\\tau_1$ is carried by $\\tau_2$. If $\\tau_1 \\prec \\tau_2$ with support map given by a sequence of folds we also say $\\tau_1$ folds to $\\tau_2$.\n \\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.66\\textwidth]{foldgood.pdf}\n \\caption{The two possible pictures of a fold. The fold on the left is a left fold and the one on the right is a right fold. Colors indicate the image of each branch on the top after folding.}\n \\label{fig:fold}\n\\end{figure}\n \\par\n A train track $\\tau$ is an \\textit{\\textbf{invariant}} track for a map $f: S \\to S$ with support map $\\sigma$ if $f(\\tau)$ is carried by $\\tau$ with support map $\\sigma$.\n If the transition matrix for $f(\\tau) \\prec_\\sigma \\tau$ is irreducible, we also say $\\tau$ is \\textit{\\textbf{irreducible}}.\nBy smoothly isotoping $\\tau$ if necessary, we may assume it is disjoint from the singularities of $\\Lambda^u$. This will be convenient later when we remove the singularities of $\\Lambda^u$. The following theorem due to Agol is the starting point of the construction of the veering triangulation of the mapping torus of a pseudo-Anosov $f$:\n\n\\par Note that the referenced theorem is stated in terms of \\textit{splits} rather than folds, which are the combinatorial inverse of folds. \n Additionally, the referenced theorem is stronger than what we state here because the splitting sequence is shown to be canonical up to ``commuting maximal splits\": at each step in the sequence, the train track is split at a branch carrying maximal measure of the invariant lamination, and the splits of maximal branches commute if there is not a unique branch carrying maximal measure.\n This theorem motivates the following definition:\n\\begin{defn}\n An invariant train track $\\tau$ for a pseudo-Anosov map $f$ is called \\textit{\\textbf{veering}} if it \n is generic and $\\tau$ can be obtained from $f(\\tau)$ by a sequence of folds, in which case there is a sequence of train tracks \n $$f(\\tau) = \\tau_0 \\leftharpoonup \\tau_1 \\leftharpoonup \\ldots \\leftharpoonup \\tau_n = \\tau$$ \n where $\\tau_i$ is carried by $\\tau_{i+1}$ with support map consisting of a single fold.\n\\end{defn}\n\\par In \\Cref{clvts} we will see that a veering train track as defined here is sufficient to produce the veering triangulation of the fully punctured mapping torus of $f$. While veering train tracks are not necessarily irreducible, they will be the beginning of our construction of irreducible tracks.\n\\par\n Given two train tracks $\\tau_1$, $\\tau_2$ on $S$, a (surface) \\textit{\\textbf{train track map}} is a map $t: S \\to S$ so that $t(\\tau_1)$ is contained in $\\tau_2$, and the restriction of $t$ to $\\tau_1$ is \n such that for any train path $p\\colon [0,1] \\to \\tau_1$, $t \\circ p\\colon [0,1] \\to \\tau_2$ is also a train path. In other words, a train track map is like a combinatorial support map except that it does not have to be homotopic to the identity map. \n \\par The following lemma will be useful for producing support maps:\n\\begin{lemma}\\label{supplemma}\n Let $t\\colon S \\to S$ be a train track map taking $\\tau_1$ to $\\tau_2$, and $g\\colon S \\to S$ be a diffeomorphism.\n If $t$ is homotopic to $g$, then $g(\\tau_1) \\prec \\tau_2$ with support map given by $t \\circ g^{-1}$.\n\\end{lemma}\n\\begin{proof}\n Since $t$ and $g$ are homotopic, $t \\circ g^{-1}$ is homotopic to the identity. Since $t(\\tau_1) \\subseteq \\tau_2$, $(t \\circ g^{-1})(g(\\tau_1)) = t(\\tau_1) \\subseteq \\tau_2$. Let $p \\in g(\\tau_1)$. Since $g$ is a diffeomorphism, the restriction of the differential \n $$d_p g^{-1}_{|g(\\tau_1)}\\colon T_p g(\\tau_1) \\to T_{g^{-1}(p)} \\tau_1$$\n is an isomorphism. Since $t$ is a train track map, $d_{g^{-1}(p)} t_{| \\tau_1}$ is an isomorphism, so the composition\n $$d_p (t \\circ g^{-1})_{|g(\\tau_1)} = (d_{g^{-1}(p)} t_{| \\tau_1}) \\circ (d_p g^{-1}_{|g(\\tau_1)})\\colon T_p g(\\tau_1) \\to T_{(t \\circ g^{-1})(p)} \\tau_2$$\n is also an isomorphism.\n\\end{proof}", + "post_theorem_intro_text_len": 3015, + "post_theorem_intro_text": "\\par In slightly more detail, \ngiven a boundaryless surface with negative Euler characteristic\nand a train track folding sequence (equivalently, splitting sequence) for a pseudo-Anosov $f\\colon S \\to S$, one can construct the veering triangulation of the mapping torus of $f$, whose\nflow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, we show that the only obstructions to irreducibility are branches which are dual to infinitesimal cycles of the flow graph and characterize the structure of veering tracks near these branches. This local picture allows us to modify the track by contracting the obstructing branches, and recover a support map for the modified train track with irreducible transition matrix.\n\n\\subsection{Connections to literature}\\label{discuss}\n\n\\par Work of Penner-Papadopoulos \\cite[Theorem 4.1]{PP} is sometimes cited as producing $f$-invariant generic train tracks with irreducible transition matrix for a pseudo-Anosov $f$, however it was pointed out to us by Chris Leininger that this construction seems to only produce train tracks which are $f^n$-invariant with irreducible transition matrix for some $n$ possibly greater than one.\n\\par In the setting of \\cite[Theorem 4.1]{PP},\nthe support map $\\sigma$ produced for the $f$-invariant train track $\\tau$ is not required to map switches to switches, rather the transition matrix is defined by choosing test points in the interiors of branches and counting the number of times the image of each branch hits each test point. Since switches don't have to map to switches, a branch $b$ of $\\tau$ could e.g. be mapped by $\\sigma \\circ f$ over itself and a small piece of an adjacent branch. Then the transition matrix for $f$ will record that $b$ only maps over itself. However, for some $n$, $(\\sigma \\circ f)^n(b)$ will be mapped over all of the adjacent branch, so the transition matrix for $(\\sigma \\circ f)^n$ will record that $b$ maps over itself and the adjacent branch. Hence, it's possible that $f^n(\\tau) \\prec \\tau$ has irreducible transition matrix while $f(\\tau) \\prec \\tau$ does not.\n\n\\par Returning to discussion of the present work, we note that much of the theory of train tracks assumes that the tracks considered are \\textit{\\textbf{generic}}, i.e. trivalent, since this simplification can often be made without loss of generality. In particular, the train tracks associated to veering triangulations are always generic. However, if the flow graph for a pseudo-Anosov $f$ is not irreducible, \\Cref{mainthm} necessarily produces train tracks which are not generic. \nThis motivates the following question:\n\n\\begin{qn}\n For all $f\\colon S \\to S$ pseudo-Anosov, does there exist a \\textit{generic} invariant train track for $f$ with irreducible transition matrix?\n\\end{qn}\n\n\\par \\textbf{Acknowledgements.} I would like to thank Sam Taylor for his generous guidance and feedback. I would also like to thank Chi Cheuk Tsang for helpful conversations and comments on a draft of this paper.", + "sketch": "Given a boundaryless surface with negative Euler characteristic and a train track folding (splitting) sequence for a pseudo-Anosov $f\\colon S\\to S$, one constructs the veering triangulation of the mapping torus of $f$, whose flow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, one shows that the only obstructions to irreducibility are branches dual to infinitesimal cycles of the flow graph, and characterizes the local structure of veering tracks near these branches. Using this local picture, one modifies the track by contracting the obstructing branches, and recovers a support map for the modified train track with irreducible transition matrix.", + "expanded_sketch": "Given a boundaryless surface with negative Euler characteristic and a train track folding (splitting) sequence for a pseudo-Anosov $f\\colon S\\to S$, one constructs the veering triangulation of the mapping torus of $f$, whose flow graph encodes the transition matrix for this train track. Using the work of \\cite{AT}, one shows that the only obstructions to irreducibility are branches dual to infinitesimal cycles of the flow graph, and characterizes the local structure of veering tracks near these branches. Using this local picture, one modifies the track by contracting the obstructing branches, and recovers a support map for the modified train track with irreducible transition matrix.", + "expanded_theorem": "\\label{mainthm}\n Every pseudo-Anosov homeomorphism has an invariant train track whose transition matrix is irreducible.,", + "theorem_type": [ + "Universal", + "Existence" + ], + "mcq": { + "question": "Let $f\\colon S\\to S$ be a pseudo-Anosov homeomorphism of a surface. A train track $\\tau$ is called invariant for $f$ if $f(\\tau)$ is carried by $\\tau$. If $f(\\tau)$ has branches $\\{b_j^1\\}$, $\\tau$ has branches $\\{b_i^2\\}$, and the carrying is given by a combinatorial support map, the associated transition matrix is the matrix $T=[t_{i,j}]$ where $t_{i,j}$ is the number of times the image of $b_j^1$ passes over $b_i^2$. A square nonnegative matrix is irreducible if for every pair of indices $i,j$, some power of the matrix has positive $(i,j)$-entry. Which statement holds for every pseudo-Anosov homeomorphism $f$?", + "correct_choice": { + "label": "A", + "text": "There exists an invariant train track $\\tau$ for $f$ such that the associated transition matrix is irreducible." + }, + "choices": [ + { + "label": "B", + "text": "There exists a generic invariant train track $\\tau$ for $f$ such that the associated transition matrix is irreducible." + }, + { + "label": "C", + "text": "There exists an invariant train track $\\tau$ for $f$ such that the associated transition matrix is nonzero." + }, + { + "label": "D", + "text": "For every invariant train track $\\tau$ for $f$, the associated transition matrix is irreducible." + }, + { + "label": "E", + "text": "There exists an invariant train track $\\tau$ for $f$ such that, after replacing $f$ by some iterate $f^n$ with $n\\ge 1$, the associated transition matrix is irreducible for $f^n$." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "B" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "extra genericity conclusion", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "finiteness", + "tampered_component": "dropped irreducibility requirement", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "existential construction versus universal claim", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "need for irreducibility for $f$ itself, not merely an iterate", + "template_used": "quantifier_dependence" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the relevant concepts but does not explicitly state or trivially reveal the correct theorem. The correct option is not leaked by wording in the stem; the student still has to identify which quantified statement is valid." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recognition question: the correct choice is essentially the standard existence statement with the key property 'irreducible.' However, the alternatives introduce meaningful variations in quantifiers and hypotheses, so it is not a pure verbatim restatement." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is required to distinguish existential from universal claims and exact versus weakened/modified conclusions. Still, the item mainly tests recall/recognition of a known theorem rather than substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong: one is a weaker true statement, others are plausible strengthenings or quantifier distortions ('generic,' 'for every,' 'after an iterate'). These reflect common mathematical failure modes and are clearly distinct." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no real answer leakage, but it leans more toward theorem recall than deep generative reasoning." + } + }, + { + "id": "2511.15135v1", + "paper_link": "http://arxiv.org/abs/2511.15135v1", + "theorems_cnt": 4, + "theorem": { + "env_name": "theorem", + "content": "\\label{main-2} Let $R$ be a finite family of slopes containing $0,1,\\infty$ of some cardinality $k+3$, and let $s$ be a slope not lying in $R$.\n\\begin{itemize}\n \\item[(i)] (Three slopes) If $k=0$, then\n\\begin{equation}\\label{2ab} 2 - \\frac{c_2}{D(R;s)} \\leq \\SD\\left(R; s\\right) \\leq 2 - \\frac{c_1}{D(R;s)} \n\\end{equation}\n for some absolute constants $c_2 > c_1 > 0$.\n \\item[(ii)] (Many slopes) In general, one has\n \\begin{equation}\\label{sam} 2 - \\frac{C_k \\log(2+D(R;s))}{D(R;s)} \\leq \\SD(R; s) \\leq 2 - \\frac{c_k}{D(R;s)^{k+1}} \n \\end{equation}\n for some absolute constants $c_k, C_k > 0$.\n\\end{itemize}", + "start_pos": 73233, + "end_pos": 73889, + "label": "main-2" + }, + "ref_dict": { + "main-sec": "\\label{main-sec}\n\nWe now prove Theorem \\ref{main-2}. We begin with part (i). Write $s=a/b$, where $b \\geq 1$ and $a \\neq 0,b$ is coprime to $b$, then $D(R;s) \\asymp \\log(2+|a|+|b|)$. Let $N$ be the", + "rat": "\\begin{equation}\\label{rat}\n s = \\frac{P(r_1,\\dots,r_k)}{Q(r_1,\\dots,r_k)}\n \\end{equation}", + "sam": "\\begin{equation}\\label{sam} 2 - \\frac{C_k \\log(2+D(R;s))}{D(R;s)} \\leq \\SD(R; s) \\leq 2 - \\frac{c_k}{D(R;s)^{k+1}} \n \\end{equation}", + "2ab": "\\begin{equation}\\label{2ab} 2 - \\frac{c_2}{D(R;s)} \\leq \\SD\\left(R; s\\right) \\leq 2 - \\frac{c_1}{D(R;s)} \n\\end{equation}", + "infr": "\\begin{equation}\\label{infr}\n\\inf_R \\SD(R;-1) = 1\n\\end{equation}", + "up": "\\begin{equation}\\label{up}\n \\SD\\left(\\{0,1,\\infty\\};\\frac{a}{b}\\right) \\geq 2 - \\frac{c_\\alpha+o(1)}{\\log b}\n \\end{equation}", + "fig:kak": "\\begin{figure}\n \\centering\n\\centerline{\\includegraphics[width=\\linewidth]{best.png}}\n \\caption{Lower bounds on $\\SD(\\{0,1,\\infty\\}; s)$ obtained by \\texttt{AlphaEvolve} for various $s=a/b$, plotted against $2 - 0.16 / \\log(|a|+b|)$. The horizontal axis is $|a|+|b|$ (so in some cases, multiple fractions $a/b$ are plotted on a single vertical line).}\n \\label{fig:kak}\n\\end{figure}", + "cts-sec": "\\begin{equation}\\label{dxy}\nd[X;Y], d[Y;Y] \\ll \\log K.\n\\end{equation}\n\nFor $a \\in \\Q$, define $g(a) \\coloneqq \\H[Y-aY'] - \\H[Y]$, where $Y'$ is an independent copy of $Y$. From \\eqref{hx}, \\eqref{ruzsa}, \\eqref{ruzsa-diff} we have $g(r) \\ll \\log K$ for $r \\in R \\backslash \\{\\infty\\}$. By Proposition \\ref{dilate}(iv), we conclude that $g(s) \\ll D^{k+1} \\log K$, thus $d[Y; sY] \\ll D^{k+1} \\log K$. Combining with \\eqref{dxy} and \\eqref{ruzsa}, \\eqref{ruzsa-diff}, we obtain \\eqref{dxsy} as required.\n\n\\section{A continuous limit}\\label{cts-sec}\n\nIn this section we establish \\Cref{cts-limit}. Let $\\alpha$, $a$, $b$, $f$, $f_0, f_1, f_\\infty$ be as in that theorem. We allow implied constants in the asymptotic notation to depend on $\\alpha, f$. We will establish the lower bound\n\\begin{equation}\\label{sd-targ}\n \\SD\\left(\\{0,1,\\infty\\},\\frac{a}{b}\\right) \\geq 2 - \\frac{h_2(f_\\alpha) - \\max(h(f_0), h(f_\\infty), h(f_1))+o(1)}{\\log b}.\n \\end{equation}", + "hf": "\\begin{equation}\\label{hf}\n h_2(f_{(\\alpha)}) - \\max(h(f_0), h(f_\\infty), h(f_\\alpha))\n \\end{equation}", + "sdf-upper": "\\begin{equation}\\label{sdf-upper}\n\\SD(R;s) \\leq 2\n\\end{equation}" + }, + "pre_theorem_intro_text_len": 7663, + "pre_theorem_intro_text": "\\subsection{The arithmetic Kakeya conjecture}\n\nDefine a \\emph{slope} to be an element $r$ of the projective rational\\footnote{One could also formulate the arithmetic Kakeya conjecture in other fields than the rationals, and in fact most of the results here apply to arbitrary infinite fields, with minor modifications in the case where the field has positive characteristic. But in this paper we shall restrict attention for simplicity to the rational case, which is the case of interest for applications to the Kakeya problem.} line $\\Q \\cup \\{\\infty\\}$. We then define the projection operators $\\pi_r \\colon \\Q \\times \\Q \\to \\Q$ by setting\n$$ \\pi_r(x,y) \\coloneqq x + ry$$\nfor $r \\neq \\{\\infty\\}$, and\n$$ \\pi_\\infty(x,y) \\coloneqq y.$$\nGiven a finite set $R$ of slopes and a further slope $s$ not in $R$, we define the \\emph{sum-difference constant} $\\SD(R;s)$ to be the least exponent such that the bound\n\\begin{equation}\\label{sdef}\n\\H[ \\pi_s(X,Y) ] \\leq \\SD(R;s) \\max_{r \\in R} \\H[\\pi_r(X,Y)]\n\\end{equation}\nholds for $\\Q$-valued random variables\\footnote{In this paper, all random variables are understood to be discrete, and in fact take only finitely many values.} $X,Y$ (not necessarily independent), where\n$$ \\H[X] \\coloneqq \\sum_{x} \\h(\\P(X=x))$$\nis the Shannon entropy of a random variable $X$, with $\\h(t) \\coloneqq t \\log \\frac{1}{t}$ using the convention $\\h(0) \\coloneqq 0$. Thus for instance\n$$\\H[ X-Y ] \\leq \\SD(\\{0,1,\\infty\\};-1) \\max\\left( \\H[X], \\H[X+Y], \\H[Y] \\right)$$\nwhenever $X,Y$ are (possibly dependent) $\\Q$-valued random variables.\n\nThe quantity $\\SD(R;s)$ can equivalently be defined as the least exponent such that the bound\n\\begin{equation}\\label{pise}\n|\\pi_{s}(E)| \\leq (\\max_{r \\in R} |\\pi_s(E)|)^{\\SD(R;s)}\n\\end{equation}\nfor all finite non-empty $E \\subset \\Q$, where $|A|$ denotes the cardinality of a finite set $A$; see \\cite{green}. However, it will be convenient in this paper to work with the entropy formulation, in order to take advantage of the ``entropic Pl\\\"unnecke--Ruzsa calculus'' that are founded on the Shannon entropy inequalities.\n\nIt is easy to see that one has the projective invariance\n$$ \\SD(\\phi(R); \\phi(s)) = \\SD(R; s)$$\nfor any projective transformation $\\phi \\colon \\Q \\cup \\{\\infty\\} \\to \\Q \\cup \\{\\infty\\}$, that is to say a map of the form $\\phi(r) \\coloneqq \\frac{ar+b}{cr+d}$ for some $a,b,c,d \\in \\Q$ with $ad-bc \\neq 0$, with the usual conventions when $r$ is infinite or $cr+d$ vanishes. For instance, by using a dilation transformation, we have\n$$ \\SD(\\{0,1,\\infty\\}; s) = \\SD(\\{0,-1/s,\\infty\\}; -1)$$\nfor any slope $s$ other than $0,1,\\infty$.\nIn the literature it is conventional to use this $3$-transitive projective symmetry to normalize $s = -1$ and $0, \\infty \\in R$ (assuming that $|R|\\geq 2$ of course), though in this paper it will be more convenient to adopt the normalization $0,1,\\infty \\in R$ (assuming $|R| \\geq 3$).\n\nIt is easy to see that $\\SD(R;s)=\\infty$ when $|R|<2$.\nFrom the entropy inequality $\\H[X-Y] \\leq \\H[X,Y] \\leq \\H[X]+\\H[Y]$ we see that $\\SD(R;-1) \\leq 2$ when $0,\\infty \\in S$, and it is easy to see (using the uniform distribution on a long arithmetic progression) that we have equality when $R = \\{0,\\infty\\}$. By projective invariance, this implies that $\\SD(R;s)=2$ whenever $|R|=2$. As these quantities are clearly non-decreasing in $R$, we then have the trivial bound\n\\begin{equation}\\label{sdf-upper}\n\\SD(R;s) \\leq 2\n\\end{equation}\nfor $|R| \\geq 2$. \n\nImprovements upon \\eqref{sdf-upper} directly lead to improved upper bounds on the dimension of Kakeya and Nikodym sets in high dimensions. Indeed, in \\cite{bourgain} it was observed that a bound of the form $\\SD(R;-1) \\leq \\alpha$ implies that Kakeya and Nikodym sets in dimension $d$ have (upper) Minkowski dimension\\footnote{It was recently observed by Thomas Bloom (private communication) that, by combining Bourgain's arguments with the recent quantitative progress on Szemer\\'edi's theorem by Leng, Sawhney, and Sah \\cite{lss}, that one also obtains this bound for the Hausdorff dimension as well. For the corresponding results for packing dimension, see \\cite{cowen}.} at most $\\frac{d-1}{\\alpha}+1$. In particular, if one can establish the \\emph{arithmetic Kakeya conjecture}\n\\begin{equation}\\label{infr}\n\\inf_R \\SD(R;-1) = 1\n\\end{equation}\nthen this would imply that Kakeya and Nikodym sets in $\\R^d$ have full Minkowski and Hausdorff dimension for all $d$. This is currently only known for $d \\leq 3$ \\cite{wang}. We refer the reader to \\cite{green}, \\cite{cowen}, \\cite{pohoata} for a discussion of this conjecture (and other equivalent forms of it), and its connection with other variants of the Kakeya conjecture. \n\nNontrivial progress towards the arithmetic Kakeya conjecture was first obtained in \\cite{bourgain}, who in our notation showed that \n$$ \\SD(\\{0,1,\\infty\\};-1) \\leq 2 - \\frac{1}{13} = 1.923\\dots,$$\nand used this to obtain new bounds on the Kakeya conjecture in high dimensions.\nFurther improvements were then obtained in \\cite{katz-tao}, \\cite{katz-tao-new}. For instance, it is known that\n$$ 1.77898 \\leq \\SD(\\{0,1,\\infty\\};-1) \\leq 2 - \\frac{1}{6} = 1.833\\dots$$\nand\n$$ 1.668 \\leq \\SD(\\{0,1,2,\\infty\\};-1) \\leq 2 - \\frac{1}{4} = 1.75,$$\nwith the upper bounds established in \\cite{katz-tao}, and the lower bounds in \\cite{lemm}, \\cite{gdm} respectively.\nAt present, the best upper bound known towards \\eqref{infr} is\n$$\\inf_R \\SD(R;-1) \\leq 1.67513\\dots;$$\nsee \\cite{katz-tao-new}. \n\n\\subsection{Asymptotic behavior}\n\nInformally, the arithmetic Kakeya conjecture asserts that in the asymptotic regime where the number of slopes $R$ is large, the constants $\\SD(R;s)$ converge to $1$. Here we consider a complementary regime, in which the number of slopes $R$ is fixed, but we instead let the elements of $R$ (or $s$) vary. Our main results assert, roughly speaking, that the behavior of these constants is determined by the \\emph{rational complexity} of $s$ relative to $R$, with the constants approaching $2$ as it becomes harder to express $s$ in terms of a rational expression of the $R$. We give a (slightly artificial) definition of this quantity, restricting attention to the normalized setting $\\{0,1,\\infty\\} \\subset R$ for simplicity.\n\n\\begin{definition}[Rational complexity] Given a family of slopes $R = \\{0,1,\\infty,r_1,\\dots,r_k\\}$ containing $0,1,\\infty$ and a further slope $s$ not in $R$, we define the \\emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form\n\\begin{equation}\\label{rat}\n s = \\frac{P(r_1,\\dots,r_k)}{Q(r_1,\\dots,r_k)}\n \\end{equation}\nwhere $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.\n\\end{definition}\n\nInformally, if the complexity of $s$ with respect to $R$ is equal to $D$, then $s$ can be expressed in terms of the slopes in $R$ by a rational expression whose length (when expressed as string of characters) is comparable to $D$. The rational complexity is reminiscent of the \\emph{arithmetic circuit complexity} of $s$ in terms of $R$, but with the key difference that the circuit must take the specific rational form \\eqref{rat}.\n\n\\begin{example} $D(\\{0,1,\\infty\\};s)$ is the least natural number for which one can express $s$ as $a/b$ where $a,b$ are integers of magnitude at most $2^D$. In particular, if $a,b$ are coprime then\n$$D \\left( \\{0,1,\\infty\\};\\frac{a}{b} \\right) \\asymp \\log(2+|a|+|b|).$$\n\\end{example}\n\nOur main result, proven in Section \\ref{main-sec}, is then as follows.", + "context": "Define a \\emph{slope} to be an element $r$ of the projective rational\\footnote{One could also formulate the arithmetic Kakeya conjecture in other fields than the rationals, and in fact most of the results here apply to arbitrary infinite fields, with minor modifications in the case where the field has positive characteristic. But in this paper we shall restrict attention for simplicity to the rational case, which is the case of interest for applications to the Kakeya problem.} line $\\Q \\cup \\{\\infty\\}$. We then define the projection operators $\\pi_r \\colon \\Q \\times \\Q \\to \\Q$ by setting\n$$ \\pi_r(x,y) \\coloneqq x + ry$$\nfor $r \\neq \\{\\infty\\}$, and\n$$ \\pi_\\infty(x,y) \\coloneqq y.$$\nGiven a finite set $R$ of slopes and a further slope $s$ not in $R$, we define the \\emph{sum-difference constant} $\\SD(R;s)$ to be the least exponent such that the bound\n\\begin{equation}\\label{sdef}\n\\H[ \\pi_s(X,Y) ] \\leq \\SD(R;s) \\max_{r \\in R} \\H[\\pi_r(X,Y)]\n\\end{equation}\nholds for $\\Q$-valued random variables\\footnote{In this paper, all random variables are understood to be discrete, and in fact take only finitely many values.} $X,Y$ (not necessarily independent), where\n$$ \\H[X] \\coloneqq \\sum_{x} \\h(\\P(X=x))$$\nis the Shannon entropy of a random variable $X$, with $\\h(t) \\coloneqq t \\log \\frac{1}{t}$ using the convention $\\h(0) \\coloneqq 0$. Thus for instance\n$$\\H[ X-Y ] \\leq \\SD(\\{0,1,\\infty\\};-1) \\max\\left( \\H[X], \\H[X+Y], \\H[Y] \\right)$$\nwhenever $X,Y$ are (possibly dependent) $\\Q$-valued random variables.\n\nIt is easy to see that $\\SD(R;s)=\\infty$ when $|R|<2$.\nFrom the entropy inequality $\\H[X-Y] \\leq \\H[X,Y] \\leq \\H[X]+\\H[Y]$ we see that $\\SD(R;-1) \\leq 2$ when $0,\\infty \\in S$, and it is easy to see (using the uniform distribution on a long arithmetic progression) that we have equality when $R = \\{0,\\infty\\}$. By projective invariance, this implies that $\\SD(R;s)=2$ whenever $|R|=2$. As these quantities are clearly non-decreasing in $R$, we then have the trivial bound\n\\begin{equation}\\label{sdf-upper}\n\\SD(R;s) \\leq 2\n\\end{equation}\nfor $|R| \\geq 2$.\n\nImprovements upon \\eqref{sdf-upper} directly lead to improved upper bounds on the dimension of Kakeya and Nikodym sets in high dimensions. Indeed, in \\cite{bourgain} it was observed that a bound of the form $\\SD(R;-1) \\leq \\alpha$ implies that Kakeya and Nikodym sets in dimension $d$ have (upper) Minkowski dimension\\footnote{It was recently observed by Thomas Bloom (private communication) that, by combining Bourgain's arguments with the recent quantitative progress on Szemer\\'edi's theorem by Leng, Sawhney, and Sah \\cite{lss}, that one also obtains this bound for the Hausdorff dimension as well. For the corresponding results for packing dimension, see \\cite{cowen}.} at most $\\frac{d-1}{\\alpha}+1$. In particular, if one can establish the \\emph{arithmetic Kakeya conjecture}\n\\begin{equation}\\label{infr}\n\\inf_R \\SD(R;-1) = 1\n\\end{equation}\nthen this would imply that Kakeya and Nikodym sets in $\\R^d$ have full Minkowski and Hausdorff dimension for all $d$. This is currently only known for $d \\leq 3$ \\cite{wang}. We refer the reader to \\cite{green}, \\cite{cowen}, \\cite{pohoata} for a discussion of this conjecture (and other equivalent forms of it), and its connection with other variants of the Kakeya conjecture.\n\n\\begin{definition}[Rational complexity] Given a family of slopes $R = \\{0,1,\\infty,r_1,\\dots,r_k\\}$ containing $0,1,\\infty$ and a further slope $s$ not in $R$, we define the \\emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form\n\\begin{equation}\\label{rat}\n s = \\frac{P(r_1,\\dots,r_k)}{Q(r_1,\\dots,r_k)}\n \\end{equation}\nwhere $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.\n\\end{definition}\n\n\\begin{example} $D(\\{0,1,\\infty\\};s)$ is the least natural number for which one can express $s$ as $a/b$ where $a,b$ are integers of magnitude at most $2^D$. In particular, if $a,b$ are coprime then\n$$D \\left( \\{0,1,\\infty\\};\\frac{a}{b} \\right) \\asymp \\log(2+|a|+|b|).$$\n\\end{example}\n\nOur main result, proven in Section \\ref{main-sec}, is then as follows.\n\n\\label{main-sec}\n\nWe now prove Theorem \\ref{main-2}. We begin with part (i). Write $s=a/b$, where $b \\geq 1$ and $a \\neq 0,b$ is coprime to $b$, then $D(R;s) \\asymp \\log(2+|a|+|b|)$. Let $N$ be the", + "full_context": "Define a \\emph{slope} to be an element $r$ of the projective rational\\footnote{One could also formulate the arithmetic Kakeya conjecture in other fields than the rationals, and in fact most of the results here apply to arbitrary infinite fields, with minor modifications in the case where the field has positive characteristic. But in this paper we shall restrict attention for simplicity to the rational case, which is the case of interest for applications to the Kakeya problem.} line $\\Q \\cup \\{\\infty\\}$. We then define the projection operators $\\pi_r \\colon \\Q \\times \\Q \\to \\Q$ by setting\n$$ \\pi_r(x,y) \\coloneqq x + ry$$\nfor $r \\neq \\{\\infty\\}$, and\n$$ \\pi_\\infty(x,y) \\coloneqq y.$$\nGiven a finite set $R$ of slopes and a further slope $s$ not in $R$, we define the \\emph{sum-difference constant} $\\SD(R;s)$ to be the least exponent such that the bound\n\\begin{equation}\\label{sdef}\n\\H[ \\pi_s(X,Y) ] \\leq \\SD(R;s) \\max_{r \\in R} \\H[\\pi_r(X,Y)]\n\\end{equation}\nholds for $\\Q$-valued random variables\\footnote{In this paper, all random variables are understood to be discrete, and in fact take only finitely many values.} $X,Y$ (not necessarily independent), where\n$$ \\H[X] \\coloneqq \\sum_{x} \\h(\\P(X=x))$$\nis the Shannon entropy of a random variable $X$, with $\\h(t) \\coloneqq t \\log \\frac{1}{t}$ using the convention $\\h(0) \\coloneqq 0$. Thus for instance\n$$\\H[ X-Y ] \\leq \\SD(\\{0,1,\\infty\\};-1) \\max\\left( \\H[X], \\H[X+Y], \\H[Y] \\right)$$\nwhenever $X,Y$ are (possibly dependent) $\\Q$-valued random variables.\n\nIt is easy to see that $\\SD(R;s)=\\infty$ when $|R|<2$.\nFrom the entropy inequality $\\H[X-Y] \\leq \\H[X,Y] \\leq \\H[X]+\\H[Y]$ we see that $\\SD(R;-1) \\leq 2$ when $0,\\infty \\in S$, and it is easy to see (using the uniform distribution on a long arithmetic progression) that we have equality when $R = \\{0,\\infty\\}$. By projective invariance, this implies that $\\SD(R;s)=2$ whenever $|R|=2$. As these quantities are clearly non-decreasing in $R$, we then have the trivial bound\n\\begin{equation}\\label{sdf-upper}\n\\SD(R;s) \\leq 2\n\\end{equation}\nfor $|R| \\geq 2$.\n\nImprovements upon \\eqref{sdf-upper} directly lead to improved upper bounds on the dimension of Kakeya and Nikodym sets in high dimensions. Indeed, in \\cite{bourgain} it was observed that a bound of the form $\\SD(R;-1) \\leq \\alpha$ implies that Kakeya and Nikodym sets in dimension $d$ have (upper) Minkowski dimension\\footnote{It was recently observed by Thomas Bloom (private communication) that, by combining Bourgain's arguments with the recent quantitative progress on Szemer\\'edi's theorem by Leng, Sawhney, and Sah \\cite{lss}, that one also obtains this bound for the Hausdorff dimension as well. For the corresponding results for packing dimension, see \\cite{cowen}.} at most $\\frac{d-1}{\\alpha}+1$. In particular, if one can establish the \\emph{arithmetic Kakeya conjecture}\n\\begin{equation}\\label{infr}\n\\inf_R \\SD(R;-1) = 1\n\\end{equation}\nthen this would imply that Kakeya and Nikodym sets in $\\R^d$ have full Minkowski and Hausdorff dimension for all $d$. This is currently only known for $d \\leq 3$ \\cite{wang}. We refer the reader to \\cite{green}, \\cite{cowen}, \\cite{pohoata} for a discussion of this conjecture (and other equivalent forms of it), and its connection with other variants of the Kakeya conjecture.\n\n\\begin{definition}[Rational complexity] Given a family of slopes $R = \\{0,1,\\infty,r_1,\\dots,r_k\\}$ containing $0,1,\\infty$ and a further slope $s$ not in $R$, we define the \\emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form\n\\begin{equation}\\label{rat}\n s = \\frac{P(r_1,\\dots,r_k)}{Q(r_1,\\dots,r_k)}\n \\end{equation}\nwhere $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.\n\\end{definition}\n\n\\begin{example} $D(\\{0,1,\\infty\\};s)$ is the least natural number for which one can express $s$ as $a/b$ where $a,b$ are integers of magnitude at most $2^D$. In particular, if $a,b$ are coprime then\n$$D \\left( \\{0,1,\\infty\\};\\frac{a}{b} \\right) \\asymp \\log(2+|a|+|b|).$$\n\\end{example}\n\nOur main result, proven in Section \\ref{main-sec}, is then as follows.\n\n\\label{main-sec}\n\nWe now prove Theorem \\ref{main-2}. We begin with part (i). Write $s=a/b$, where $b \\geq 1$ and $a \\neq 0,b$ is coprime to $b$, then $D(R;s) \\asymp \\log(2+|a|+|b|)$. Let $N$ be the\n\n\\begin{definition}[Rational complexity] Given a family of slopes $R = \\{0,1,\\infty,r_1,\\dots,r_k\\}$ containing $0,1,\\infty$ and a further slope $s$ not in $R$, we define the \\emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form\n\\begin{equation}\\label{rat}\n s = \\frac{P(r_1,\\dots,r_k)}{Q(r_1,\\dots,r_k)}\n \\end{equation}\nwhere $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.\n\\end{definition}\n\nOur main result, proven in Section \\ref{main-sec}, is then as follows.\n\n\\begin{figure}\n \\centering\n\\centerline{\\includegraphics[width=\\linewidth]{best.png}}\n \\caption{Lower bounds on $\\SD(\\{0,1,\\infty\\}; s)$ obtained by \\texttt{AlphaEvolve} for various $s=a/b$, plotted against $2 - 0.16 / \\log(|a|+b|)$. The horizontal axis is $|a|+|b|$ (so in some cases, multiple fractions $a/b$ are plotted on a single vertical line).}\n \\label{fig:kak}\n\\end{figure}\n\n\\begin{theorem}[Continuous limit]\\label{cts-limit} Let $\\alpha$ be a real number, and let $a,b$ be coprime integer parameters with $b \\to \\infty$ and $a/b \\to \\alpha$. Then\n\\begin{equation}\\label{up}\n \\SD\\left(\\{0,1,\\infty\\};\\frac{a}{b}\\right) \\geq 2 - \\frac{c_\\alpha+o(1)}{\\log b}\n \\end{equation}\n as $b\\to \\infty$ and $a/b \\to \\alpha$, where $0 < c_\\alpha < \\infty$ is the infimum of all quantities\n\\begin{equation}\\label{hf}\n h_2(f_{(\\alpha)}) - \\max(h(f_0), h(f_\\infty), h(f_\\alpha))\n \\end{equation}\nwhere $f \\colon \\R^2 \\to \\R^+$ ranges over smooth compactly supported functions with total mass $\\int_{\\R^2} f(x,y)\\ dx dy$ equal to $1$, $f_{(\\alpha)} \\colon \\R \\times [0,1) \\to \\R^+$ is a ``folded'' version\n\\begin{equation}\\label{falpha-def}\n f_{(\\alpha)}(x,y) \\coloneqq \\sum_{j \\in \\Z} f(x-\\alpha j, y+j)\n \\end{equation}\nof $f$, $f_0, f_\\infty, f_1 \\colon \\R \\to \\R^+$ are the projections\n\\begin{align}\nf_0(x) &\\coloneqq \\int_\\R f(x,y)\\ dy\\label{f0-def}\\\\\nf_\\infty(y) &\\coloneqq \\int_\\R f(x,y)\\ dx\\label{finfty-def}\\\\\nf_1(z) &\\coloneqq \\int_\\R f(x,z- x)\\ dx,\\label{f1-def}\n\\end{align}\nthe two-dimensional differential entropy $h_2(f_{(\\alpha)})$ of $f_{(\\alpha)}$ is defined as\n$$h_2(f_{(\\alpha)}) \\coloneqq \\int_0^1 \\int_\\R \\h(f_{(\\alpha)}(x,y))\\ dx dy,$$\nand the one-dimensional differential entropies $h(f_r)$ for $r=0,1,\\infty$ is defined as\n$$ h(f_r) \\coloneqq \\int_\\R \\h(f_r(x))\\ dx.$$\n\\end{theorem}\n\n\\begin{proposition}\\label{dilate} Let $X$ be an $\\Q$-valued random variable, and for any $a \\in \\Q$, let $g(a)$ denote the quantity $g(a) \\coloneqq \\H[X-aX'] - \\H[X]$, where $X'$ is an independent copy of $X$.\n\\begin{itemize}\n \\item[(i)] $g(0) = 0$ and $g(1)=d[X;X]$.\n \\item[(ii)] For any $a \\in \\Q$, we have $g(-a) \\leq 3g(a)$, and if $a$ is non-zero, $g(a^{-1}) = g(a)$.\n \\item[(iii)] For any $k \\geq 1$ and $a_1,\\dots,a_k \\in \\Q$, we have $g(a_1 \\dots a_k) \\leq g(a_1)+ \\dots + g(a_k)$ and $g(a_1 + \\dots + a_k) \\leq g(a_1)+\\dots+g(a_k)+(k-1) g(1)$.\n \\item[(iv)] If $a$ is a non-zero integer, then $g(a) \\leq (4 + 10 \\lfloor \\log_2 |a| \\rfloor) g(1)$.\n \\item[(v)] If $R$ is a finite set of slopes containing $0,1,\\infty$ of cardinality $k+3$, then\n $$ g(a) \\ll D(R;a)^{k+1} \\max_{r \\in R \\backslash \\{\\infty\\}} g(r).$$\n\\end{itemize}\n\\end{proposition}\n\nIn this section we establish \\Cref{cts-limit}. Let $\\alpha$, $a$, $b$, $f$, $f_0, f_1, f_\\infty$ be as in that theorem. We allow implied constants in the asymptotic notation to depend on $\\alpha, f$. We will establish the lower bound\n\\begin{equation}\\label{sd-targ}\n \\SD\\left(\\{0,1,\\infty\\},\\frac{a}{b}\\right) \\geq 2 - \\frac{h_2(f_\\alpha) - \\max(h(f_0), h(f_\\infty), h(f_1))+o(1)}{\\log b}.\n \\end{equation}\nComparing this with Theorem \\ref{main-2}, we conclude that the expression $h_2(f_\\alpha) - \\max(h(f_0), h(f_\\infty), h(f_1))$ is bounded from below, hence $c_\\alpha > 0$; by taking an arbitrary test function for $f$ we also see that $c_\\alpha < \\infty$. Taking $h_2(f_\\alpha) - \\max(h(f_0), h(f_\\infty), h(f_1))$ arbitrarily close to $c_\\alpha$, we obtain the desired claim \\eqref{up}.\n\nIt remains to establish \\eqref{sd-targ}.\nBy the Poisson summation formula, the rapid decrease of the Fourier transform of the smooth compactly supported $f$, and the mass one hypothesis, we have\n\\begin{equation}\\label{bib}\n \\frac{1}{b^2} \\sum_{n,m \\in \\Z} f\\left(\\frac{n}{b}, \\frac{m}{b}\\right) = 1 + \\eps\n \\end{equation}\nfor some $\\eps = O(1/b)$; in fact one can get much better decay than this, but for our purposes any decay faster than $1/\\log b$ will suffice. We then take $X,Y$ to be supported on the grid $\\Z^2$ with probability distribution\n$$ \\P((X,Y) = (n,m)) \\coloneqq \\frac{1}{(1+\\eps) b^2} \\left(\\frac{n}{b}, \\frac{m}{b}\\right)$$\nthus $(X,Y)$ takes values in a ball of radius $O(b)$ and\n\\begin{equation}\\label{pnx}\n\\P((X,Y) = (n,m)) = \\frac{1}{b^2} f\\left(\\frac{n}{b}, \\frac{m}{b}\\right) + O\\left( \\frac{1}{b^3} \\right).\n\\end{equation}\nBy Bezout's theorem, any integer can be uniquely written in the form $bn+am$ for some $0 \\leq m < b$, and any other representation of the form $bn'+am'$ takes the form $b(n-aj) + a(m+bj)$. Thus\n$$ \\P(bX+aY = bn+am) = \\sum_{j \\in\\Z} \\P( (X,Y) = (n-aj, m+bj) ).$$\nThere are only $O(1)$ values of $j$ for which this sum is non-zero, so from \\eqref{pnx} one has\n$$ \\P(bX+aY = bn+m) = \\frac{1}{b^2} f_{(a/b)}\\left(\\frac{n}{b}, \\frac{m}{b}\\right) + O\\left( \\frac{1}{b^3} \\right).$$\nwhere $f_{(a,b)}: \\R \\times [0,1) \\to \\R^+$ is the function\n$$f_{(a/b)}(x,y) \\coloneqq \\sum_{j \\in \\Z} f\\left(x-\\frac{a}{b} j, y+j\\right).$$\nApplying the entropy function $\\h$, we conclude that\n$$ \\h(\\P(bX+aY = bn+am)) = \\frac{2 \\log b}{b^2} f_{(a/b)}\\left(\\frac{n}{b}, \\frac{m}{b}\\right) +\n\\frac{1}{b^2} \\h\\left(f_{(a/b)}\\left(\\frac{n}{b}, \\frac{m}{b}\\right)\\right) + O\\left( \\frac{\\log b}{b^3} \\right).$$\nFrom \\eqref{bib} one has\n$$ \\frac{1}{b^2} \\sum_{n \\in\\Z; 0 \\leq m < b} f_{(a/b)}\\left(\\frac{n}{b}, \\frac{m}{b}\\right) = 1 + \\eps = 1 + O\\left(\\frac{1}{b} \\right)$$\nand from (uniform) Riemann integrability of the $f_{(a/b)}$ one has\n$$ \\frac{1}{b^2} \\sum_{n \\in\\Z; 0 \\leq m < b} \\h\\left(f_{(a/b)}\\left(\\frac{n}{b}, \\frac{m}{b}\\right)\\right) = h_2(f_{(a/b)}) + o(1).$$\nFinally, from dominated convergence one has\n$$ h_2(f_{(a/b)}) = h_2(f_{(\\alpha)})+o(1)$$\nso we conclude that\n$$ \\H\\left[\\pi_{a/b}(X,Y)\\right] = \\H[bX+aY] = 2 \\log b + h_2(f_{(\\alpha)}) + o(1).$$\nIn a similar vein, from another application of Poisson summation and \\eqref{f0-def} we see that\n$$ \\P(X=n) = \\frac{1}{b} f_0\\left(\\frac{n}{b}\\right) + O\\left(\\frac{1}{b^2}\\right)$$\nfor any integer $n$, hence\n$$ \\h(\\P(X=n)) = \\frac{\\log b}{b} f_0\\left(\\frac{n}{b}\\right) + \\frac{1}{b} \\h\\left(f_0 \\left(\\frac{n}{b}\\right)\\right) + O\\left(\\frac{\\log b}{b^2}\\right).$$\nSince $X = O(b)$, one can sum using Riemann integrability and \\eqref{bib} to conclude that\n$$ \\H[\\pi_0(X,Y)] = \\H[X]= \\log b + h(f_0) + o(1).$$\nSimilar arguments give\n$$ \\H[\\pi_\\infty(X,Y)] = \\H[Y] = \\log b + h(f_\\infty) + o(1)$$\nand\n$$ \\H[\\pi_1(X,Y)] = \\H[X+Y] = \\log b + h(f_1) + o(1).$$\nComparing this with \\eqref{sdef}, we obtain \\eqref{sd-targ}.", + "post_theorem_intro_text_len": 6567, + "post_theorem_intro_text": "\\begin{figure}\n \\centering\n\\centerline{\\includegraphics[width=\\linewidth]{best.png}}\n \\caption{Lower bounds on $\\SD(\\{0,1,\\infty\\}; s)$ obtained by \\texttt{AlphaEvolve} for various $s=a/b$, plotted against $2 - 0.16 / \\log(|a|+b|)$. The horizontal axis is $|a|+|b|$ (so in some cases, multiple fractions $a/b$ are plotted on a single vertical line).}\n \\label{fig:kak}\n\\end{figure}\n\nThe logarithmic convergence in \\eqref{2ab} was suggested to us by experiments in \\cite{gdm} using \\texttt{AlphaEvolve} to obtain lower bounds for $\\SD(\\{0,1,\\infty\\}; s)$ for various slopes $s$. This data was of low accuracy, as \\texttt{AlphaEvolve} could only provide lower bounds and not upper bounds for these quantities; nevertheless, a logarithmic decay was numerically evident (see Figure \\ref{fig:kak}), and furthermore the approximate discrete gaussian shape of the joint distributions of the random variables $X,Y$ obtained by this tool (see \\cite[Figure 18]{gdm}) suggested an approach to make the lower bound in \\eqref{2ab} rigorous. Once this was accomplished, the author was also able to obtain the matching upper bound in \\eqref{2ab} using the entropic Pl\\\"unnecke--Ruzsa calculus. A modification of these arguments, with some inefficiencies, then gives \\eqref{sam}. We tentatively conjecture that the bounds in \\eqref{sam} can be improved to be of the form \\eqref{2ab} for all $k$, not just $k=0$ (possibly after some slight adjustments to the definition of rational complexity).\n\nIn the limit as $a/b$ converges to some real number $\\alpha$ and $b \\to \\infty$, we can obtain a more precise lower bound (also suggested by the aforementioned \\texttt{AlphaEvolve} numerics) as follows.\n\n\\begin{theorem}[Continuous limit]\\label{cts-limit} Let $\\alpha$ be a real number, and let $a,b$ be coprime integer parameters with $b \\to \\infty$ and $a/b \\to \\alpha$. Then\n\\begin{equation}\\label{up}\n \\SD\\left(\\{0,1,\\infty\\};\\frac{a}{b}\\right) \\geq 2 - \\frac{c_\\alpha+o(1)}{\\log b}\n \\end{equation}\n as $b\\to \\infty$ and $a/b \\to \\alpha$, where $0 < c_\\alpha < \\infty$ is the infimum of all quantities\n\\begin{equation}\\label{hf}\n h_2(f_{(\\alpha)}) - \\max(h(f_0), h(f_\\infty), h(f_\\alpha))\n \\end{equation}\nwhere $f \\colon \\R^2 \\to \\R^+$ ranges over smooth compactly supported functions with total mass $\\int_{\\R^2} f(x,y)\\ dx dy$ equal to $1$, $f_{(\\alpha)} \\colon \\R \\times [0,1) \\to \\R^+$ is a ``folded'' version\n\\begin{equation}\\label{falpha-def}\n f_{(\\alpha)}(x,y) \\coloneqq \\sum_{j \\in \\Z} f(x-\\alpha j, y+j)\n \\end{equation}\nof $f$, $f_0, f_\\infty, f_1 \\colon \\R \\to \\R^+$ are the projections\n\\begin{align}\nf_0(x) &\\coloneqq \\int_\\R f(x,y)\\ dy\\label{f0-def}\\\\\nf_\\infty(y) &\\coloneqq \\int_\\R f(x,y)\\ dx\\label{finfty-def}\\\\\nf_1(z) &\\coloneqq \\int_\\R f(x,z- x)\\ dx,\\label{f1-def}\n\\end{align}\nthe two-dimensional differential entropy $h_2(f_{(\\alpha)})$ of $f_{(\\alpha)}$ is defined as\n$$h_2(f_{(\\alpha)}) \\coloneqq \\int_0^1 \\int_\\R \\h(f_{(\\alpha)}(x,y))\\ dx dy,$$\nand the one-dimensional differential entropies $h(f_r)$ for $r=0,1,\\infty$ is defined as\n$$ h(f_r) \\coloneqq \\int_\\R \\h(f_r(x))\\ dx.$$\n\\end{theorem}\n\nWe establish this result in \\Cref{cts-sec}.\nWe tentatively conjecture that the upper bound in \\eqref{up} is in fact an asymptotic equality, so that the asymptotic behavior of\n$ \\SD\\left(\\{0,1,\\infty\\}, \\frac{a}{b}\\right)$ is controlled not only by the rational complexity (as represented by the $\\log b$ denominator), but also by the variational quantity $c_\\alpha$ appearing in the numerator. It is not clear whether this quantity $c_\\alpha$ can be computed exactly; numerically, two-dimensional gaussians are reasonably good candidates for $f$, but in practice they do not extremize the functional \\eqref{hf} precisely.\n\n\\subsection{Notation}\n\nWe use the asymptotic notation $X = O(Y)$, $X \\ll Y$, or $Y \\gg X$ to denote the assertion that $|X| \\leq CY$ for some absolute constant $C$; if we need this implied constant $C$ to depend on some fixed quantities (such as the number $k$ of slopes), we will indicate this in the text.\n\nWhen there is no possibility of ambiguity, we omit parentheses from pairs $(X,Y)$ of random variables, for instance abbreviating $\\H[(X,Y)]$ as $\\H[X,Y]$. Given a random variable $Y$ taking values in some set $S$ and some function $f \\colon S \\to \\R$, we define the expectation\n$$ \\E_{Y=y} f(y) \\coloneqq \\sum_y \\P(Y=y) f(y)$$\nwhere $y$ ranges over the essential range of $Y$. While this expression could also be abbreviated as $\\E f(Y)$, it will be notationally useful to distinguish between the random variable $Y$ and the possible values $y$ that this variable could take. For instance, with this notation, the \\emph{conditional entropy} $\\H[X|Y]$ of one random variable $X$ with respect to another $Y$ can now be defined by the formula\n$$ \\H[X|Y] \\coloneqq \\E_{Y=y} \\H[ X | Y = y ]$$\nwhere $(X|Y=y)$ is $X$ conditioned to the event $Y=y$ (again, we omit parentheses when there is no possibility of ambiguity). The chain rule asserts that $\\H[X|Y]$ can also be expressed by the formula\n$$ \\H[X|Y] = \\H[X,Y] - \\H[Y].$$\nThe \\emph{mutual information} $\\I(X:Y)$ between two random variables is given by the formula\n$$ \\I(X:Y) = \\H[X] - \\H[X|Y] = \\H[Y] - \\H(Y|X) = \\H[X] + \\H[Y] - \\H[X,Y].$$\nAs is well known, $\\I(X:Y)$ is non-negative, and vanishes precisely when $X,Y$ are independent. Equivalently, one has the subadditivity property\n$$ \\H[X,Y] \\leq \\H[X] + \\H[Y]$$\nwith equality precisely when $X,Y$ are independent.\n\nWe also define the conditional mutual information\n$$ \\I[X:Y|Z] \\coloneqq \\E_{Z=z} \\I[(X|Z=z):(Y|Z=z)].$$\nClearly, $\\I[X:Y|Z]$ is non-negative, and vanishes precisely when $X,Y$ are independent conditionally on $Z$.\nFrom the chain rule we have\n\\begin{equation}\\label{i-split}\n\\begin{split}\n\\I[X:Y|Z] &= \\H[X|Z] - \\H[X|Y,Z] \\\\\n&= \\H[Y|Z] - \\H[Y|X,Z] \\\\\n&= \\H[X|Z] + \\H[Y|Z] - \\H[X,Y|Z]. \n\\end{split}\n\\end{equation}\n\n\\subsection{Acknowledgments}\n\nThe author was supported by the James and Carol Collins Chair, the Mathematical Analysis \\& Application Research Fund, and by NSF grants DMS-2347850, and is particularly grateful to recent donors to the Research Fund. He particularly thanks his coauthors Bogdan Georgiev, Javier G\\'omez-Serrano, and Adam Zsolt Wagner for the highly productive and enjoyable collaboration \\cite{gdm}, and for generously sharing the outputs of that collaboration for the purposes of writing the current paper.\n\nWhile some of the results proven here were suggested by the outcome of AI-assisted experiments, the arguments in this paper are completely human-generated.", + "sketch": "The post-theorem introduction does not give a step-by-step proof, but it does outline the methods used to establish the bounds in Theorem~\\ref{main-2}. For the three-slope case \\eqref{2ab}, the “logarithmic convergence” was “suggested… by experiments… using \\texttt{AlphaEvolve} to obtain lower bounds,” and the “approximate discrete gaussian shape of the joint distributions of the random variables $X,Y$… suggested an approach to make the lower bound in \\eqref{2ab} rigorous.” “Once this was accomplished, the author was also able to obtain the matching upper bound in \\eqref{2ab} using the entropic Pl\\\"unnecke--Ruzsa calculus.” Finally, “a modification of these arguments, with some inefficiencies, then gives \\eqref{sam}.”", + "expanded_sketch": "The post-theorem introduction does not give a step-by-step proof, but it does outline the methods used to establish the bounds in the main theorem. For the three-slope case\n\\begin{equation}\\label{2ab} 2 - \\frac{c_2}{D(R;s)} \\leq \\SD\\left(R; s\\right) \\leq 2 - \\frac{c_1}{D(R;s)} \n\\end{equation}\n, the “logarithmic convergence” was “suggested… by experiments… using \\texttt{AlphaEvolve} to obtain lower bounds,” and the “approximate discrete gaussian shape of the joint distributions of the random variables $X,Y$… suggested an approach to make the lower bound in the equation above rigorous.” “Once this was accomplished, the author was also able to obtain the matching upper bound in the equation above using the entropic Pl\\\"unnecke--Ruzsa calculus.” Finally, “a modification of these arguments, with some inefficiencies, then gives\n\\begin{equation}\\label{sam} 2 - \\frac{C_k \\log(2+D(R;s))}{D(R;s)} \\leq \\SD(R; s) \\leq 2 - \\frac{c_k}{D(R;s)^{k+1}} \n \\end{equation}\n.”", + "expanded_theorem": "\\label{main-2} Let $R$ be a finite family of slopes containing $0,1,\\infty$ of some cardinality $k+3$, and let $s$ be a slope not lying in $R$.\n\\begin{itemize}\n \\item[(i)] (Three slopes) If $k=0$, then\n\\begin{equation}\\label{2ab} 2 - \\frac{c_2}{D(R;s)} \\leq \\SD\\left(R; s\\right) \\leq 2 - \\frac{c_1}{D(R;s)} \n\\end{equation}\n for some absolute constants $c_2 > c_1 > 0$.\n \\item[(ii)] (Many slopes) In general, one has\n \\begin{equation}\\label{sam} 2 - \\frac{C_k \\log(2+D(R;s))}{D(R;s)} \\leq \\SD(R; s) \\leq 2 - \\frac{c_k}{D(R;s)^{k+1}} \n \\end{equation}\n for some absolute constants $c_k, C_k > 0$.\n\\end{itemize}", + "theorem_type": [ + "Inequality or Bound", + "Universal" + ], + "mcq": { + "question": "Let a slope mean an element of \\(\\mathbb{Q}\\cup\\{\\infty\\}\\). For each slope \\(r\\), define \\(\\pi_r:\\mathbb{Q}\\times\\mathbb{Q}\\to\\mathbb{Q}\\) by \\(\\pi_r(x,y)=x+ry\\) if \\(r\\neq\\infty\\), and \\(\\pi_\\infty(x,y)=y\\). For a finite family of slopes \\(R\\) and a slope \\(s\\notin R\\), let \\(\\mathrm{SD}(R;s)\\) be the least exponent such that\n\\[\nH[\\pi_s(X,Y)]\\le \\mathrm{SD}(R;s)\\max_{r\\in R} H[\\pi_r(X,Y)]\n\\]\nfor all finitely supported \\(\\mathbb{Q}\\)-valued random variables \\(X,Y\\) (not necessarily independent), where \\(H[Z]=\\sum_z h(\\mathbb{P}(Z=z))\\) is Shannon entropy and \\(h(t)=t\\log(1/t)\\) with \\(h(0)=0\\). Now let \\(R=\\{0,1,\\infty,r_1,\\dots,r_k\\}\\subset \\mathbb{Q}\\cup\\{\\infty\\}\\) be a finite family of slopes of cardinality \\(k+3\\), and let \\(s\\notin R\\). Define the rational complexity \\(D(R;s)\\) to be the least natural number \\(D\\) for which\n\\[\ns=\\frac{P(r_1,\\dots,r_k)}{Q(r_1,\\dots,r_k)}\n\\]\nfor some polynomials \\(P,Q\\) of degree at most \\(D\\) with integer coefficients of magnitude at most \\(2^D\\), with \\(Q(r_1,\\dots,r_k)\\neq 0\\). Which statement holds for every such \\(R\\) and \\(s\\)?", + "correct_choice": { + "label": "A", + "text": "If \\(k=0\\) (so \\(R=\\{0,1,\\infty\\}\\)), then there exist absolute constants \\(c_2>c_1>0\\) such that\n\\[\n2-\\frac{c_2}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_1}{D(R;s)}.\n\\]\nMore generally, for arbitrary \\(k\\), there exist constants \\(c_k,C_k>0\\) depending only on \\(k\\) such that\n\\[\n2-\\frac{C_k\\log(2+D(R;s))}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_k}{D(R;s)^{k+1}}.\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "If \\(k=0\\) (so \\(R=\\{0,1,\\infty\\}\\)), then there exist absolute constants \\(c_2>c_1>0\\) such that\n\\[\n2-\\frac{c_2\\log(2+D(R;s))}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_1\\log(2+D(R;s))}{D(R;s)}.\n\\]\nMore generally, for arbitrary \\(k\\), there exist constants \\(c_k,C_k>0\\) depending only on \\(k\\) such that\n\\[\n2-\\frac{C_k}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_k}{D(R;s)^{k+1}}.\n\\]" + }, + { + "label": "C", + "text": "For arbitrary \\(k\\), there exist constants \\(c_k,C_k>0\\) depending only on \\(k\\) such that\n\\[\n2-\\frac{C_k\\log(2+D(R;s))}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2.\n\\]\nIn particular, when \\(k=0\\), one has \\(2-\\frac{C_0\\log(2+D(R;s))}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2\\)." + }, + { + "label": "D", + "text": "If \\(k=0\\) (so \\(R=\\{0,1,\\infty\\}\\)), then there exist absolute constants \\(c_2>c_1>0\\) such that\n\\[\n2-\\frac{c_2}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_1}{D(R;s)^2}.\n\\]\nMore generally, for arbitrary \\(k\\), there exist constants \\(c_k,C_k>0\\) depending only on \\(k\\) such that\n\\[\n2-\\frac{C_k\\log(2+D(R;s))}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_k}{D(R;s)^k}.\n\\]" + }, + { + "label": "E", + "text": "If \\(k=0\\) (so \\(R=\\{0,1,\\infty\\}\\)), then there exist absolute constants \\(c_2>c_1>0\\) such that\n\\[\n2-\\frac{c_2}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c_1}{D(R;s)}.\n\\]\nMore generally, there exist absolute constants \\(c,C>0\\), independent of \\(k\\), \\(R\\), and \\(s\\), such that for every \\(k\\)\n\\[\n2-\\frac{C\\log(2+D(R;s))}{D(R;s)}\\le \\mathrm{SD}(R;s)\\le 2-\\frac{c}{D(R;s)^{k+1}}.\n\\]" + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "special k=0 logarithmic-vs-linear behavior", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "nontrivial upper bound improvement below 2", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "upper-bound exponent in D for many slopes and three-slope matching rate", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "dependence of constants on k", + "template_used": "quantifier_dependence" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem introduces definitions and asks which global bound is true; it does not explicitly state or strongly hint at the correct asymptotic form." + }, + "TAS": { + "score": 2, + "justification": "This is not a direct restatement of something already asserted in the stem. The respondent must distinguish among several competing formulations with different rates, logarithmic factors, and quantifier dependence." + }, + "GPS": { + "score": 1, + "justification": "There is some reasoning pressure because the choices differ in subtle mathematical ways, but the item mainly tests precise theorem recall/recognition rather than derivation or substantial generative reasoning from the definitions alone." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic confusions: log-vs-linear decay, exponent shifts, weaker-but-true bounds, and dependence of constants on k." + }, + "total_score": 7, + "overall_assessment": "A strong MCQ in terms of no answer leakage and high-quality distractors, though it functions more as a theorem-recognition item than a deep generative-reasoning question." + } + }, + { + "id": "2511.14012v1", + "paper_link": "http://arxiv.org/abs/2511.14012v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "theorem", + "content": "\\label{main theorem}\n \tLet $q \\ge 3$ be fixed. Uniformly for \n $\n \t\\tau \\le \\log n + \\log_2 n - \\theta(n),\n \t$\n \twhere $2\\leq \\theta(n)\\ll \\log_3 n$ is any function tending arbitrarily slowly to infinity as $n \\to \\infty$, we have\n \t\\[\n \t\\phi_n(\\tau)\n \t:= \\frac{1}{|\\mathcal{H}_n|}\n \t\\sum_{\\substack{D \\in \\mathcal{H}_n \\\\ L(1, \\chi_D) \\ge e^{\\gamma}\\tau}} 1\n \t= \\exp\\!\\left(\n \t-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\n \t\\,\\frac{q^{\\,\\tau - C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\n \t\\bigl(1 + o_\\theta(1)\\bigr)\n \t\\right),\n \t\\]\n \twhere the constants $C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ and $C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ depend on $\\tau$, $q$, and $\\kappa(\\tau)$, yet remain bounded as the argument varies between $1$ and $q$. \n \tFurther details on these quantities can be found in~\\cite[Theorem~1.3]{Lumley}.", + "start_pos": 7862, + "end_pos": 8744, + "label": "main theorem" + }, + "ref_dict": { + "genus": "\\begin{align}\\label{genus}\n2g=d(D)-1-\\lambda\n\\end{align}", + "theorem on resonance method": "\\begin{theorem}\\label{theorem on resonance method}\n\tLet $q \\ge 3$ and for every $\\beta > 0$, define $\\tau_{\\beta, n}\n\t= e^{\\gamma}\\!\\left(\\log n + \\log_2 n + C_2(q) - \\beta\\right),$ where\n\t\\[\n\tC_2(q)\n\t= \\frac{1}{2}\n\t- \\left(\\frac{\\pi}{4} - \\frac{\\ln 2}{2}\\right)\\frac{q}{q-1}\n\t+ \\log\\!\\left(\n\t\\frac{(q-1)\\ln q}{2q(3\\ln 2 - \\pi/2)}\n\t\\right).\n\t\\]\n\tThen we have\n\t\\[\n\\phi_n(\\beta)\n:= \\frac{1}{|\\mathcal{H}_n|}\n\\sum_{\\substack{D \\in \\mathcal{H}_n \\\\ L(1, \\chi_D) \\ge \\tau_{\\beta, n}}} 1\n\t\\ge \\exp\\!\\left(\n\t-\\frac{q^{-\\beta}\\ln q}{2}\n\t\\left(1 + o(1)\\right)\n\t\\right).\n\t\\]\n\\end{theorem}", + "main theorem": "\\begin{theorem}\\label{main theorem}\n \tLet $q \\ge 3$ be fixed. Uniformly for \n $\n \t\\tau \\le \\log n + \\log_2 n - \\theta(n),\n \t$\n \twhere $2\\leq \\theta(n)\\ll \\log_3 n$ is any function tending arbitrarily slowly to infinity as $n \\to \\infty$, we have\n \t\\[\n \t\\phi_n(\\tau)\n \t:= \\frac{1}{|\\mathcal{H}_n|}\n \t\\sum_{\\substack{D \\in \\mathcal{H}_n \\\\ L(1, \\chi_D) \\ge e^{\\gamma}\\tau}} 1\n \t= \\exp\\!\\left(\n \t-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\n \t\\,\\frac{q^{\\,\\tau - C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\n \t\\bigl(1 + o_\\theta(1)\\bigr)\n \t\\right),\n \t\\]\n \twhere the constants $C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ and $C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ depend on $\\tau$, $q$, and $\\kappa(\\tau)$, yet remain bounded as the argument varies between $1$ and $q$. \n \tFurther details on these quantities can be found in~\\cite[Theorem~1.3]{Lumley}.\n \\end{theorem}", + "full l to its truncation": "\\begin{lemma}\\label{full l to its truncation}\nFor sufficiently large $n$,\n\\[\nL(1, \\chi_D)=L(1, \\chi_D; N)\\left(1+O\\left(\\frac{1}{f(n)\\log n}\\right)\\right)\n\\]\nfor all but at most $q^{\\frac{7n}{10}}$ elements $D\\in \\mathcal{H}_n$, where $N=\\log n+\\log_2 n+3\\log f(n)$, and $f(n) \\to \\infty$ arbitrarily slowly as $n \\to \\infty$, with $f(n) \\ll \\log n$.\n\t\\end{lemma}", + "lambda": "\\begin{align}\\label{lambda}\n\\lambda= \n\\left\\{\n\\begin{array}\n[c]{ll}\n1, & \\;\\text{if\\, $d(D)$ even}, \\\\\n0, &\\; \\text{if\\, $d(D)$ odd},\n\\end{array}\n\\right.\n\\end{align}", + "character sum over nonsquare": "\\begin{lemma}\\label{character sum over nonsquare}\n\tLet $\\ell=\\ell_1 \\ell_2^2\\neq \\square\\in \\mathbb{F}_q[t]$ with $\\ell_1$ square-free. For any $\\varepsilon>0$, we have\n\t\\[\n\t\\sum_{D\\in \\mathcal{H}_n}\\chi_D(\\ell)\\ll q^{(1/2+\\varepsilon)n}\\exp\\left((d(\\ell_1))^{1-\\varepsilon}\\right)\\prod_{P\\mid \\ell_2}\\left(1+|P|^{-1/2-\\varepsilon}\\right).\n\t\\]\t\n\n\\end{lemma}" + }, + "pre_theorem_intro_text_len": 3454, + "pre_theorem_intro_text": "One of the central problems in analytic number theory is to determine the value distribution and obtain sharp bounds for the maximal size of Dirichlet $L$-functions on the line $\\operatorname{Re}(s)=1$. These quantities are deeply intertwined with fundamental arithmetic invariants such as class numbers of number fields, and are closely linked to estimates for character sums via Fourier analysis. In particular, the tail behavior of the value distribution plays a decisive role in determining the true order of magnitude of extreme values.\n\nOver the past few decades, significant progress has been made toward understanding the tails of these distributions and the maximal size of $L$-functions on $\\operatorname{Re}(s)=1$. This line of research encompasses the Riemann zeta function and Dirichlet $L$-functions in the modulus aspect (see~\\cite{AMM, AMMP, BS3, DOKIC, GSzeta}), as well as the family of quadratic Dirichlet $L$-functions $L(1,\\chi_d)$ (see~\\cite{DM, GS, Lumley}), and more generally, various families of automorphic $L$-functions (see~\\cite{Lamzouri}).\n\nAn important $\\mathrm{GL}(1)$ family of symplectic type is formed by the quadratic Dirichlet $L$-functions $L(s,\\chi_d)$, where $d$ ranges over fundamental discriminants. The analytic study of this family is notably challenging due to the lack of a natural orthogonality relation among the characters. The tail behavior of $L(1,\\chi_d)$ is closely tied to the distribution of class numbers of quadratic number fields. In this direction, Granville and Soundararajan~\\cite{GS} carried out a seminal analysis of these quantities, and following their approach, Dahl and Lamzouri~\\cite{DL} investigated an interesting subfamily, known as the \\emph{Chowla family}, corresponding to real quadratic fields.\n\n\\subsection{Distribution of $L$-functions in the hyperelliptic ensemble}\\label{large values section}\nWe now move our discussion to the setup of $L$-functions over function fields, a topic of interest in modern research in number theory; see, e.g., \\cite{AT, AK, BF, BF2, DFL, DL, FR, Floreanegative, Florea, Jung, Lumley2}.\nThis includes investigations of various statistical properties of these $L$-functions, such as moments, non-vanishing, and value distribution.\n\n Before we enunciate the main theorem of this paper, we need to present\n some basic notations on function fields.\n Let $\\mathbb{F}_{q}$ be a finite field of odd cardinality and $\\mathbb{F}_{q}[t]$ be the polynomial ring over $\\mathbb{F}_{q}$ in variable $t$. Let $D\\in \\mathbb{F}_q[t]$ be a monic square-free polynomial. The quadratic character $\\chi_D$ attached to $D$ is defined using quadratic residue symbol for $\\mathbb{F}_{q}[t]$ by $\\chi_{D}(f)=\\left(\\frac{f}{D}\\right)$ and the corresponding Dirichlet $L$-function is denoted by $L(s, \\chi_D)$. \n Let $\\mathcal{H}_n$\n be the family of all curves given in affine form by $C_P: y^2=D(t)$, where $D$ be a monic square-free polynomial of degree $n$. These curve are non-singular of genus $g$ given by \\eqref{lambda} and \\eqref{genus}.\n\n In~\\cite{Lumley}, Lumley studied the distribution of $L(1,\\chi_D)$ as $D$ ranges over $\\mathcal{H}_n$. \n The following theorem strengthens the range of uniformity for the tail of the distribution of $L(1,\\chi_D)$ established in~\\cite[Theorem~1.2]{Lumley}.\nThroughout, $\\log$ denotes the logarithm to base $q$, while $\\ln$ is the natural logarithm, and $\\log_j$ (respectively $\\ln_j$) for the $j$-fold iterated logarithm.", + "context": "One of the central problems in analytic number theory is to determine the value distribution and obtain sharp bounds for the maximal size of Dirichlet $L$-functions on the line $\\operatorname{Re}(s)=1$. These quantities are deeply intertwined with fundamental arithmetic invariants such as class numbers of number fields, and are closely linked to estimates for character sums via Fourier analysis. In particular, the tail behavior of the value distribution plays a decisive role in determining the true order of magnitude of extreme values.\n\nOver the past few decades, significant progress has been made toward understanding the tails of these distributions and the maximal size of $L$-functions on $\\operatorname{Re}(s)=1$. This line of research encompasses the Riemann zeta function and Dirichlet $L$-functions in the modulus aspect (see~\\cite{AMM, AMMP, BS3, DOKIC, GSzeta}), as well as the family of quadratic Dirichlet $L$-functions $L(1,\\chi_d)$ (see~\\cite{DM, GS, Lumley}), and more generally, various families of automorphic $L$-functions (see~\\cite{Lamzouri}).\n\nAn important $\\mathrm{GL}(1)$ family of symplectic type is formed by the quadratic Dirichlet $L$-functions $L(s,\\chi_d)$, where $d$ ranges over fundamental discriminants. The analytic study of this family is notably challenging due to the lack of a natural orthogonality relation among the characters. The tail behavior of $L(1,\\chi_d)$ is closely tied to the distribution of class numbers of quadratic number fields. In this direction, Granville and Soundararajan~\\cite{GS} carried out a seminal analysis of these quantities, and following their approach, Dahl and Lamzouri~\\cite{DL} investigated an interesting subfamily, known as the \\emph{Chowla family}, corresponding to real quadratic fields.\n\n\\subsection{Distribution of $L$-functions in the hyperelliptic ensemble}\\label{large values section}\nWe now move our discussion to the setup of $L$-functions over function fields, a topic of interest in modern research in number theory; see, e.g., \\cite{AT, AK, BF, BF2, DFL, DL, FR, Floreanegative, Florea, Jung, Lumley2}.\nThis includes investigations of various statistical properties of these $L$-functions, such as moments, non-vanishing, and value distribution.\n\nBefore we enunciate the main theorem of this paper, we need to present\n some basic notations on function fields.\n Let $\\mathbb{F}_{q}$ be a finite field of odd cardinality and $\\mathbb{F}_{q}[t]$ be the polynomial ring over $\\mathbb{F}_{q}$ in variable $t$. Let $D\\in \\mathbb{F}_q[t]$ be a monic square-free polynomial. The quadratic character $\\chi_D$ attached to $D$ is defined using quadratic residue symbol for $\\mathbb{F}_{q}[t]$ by $\\chi_{D}(f)=\\left(\\frac{f}{D}\\right)$ and the corresponding Dirichlet $L$-function is denoted by $L(s, \\chi_D)$. \n Let $\\mathcal{H}_n$\n be the family of all curves given in affine form by $C_P: y^2=D(t)$, where $D$ be a monic square-free polynomial of degree $n$. These curve are non-singular of genus $g$ given by \\eqref{lambda} and \\eqref{genus}.\n\nIn~\\cite{Lumley}, Lumley studied the distribution of $L(1,\\chi_D)$ as $D$ ranges over $\\mathcal{H}_n$. \n The following theorem strengthens the range of uniformity for the tail of the distribution of $L(1,\\chi_D)$ established in~\\cite[Theorem~1.2]{Lumley}.\nThroughout, $\\log$ denotes the logarithm to base $q$, while $\\ln$ is the natural logarithm, and $\\log_j$ (respectively $\\ln_j$) for the $j$-fold iterated logarithm.\n\n\\begin{align}\\label{genus}\n2g=d(D)-1-\\lambda\n\\end{align}\n\n\\begin{align}\\label{lambda}\n\\lambda= \n\\left\\{\n\\begin{array}\n[c]{ll}\n1, & \\;\\text{if\\, $d(D)$ even}, \\\\\n0, &\\; \\text{if\\, $d(D)$ odd},\n\\end{array}\n\\right.\n\\end{align}", + "full_context": "One of the central problems in analytic number theory is to determine the value distribution and obtain sharp bounds for the maximal size of Dirichlet $L$-functions on the line $\\operatorname{Re}(s)=1$. These quantities are deeply intertwined with fundamental arithmetic invariants such as class numbers of number fields, and are closely linked to estimates for character sums via Fourier analysis. In particular, the tail behavior of the value distribution plays a decisive role in determining the true order of magnitude of extreme values.\n\nOver the past few decades, significant progress has been made toward understanding the tails of these distributions and the maximal size of $L$-functions on $\\operatorname{Re}(s)=1$. This line of research encompasses the Riemann zeta function and Dirichlet $L$-functions in the modulus aspect (see~\\cite{AMM, AMMP, BS3, DOKIC, GSzeta}), as well as the family of quadratic Dirichlet $L$-functions $L(1,\\chi_d)$ (see~\\cite{DM, GS, Lumley}), and more generally, various families of automorphic $L$-functions (see~\\cite{Lamzouri}).\n\nAn important $\\mathrm{GL}(1)$ family of symplectic type is formed by the quadratic Dirichlet $L$-functions $L(s,\\chi_d)$, where $d$ ranges over fundamental discriminants. The analytic study of this family is notably challenging due to the lack of a natural orthogonality relation among the characters. The tail behavior of $L(1,\\chi_d)$ is closely tied to the distribution of class numbers of quadratic number fields. In this direction, Granville and Soundararajan~\\cite{GS} carried out a seminal analysis of these quantities, and following their approach, Dahl and Lamzouri~\\cite{DL} investigated an interesting subfamily, known as the \\emph{Chowla family}, corresponding to real quadratic fields.\n\n\\subsection{Distribution of $L$-functions in the hyperelliptic ensemble}\\label{large values section}\nWe now move our discussion to the setup of $L$-functions over function fields, a topic of interest in modern research in number theory; see, e.g., \\cite{AT, AK, BF, BF2, DFL, DL, FR, Floreanegative, Florea, Jung, Lumley2}.\nThis includes investigations of various statistical properties of these $L$-functions, such as moments, non-vanishing, and value distribution.\n\nBefore we enunciate the main theorem of this paper, we need to present\n some basic notations on function fields.\n Let $\\mathbb{F}_{q}$ be a finite field of odd cardinality and $\\mathbb{F}_{q}[t]$ be the polynomial ring over $\\mathbb{F}_{q}$ in variable $t$. Let $D\\in \\mathbb{F}_q[t]$ be a monic square-free polynomial. The quadratic character $\\chi_D$ attached to $D$ is defined using quadratic residue symbol for $\\mathbb{F}_{q}[t]$ by $\\chi_{D}(f)=\\left(\\frac{f}{D}\\right)$ and the corresponding Dirichlet $L$-function is denoted by $L(s, \\chi_D)$. \n Let $\\mathcal{H}_n$\n be the family of all curves given in affine form by $C_P: y^2=D(t)$, where $D$ be a monic square-free polynomial of degree $n$. These curve are non-singular of genus $g$ given by \\eqref{lambda} and \\eqref{genus}.\n\nIn~\\cite{Lumley}, Lumley studied the distribution of $L(1,\\chi_D)$ as $D$ ranges over $\\mathcal{H}_n$. \n The following theorem strengthens the range of uniformity for the tail of the distribution of $L(1,\\chi_D)$ established in~\\cite[Theorem~1.2]{Lumley}.\nThroughout, $\\log$ denotes the logarithm to base $q$, while $\\ln$ is the natural logarithm, and $\\log_j$ (respectively $\\ln_j$) for the $j$-fold iterated logarithm.\n\n\\begin{align}\\label{genus}\n2g=d(D)-1-\\lambda\n\\end{align}\n\n\\begin{align}\\label{lambda}\n\\lambda= \n\\left\\{\n\\begin{array}\n[c]{ll}\n1, & \\;\\text{if\\, $d(D)$ even}, \\\\\n0, &\\; \\text{if\\, $d(D)$ odd},\n\\end{array}\n\\right.\n\\end{align}\n\nBefore we enunciate the main theorem of this paper, we need to present\n some basic notations on function fields.\n Let $\\mathbb{F}_{q}$ be a finite field of odd cardinality and $\\mathbb{F}_{q}[t]$ be the polynomial ring over $\\mathbb{F}_{q}$ in variable $t$. Let $D\\in \\mathbb{F}_q[t]$ be a monic square-free polynomial. The quadratic character $\\chi_D$ attached to $D$ is defined using quadratic residue symbol for $\\mathbb{F}_{q}[t]$ by $\\chi_{D}(f)=\\left(\\frac{f}{D}\\right)$ and the corresponding Dirichlet $L$-function is denoted by $L(s, \\chi_D)$. \n Let $\\mathcal{H}_n$\n be the family of all curves given in affine form by $C_P: y^2=D(t)$, where $D$ be a monic square-free polynomial of degree $n$. These curve are non-singular of genus $g$ given by \\eqref{lambda} and \\eqref{genus}.\n\n\\subsection{Truncating the Euler product of $L(1, \\chi_D)$}\nWe show that $L(1, \\chi_D)$ can be well approximated by its short Euler product for almost all $D \\in \\mathcal{H}_n$. Let \n\\begin{align}\\label{short euler product}\nL(1,\\chi_D; M):= \\prod_{d(P)\\le M}\\left(1-\\frac{\\chi_D(P)}{|P|}\\right)^{-1}=\\sum_{\\substack{f\\in \\mathcal{M}\\\\ P\\mid f \\implies d(P)\\leq M}}\\frac{\\chi_D(f)}{|f|}.\n\\end{align} \n Let $M:=3\\log n$. From \\cite[Lemma 2.2]{Lumley}, for $D\\in \\mathcal{H}_n$, we have\n \\begin{align}\\label{asym for L(1)}\n L(1,\\chi_{D})=L(1,\\chi_D; M)\\left(1+O\\left(\\frac{1}{n^{1/2}\\log n}\\right)\\right).\n \\end{align}\nThe following lemma provides a better approximation on the truncation length and the associated error chosen optimally for the proof of Theorem~\\ref{main theorem}.\n\\begin{lemma}\\label{full l to its truncation}\nFor sufficiently large $n$,\n\\[\nL(1, \\chi_D)=L(1, \\chi_D; N)\\left(1+O\\left(\\frac{1}{f(n)\\log n}\\right)\\right)\n\\]\nfor all but at most $q^{\\frac{7n}{10}}$ elements $D\\in \\mathcal{H}_n$, where $N=\\log n+\\log_2 n+3\\log f(n)$, and $f(n) \\to \\infty$ arbitrarily slowly as $n \\to \\infty$, with $f(n) \\ll \\log n$.\n \\end{lemma}\n\\begin{proof}\n By using \\eqref{asym for L(1)}, we have\n \\[\n L(1, \\chi_D) = \n L(1, \\chi_D; N)\n \\exp \\!\\bigg(\n \\sum_{\\substack{N < d(P) \\le M}} \n \\bigg(\\frac{\\chi_D(P)}{|P|} \n + O\\!\\left( \\frac{1}{|P|^2} \\right)\n \\bigg)\\bigg)\n \\bigg( 1 + O\\!\\bigg( \\frac{1}{n^{\\tfrac{1}{2}} \\log n} \\bigg) \\bigg).\n \\]\n Next, we identify a density zero subfamily of $\\mathcal{H}_n$ for which\n \\begin{align}\\label{DP bound} \n \\Bigg|\\sum_{N < d(P) \\le M} \\frac{\\chi_D(P)}{|P|}\\Bigg| \\ge \\frac{1}{h(n)}\n \\end{align}\n holds, where $\n \\log n \\ll h(n) \\to \\infty$ as $n \\to \\infty.$\n It is worth noting that $h(n)$ must grow faster than $\\log n$ chosen appropriately later in order for the saddle point analysis to yield an asymptotic formula exhibiting double exponential decay (see Section~\\ref{spa}).\n\nThe additional factor $1/\\sqrt{\\log n}$ in the choice of $c$ is essential for proving Theorem \\ref{theorem on resonance method}.\nHence, using \\eqref{asym for L(1)}, we can that there exists a $D\\in \\mathcal{H}_n$ such that \n\\[\nL(1,\\chi_D)\\ge e^{\\gamma}\\left(\\log n +\\log_2 n+C_2(q)-\\beta+o(1)\\right).\n\\]\nWe use \\eqref{asym for L(1)}, \\eqref{ratio final}, \\eqref{constant c} to obtain \n\\begin{align*}\n&\\bigg(\\tau_{\\beta, n}+\\frac{1}{2\\sqrt{\\log n}}\\bigg)\\sum_{D\\in \\mathcal{H}_n}R_D^2\\leq \\sum_{D\\in \\mathcal{H}_n}L(1, \\chi_D)R_D^2\\\\\n&= \\sum_{\\substack{D\\in \\mathcal{H}_n\\\\ L(1, \\chi_D)< \\tau_{\\beta, n}}}L(1, \\chi_D)R_D^2+\\sum_{\\substack{D\\in \\mathcal{H}_n\\\\ L(1, \\chi_D)\\geq \\tau_{\\beta, n}}}L(1, \\chi_D)R_D^2< \\tau_{\\beta, n} \\sum_{D\\in \\mathcal{H}_n}R_D^2+ \\sum_{\\substack{D\\in \\mathcal{H}_n\\\\ L(1, \\chi_D)\\geq \\tau_{\\beta, n}}}L(1, \\chi_D)R_D^2.\n\\end{align*}\nFrom the above computation of $S_2$ together with \\eqref{scx} gives us\n\\[\n\\sum_{\\substack{D\\in \\mathcal{H}_n\\\\ L(1, \\chi_D)\\geq \\tau_{\\beta, n}}}L(1, \\chi_D)R_D^2> \\frac{e^{\\gamma}}{2 \\sqrt{\\log n}}\\sum_{D\\in \\mathcal{H}_n}R_D^2> \\frac{e^{\\gamma}}{2}\\frac{q^{(1+c')n\\left(1+O(\\log_2 n/\\log n)\\right)}}{\\sqrt{\\log n}}.\n\\]\nOn the other hand, using \\cite[Proposition 1.4]{Lumley} that\n\\[\n\\max_{D\\in \\mathcal{H}_n}L(1, \\chi_D)\\leq 2e^{\\gamma}\\log n+O_q(1),\n\\]\nwe have from \\eqref{RD},\n\\begin{align*}\n\\sum_{\\substack{D\\in \\mathcal{H}_n\\\\ L(1, \\chi_D)\\geq \\tau_{\\beta, n}}}L(1, \\chi_D)R_D^2&\\leq |\\mathcal{H}_n| \\left(\\max_{D\\in \\mathcal{H}_n} R_D^2 \\cdot \\max_{D\\in \\mathcal{H}_n}L(1, \\chi_D)\\right)\\phi_n(\\beta)\\\\\n&\\leq \\left(2e^{\\gamma}\\log n\\right) q^{\\left(1+\\frac{2c\\zeta_{\\mathbb{A}}(2)}{\\ln q}\\right)\\left(1+O\\left(\\frac{\\log_2 n}{\\log n}\\right)\\right)n} \\phi_n(\\beta).\n\\end{align*}\nComparing the above lower and upper bound, and using $c$ as expressed in \\eqref{constant c}, we finally conclude that\n\\[\n\\phi_n(\\beta)> \\frac{1}{4(\\log n)^{3/2}}q^{-\\frac{c(3\\ln 2-\\pi/2)\\zeta_{\\mathbb{A}}(2)}{\\ln q}\\left(1+O\\left(\\log_2 n/\\log n\\right)\\right)}\\geq e^{-\\frac{q^{-\\beta}\\left(1+O\\left(1/\\sqrt{\\log n}\\right)\\right)\\ln q}{2}}.\n\\]\n\\end{proof}\n\nFrom the estimation of $\\mathcal{S}_{c, N}$ and $\\mathcal{R}_{c, N}$, we finally obtain \n\\begin{align}\\label{ratio}\n\\frac{S_1}{S_2}\\ge \\frac{|\\mathcal{H}_n| \\mathcal{E}(N) \\mathcal{E}_1(N) \\prod_{d(P)< N} (1-|P|^{-1}) + O\\left(q^{\\left(\\frac12+\\frac{2c\\zeta_{\\mathbb{A}}(2)}{\\ln q}+\\frac{11\\varepsilon}{6}\\right)n}\\right)}{|\\mathcal{H}_n| \\mathcal{E}(N) \\mathcal{E}_2(N)\\prod_{d(P) 0$, define $\\tau_{\\beta, n}\n\t= e^{\\gamma}\\!\\left(\\log n + \\log_2 n + C_2(q) - \\beta\\right),$ where\n\t\\[\n\tC_2(q)\n\t= \\frac{1}{2}\n\t- \\left(\\frac{\\pi}{4} - \\frac{\\ln 2}{2}\\right)\\frac{q}{q-1}\n\t+ \\log\\!\\left(\n\t\\frac{(q-1)\\ln q}{2q(3\\ln 2 - \\pi/2)}\n\t\\right).\n\t\\]\n\tThen we have\n\t\\[\n\\phi_n(\\beta)\n:= \\frac{1}{|\\mathcal{H}_n|}\n\\sum_{\\substack{D \\in \\mathcal{H}_n \\\\ L(1, \\chi_D) \\ge \\tau_{\\beta, n}}} 1\n\t\\ge \\exp\\!\\left(\n\t-\\frac{q^{-\\beta}\\ln q}{2}\n\t\\left(1 + o(1)\\right)\n\t\\right).\n\t\\]\n\\end{theorem}\nThe best previously known unconditional result in this direction over number fields was due to Granville and Soundararajan~\\cite[Theorem~5b]{GS}, while the conditional analogue under the GRH was obtained by the author and Maiti~\\cite[Theorem~2]{DM}.\n\nDuring the proof of Theorem~\\ref{theorem on resonance method}, we also observed that\n\\[\n\\max_{D\\in \\mathcal{H}_n} L(1,\\chi_D)\n\\ge e^{\\gamma}\\Bigl(\\log n + \\log_2 n + C_2(q) + o(1)\\Bigr),\n\\]\nwhere \\(C_2(q)\\) is as defined in the statement of the theorem. This result refines~\\cite[Theorem~1.6]{Lumley} by providing an explicit value for the constant \\(C_2(q)\\). More strikingly, one has \\(C_2(q) > 0\\) for \\(q > 10\\); for instance, \\(C_2(17) \\approx 0.04\\). This is the first time a positive constant is observed, showing that the maximum of \\(L(1,\\chi_D)\\) can exceed the range \\(\\log n + \\log_2 n\\) (see~\\cite[Conjecture~2]{MV}).\n\n\\subsubsection{Applications} For a monic square-free polynomial $D \\in \\mathbb{F}_q[t]$, define \n\\[\nh_D = \\lvert \\mathrm{Pic}(\\mathcal{O}_D) \\rvert,\n\\]\nwhere $\\mathrm{Pic}(\\mathcal{O}_D)$ denotes the Picard group of the ring of integers $\\mathcal{O}_D \\subset \\mathbb{F}_q[t](\\sqrt{D(t)})$. \nArtin~\\cite{Artin} established a class number formula over the hyperelliptic ensemble that connects $h_D$ and $L(1, \\chi_D)$:\n\\begin{equation*}\nL(1, \\chi_D) = \\frac{\\sqrt{q}}{\\sqrt{|D|}}\\, h_D = q^{-g} h_D \n\\quad \\text{for } D \\in \\mathcal{H}_{2g+1}.\n\\end{equation*}\nFrom Theorem~\\ref{main theorem}, we directly extend the range of uniformity in \\cite[Corollary~1.8]{Lumley}, showing that the tail of the distribution of large values of $h_D$ over $\\mathcal{H}_{2g+1}$ decays doubly exponentially. An analysis towards maximal value for $h_D$ follows from Theorem~\\ref{theorem on resonance method}.\n\nFor $n = 2g + 2$ and $D \\in \\mathcal{H}_{2g+2}$, Artin also proved that\n\\begin{equation*}\nL(1, \\chi_D) = \\frac{q - 1}{\\sqrt{|D|}}\\, h_D R_D,\n\\end{equation*}\nwhere $R_D$ denotes the regulator of $\\mathcal{O}_D$ (see \\cite[Chapter 14]{ROS}). \nAnalogous results hold for the tail of the distribution of $h_D R_D$ as $D$ varies over $\\mathcal{H}_{2g+2}$, improving~\\cite[Corollary~1.9]{Lumley} in a similar manner.\n \\subsection{Essence of the paper} We employ two distinct methods to prove Theorems~\\ref{main theorem} and~\\ref{theorem on resonance method}. \n The study of the distribution of $L(1, \\chi_d)$ over number fields through an underlying probabilistic model was initiated by Granville and Soundararajan~\\cite{GS}. \n Lumley~\\cite{Lumley} studied the tail of the distribution for $L(1, \\chi_D)$ over function fields from the corresponding probabilistic model (see \\cite[eq.~(1.7)]{Lumley}) and computing large moments og $L(1, \\chi_D)$ over the family. \n To extend the range of uniformity stated in Theorem~\\ref{main theorem}, we able to compute much higher moments for the short Euler product $L(1, \\chi_D; N)$ and then utilize the saddle-point method. \n The Lemma~\\ref{full l to its truncation} provides an optimal way to approximate $L(1, \\chi_D)$ by $L(1, \\chi_D; N)$, allowing us to taken a shorter truncation length $N$. \n This step relies crucially on a large sieve type estimate together with Chebyshev’s inequality. \n\n To prove Theorem~\\ref{theorem on resonance method}, we employ the long resonance method for quadratic Dirichlet $L$-functions over number fields, as developed by the author and Maiti in~\\cite{DM}. \n The character sum estimate in Lemma~\\ref{character sum over nonsquare} plays a central role in both results: it extends the range of uniformity as in Theorem \\ref{main theorem} via computing larger moments of $L(1, \\chi_D; N)$ over the family, and it enables the control of long resonators in Theorem~\\ref{theorem on resonance method}. These ideas appear difficult to implement for higher degree $L$-functions but may still be applicable to families of $\\mathrm{GL}(1)$ $L$-functions associated with higher order characters.", + "sketch": "To extend the range of uniformity stated in Theorem~\\ref{main theorem}, we are able to compute much higher moments for the short Euler product $L(1, \\chi_D; N)$ and then utilize the saddle-point method. The Lemma~\\ref{full l to its truncation} provides an optimal way to approximate $L(1, \\chi_D)$ by $L(1, \\chi_D; N)$, allowing us to taken a shorter truncation length $N$. This step relies crucially on a large sieve type estimate together with Chebyshev’s inequality. The character sum estimate in Lemma~\\ref{character sum over nonsquare} plays a central role: it extends the range of uniformity as in Theorem~\\ref{main theorem} via computing larger moments of $L(1, \\chi_D; N)$ over the family.", + "expanded_sketch": "To extend the range of uniformity stated in the main theorem, we are able to compute much higher moments for the short Euler product $L(1, \\chi_D; N)$ and then utilize the saddle-point method. \n\nFor sufficiently large $n$,\n\\[\nL(1, \\chi_D)=L(1, \\chi_D; N)\\left(1+O\\left(\\frac{1}{f(n)\\log n}\\right)\\right)\n\\]\nfor all but at most $q^{\\frac{7n}{10}}$ elements $D\\in \\mathcal{H}_n$, where $N=\\log n+\\log_2 n+3\\log f(n)$, and $f(n) \\to \\infty$ arbitrarily slowly as $n \\to \\infty$, with $f(n) \\ll \\log n$. This provides an optimal way to approximate $L(1, \\chi_D)$ by $L(1, \\chi_D; N)$, allowing us to taken a shorter truncation length $N$. This step relies crucially on a large sieve type estimate together with Chebyshev’s inequality. \n\nLet $\\ell=\\ell_1 \\ell_2^2\\neq \\square\\in \\mathbb{F}_q[t]$ with $\\ell_1$ square-free. For any $\\varepsilon>0$, we have\n\\[\n\\sum_{D\\in \\mathcal{H}_n}\\chi_D(\\ell)\\ll q^{(1/2+\\varepsilon)n}\\exp\\left((d(\\ell_1))^{1-\\varepsilon}\\right)\\prod_{P\\mid \\ell_2}\\left(1+|P|^{-1/2-\\varepsilon}\\right).\n\\]\nThis character sum estimate plays a central role: it extends the range of uniformity in establishing the main theorem via computing larger moments of $L(1, \\chi_D; N)$ over the family.", + "expanded_theorem": "\\label{main theorem}\n \tLet $q \\ge 3$ be fixed. Uniformly for \n$\n \t\\tau \\le \\log n + \\log_2 n - \\theta(n),\n \t$\n \twhere $2\\leq \\theta(n)\\ll \\log_3 n$ is any function tending arbitrarily slowly to infinity as $n \\to \\infty$, we have\n \t\\[\n \t\\phi_n(\\tau)\n \t:= \\frac{1}{|\\mathcal{H}_n|}\n \t\\sum_{\\substack{D \\in \\mathcal{H}_n \\\\ L(1, \\chi_D) \\ge e^{\\gamma}\\tau}} 1\n \t= \\exp\\!\\left(\n \t-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\n \t\\,\\frac{q^{\\,\\tau - C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\n \t\\bigl(1 + o_\\theta(1)\\bigr)\n \t\\right),\n \t\\]\n \twhere the constants $C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ and $C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)$ depend on $\\tau$, $q$, and $\\kappa(\\tau)$, yet remain bounded as the argument varies between $1$ and $q$. \n \tFurther details on these quantities can be found in~\\cite[Theorem~1.3]{Lumley}.", + "theorem_type": [ + "Asymptotic or Limit", + "Universal" + ], + "mcq": { + "question": "Let \\(\\mathbb F_q\\) be a finite field of odd cardinality with fixed \\(q\\ge 3\\), and let \\(\\mathcal H_n\\) denote the set of monic square-free polynomials \\(D\\in \\mathbb F_q[t]\\) of degree \\(n\\) (equivalently, the hyperelliptic ensemble \\(y^2=D(t)\\)). For each \\(D\\in\\mathcal H_n\\), let \\(\\chi_D(f)=\\left(\\frac{f}{D}\\right)\\) be the quadratic character modulo \\(D\\), and let \\(L(s,\\chi_D)\\) be the associated Dirichlet \\(L\\)-function. Define the tail distribution\n\\[\n\\phi_n(\\tau):=\\frac{1}{|\\mathcal H_n|}\\sum_{\\substack{D\\in\\mathcal H_n\\\\ L(1,\\chi_D)\\ge e^{\\gamma}\\tau}}1,\n\\]\nwhere \\(\\gamma\\) is Euler's constant, \\(\\log\\) denotes logarithm base \\(q\\), \\(\\log_j\\) denotes the \\(j\\)-fold iterated base-\\(q\\) logarithm, and \\(\\{x\\}\\) denotes the fractional part of \\(x\\). For any function \\(\\theta(n)\\) with \\(2\\le \\theta(n)\\ll \\log_3 n\\) and \\(\\theta(n)\\to\\infty\\) arbitrarily slowly as \\(n\\to\\infty\\), which statement holds uniformly for all \\(\\tau\\le \\log n+\\log_2 n-\\theta(n)\\)?", + "correct_choice": { + "label": "A", + "text": "\\[\n\\phi_n(\\tau)=\\exp\\!\\left(-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\,\\frac{q^{\\,\\tau-C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\\bigl(1+o_\\theta(1)\\bigr)\\right),\n\\]\nwhere \\(C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\) and \\(C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\) may depend on \\(\\tau\\), \\(q\\), and \\(\\kappa(\\tau)\\), but remain bounded as their argument varies between \\(1\\) and \\(q\\)." + }, + "choices": [ + { + "label": "B", + "text": "\\[\n\\phi_n(\\tau)=\\exp\\!\\left(-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\,\\frac{q^{\\,\\tau-C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\\bigl(1+o_\\theta(1)\\bigr)\\right),\n\\]\nwhere \\(C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\) and \\(C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\) depend only on \\(q\\) and remain bounded as their argument varies between \\(1\\) and \\(q\\)." + }, + { + "label": "C", + "text": "\\[\n\\phi_n(\\tau)=\\exp\\!\\left(-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\,\\frac{q^{\\,\\tau-C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\\,(1+o(1))\\right),\n\\]\nfor some bounded functions \\(C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\) and \\(C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\) as their argument varies between \\(1\\) and \\(q\\)." + }, + { + "label": "D", + "text": "\\[\n\\phi_n(\\tau)=\\exp\\!\\left(-\\,C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\,\\frac{q^{\\,\\tau-C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)}}{\\tau}\\bigl(1+o_\\theta(1)\\bigr)\\right),\n\\]\nuniformly for all \\(\\tau\\le \\log n+\\log_2 n+\\theta(n)\\), where \\(C_0\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\) and \\(C_1\\!\\left(q^{\\{\\log \\kappa(\\tau)\\}}\\right)\\) may depend on \\(\\tau\\), \\(q\\), and \\(\\kappa(\\tau)\\), but remain bounded as their argument varies between \\(1\\) and \\(q\\)." + }, + { + "label": "E", + "text": "\\[\n\\phi_n(\\tau)=\\exp\\!\\left(-\\,C_1\\!\\left(q^{\\log \\kappa(\\tau)}\\right)\\,\\frac{q^{\\,\\tau-C_0\\!\\left(q^{\\log \\kappa(\\tau)}\\right)}}{\\tau}\\bigl(1+o_\\theta(1)\\bigr)\\right),\n\\]\nwhere \\(C_0\\!\\left(q^{\\log \\kappa(\\tau)}\\right)\\) and \\(C_1\\!\\left(q^{\\log \\kappa(\\tau)}\\right)\\) may depend on \\(\\tau\\), \\(q\\), and \\(\\kappa(\\tau)\\), yet remain bounded as the argument varies between \\(1\\) and \\(q\\)." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "quantifier_dependence", + "tampered_component": "dependence of bounded constants on \\(\\tau\\) and \\(\\kappa(\\tau)\\)", + "template_used": "quantifier_dependence" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "dropped the specific \\(o_\\theta(1)\\) uniformity/dependence information while keeping the same main asymptotic shape", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "counting_estimate", + "tampered_component": "upper range of uniformity in \\(\\tau\\) beyond \\(\\log n+\\log_2 n-\\theta(n)\\)", + "template_used": "boundary_range" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "periodic dependence through fractional part \\(\\{\\log \\kappa(\\tau)\\}\\)", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly reveal the correct option. It sets up the theorem context and asks which asymptotic holds, but the exact dependence on the fractional part, the error term, and the range of uniformity are not given away." + }, + "TAS": { + "score": 1, + "justification": "This is very close to asking for the precise statement of a known theorem, so it is partly a restatement task. However, it is not completely tautological because the options differ in subtle but meaningful ways involving uniformity, parameter dependence, and periodicity." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish the exact theorem statement from weaker or false variants, especially regarding the range in tau and the role of {log kappa(tau)}. Still, the item mainly tests recall/recognition of a technical result rather than substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong: they are mathematically close to the correct statement, target realistic failure modes (weakened error term, incorrect dependence of constants, overextended uniformity range, and loss of periodic fractional-part structure), and are clearly distinct." + }, + "total_score": 6, + "overall_assessment": "A technically well-constructed theorem-recognition MCQ with strong distractors and little answer leakage, but it primarily tests precise recall of a result rather than deeper generative reasoning." + } + }, + { + "id": "2511.12595v2", + "paper_link": "http://arxiv.org/abs/2511.12595v2", + "theorems_cnt": 3, + "theorem": { + "env_name": "theorem", + "content": "\\label{open}\nLet $[a_1,b_1],\\cdots,[a_k,b_k]$ be $k$ disjoint intervals. Then as $g\\to \\infty$, the vector of random variables\n$$(N_{g,[a_1,b_1]},\\cdots,N_{g,[a_k,b_k]}):\\H_g(1^{2g-2})\\to \\mathbb{N}_0^k$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=8\\pi(b_i^2-a_i^2)$$\nfor $i=1,\\cdots,k$. That is,\n$$\\mathbb{P}(N_{g,[a_1,b_1]}=n_1,\\cdots,N_{g,[a_k,b_k]}=n_k)=\\prod_{i=1}^k\\frac{\\lambda_{[a_i,b_i]}^{n_i}e^{-\\lambda_{[a_i,b_i]}}}{n_i!}$$", + "start_pos": 23973, + "end_pos": 24547, + "label": "open" + }, + "ref_dict": { + "claim": "\\begin{theorem}[The method of moment from \\cite{bollobas2001random}, Theorem 1.23]\\label{moment}\nLet $\\left\\{(\\Omega_i,\\mathbb{P}_i)\\right\\}_{i\\in \\mathbb{N}}$ be a sequence of probability spaces. \nFor $m\\in \\mathbb{N}$, let $N_{1,i},\\cdots,N_{m,i}:\\Omega_i\\to \\mathbb{N}_0$ be random variables for all $i\\in \\mathbb{N}$\nand suppose there exist $\\lambda_1,\\cdots,\\lambda_m\\in(0,\\infty)$ such that:\n$$\\lim_{i\\to \\infty}\\mathbb{E}[(N_{1,i})_{k_1}\\cdots(N_{m,i})_{k_m}]=\\lambda_1^{k_1}\\cdots\\lambda_m^{k_m}$$\nfor all $k_1\\cdots k_m\\in \\mathbb{N}$. Then\n$$\\lim_{i\\to \\infty}\\mathbb{P}[(N_{1,i})=n_1,\\cdots,(N_{m,i})=n_m]=\\prod^k_{j=1}\\frac{\\lambda_j^{n_j} e^{-\\lambda_j}}{n_j!}.$$\nThat is, $(N_{1,i},\\cdots,N_{m,i}):\\Omega_i \\to \\mathbb{N}^m_0$ converges jointly in distribution to a vector which is independently Poisson distributed with means \n$\\lambda_1,\\cdots,\\lambda_m$.\n\n\\end{theorem}\n\n\\section{Some claims}\\label{claim}\nBefore the proof of the main theorem, we have to make some claims which will simplify the situation.\nWe consider the stratum whose zero order is $O(1)$, and it inplies $|\\kappa|=O(g)$. \n\n\\begin{clm} \\label{clm1}\nA closed saddle connection is non-separable.\n\\end{clm}\n\\begin{proof}\nFor a closed connection $\\gamma$ of $(X,\\omega)$, if it's separable, then it cuts $X$ into $X_1$ and $X_2$, and $\\partial X_1=\\partial X_2=\\gamma$.\nBy Stokes theorem, $\\int_\\gamma \\omega=0$, but since $\\gamma$ is a saddle connection its length is $|\\int_\\gamma \\omega|$, which is a contradiction.\n\\end{proof}\n\n\\begin{clm} \\label{clm2}\nThe probability measure of the subset of $\\H_g(\\kappa)$ which has a closed saddle connection involving $q$ cylinders of multiplicity $p$ \ngoes to $0$ as $g\\to \\infty$, where $p>1$ or $q>0$.\n\\end{clm}\n\n\\begin{proof}\nFrom Theorem \\ref{Siegel of closed}, \nthe Siegel-Veech constant of configuration with closed saddle connection involving $q$ cylinders of multiplicity $p$ is $O(\\frac{1}{g^{2p+q-2}})$ . \nDenote by $\\mathcal{C}^{p,q}_g(\\kappa)[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ the subset of $\\H_g(\\kappa)$ consisting of surfaces which have a closed saddle connection \nof length in the interval $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$\ninvolving $q$ cylinders of multiplicity $p$.\nThen for $p\\geq 2$ or $q\\geq 1$, we have\n $$\\V(\\mathcal{C}^{p,q}_g(\\kappa)[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}])\\leq |\\kappa|O(\\frac{1}{g^{q}})\\frac{b^2-a^2}{g}\\V(\\H_g(\\kappa))=O(\\frac{1}{g})\\V(\\H_g(\\kappa)$$\nwhere the coefficient $|\\kappa|$ is the selection of the zero that the closed saddle connection goes through. \nSo when $g\\to \\infty$ its probability measure goes to $0$ \nand it suffices to consider the closed saddle connection with multiplicity $1$ and no cylinders around it.\n\\end{proof}\n\n\\begin{clm} \\label{clm3}\nThe angle of a closed saddle connection at the zero which it connects is odd multiples of $\\pi$.\n\\end{clm}\n\n\\begin{proof}\nFor an Abelian differential $(X,\\omega)$, if it has a closed saddle connection $\\gamma$ connecting a zero $p$, if its angle at $p$ is $2k\\pi$,\nthen the holonomy of $\\gamma$ and $-\\gamma$ have same direction. But\n$$\\int_\\gamma \\omega=-\\int_{-\\gamma} \\omega.$$\nIt implies $\\int_\\gamma \\omega=0$, which is a contradiction.\n\\end{proof}\n\n\\begin{clm} \\label{clm4}\nIf a closed saddle connection has angle $\\pi$ at one side, it must have a cylinder at the side.\n\\end{clm}\n\n\\begin{proof}\nLet $\\gamma$ be a colsed saddle connection on $(X,\\omega)$ which has angle $\\pi$ at one side. Since $(X,\\omega)$ is oriented, its normal bundle of $\\gamma$ at the side is oriented.\nConsider the exponential map from the bundle to $(X,\\omega)$, which is well-defined on $\\gamma \\times [0,s]$ for some $s$ small sufficiently and the image is a cylinder.\n\n\\end{proof}\n\nCombine claim $2,3$ and $4$, we have that for a closed saddle connection $\\gamma$ on $(X,\\omega)$ in principal stratum, \nsince its total angle around the zero is $4\\pi$ and has to be divided into odd multiples of $\\pi$, \nit must have angle $\\pi$ at one side of $\\gamma$ and $3\\pi$ at another.\nSo there is a cyliner with $\\gamma$ as a boundary. As for the other boundary, if it consists of some open saddle connections, \nthen these saddle connections have angles $\\pi$. That means there are at least two non-homologous saddle connections have angle $\\pi$,\nwhich occurs on a set of measure zero. \nSo we only need to consider the cylinder bounded by curves homologous to $\\gamma$.\nThen from Claim \\ref{clm2} the probability measure of $\\mathcal{C}_g(1^{2g-2})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ has limit\n $$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g(1^{2g-2})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}])}{\\V(\\H_g(1^{2g-2})}=O(\\frac{1}{g})\\to 0.$$\n\nSo we need to consider the question on stratum with higher order zeros. Next we will consider $\\H_g(2^{g-1})$ firstly.\n\n\\section{Surgery}\\label{surgery}\nThis section we will introduce some surgeries that, when combined, can collapse a closed saddle connection.\n\\subsection{Open up a higher-order zero}\nMasur, Rafi and Randecker introduce a surgery in \\cite{masur2024lengths} which is a variation of a surgery from \\cite{eskin2003moduli}. \nThis surgery can collapse a saddle connection which is not closed and has multiplicity $1$. \n\nLet $(X,\\omega)$ be an Abelian differential, $\\sigma$ be an open saddle connection with endpoints $v_1$ and $v_2$, \nwhose orders are $n_1$ and $n_2$ respectively.\nSince the total angle at $v_1$ is $2(n_1+1)\\pi$, \none can extend $\\sigma$ from $v_1$ along its direction to $v_3$ and denote by $\\sigma'$ the geodesic segment $v_1v_3$ \nsuch that $\\ell_\\omega(\\sigma)=\\ell_\\omega(\\sigma')$\nand the angle between $\\sigma$ and $\\sigma'$ is\n$2k_1\\pi$ and $2k_2\\pi$, where $k_1+k_2=n_1+1$. \nIf $\\sigma'$ does not go through other zeros of $(X,\\omega)$, \nwhich means $\\sigma'$ is a ray from $v_1$, the following surgery can be carried out.\n\nCut along $\\sigma+\\sigma'$ and denote the two copies of $\\sigma+\\sigma'$ and $v_1$ by $\\sigma^{\\pm}+(\\sigma')^{\\pm}$ and $v_1^{\\pm}$. \nThen glue $\\sigma^+$ and $(\\sigma')^+$, $\\sigma^-$ and $(\\sigma')^-$. \nThis surgery reduces the order of $v_1$ and constructs a new Abelian differential $(X',\\omega')$. \nOn $(X',\\omega')$, the total angles at $v_1^+$, $v_1^-$ and $v_2$ are $2k_1\\pi$, $2k_2\\pi$, and $2(n_2+2)\\pi$ respectively,\nthat is, the orders of $v_1^+$, $v_1^-$ and $v_2$ are $k_1-1$, $k_2-1$, and $n_2+1$.\nParticularly, if $v_1$ is a simple zero, the surgery collapses the saddle connection. We call this surgery \\emph{collapsing surgery}.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.6\\linewidth]{collapsing}\n \\caption{The collapsing surgery}\n \\label{collapsing}\n\\end{figure}\n\nIn our situation, we need to use the inverse of collapsing surgery which we call \\emph{opening surgery}.\nFrom Claim \\ref{clm2}, it suffices to consider the subset $\\mathcal{C}^{1,0}_g(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nFor an Abelian differential $(X,\\omega)$ in $\\mathcal{C}^{1,0}_g(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$, \nchoose a closed saddle connection $\\gamma$ with length in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ at a double zero $p$.\nWe can reverse the surgery introduced above as follows.\n\nSince the total angle at $p$ is $6\\pi$, from Claim \\ref{clm3} and Claim \\ref{clm4}, the angles on both side of $\\gamma$ at $p$ must be $3\\pi$.\nFix the orientation of $\\gamma$, choose two rays $\\sigma^+$ and $\\sigma^-$ from $p$ such that \n$\\ell_\\omega(\\sigma^+)=\\ell_\\omega(\\sigma^-)=\\ell_\\omega(\\gamma)$,\nand the angle between $\\gamma$ and $\\sigma^+$\nis $\\pi$, the angle between $\\gamma$ and $\\sigma^-$ is $-\\pi$, where the sign is consistent with the orientation of the surface.\nDenote by $q^+$ and $q^-$ the other endpoints of $\\sigma^+$ and $\\sigma^-$.\nThen cut along $\\sigma^+ + \\sigma^-$ and glue the two copies of $\\sigma^+$ and $\\sigma^-$.\nAfter the surgery, the two copies of $p$ become a regular and a simple zero, $q^+$ and $q^-$ are glued to become a new simple zero $q$.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.6\\linewidth]{openinglocally}\n \\caption{The opening surgery}\n \\label{opening}\n\\end{figure}\n\nThis surgery is the inverse of the collapsing surgery above: the double zero is replaced by two simple zeros,\nand it can be carried out except for the surgery locus goes through some zero, \nthat is, it has two non-homologous saddle connections with angle $2\\pi$.\nFrom period mapping, such subset has measure zero. We call the subset on which the surgery can be carried out \\emph{permissible set}, \nand denote it by $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$, from the discussion above we have \n\\begin{proposition}\n$\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ is a full measure subset of $\\mathcal{C}^{1,0}_g(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\n\\end{proposition}\n\nBy the surgery of collapsing this map is one-to-one and denote the opening surgery by $F_1$.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.6\\linewidth]{opening}\n \\caption{The resulting surface}\n \\label{opening surface}\n\\end{figure}\n\n\\begin{remark}\nWe can also choose $\\sigma^{\\pm}$ to make the angle between $\\gamma$ and them are $2\\pi$, \nit suffices to make sure the opening operation can obtain an abelian differential and a smooth loop homotopic to $\\gamma$.\n\\end{remark}\n\\subsection{Move zero along closed curve and pinch}\nWe have constructed the mapping $F_1$, \nwhich takes a closed saddle connection $\\gamma$ at a double zero to two saddle connections $\\gamma_1$ and $\\gamma_2$\nsharing same endpoints $p$ and $q$ which are both simple zeros.\nMoreover, the angles between the two saddle connections at the two simple zeros (on both sides) are all $2\\pi$, \nand if we denote by $\\ell_\\omega(\\gamma)=L$, we have $\\ell_{\\omega'}(\\gamma_1)=L$, $\\ell_{\\omega'}(\\gamma_2)=2L$, where $(X',\\omega')=F_1(X,\\omega)$.\nIf we want to collapse $\\gamma_1$ and $\\gamma_2$ simultaneously, we have to make a surgery to move $q$ along $\\gamma_2$ to adjust the length of $\\gamma_2$.\n\n\\subsubsection{Move the zero locally by period mapping.}\n\nFirst we need to choose a special basis of $H^1(X,\\Sigma;\\mathbb{C})$, where $\\Sigma=\\{p,q,p_2,\\cdots,p_{g-1}\\}$\nis the zero set of $(X',\\omega')=F_1(X,\\omega)$.\nFrom Claim \\ref{clm1}, the homology class $[\\gamma]=[\\gamma_1+\\gamma_2]$ is non-separable.\nSo we can choose a basis of $H^1(X,\\mathbb{C})$\n$$(\\alpha_1,\\beta_1,\\cdots,\\alpha_g,\\beta_g),$$\nwhere $\\alpha_1=[\\gamma]$.\n\nNext we choose the relative homology class\n$$(pq,pp_2\\cdots,pp_{g-1}),$$\nwhere $[\\gamma_1]=pq$, and $pp_i$ is freely homotopic to $0$ in $H^1(X,\\mathbb{C})$. \nTogether they compose a basis of $H^1(X,\\Sigma;\\mathbb{C})$, \nand their holonomy gives a period mapping around $(X',\\omega')$ locally.\n\nLet $\\omega'(\\alpha_1)=(x_1,y_1)$, $\\omega'(\\beta_1)=(x_2,y_2)$, $\\omega'(\\alpha_2)=(x_3,y_3)$\n$\\omega'(\\beta_2)=(x_4,y_4)$.\nNow choose a relative cohomology class $\\upsilon \\in H^1(X,\\Sigma;\\mathbb{C})$ ,\nwhich can be considered as a tangent vector in $T_{\\H_g(1,1,\\cdots,2)}(X',\\omega')$ such that\n\\begin{equation}\\label{moving}\n\\upsilon(\\alpha_1)=(-x_1,-y_1), \\upsilon(\\alpha_2)=(-\\frac{y_2x_1}{y_4},\\frac{x_2y_1}{x_4})\n\\end{equation}\nand on the other basis we assign zero to $\\upsilon$.\n\n\\begin{remark}\\label{explain}\nHere $\\upsilon(\\alpha_1)$ is chosen to guarantee that along the curve $(X',\\omega')+t\\upsilon$ in moduli space the zero moves along $\\gamma$, \nand $\\upsilon(\\alpha_2)$ is chosen to guarantee the resulting surface has area $1$, since the area of $(X,\\omega)$ can be written by\n$$\\int_X|\\omega|^2=\\frac{i}{2}\\sum_i(\\int_{\\alpha_i}\\omega \\int_{\\beta_i}\\overline{\\omega}-\\int_{\\beta_i}\\omega \\int_{\\alpha_i}\\overline{\\omega}).$$\n\\end{remark}\n\nConsider the curve in $\\H_g(1,1,2,,\\cdots,2)$: $(X',\\omega')+t\\upsilon, t\\in [0,1]$. \nIf $\\gamma_1$ and $\\gamma_2$ don't degenerate along the curve, then we obtain \n$$F_2:\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}] \\to \\H_g(1^2,2^{g-1})$$\nsuch that $F_2[(X,\\omega)]=F_1[(X,\\omega)]+\\upsilon$. \nFrom the construction above the resulting surface has two saddle connections \nin the relative homology class $\\gamma_1$ and $\\gamma_2$ with equal length.\n\n\\subsubsection{$F_2$ is well-defined almost everywhere}\nWe have defined a mapping if the saddle connections $\\gamma_1$ and $\\gamma_2$ are preserved along the curve $(X',\\omega')+t\\upsilon, t\\in [0,1]$\n on $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})$.\nNext we will see the mapping can be defined for almost every translation surface in $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})$.\n\nFrom above we know the moving surgery can be realized except for some of $\\gamma_1$ or $\\gamma_2$ degenerate along $(X',\\omega')+t\\upsilon, t\\in [0,1]$.\nSuppose $T$ is the first time when the geodesic in $[\\gamma_1]$ can be represented by $pp_i-p_ip_j-\\cdots-p_kq$,\nsince the saddle connection is smooth on $(X',\\omega')+t\\upsilon,t\\in [0,T]$, the corner of $pp_i-p_ip_j-\\cdots-p_kq$ must be $\\pi$. \nBut the holonomy of $pp_i$ is not changed on $(X',\\omega')+t\\upsilon,t\\in [0,T]$\nThis implies there exists some $pp_i$ such that its holonomy on $(X',\\omega')$ has the same direction with $p_0p_1$. \nAnd by the relation of holonomy under $F_1$, this implies $p_0p_i$ has the same direction with the closed curve on $(X,\\omega)$.\nUnder period mapping, on each local chart this is a measure-zero set.\nSo all such $(X,\\omega)$ is a subset of measure zero in $ \\H_g(2^{g-1})$, also in $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\n\n\\subsubsection{Pinching}\nOn the subset of $\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ where $F_2$ is well-defined,\nwe can cut along $\\gamma_1+\\gamma_2$ and denote the two copies of $p$ and $q$ by $q^+$, $q^-$, $p^+$, $p^-$, \nthe two copies of $\\gamma_1$ and $\\gamma_2$ by $\\gamma^+_1$, $\\gamma^-_1$, $\\gamma^+_2$, $\\gamma^-_2$.\nNotice that the total angles at $q^+$, $q^-$, $p^+$, $p^-$ are all $2\\pi$, \nwhich means $\\gamma^\\pm_1$ have same direction with $\\gamma^\\pm_2$,\nthen glue $\\gamma^+_1$ with $\\gamma^+_2$, $\\gamma^-_1$ with $\\gamma^-_2$, which make $q^+$, $q^-$, $p^+$, $p^-$ become regular points.\nSince $\\gamma_1+\\gamma_2$ is non-separable, this surgery reduces the genus by one, and obtains a new abelian differential \non which we mark two points $p^+$ and $p^-$ so as to find its inverse. \nConsidering the choice of which zero is selected after applying the inverse map, we finally obtain:\n$$F_3: \\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]\\to M\\times \\H_{g-1}(2^{g-2},0,0)\\times A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]},$$\nwhere $M$ is the combinatorial data of choosing one from $g-1$ zeros.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.6\\linewidth]{movingandcut}\n \\caption{Moving and cut}\n \\label{moving and cut}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.6\\linewidth]{pinching}\n \\caption{Pinching}\n \\label{pinching}\n\\end{figure}\n\n\\subsubsection{$F_3$ is measure-preserving.}\nThis surgery can be inverse except for some directions which has saddle connections, which is a measure zero subset $A_0(X,\\omega)$\nin $A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]}$ for every surface in $\\H_{g-1}(2^{g-2},0,0)$: \nchoose a vector $\\kappa$ in $A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]}$, \nconsider the two rays from $p^+$ and $p^-$ whose holonomy is $\\kappa$, if the rays exist, then cut the surface along the two rays\nto obtain two closed loci and glue them. \nLet \n$$\\tilde{\\H}=\\{(X,\\omega, \\tau): (X,\\omega)\\in \\H_{g-1}(2^{g-2},0,0), \\tau \\in A_0(X,\\omega)\\},$$\nwhich is a measure zero subset of $\\H_{g-1}(2^{g-2},0,0)\\times A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]}$.\nSo we have \n$$F_3( \\mathcal{C}^{1,0}_{g,Per}(2^{g-1}))\\subset [\\H_{g-1}(2^{g-2},0,0)\\times (A_{[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]})]\\setminus \\tilde{\\H}.$$\n\nMoreover, under local chart determined by the basis we choose,\n$$F_3(z_1,\\cdots,z_n)=(z_1,z_i+\\upsilon(z_i)),$$ \nwhose Jacobian has deteminant one and thus is measure-preserving.\n\n\\section{Collapse closed saddle connections simultaneously}\nFrom the construction above we can collapse one closed saddle connection on some $(X,\\omega)\\in \\mathcal{C}^{1,0}_{g}(2^{g-1}))$ and obtain a new surface\n$(X',\\omega')\\in \\H_{g-1}(2^{g-2},0,0)$. if we want to collapse any $k$ closed saddle connections, \nwe have to ensure the locus of surgery for these saddle connections are disjoint.\nFor a given interval $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$,\ndefine the \\emph{exception set} $\\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ to be the subset of \n$\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ \nsuch that on every $(X,\\omega)\\in \\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ there are two closed saddle connections with length in\n$[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ whose surgery loci intersect.\nFirst we will show that the measure of exception set goes to zero as genus goes to infinity.\n\n\\subsection{The measure of exception set}\n\nFor $(X,\\omega)\\in \\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$, there are three situations: \n\\begin{itemize}\n \\item[1] There exist two closed saddle connections on $(X,\\omega)$ with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$\n that do not share a zero. And the loci of opening surgery intersect or the two closed saddle connections intersect.\n \\item[2] There exist two closed saddle connections on $(X,\\omega)$ with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ sharing one zero.\n\\end{itemize}\n\nNote that in the first situation, we can find a curve connecting the two zeros with length no more than $\\frac{2b}{\\sqrt{g}}$.\n\nFix $B\\in \\mathbb{R}^+$, define $\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$ be the set of $(X,\\omega)\\in \\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[0,\\frac{B}{\\sqrt{g}}]$ \nsuch that there exist two closed saddle connections on $(X,\\omega)$\nthat do not share a zero, moreover the lengths of the two closed saddle connections and the distance between the two zeros are less than $\\frac{B}{\\sqrt{g}}$.\nDefine $\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$ be the set\nof $(X,\\omega)\\in \\mathcal{C}^{1,0}_{g,Per}(2^{g-1})[0,\\frac{B}{\\sqrt{g}}]$ such that there exists two closed saddle connections with lengths less than $\\frac{B}{\\sqrt{g}}$ sharing one zero.\nObviously for $B\\geq 2b$, we have \n$$ \\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]\\subset \\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}}) \\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}).$$\n\nWe need to compute the measure of $\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$ and $\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$ when $g\\to \\infty$.\nFirst consider $\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$, we have\n\\begin{proposition}\\label{chain}\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))} = 0.$$\n\\end{proposition}\n\n\\begin{proof}\nFor $(X,\\omega) \\in \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$, \nlet $p$ be a zero on $(X,\\omega)$ and $\\gamma_1, \\gamma_2$ be the closed saddle connections with lengths less than $\\frac{B}{\\sqrt{g}}$ at $p$.\n\nAs above we can collapse $\\gamma_1$ and get a new translation surface in $\\H_{g-1}(2^{g-2},0,0)$. \nAfter the surgery $\\gamma_2$ will become a segment connecting two marked regular points which are from the pinching surgery.\nSo the image of mapping $F_3$ on $\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}})$ is in \n$\\H_{g-1,\\frac{B}{\\sqrt{g}}}(2^{g-2},0,0)\\times D(\\frac{B}{\\sqrt{g}})$, \nwhere $\\H_{g-1,\\frac{B}{\\sqrt{g}}}(2^{g-2},0,0)$ is the subset that there exists a segment connnecting the two marked points of length less than \n$\\frac{B}{\\sqrt{g}}$ and $D(\\frac{B}{\\sqrt{g}})$ is the disk of radius $\\frac{B}{\\sqrt{g}}$.\nFrom \\cite[Theorem 1.2]{aggarwal2019large}, the Siegel-Veech constant of saddle connections connecting two fix zeros of order $m_1$ and $m_2$ is\n$$c=(m_1+1)(m_2+1)(1+O(\\frac{1}{g})).$$\nSo we have\n$$\\V(\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))\\leq (g-1)\\frac{B^2}{g}\\V(\\H_{g-1,\\frac{B}{\\sqrt{g}}}(2^{g-2},0,0)) \\leq c(g-1)\\frac{B^4}{g^2}\\V(\\H_{g-1}(2^{g-2},0,0)),$$\nwhere the coefficient $g-1$ is the choice of $p$.\nThen by \\ref{Volume}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}\\leq \\lim_{g\\to \\infty}3c\\frac{B^4}{g}=0$$\n\\end{proof}\n\nFor $\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$ we also have\n\\begin{proposition}\\label{intersect}\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))} = 0.$$\n\\end{proposition}\n\n\\begin{proof}\nFor $(X,\\omega) \\in \\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$, let $p_1, p_2$ be the two zeros on $(X,\\omega)$ with distance less than $\\frac{B}{\\sqrt{g}}$ \nand $\\gamma_1, \\gamma_2$ be the two closed saddle connections connecting $p_1, p_2$.\nFor every $k\\in \\mathbb{N}^+$, denote by $\\mathcal{C}_g^{1,k}(\\frac{B}{\\sqrt{g}})$ the subset of $\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})$\nthat the geodesic between $p_1$ and $p_2$ is a concatenation of $k$ open saddle connections.\nTheir lengths are less than $\\frac{B}{\\sqrt{g}}$.\nChoose the shortest saddle connection, using the surgery in \\cite{masur2024lengths} we can collapse it and get a new translation surface in \n$\\H_g(3,1,2^{g-3})$. This is because the locus, that is the extension of the shortest saddle connection can not intersect the whole geodesic.\nRepeat the surgery until $p_1$ and $p_2$ are collapsed to one zero and we get a map\n$$F:\\mathcal{C}_g^{1,k}(\\frac{B}{\\sqrt{g}})\\to \\mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}})\\times D^k(\\frac{B}{\\sqrt{g}}),$$\nwhere $\\mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}})$ is the subset of $\\H_g(2+k,1^k,2^{g-2-k})$ such that there are\ntwo closed saddle connections $\\gamma_1, \\gamma_2$ connecting the zero of order $2+k$ with lengths less than $\\frac{B}{\\sqrt{g}}$ .\n\nSimilarly one can collapse $\\gamma_1$ and the other will become a saddle connection connecting two zeros. \nNote that the order is larger than $2$, although the collapsing surgery can also be realized, we need to require the angles \non the two sides of the closed saddle connections collapsed, which decide the order of zeros after collapsing. \nWe will explain this situation in detail in Section \\ref{general case}.\n\nLet $\\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}})$ be the subset of \n$\\mathcal{C}_g^{2}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}})$ such that \nthe two angles on both sides of $\\gamma_1$ are $(2b'+1)\\pi$ and $(2b''+1)\\pi$, where $b'+b''=2+k$.\nThen after collapsing we get a map\n$$F':\\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}}) \\to \\H_{g-1,\\frac{B}{\\sqrt{g}}}(b'-1,b''-1,1^k,2^{g-2-k})\\times D(\\frac{B}{\\sqrt{g}}).$$\n\nSimilar to Proposition \\ref{chain}, we have\n$$\\V(\\mathcal{C}_g^{2,b',b''}(2+k,1^k,2^{g-2-k})(\\frac{B}{\\sqrt{g}}))\\leq b'b''\\frac{B^4}{g^2}\\V(\\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k})).$$\n\nSum all $k$ and $(b',b'')$ we have\n$$\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}}))\\leq (g-1)(g-2)\\sum_k \\sum_{(b',b'')}b'b''\\frac{B^4}{g^2}\\V(\\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k}))(\\frac{B^2}{g})^k,$$\nwhere the coefficient $(g-1)(g-2)$ is the choice of $p_1$ and $p_2$.\nThen when $g\\to \\infty$ we have\n$$\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}\\leq\n\\sum_k \\sum_{(b',b'')}b'b''B^4\\frac{\\V(\\H_{g-1}(b'-1,b''-1,1^k,2^{g-2-k}))}{\\V(\\H_g(2^{g-1}))}(\\frac{B^2}{g})^k=O(\\frac{1}{g}).$$\n\\end{proof}\n\nCombine the two proposition we have\n\\begin{corollary}\n$$\\frac{\\V(\\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}])}{\\V(\\H_g(2^{g-1}))}=O(\\frac{1}{g}).$$\n\\end{corollary}\n\n\\subsection{Collapsing simultaneously}\nNow fix a positive integer $K$ and $n_1,\\cdots,n_k$ a partition of $K$.\nWe want to compute the limit of the expectation\n$$(L_{g,[a_1,b_1]})_{n_1}\\cdots (L_{g,[a_k,b_k]})_{n_k},$$\nfor which we will collapse $K$ closed saddle connections in order.\n\nDefine $\\hat{\\mathcal{C}}^{1,0}_{g}(2^{g-1})=(X,\\omega,\\Gamma_1,\\cdots,\\Gamma_k)$, \nwhere $(X,\\omega)\\in \\mathcal{C}^{1,0}_{g}(2^{g-1})$, \nand $\\Gamma_i$ is an ordered list of $n_i$ closed saddle connections $(\\gamma_{1,i},\\cdots,\\gamma_{n_i,i})$ \nwith lengths in $[\\frac{a_i}{\\sqrt{g}},\\frac{b_i}{\\sqrt{g}}]$. \nSuppose $a_1\\leq b_1 \\leq a_2\\leq b_2 \\cdots \\leq a_k \\leq b_k$, let $B=b_k$.\nThen we have\n$$\\bigcup_i \\mathcal{C}^{exc}_{g,Per}(2^{g-1})[\\frac{a_i}{\\sqrt{g}},\\frac{b_i}{\\sqrt{g}}]\\subset \\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}).$$\n\nLet $\\mathcal{C}'_{g,Per}=\\mathcal{C}^{1,0}_{g,Per}(2^{g-1})\\setminus (\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))$,\nwe will adjust the surgery from last section on $\\mathcal{C}'_{g,Per}$ to collapse the closed saddle connections in\n$\\Gamma_1,\\cdots,\\Gamma_k$ in order.\nThe opening and pinching operations are same to the situation of one curve. The only modification is the equation \\ref{moving}:\nby remark \\ref{explain}, we need to choose the moving vector $\\upsilon$ to ensure the area unchanged, \nhence we can choose a basis containing $\\gamma'_1,\\gamma_1\\cdots,\\gamma'_K,\\gamma_K$, \nwhere $\\gamma'_i$ is a saddle connection from opening $\\gamma_i$. \nThen the moving vector $\\upsilon$ can be constructed like equation \\ref{moving}: after defining the value on $\\gamma_i$, \nwe can choose another relative homotopy class and give a value of $\\upsilon$ on it to ensure the area unchanged.\nThis means we open and move the closed saddle connections simultaneously and then pinch them.\n\nFinally we obtain a mapping\n$$\\hat{F}: \\hat{\\mathcal{C}}'_{g,Per}(2^{g-1}) \\to M_K\\cdot \\tilde{\\H}_{g-K}(2^{g-1-K},0^{2K}) \\times \\prod^k_{i=1} (A_{[\\frac{a_i}{\\sqrt{g}},\\frac{b_i}{\\sqrt{g}}]})^{n_i},$$\nwhere $\\hat{\\mathcal{C}}'_{g,Per}(2^{g-1})$ is defined as $\\hat{\\mathcal{C}}^{1,0}_{g}(2^{g-1})$\nand $M_K$ is a combinatorial data consisting of choosing $K$ ordered zeros from $g-1$ zeros to label. As the discussion above, \nwe have \n\\begin{equation*}\n\\begin{aligned}\n&\\int_{\\hat{\\mathcal{C}}'_{g,Per}(2^{g-1})}(L_{g,[a_1,b_1]})_{n_1}\\cdots (L_{g,[a_k,b_k]})_{n_k}d\\mu_{MV}\\\\&=\\V(\\hat{\\mathcal{C}}'_{g}(2^{g-1})))\n\\\\&=|M_K|\\prod^k_{i=1}[\\frac{\\pi(b^2-a^2)}{g}]^{n_i}\\V(\\H_{g-k}(2^{g-1-K}))\\\\\n&\\to \\prod^k_{i=1}[\\pi(b_i^2-a_i^2)]^{n_i}\\V(\\H_{g-K}(2^{g-1-K})), g\\to \\infty,\n\\end{aligned}\n\\end{equation*}\nwhere the limit is because $|M_K|=(g-1)\\cdots (g-K)$ and\n$$\\lim_{g\\to \\infty}\\frac{(g-1)\\cdots (g-K)}{g^{n_1+\\cdots+n_k}}=\\frac{(g-1)\\cdots (g-K)}{g^K}=1.$$\n\nLet $L'_{g,[a_i,b_i]}$ be the restriction of $L_{g,[a_i,b_i]}$ on $\\mathcal{C}'_{g,Per}(2^{g-1})$.\nThen the limit of factorial moment of $(L'_{g,[a_i,b_i]})_i$ is\n$$\\lim_{g\\to \\infty}\\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\\lim_{g\\to \\infty}\\frac{\\V(\\H_{g-K}(2^{g-1-K}))}{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}\\prod^k_{i=1}[\\pi(b_i^2-a_i^2)]^{n_i}.$$\nFrom Proposition \\ref{chain} and \\ref{intersect}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}=1.$$\n\nFrom Theorem \\ref{Volume}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\H_{g-K}(2^{g-1-K}))}{\\V(\\H_{g}(2^{g-1}))}=3^K.$$\nSo we have\n$$\\lim_{g\\to \\infty}\\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\\prod^k_{i=1}[3\\pi(b_i^2-a_i^2)]^{n_i}.$$\nBy theorem \\ref{moment}, $(L'_{g,[a_1,b_1]},\\cdots,L'_{g,[a_k,b_k]})$ converges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\nFrom the property of Poisson distribution, we have\n$$\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)=\\prod_{i=1}^k\\frac{\\lambda_{[a_i,b_i]}^{n_i}e^{-\\lambda_{[a_i,b_i]}}}{n_i!}.$$\nOn the other hand \n$$\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)=\\frac{\\V(\\{(X,\\omega):L'_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})}{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}.$$\nNote that \n\\begin{equation*}\n\\begin{aligned}\n&\\V(\\{(X,\\omega):L_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})-\n\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))\\\\ &\\leq \\V(\\{(X,\\omega):L'_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})\\\\&\\leq\n\\V(\\{(X,\\omega):L_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\}).\n\\end{aligned}\n\\end{equation*}\nFrom Proposition \\ref{chain} and \\ref{intersect}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}=1,\\\\ \\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}=0.$$\nSo\n\\begin{equation*}\n\\begin{aligned}\n&\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k)-\n\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}\\\\ &\\leq \\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}\\\\&\\leq\n\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k).\n\\end{aligned}\n\\end{equation*}\n\nLet $g \\to \\infty$ we have\n$$\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k)=\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k).$$\nSo $(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]})$ also converges jointly in distribution to a vector of random variables \nwith Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\nSince we don't need to deal with simple zeros in the collapsing surgery, \nsamilarly to the proof of theorem \\ref{closed}, we can prove Corollary \\ref{general}.\n\n\\section{General stratum}\\label{general case}\nThis section we consider the stratum $\\H_g(m^{O(g)},1^{2g-2-mO(g)})$, where $m\\geq 3$.\n\nIf a stratum has zeros of order more than $3$, the angles at both sides are not certain. \nBut we can consider some fixed configuration of closed saddle connection.\nFor a configuration $(J,b'_k,b''_k,a'_i,a''_i)$ of closed saddle connection.\nWe have known it suffices to consider multiplicity $1$ without cylinders, then the configuration become $(b',b'')$, where $b'+b''=m$, \nwhich means the angles of the closed saddle connection are $(2b'+1)\\pi$ on one side and $(2b''+1)\\pi$ on the other side.\nNow we choose all such configuration at each $m$-order zeros.\n\nFor $k$ such closed saddle connections, we can use the surgery in Section \\ref{surgery} to collapse them. \nNote that in this situation the opening surgery will replace the zero of order $m$ to two zeros of order $1$ and $m-1$, \nand the angles at both sides of the zero of $m-1$ order are $(2b')\\pi$ and $(2b'')\\pi$. Then after moving and pinching, \nthe simple zero become two regular points and the zero of $m-1$ order become two zeros of order $b'-1$ and $b''-1$.\nFinally we obtain a mapping\n$$\\hat{F}: \\hat{\\mathcal{C}}'_{g,Per}(m^{O(g)},1^{O'(g)})\\to \\tilde{\\H}_{g-K}((b'-1)^K,(b''-1)^K,m^{O(g)-K},1^{O'(g)})\\times \\prod^k_{i=1} (A_{[\\frac{a_i}{\\sqrt{g}},\\frac{b_i}{\\sqrt{g}}]})^{n_i},$$\nwhere $O'(g)=2g-2-mO(g)$.\nAgain we have\n$$\\lim_{g\\to \\infty}\\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\\prod^k_{i=1}(\\frac{(m+1)\\pi(b_i^2-a_i^2)}{b'b''})^{n_i}.$$\n\nAnd similar to the proof of theorem \\ref{closed} we have\n\\begin{theorem}\nFor the stratum $\\H(m^{O(g)},1^{2g-2-mO(g)})$, a configuration $(b',b'')$ and $a,b\\in \\mathbb{R}$,\nlet the random variable\n$$L_{\\mathcal{C},g,[a,b]}:\\H(m^{O(g)},1^{2g-2-mO(g)}) \\to \\mathbb{N}_0$$ \nbe the number of closed saddle connections on $(X,\\omega)$ satisfying the configuration with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nThen for disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{\\mathcal{C},g,[a_1,b_1]},\\cdots,L_{\\mathcal{C},g,[a_k,b_k]}):\\H(m^{O(g)},1^{2g-2-mO(g)}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=\\frac{(m+1)\\pi(b_i^2-a_i^2)}{b'b''}.$$\n\n\\end{theorem}", + "closed": "\\begin{theorem}\\label{closed}\nFor the stratum $\\H(2^{g-1})$ and disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]}):\\H(2^{g-1}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 1963, + "pre_theorem_intro_text": "In 2024, Masur, Rafi and Randecker study the distribution of open saddle connections on a random translation surface in \\cite{masur2024lengths}. \nIn this paper we study the distribution of closed saddle connections.\n\nA translation surface can be denoted by $(X,\\omega)$ where $X$ is a closed Riemann surface of genus $g$ and $\\omega$ is an Abelian differential on $X$.\n$(X,\\omega)$ has a flat structure defined by the conical metric $|\\omega|$, with singularities at the zeros of $\\omega$.\nIf an $|\\omega|$-geodesic of $(X,\\omega)$ connects two zeros (or one zero to itself) of $\\omega$ and has no other zeros in its interior,\nwe call it an open (or closed) saddle connection.\n\nLet $\\H_g$ be the moduli space of unit-area translation surfaces of genus $g$.\n$\\H_g$ has a natural stratification according to the zero order of Abelian differentials.\nParticularly, the stratum consisting of all abelian differentials in $\\H_g$ which have only simple zeros is called principal stratum, \ndenoted by $\\H_g(1^{2g-2})$. For convenience, if a stratum consists of Abelian differentials on which the number of zeros of order $d_i$ is $\\kappa_i$\nwe denote it by \n$$\\H_g(d_1^{\\kappa_1},\\cdots,d_k^{\\kappa_k})$$\nwhere $\\sum^{k}_{i=1}\\kappa_id_i=2g-2$.\n\nTogether with a finite natural measure which is introduced by Masur and Veech denoted by $\\mu_{MV}$, \nevery stratum $\\H_g(\\kappa)$ of $\\H_g$ becomes a probability space\nwith probability measure $\\frac{\\mu_{MV}}{\\operatorname{Vol}{\\H_g(\\kappa)}}$, where $\\operatorname{Vol}{\\H_g(\\kappa)}$ is the measure of total stratum, \nwhich we call the Masur-Veech volume of $\\H_g(\\kappa)$.\n\nGiven $(X,\\omega)\\in \\H_g(1^{2g-2})$ and $a,b\\in \\mathbb{R}^+$,\nlet $N_{g,[a,b]}(X,\\omega)$ be the number of open saddle connections with lengths in the interval $[\\frac{a}{g},\\frac{b}{g}]$ on $(X,\\omega)$.\nThen $N_{g,[a,b]}:\\H_g(1^{2g-2})\\to \\mathbb{N}_0$ becomes a random variable.\nIn \\cite{masur2024lengths}, Masur, Rafi and Randecker proved :", + "context": "In 2024, Masur, Rafi and Randecker study the distribution of open saddle connections on a random translation surface in \\cite{masur2024lengths}. \nIn this paper we study the distribution of closed saddle connections.\n\nA translation surface can be denoted by $(X,\\omega)$ where $X$ is a closed Riemann surface of genus $g$ and $\\omega$ is an Abelian differential on $X$.\n$(X,\\omega)$ has a flat structure defined by the conical metric $|\\omega|$, with singularities at the zeros of $\\omega$.\nIf an $|\\omega|$-geodesic of $(X,\\omega)$ connects two zeros (or one zero to itself) of $\\omega$ and has no other zeros in its interior,\nwe call it an open (or closed) saddle connection.\n\nLet $\\H_g$ be the moduli space of unit-area translation surfaces of genus $g$.\n$\\H_g$ has a natural stratification according to the zero order of Abelian differentials.\nParticularly, the stratum consisting of all abelian differentials in $\\H_g$ which have only simple zeros is called principal stratum, \ndenoted by $\\H_g(1^{2g-2})$. For convenience, if a stratum consists of Abelian differentials on which the number of zeros of order $d_i$ is $\\kappa_i$\nwe denote it by \n$$\\H_g(d_1^{\\kappa_1},\\cdots,d_k^{\\kappa_k})$$\nwhere $\\sum^{k}_{i=1}\\kappa_id_i=2g-2$.\n\nTogether with a finite natural measure which is introduced by Masur and Veech denoted by $\\mu_{MV}$, \nevery stratum $\\H_g(\\kappa)$ of $\\H_g$ becomes a probability space\nwith probability measure $\\frac{\\mu_{MV}}{\\operatorname{Vol}{\\H_g(\\kappa)}}$, where $\\operatorname{Vol}{\\H_g(\\kappa)}$ is the measure of total stratum, \nwhich we call the Masur-Veech volume of $\\H_g(\\kappa)$.\n\nGiven $(X,\\omega)\\in \\H_g(1^{2g-2})$ and $a,b\\in \\mathbb{R}^+$,\nlet $N_{g,[a,b]}(X,\\omega)$ be the number of open saddle connections with lengths in the interval $[\\frac{a}{g},\\frac{b}{g}]$ on $(X,\\omega)$.\nThen $N_{g,[a,b]}:\\H_g(1^{2g-2})\\to \\mathbb{N}_0$ becomes a random variable.\nIn \\cite{masur2024lengths}, Masur, Rafi and Randecker proved :", + "full_context": "In 2024, Masur, Rafi and Randecker study the distribution of open saddle connections on a random translation surface in \\cite{masur2024lengths}. \nIn this paper we study the distribution of closed saddle connections.\n\nA translation surface can be denoted by $(X,\\omega)$ where $X$ is a closed Riemann surface of genus $g$ and $\\omega$ is an Abelian differential on $X$.\n$(X,\\omega)$ has a flat structure defined by the conical metric $|\\omega|$, with singularities at the zeros of $\\omega$.\nIf an $|\\omega|$-geodesic of $(X,\\omega)$ connects two zeros (or one zero to itself) of $\\omega$ and has no other zeros in its interior,\nwe call it an open (or closed) saddle connection.\n\nLet $\\H_g$ be the moduli space of unit-area translation surfaces of genus $g$.\n$\\H_g$ has a natural stratification according to the zero order of Abelian differentials.\nParticularly, the stratum consisting of all abelian differentials in $\\H_g$ which have only simple zeros is called principal stratum, \ndenoted by $\\H_g(1^{2g-2})$. For convenience, if a stratum consists of Abelian differentials on which the number of zeros of order $d_i$ is $\\kappa_i$\nwe denote it by \n$$\\H_g(d_1^{\\kappa_1},\\cdots,d_k^{\\kappa_k})$$\nwhere $\\sum^{k}_{i=1}\\kappa_id_i=2g-2$.\n\nTogether with a finite natural measure which is introduced by Masur and Veech denoted by $\\mu_{MV}$, \nevery stratum $\\H_g(\\kappa)$ of $\\H_g$ becomes a probability space\nwith probability measure $\\frac{\\mu_{MV}}{\\operatorname{Vol}{\\H_g(\\kappa)}}$, where $\\operatorname{Vol}{\\H_g(\\kappa)}$ is the measure of total stratum, \nwhich we call the Masur-Veech volume of $\\H_g(\\kappa)$.\n\nGiven $(X,\\omega)\\in \\H_g(1^{2g-2})$ and $a,b\\in \\mathbb{R}^+$,\nlet $N_{g,[a,b]}(X,\\omega)$ be the number of open saddle connections with lengths in the interval $[\\frac{a}{g},\\frac{b}{g}]$ on $(X,\\omega)$.\nThen $N_{g,[a,b]}:\\H_g(1^{2g-2})\\to \\mathbb{N}_0$ becomes a random variable.\nIn \\cite{masur2024lengths}, Masur, Rafi and Randecker proved :\n\nTogether with a finite natural measure which is introduced by Masur and Veech denoted by $\\mu_{MV}$, \nevery stratum $\\H_g(\\kappa)$ of $\\H_g$ becomes a probability space\nwith probability measure $\\frac{\\mu_{MV}}{\\V{\\H_g(\\kappa)}}$, where $\\V{\\H_g(\\kappa)}$ is the measure of total stratum, \nwhich we call the Masur-Veech volume of $\\H_g(\\kappa)$.\n\nTheir method is based on the relationship between Poisson distribution and its factorial moment \\cite{bollobas2001random}, \nwhich was used first in the work of Mirzakhani–Petri \\cite{mirzakhani2019lengths} to study the distribution of closed hyperbolic geodesics on random hyperbolic surfaces in Teichm\\\"uller space with respect to Weil-Peterson measure.\n\n\\begin{theorem}\\label{closed}\nFor the stratum $\\H(2^{g-1})$ and disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]}):\\H(2^{g-1}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\\end{theorem}\n\\begin{remark}\nWe will expalin the question is trivial for the principal stratum in Section \\ref{claim}: \nthe probability measure of $\\mathcal{C}_g(1^{2g-2})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ is zero when $g\\to \\infty$.\nFor principal stratum, the length interval to be considered should be $[a,b]$.\n\\end{remark}\n\nRecall that a random variable $X:\\Omega \\to \\mathbb{N}_0$ is Poisson distributed with mean $\\lambda$ if\n$$\\mathbb{P}(X=k)=\\frac{\\lambda^k e^{-\\lambda}}{k!}.$$\nThe following theorem from \\cite{bollobas2001random} gives the relationship between factorial moment and distribution for Poisson distribution:\n\\begin{theorem}[The method of moment from \\cite{bollobas2001random}, Theorem 1.23]\\label{moment}\nLet $\\left\\{(\\Omega_i,\\mathbb{P}_i)\\right\\}_{i\\in \\mathbb{N}}$ be a sequence of probability spaces. \nFor $m\\in \\mathbb{N}$, let $N_{1,i},\\cdots,N_{m,i}:\\Omega_i\\to \\mathbb{N}_0$ be random variables for all $i\\in \\mathbb{N}$\nand suppose there exist $\\lambda_1,\\cdots,\\lambda_m\\in(0,\\infty)$ such that:\n$$\\lim_{i\\to \\infty}\\mathbb{E}[(N_{1,i})_{k_1}\\cdots(N_{m,i})_{k_m}]=\\lambda_1^{k_1}\\cdots\\lambda_m^{k_m}$$\nfor all $k_1\\cdots k_m\\in \\mathbb{N}$. Then\n$$\\lim_{i\\to \\infty}\\mathbb{P}[(N_{1,i})=n_1,\\cdots,(N_{m,i})=n_m]=\\prod^k_{j=1}\\frac{\\lambda_j^{n_j} e^{-\\lambda_j}}{n_j!}.$$\nThat is, $(N_{1,i},\\cdots,N_{m,i}):\\Omega_i \\to \\mathbb{N}^m_0$ converges jointly in distribution to a vector which is independently Poisson distributed with means \n$\\lambda_1,\\cdots,\\lambda_m$.\n\nFrom Theorem \\ref{Volume}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\H_{g-K}(2^{g-1-K}))}{\\V(\\H_{g}(2^{g-1}))}=3^K.$$\nSo we have\n$$\\lim_{g\\to \\infty}\\mathbb{E}[(L'_{g,[a_1,b_1]})_{n_1}\\cdots (L'_{g,[a_k,b_k]})_{n_k}]=\\prod^k_{i=1}[3\\pi(b_i^2-a_i^2)]^{n_i}.$$\nBy theorem \\ref{moment}, $(L'_{g,[a_1,b_1]},\\cdots,L'_{g,[a_k,b_k]})$ converges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\nFrom the property of Poisson distribution, we have\n$$\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)=\\prod_{i=1}^k\\frac{\\lambda_{[a_i,b_i]}^{n_i}e^{-\\lambda_{[a_i,b_i]}}}{n_i!}.$$\nOn the other hand \n$$\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)=\\frac{\\V(\\{(X,\\omega):L'_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})}{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}.$$\nNote that \n\\begin{equation*}\n\\begin{aligned}\n&\\V(\\{(X,\\omega):L_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})-\n\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))\\\\ &\\leq \\V(\\{(X,\\omega):L'_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\})\\\\&\\leq\n\\V(\\{(X,\\omega):L_{g,[a_i,b_i]}(X,\\omega)=n_i,i=1,\\cdots,k\\}).\n\\end{aligned}\n\\end{equation*}\nFrom Proposition \\ref{chain} and \\ref{intersect}, we have\n$$\\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}=1,\\\\ \\lim_{g\\to \\infty}\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}=0.$$\nSo\n\\begin{equation*}\n\\begin{aligned}\n&\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k)-\n\\frac{\\V(\\mathcal{C}_g^{1}(\\frac{B}{\\sqrt{g}})\\cup \\mathcal{C}_g^{2}(\\frac{B}{\\sqrt{g}}))}{\\V(\\H_g(2^{g-1}))}\\\\ &\\leq \\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k)\\frac{\\V(\\mathcal{C}'_{g,Per}(2^{g-1}))}{\\V(\\H_g(2^{g-1}))}\\\\&\\leq\n\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k).\n\\end{aligned}\n\\end{equation*}\n\nLet $g \\to \\infty$ we have\n$$\\mathbb{P}(L_{g,[a_1,b_1]}=n_1,\\cdots,L_{g,[a_k,b_k]}=n_k)=\\mathbb{P}(L'_{g,[a_1,b_1]}=n_1,\\cdots,L'_{g,[a_k,b_k]}=n_k).$$\nSo $(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]})$ also converges jointly in distribution to a vector of random variables \nwith Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\nAnd similar to the proof of theorem \\ref{closed} we have\n\\begin{theorem}\nFor the stratum $\\H(m^{O(g)},1^{2g-2-mO(g)})$, a configuration $(b',b'')$ and $a,b\\in \\mathbb{R}$,\nlet the random variable\n$$L_{\\mathcal{C},g,[a,b]}:\\H(m^{O(g)},1^{2g-2-mO(g)}) \\to \\mathbb{N}_0$$ \nbe the number of closed saddle connections on $(X,\\omega)$ satisfying the configuration with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nThen for disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{\\mathcal{C},g,[a_1,b_1]},\\cdots,L_{\\mathcal{C},g,[a_k,b_k]}):\\H(m^{O(g)},1^{2g-2-mO(g)}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=\\frac{(m+1)\\pi(b_i^2-a_i^2)}{b'b''}.$$", + "post_theorem_intro_text_len": 2238, + "post_theorem_intro_text": "Their method is based on the relationship between Poisson distribution and its factorial moment \\cite{bollobas2001random}, \nwhich was used first in the work of Mirzakhani–Petri \\cite{mirzakhani2019lengths} to study the distribution of closed hyperbolic geodesics on random hyperbolic surfaces in Teichm\\\"uller space with respect to Weil-Peterson measure.\n\nAs for translation surface, to use the method, \n\\cite{masur2024lengths} introduces a surgery to collapse the open saddle connections to obtain a new general translation surface in a new stratum.\nThis surgery does not work for closed saddle connections and they proposed the distribution question in the situation of closed saddle connections,\nwhich inspires this work.\nThis paper gives a new surgery to collapse a closed saddle connection, which ensures the same method can be used for closed saddle connections.\n\nLet $L_{g,[a,b]}(X,\\omega)$ be the number of closed saddle connections on $(X,\\omega)$ with lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nDenote by $\\mathcal{C}_g(\\kappa)[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ the subset of $\\H_g(\\kappa)$ consisting of translation\nsurfaces on which there exist closed saddle connections\nwith lengths in $[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$.\nThe main result of this paper is the following theorem \n\n\\begin{theorem}\\label{closed}\nFor the stratum $\\H(2^{g-1})$ and disjoint intervals $[a_1,b_1],\\cdots,[a_k,b_k]$, when $g\\to \\infty$, the random variable sequence\n$$(L_{g,[a_1,b_1]},\\cdots,L_{g,[a_k,b_k]}):\\H(2^{g-1}) \\to \\mathbb{N}^k_0$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=3\\pi(b_i^2-a_i^2).$$\n\\end{theorem}\n\\begin{remark}\nWe will expalin the question is trivial for the principal stratum in Section \\ref{claim}: \nthe probability measure of $\\mathcal{C}_g(1^{2g-2})[\\frac{a}{\\sqrt{g}},\\frac{b}{\\sqrt{g}}]$ is zero when $g\\to \\infty$.\nFor principal stratum, the length interval to be considered should be $[a,b]$.\n\\end{remark}\n\n\\begin{corollary}\\label{general}\nLet $\\H_g(2^{O(g)},1^{2g-2-2O(g)})$ be a stratum with $O(g)$ double zeros, then the result in Theorem \\ref{closed} is also true.\n\\end{corollary}", + "sketch": "Their method is based on the relationship between Poisson distribution and its factorial moment \\cite{bollobas2001random}, which was used first in the work of Mirzakhani--Petri \\cite{mirzakhani2019lengths}. For translation surfaces, \\cite{masur2024lengths} introduces a surgery to collapse the open saddle connections to obtain a new general translation surface in a new stratum. \"This surgery does not work for closed saddle connections\"; the paper \"gives a new surgery to collapse a closed saddle connection, which ensures the same method can be used for closed saddle connections.\"", + "expanded_sketch": "Their method is based on the relationship between Poisson distribution and its factorial moment \\cite{bollobas2001random}, which was used first in the work of Mirzakhani--Petri \\cite{mirzakhani2019lengths}. For translation surfaces, \\cite{masur2024lengths} introduces a surgery to collapse the open saddle connections to obtain a new general translation surface in a new stratum. \"This surgery does not work for closed saddle connections\"; the paper \"gives a new surgery to collapse a closed saddle connection, which ensures the same method can be used for closed saddle connections.\"", + "expanded_theorem": "\\label{open}\nLet $[a_1,b_1],\\cdots,[a_k,b_k]$ be $k$ disjoint intervals. Then as $g\\to \\infty$, the vector of random variables\n$$(N_{g,[a_1,b_1]},\\cdots,N_{g,[a_k,b_k]}):\\H_g(1^{2g-2})\\to \\mathbb{N}_0^k$$\nconverges jointly in distribution to a vector of random variables with Poisson distributions of means $\\lambda_{[a_i,b_i]}$, where\n$$\\lambda_{[a_i,b_i]}=8\\pi(b_i^2-a_i^2)$$\nfor $i=1,\\cdots,k$. That is,\n$$\\mathbb{P}(N_{g,[a_1,b_1]}=n_1,\\cdots,N_{g,[a_k,b_k]}=n_k)=\\prod_{i=1}^k\\frac{\\lambda_{[a_i,b_i]}^{n_i}e^{-\\lambda_{[a_i,b_i]}}}{n_i!}$$,", + "theorem_type": [ + "Asymptotic or Limit", + "Universal" + ], + "mcq": { + "question": "Let H_g(1^{2g-2}) be the principal stratum of unit-area translation surfaces of genus g, consisting of pairs (X, omega) where X is a closed Riemann surface of genus g and omega is an Abelian differential with only simple zeros, equipped with the normalized Masur-Veech probability measure. For a,b > 0 and (X, omega) in H_g(1^{2g-2}), let N_{g,[a,b]}(X, omega) be the number of open saddle connections on (X, omega) (that is, |omega|-geodesic segments joining two zeros of omega and having no zeros in their interior) whose lengths lie in [a/g, b/g]. For fixed disjoint intervals [a_1,b_1], ..., [a_k,b_k] in R_{>0}, which statement holds as g -> infinity for the random vector (N_{g,[a_1,b_1]}, ..., N_{g,[a_k,b_k]})?", + "correct_choice": { + "label": "A", + "text": "The vector (N_{g,[a_1,b_1]}, ..., N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means lambda_[a_i,b_i] = 8*pi*(b_i^2 - a_i^2) for i = 1, ..., k. Equivalently, for every n_1, ..., n_k in N_0, lim_{g -> infinity} P(N_{g,[a_1,b_1]} = n_1, ..., N_{g,[a_k,b_k]} = n_k) = product_{i=1}^k (lambda_[a_i,b_i]^{n_i} e^{-lambda_[a_i,b_i]} / n_i!)." + }, + "choices": [ + { + "label": "B", + "text": "The vector (N_{g,[a_1,b_1]}, ..., N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means lambda_[a_i,b_i] = 8*pi*(b_i - a_i) for i = 1, ..., k. Equivalently, for every n_1, ..., n_k in N_0, lim_{g -> infinity} P(N_{g,[a_1,b_1]} = n_1, ..., N_{g,[a_k,b_k]} = n_k) = product_{i=1}^k (lambda_[a_i,b_i]^{n_i} e^{-lambda_[a_i,b_i]} / n_i!)." + }, + { + "label": "C", + "text": "For each fixed i = 1, ..., k, the random variable N_{g,[a_i,b_i]} converges in distribution, as g -> infinity, to a Poisson random variable with mean 8*pi*(b_i^2 - a_i^2)." + }, + { + "label": "D", + "text": "The vector (N_{g,[a_1,b_1]}, ..., N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of Poisson random variables with means lambda_[a_i,b_i] = 8*pi*(b_i^2 - a_i^2) for i = 1, ..., k, but the limiting variables need not be independent; equivalently, only the marginal laws are asymptotically Poisson." + }, + { + "label": "E", + "text": "The vector (N_{g,[a_1,b_1]}, ..., N_{g,[a_k,b_k]}) converges jointly in distribution to a vector of independent Poisson random variables with means lambda_[a_i,b_i] = 8*pi*(b_i^2 - a_i^2), provided the intervals [a_1,b_1], ..., [a_k,b_k] are pairwise disjoint and satisfy b_i < a_{i+1} with all endpoints bounded away from 0 independently of g." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "B" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "counting_estimate", + "tampered_component": "quadratic-length-intensity", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "finiteness", + "tampered_component": "joint-independence conclusion", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "finiteness", + "tampered_component": "factorial-moment-implies-independence", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "extra-endpoint-separation hypothesis", + "template_used": "stronger_trap" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the objects carefully but does not explicitly state the limiting law, independence, or the exact intensity, so the correct answer is not leaked." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question: it essentially asks for the precise asymptotic statement. However, it is not a pure verbatim restatement because the choices force discrimination between nearby variants (joint vs marginal convergence, independence, exact mean, extra hypotheses)." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to identify the strongest correct conclusion among plausible alternatives, but the task mainly tests recognition of the exact theorem rather than substantial derivation or synthesis from first principles." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic failure modes: wrong scaling of the mean, confusing marginal Poisson limits with joint independence, weakening independence, and adding unnecessary technical assumptions." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with good distractors and little answer leakage, but it leans more toward precise recall of a known result than toward genuinely generative mathematical reasoning." + } + }, + { + "id": "2511.12549v2", + "paper_link": "http://arxiv.org/abs/2511.12549v2", + "theorems_cnt": 2, + "theorem": { + "env_name": "Theorem", + "content": "\\label{th:main} \nThe trivariate statistic {\\rm (}$\\ides$, $\\des$, $\\maj${\\rm )} are equidistributed over the set of $n$-Andr\\'e I permutations, $n$-Andr\\'e II permutations and $n$-simsun permutations, i.e.,\n \\begin{align*}\n\\sum_{\\sigma \\in \\AndI_n} s^{\\ides(\\sigma)}t^{\\des(\\sigma)} q^{\\maj(\\sigma)}\n\t&=\\sum_{\\sigma \\in \\AndII_n} s^{\\ides(\\sigma)}t^{\\des(\\sigma)} q^{\\maj(\\sigma)}\\\\[5pt]\n &=\\sum_{\\sigma \\in \\RS_{n}} s^{\\ides(\\sigma)}t^{\\des(\\sigma)} q^{\\maj(\\sigma)}.\n\\end{align*}", + "start_pos": 12190, + "end_pos": 12711, + "label": "th:main" + }, + "ref_dict": { + "thm:ideseq:3": "\\begin{Theorem}\\label{thm:ideseq:3}\nLet $\\sigma \\in \\mathfrak{S}_j$ and $\\tau\\in \\mathfrak{S}_k$\nbe two permutations with \n$\\ides(\\sigma)=j'$ and $\\ides(\\tau)=k'$. Then \n\\begin{align} \\label{eqn:ideseq:pfc}\n\\sum_{\\mu \\in \\sigma \\lozenge \\tau} t^{\\ides(\\mu)}&=\\sum_{i\\geq 1} \\binom{k-k'+j'+1}{i+j'-k'} \\binom{j-j'+k'-1}{i-1} t^{i+j'},\\\\[5pt]\n\\sum_{\\mu \\in \\sigma \\vartriangle \\tau} t^{\\ides(\\mu)}&=\\sum_{i\\geq 1} \\binom{k-k'+j'}{i+j'-k'} \\binom{j-j'+k'-1}{i-1} t^{i+j'}, \\label{eqn:ideseq:pfccc} \\\\[5pt]\n\\sum_{\\mu \\in \\sigma \\triangledown \\tau} t^{\\ides(\\mu)}&=\\sum_{i\\geq 1} \\binom{k-k'+j'}{i+j'-k'} \\binom{j-j'+k'-1}{i-1} t^{i+j'}. \\label{eqn:ideseq:pfcc}\n\\end{align} \n\n\\end{Theorem}", + "th:equi-shape": "\\begin{Theorem}\\label{th:equi-shape} For $n\\geq 1$ and any unlabeled binary tree $T \\in \\URL_n$, we have \n \\begin{align}\n \t \\sum_{\\sigma \\in \\AndI(T)} s^{\\ides(\\sigma)} \\overset{(a)}{=}\\sum_{\\sigma \\in \\AndII(T)} s^{\\ides(\\sigma)} \\overset{(b)}{=}\\sum_{\\sigma \\in \\RS(T)} s^{\\ides(\\sigma)}.\n\\end{align}\n\\end{Theorem}", + "th:equi-shape-des": "\\begin{Proposition} \\label{th:equi-shape-des} Let $\\sigma$ be a permutation, and let $T_\\sigma=\\Psi(\\sigma)$ be the increasing binary tree corresponding to $\\sigma$ under the bijection $\\Psi$. Then the descent set $\\Des(\\sigma)$ of $\\sigma$ is completely determined by the shape of $T_\\sigma$. \n\\end{Proposition}", + "thm:shuffspe": "\\begin{Theorem} \\label{thm:shuffspe} Assume that $\\delta \\in \\mathcal{S}_m$ and $\\pi \\in \\mathcal{S}_n$ are two disjoint permutations, where $\\des(\\delta) = r$ and $\\des(\\pi) = s$. Moreover, $\\delta_1<\\delta_2$ and all of the elements of $\\delta$ are larger than the elements of $\\pi$. Then \n\\begin{align*}\n (1)& \\sum_{\\substack{\\alpha \\in \\pi \\shuffle_l \\delta \\\\ \\mathrm{des}(\\alpha) = d}} q^{\\mathrm{maj}(\\alpha)}= {m-r+s \\brack d-r} {n-s+r-1\\brack d-s-1} \\times q^{\\mathrm{maj}(\\delta) + \\mathrm{maj}(\\pi) + (d - s)(d - r)}, \\\\[5pt]\n (2)& \\sum_{\\substack{\\alpha \\in \\pi \\shuffle_{ls} \\delta \\\\ \\mathrm{des}(\\alpha) = d}} q^{\\mathrm{maj}(\\alpha)}= {m-r+s-1 \\brack d-r} {n-s+r-1\\brack d-s-1} \\times q^{\\mathrm{maj}(\\delta) + \\mathrm{maj}(\\pi) + (d - s)(d - r)}, \\\\[5pt] \n (3) & \\sum_{\\substack{\\alpha \\in \\pi \\shuffle_{ll} \\delta \\\\ \\mathrm{des}(\\alpha) = d}} q^{\\mathrm{maj}(\\alpha)}= {m-r+s-1 \\brack d-r} {n-s+r-1\\brack d-s-1} \\times q^{\\mathrm{maj}(\\delta) + \\mathrm{maj}(\\pi) + (d- s+1)(d - r)}. \n\\end{align*}\n\\end{Theorem}", + "th:main": "\\begin{Theorem}\\label{th:main} \nThe trivariate statistic {\\rm (}$\\ides$, $\\des$, $\\maj${\\rm )} are equidistributed over the set of $n$-Andr\\'e I permutations, $n$-Andr\\'e II permutations and $n$-simsun permutations, i.e.,\n \\begin{align*}\n\\sum_{\\sigma \\in \\AndI_n} s^{\\ides(\\sigma)}t^{\\des(\\sigma)} q^{\\maj(\\sigma)}\n\t&=\\sum_{\\sigma \\in \\AndII_n} s^{\\ides(\\sigma)}t^{\\des(\\sigma)} q^{\\maj(\\sigma)}\\\\[5pt]\n &=\\sum_{\\sigma \\in \\RS_{n}} s^{\\ides(\\sigma)}t^{\\des(\\sigma)} q^{\\maj(\\sigma)}.\n\\end{align*}\n\n\\end{Theorem}" + }, + "pre_theorem_intro_text_len": 6593, + "pre_theorem_intro_text": "In the field of enumerative combinatorics, several kinds of permutations are counted by {\\it Euler numbers}, such as {\\it alternating permutations}, {\\it Andr\\'e I and II permutations}, and {\\it simsun permutations}. {\\it Euler numbers}, denoted by $E_n$, are a sequence of integers that arise in the Taylor series expansions of $\\sec(x)+\\tan(x)$. Their combinatorial significance was cemented by the work of {Andr\\'e} in the late 19th century \\cite{andre1881permutations}. Andr\\'e proved that $E_n$ counts the number of {\\it alternating permutations} of length $n$ (see \\cite{stanley2009survey}), which are permutations $\\sigma = \\sigma_1\\sigma_2\\ldots\\sigma_n$ satisfying $\\sigma_1>\\sigma_2<\\sigma_3>\\cdots$. \n\n\\medskip\n\nAndr\\'e permutations were first introduced by Foata and Sch\\\"utzenberger and further studied by Strehl \\cite{Str74} and Foata and Strehl \\cite{FSt74, FSt76}. For clarity, we will work with permutations of length $n$ for which each permutation is a sequence of $n$ distinct integers not necessarily from 1 to $n$. The empty word $e$ and any single-letter word are defined as both {\\it Andr\\'e I permutations} and {\\it Andr\\'e II permutations}.\nFor a permutation $\\sigma=\\sigma_1\\sigma_2\\cdots \\sigma_n$ ($n\\geq 2$) of length $n$, we decompose it as $\\sigma=\\tau\\,\\min(\\sigma)\\,\\tau'$. Here $\\sigma$ is the concatenation of a left factor~$\\tau$, followed by the minimum letter $\\min(\\sigma)$, and a right factor $\\tau'$. Then, $\\sigma$ is called an {\\it Andr\\'e I permutation} (resp. {\\it Andr\\'e II permutation}) if both $\\tau$ and $\\tau'$ are Andr\\'e I permutations (resp. Andr\\'e II permutations), and the maximum letter of the subword $\\tau\\tau'$ lies in $\\tau'$ (resp. the minimum letter of $\\tau\\tau'$ lies in $\\tau'$).\n\n\\medskip\n\nThe set of all Andr\\'e I permutations on the set $[n]:=\\{1,2,\\ldots, n\\}$ is denoted by $\\AndI_{n}$ and the set of Andr\\'e II permutations on the set $[n]$ is denoted by $\\AndII_{n}$. This inductive definition immediately reveals a connection to the Euler numbers, as it can be shown that the number of Andr\\'e I permutations and Andr\\'e II permutations on the set $[n]$ are equal, i.e., $E_n=|\\AndI_n| = |\\AndII_n|$. \n\n\\smallskip\nAndr\\'e I permutations for $n\\leq 5$ are listed below:\n\n$n=1$:\\quad 1;\\qquad $n=2$:\\quad 12;\\qquad\n$n=3$:\\quad 123, 213;\n\n$n=4$:\\quad 1234, 1324, 2314, 2134, 3124;\n\n$n=5$:\\quad 12345, 12435, 13425, 23415, 13245, 14235, 34125,\n24135,\\hfil\\break\n\\indent\\hphantom{$n=5$:\\quad}23145, 21345, 41235, 31245, 21435, 32415, 41325, 31425.\n\n\\smallskip\nAndr\\'e II permutations for $n\\leq 5$ are listed below:\n\n$n=1$:\\quad 1;\\qquad $n=2$:\\quad 12;\\qquad\n$n=3$:\\quad 123, 312;\n\n$n=4$:\\quad 1234, 1423, 3412, 4123, 3124;\n\n$n=5$:\\quad 12345, 12534, 14523, 34512, 15234, 14235, 34125,\n45123,\\hfil\\break\n\\indent\\hphantom{$n=5$:\\quad}35124, 51234, 41235, 31245, 51423, 53412, 41523, 31524.\n\n\\medskip\n\n Simsun permutations were introduced by \n Rodica Simion and Sheila Sundaram in a series of studies of homology representations of the symmetric group \\cite{sundaram1995homology,sundaram1996plethysm}. To better elaborate on our results, we adopt the following definition of simsun permutations. A permutation $\\sigma = \\sigma_1\\sigma_2\\ldots\\sigma_n$ on the set $[n]$ is called a simsun permutation if $\\sigma_n=n$ and it contains no double descents, and this property is preserved after removing the elements $n, n-1, n-2, \\ldots, 1$ in order. \nFor example, it is easy to see that $\\sigma= 21473658$ is a simsun permutation since $21473658$, $2147365$, $214365$, $21435$, $2143$, $213$, $21$, $1$ have no double descents. Recall that an index $i$ (where $1 \\le i < n$) is called a {\\it descent} of a permutation $\\sigma=\\sigma_1\\ldots \\sigma_n$ if $\\sigma_i > \\sigma_{i+1}$ and an index $i$ (where $1 \\le i \\leq n-2$) is called a {\\it double descent} if $\\sigma_i > \\sigma_{i+1}>\\sigma_{i+2}$.\n\n Notably, if one removes the last element from a simsun permutation as defined here, the resulting permutation aligns with the original definition of simsun permutations due to Simion and Sundaram.\n\nThe set of all simsun permutations on the set $[n]$ is denoted by $\\RS_{n}$. A remarkable property of simsun permutations is that $|\\RS_n|=E_{n}$. The notation $\\RS_{n}$ was first adopted by Chow \nand Shiu \\cite{chow2011counting}.\n\nSimsun permutations for $n\\leq 5$ are listed below:\n\n$n=1$:\\quad 1;\\qquad $n=2$:\\quad 12; \\qquad $n=3$:\\quad 123, 213;\n\n$n=4$:\\quad 1234, 1324, 2134, 2314, 3124;\n\n $n=5$:\\quad 12345, 12435, 13245, 13425, 14235, 21345, 21435, 23145,\\hfil\\break\n\\indent\\hphantom{$n=5$:\\quad}23415, 24135, 31245, 31425, 34125, 41235, 41325, 42315. \n\n\\medskip\n\nAndr\\'e permutations and simsun permutations provide new combinatorial interpretations for the Euler numbers. \nThey play an important role in the study of $cd$-indices of simplicial Eulerian posets. For results along this line, please see \\cite{bayer2019cd,bayer1991new,He96, HR98, karu2006cd,purtill1993andre,stanley1994flag}. \nOther properties about Andr\\'e permutations and simsun permutations have been extensively studied by Barnabei et al. \\cite{barnabei2020permutations}, Chow and Shiu \\cite{chow2011counting}, Deutsch-Elizalde \\cite{deutsch2012restricted}, Disanto \\cite{Di14}, Foata and the first author \\cite{FH01} and so on. In particular, by constructing a bijection between the set of Andr\\'e I permutations and the set of simsun permutations, Chow and Shiu \\cite{chow2011counting} observed that the number of descents are equidistributed over Andr\\'e I permutations and simsun permutations. Specifically, let $\\des(\\sigma)$ denote the number of descents of $\\sigma$, they showed that \n$$\n\t \\sum_{\\sigma \\in \\AndI_n} t^{\\des(\\sigma)} =\\sum_{\\sigma \\in \\RS_{n}} t^{\\des(\\sigma)}.\n$$\n\nIn this paper, we show that the number of inverse descents are also equidistributed over Andr\\'e permutations and simsun permutations. The number of inverse descents of a permutation $\\sigma$ is simply the number of descents of its inverse permutation $\\sigma^{-1}$, namely, $\n\\mathrm{ides}(\\sigma) = \\mathrm{des}(\\sigma^{-1}).$ In fact, we show that the trivariate statistic ($\\ides$, $\\des$, $\\maj$) are equidistributed over Andr\\'e permutations and simsun permutations, where {\\it the major index} $\\maj(\\sigma)$ of $\\sigma$ is defined to be the sum of its descents of $\\sigma$. For brevity, we adopt the notation \n $n$-Andr\\'e permutations for Andr\\'e permutations on $[n]$ and $n$-simsun permutations for simsun permutations on $[n]$.\n\\medskip \n\nOur main result is as follows.", + "context": "In the field of enumerative combinatorics, several kinds of permutations are counted by {\\it Euler numbers}, such as {\\it alternating permutations}, {\\it Andr\\'e I and II permutations}, and {\\it simsun permutations}. {\\it Euler numbers}, denoted by $E_n$, are a sequence of integers that arise in the Taylor series expansions of $\\sec(x)+\\tan(x)$. Their combinatorial significance was cemented by the work of {Andr\\'e} in the late 19th century \\cite{andre1881permutations}. Andr\\'e proved that $E_n$ counts the number of {\\it alternating permutations} of length $n$ (see \\cite{stanley2009survey}), which are permutations $\\sigma = \\sigma_1\\sigma_2\\ldots\\sigma_n$ satisfying $\\sigma_1>\\sigma_2<\\sigma_3>\\cdots$.\n\nThe set of all Andr\\'e I permutations on the set $[n]:=\\{1,2,\\ldots, n\\}$ is denoted by $\\AndI_{n}$ and the set of Andr\\'e II permutations on the set $[n]$ is denoted by $\\AndII_{n}$. This inductive definition immediately reveals a connection to the Euler numbers, as it can be shown that the number of Andr\\'e I permutations and Andr\\'e II permutations on the set $[n]$ are equal, i.e., $E_n=|\\AndI_n| = |\\AndII_n|$.\n\nThe set of all simsun permutations on the set $[n]$ is denoted by $\\RS_{n}$. A remarkable property of simsun permutations is that $|\\RS_n|=E_{n}$. The notation $\\RS_{n}$ was first adopted by Chow \nand Shiu \\cite{chow2011counting}.\n\nAndr\\'e permutations and simsun permutations provide new combinatorial interpretations for the Euler numbers. \nThey play an important role in the study of $cd$-indices of simplicial Eulerian posets. For results along this line, please see \\cite{bayer2019cd,bayer1991new,He96, HR98, karu2006cd,purtill1993andre,stanley1994flag}. \nOther properties about Andr\\'e permutations and simsun permutations have been extensively studied by Barnabei et al. \\cite{barnabei2020permutations}, Chow and Shiu \\cite{chow2011counting}, Deutsch-Elizalde \\cite{deutsch2012restricted}, Disanto \\cite{Di14}, Foata and the first author \\cite{FH01} and so on. In particular, by constructing a bijection between the set of Andr\\'e I permutations and the set of simsun permutations, Chow and Shiu \\cite{chow2011counting} observed that the number of descents are equidistributed over Andr\\'e I permutations and simsun permutations. Specifically, let $\\des(\\sigma)$ denote the number of descents of $\\sigma$, they showed that \n$$\n \\sum_{\\sigma \\in \\AndI_n} t^{\\des(\\sigma)} =\\sum_{\\sigma \\in \\RS_{n}} t^{\\des(\\sigma)}.\n$$\n\nIn this paper, we show that the number of inverse descents are also equidistributed over Andr\\'e permutations and simsun permutations. The number of inverse descents of a permutation $\\sigma$ is simply the number of descents of its inverse permutation $\\sigma^{-1}$, namely, $\n\\mathrm{ides}(\\sigma) = \\mathrm{des}(\\sigma^{-1}).$ In fact, we show that the trivariate statistic ($\\ides$, $\\des$, $\\maj$) are equidistributed over Andr\\'e permutations and simsun permutations, where {\\it the major index} $\\maj(\\sigma)$ of $\\sigma$ is defined to be the sum of its descents of $\\sigma$. For brevity, we adopt the notation \n $n$-Andr\\'e permutations for Andr\\'e permutations on $[n]$ and $n$-simsun permutations for simsun permutations on $[n]$.\n\\medskip\n\nOur main result is as follows.", + "full_context": "In the field of enumerative combinatorics, several kinds of permutations are counted by {\\it Euler numbers}, such as {\\it alternating permutations}, {\\it Andr\\'e I and II permutations}, and {\\it simsun permutations}. {\\it Euler numbers}, denoted by $E_n$, are a sequence of integers that arise in the Taylor series expansions of $\\sec(x)+\\tan(x)$. Their combinatorial significance was cemented by the work of {Andr\\'e} in the late 19th century \\cite{andre1881permutations}. Andr\\'e proved that $E_n$ counts the number of {\\it alternating permutations} of length $n$ (see \\cite{stanley2009survey}), which are permutations $\\sigma = \\sigma_1\\sigma_2\\ldots\\sigma_n$ satisfying $\\sigma_1>\\sigma_2<\\sigma_3>\\cdots$.\n\nThe set of all Andr\\'e I permutations on the set $[n]:=\\{1,2,\\ldots, n\\}$ is denoted by $\\AndI_{n}$ and the set of Andr\\'e II permutations on the set $[n]$ is denoted by $\\AndII_{n}$. This inductive definition immediately reveals a connection to the Euler numbers, as it can be shown that the number of Andr\\'e I permutations and Andr\\'e II permutations on the set $[n]$ are equal, i.e., $E_n=|\\AndI_n| = |\\AndII_n|$.\n\nThe set of all simsun permutations on the set $[n]$ is denoted by $\\RS_{n}$. A remarkable property of simsun permutations is that $|\\RS_n|=E_{n}$. The notation $\\RS_{n}$ was first adopted by Chow \nand Shiu \\cite{chow2011counting}.\n\nAndr\\'e permutations and simsun permutations provide new combinatorial interpretations for the Euler numbers. \nThey play an important role in the study of $cd$-indices of simplicial Eulerian posets. For results along this line, please see \\cite{bayer2019cd,bayer1991new,He96, HR98, karu2006cd,purtill1993andre,stanley1994flag}. \nOther properties about Andr\\'e permutations and simsun permutations have been extensively studied by Barnabei et al. \\cite{barnabei2020permutations}, Chow and Shiu \\cite{chow2011counting}, Deutsch-Elizalde \\cite{deutsch2012restricted}, Disanto \\cite{Di14}, Foata and the first author \\cite{FH01} and so on. In particular, by constructing a bijection between the set of Andr\\'e I permutations and the set of simsun permutations, Chow and Shiu \\cite{chow2011counting} observed that the number of descents are equidistributed over Andr\\'e I permutations and simsun permutations. Specifically, let $\\des(\\sigma)$ denote the number of descents of $\\sigma$, they showed that \n$$\n \\sum_{\\sigma \\in \\AndI_n} t^{\\des(\\sigma)} =\\sum_{\\sigma \\in \\RS_{n}} t^{\\des(\\sigma)}.\n$$\n\nIn this paper, we show that the number of inverse descents are also equidistributed over Andr\\'e permutations and simsun permutations. The number of inverse descents of a permutation $\\sigma$ is simply the number of descents of its inverse permutation $\\sigma^{-1}$, namely, $\n\\mathrm{ides}(\\sigma) = \\mathrm{des}(\\sigma^{-1}).$ In fact, we show that the trivariate statistic ($\\ides$, $\\des$, $\\maj$) are equidistributed over Andr\\'e permutations and simsun permutations, where {\\it the major index} $\\maj(\\sigma)$ of $\\sigma$ is defined to be the sum of its descents of $\\sigma$. For brevity, we adopt the notation \n $n$-Andr\\'e permutations for Andr\\'e permutations on $[n]$ and $n$-simsun permutations for simsun permutations on $[n]$.\n\\medskip\n\nOur main result is as follows.\n\n\\begin{document}\n \\begin{abstract}\nSimsun permutations, Andr\\'e I permutations and Andr\\'e II permutations are three combinatorial models for Euler numbers. It's known that\nthe descent statistic is equidistributed \nover the set of Andr\\'e I permutations and the set of simsun permutations. In this paper, we prove that\nthe trivariate statistic $(\\ides, \\des, \\maj)$, comprising the inverse descent, descent, and major index,\nare equidistributed over these three sets. This result is equivalent to showing that the inverse descent is equidistributed over these three sets that share the same tree shape. The proof of the equidistribution of the inverse descent over the set of Andr\\'e I permutations and the set of Andr\\'e II permutations with the same tree shape reduces to establishing new refinements of Stanley's shuffle theorem. \n \\end{abstract}\n \\maketitle\n\nAndr\\'e permutations and simsun permutations provide new combinatorial interpretations for the Euler numbers. \nThey play an important role in the study of $cd$-indices of simplicial Eulerian posets. For results along this line, please see \\cite{bayer2019cd,bayer1991new,He96, HR98, karu2006cd,purtill1993andre,stanley1994flag}. \nOther properties about Andr\\'e permutations and simsun permutations have been extensively studied by Barnabei et al. \\cite{barnabei2020permutations}, Chow and Shiu \\cite{chow2011counting}, Deutsch-Elizalde \\cite{deutsch2012restricted}, Disanto \\cite{Di14}, Foata and the first author \\cite{FH01} and so on. In particular, by constructing a bijection between the set of Andr\\'e I permutations and the set of simsun permutations, Chow and Shiu \\cite{chow2011counting} observed that the number of descents are equidistributed over Andr\\'e I permutations and simsun permutations. Specifically, let $\\des(\\sigma)$ denote the number of descents of $\\sigma$, they showed that \n$$\n \\sum_{\\sigma \\in \\AndI_n} t^{\\des(\\sigma)} =\\sum_{\\sigma \\in \\RS_{n}} t^{\\des(\\sigma)}.\n$$\n\nIn this paper, we show that the number of inverse descents are also equidistributed over Andr\\'e permutations and simsun permutations. The number of inverse descents of a permutation $\\sigma$ is simply the number of descents of its inverse permutation $\\sigma^{-1}$, namely, $\n\\mathrm{ides}(\\sigma) = \\mathrm{des}(\\sigma^{-1}).$ In fact, we show that the trivariate statistic ($\\ides$, $\\des$, $\\maj$) are equidistributed over Andr\\'e permutations and simsun permutations, where {\\it the major index} $\\maj(\\sigma)$ of $\\sigma$ is defined to be the sum of its descents of $\\sigma$. For brevity, we adopt the notation \n $n$-Andr\\'e permutations for Andr\\'e permutations on $[n]$ and $n$-simsun permutations for simsun permutations on $[n]$.\n\\medskip\n\nOur main result is as follows.\n\n\\medskip\n\nTo our knowledge, even the special case of the above result for the univariate statistic ``$\\ides$\" is new:\n$$\n A_n(s):= \\sum_{\\sigma \\in \\AndI_n} s^{\\ides(\\sigma)} \n =\\sum_{\\sigma \\in \\AndII_n} s^{\\ides(\\sigma)} \n =\\sum_{\\sigma \\in \\RS_{n}} s^{\\ides(\\sigma)}.\n$$\nWe list the first values of the polynomials $A_n(s)$ below:\n\\begin{align*}\nA_1(s)&=1, \\quad A_2(s)=1,\\quad A_3(s)=s + 1,\\quad\nA_4(s)=4s + 1,\\\\\nA_5(s)&=4s^2 + 11s + 1,\\quad\nA_6(s)=2s^3 + 32s^2 + 26s + 1.\n\\end{align*}\n\n\\begin{Theorem}\\label{th:equi-shape} For $n\\geq 1$ and any unlabeled binary tree $T \\in \\URL_n$, we have \n \\begin{align}\n \\sum_{\\sigma \\in \\AndI(T)} s^{\\ides(\\sigma)} \\overset{(a)}{=}\\sum_{\\sigma \\in \\AndII(T)} s^{\\ides(\\sigma)} \\overset{(b)}{=}\\sum_{\\sigma \\in \\RS(T)} s^{\\ides(\\sigma)}.\n\\end{align}\n\\end{Theorem}\n\n\\noindent{\\it Proof of relation {\\rm (}a{\\rm )} in Theorem~ \\ref{th:equi-shape}.} We proceed by induction on $n$. For $n=1$, relation (a) clearly holds. Assume that it holds for all $p\\sigma_{i+1}\\), let \\(\\ides(\\sigma)=\\des(\\sigma^{-1})\\), and let \\(\\maj(\\sigma)\\) be the sum of all descent positions of \\(\\sigma\\). Which statement holds for every \\(n\\)?", + "correct_choice": { + "label": "A", + "text": "The joint distribution of \\((\\ides,\\des,\\maj)\\) is the same on \\(\\AndI_n\\), \\(\\AndII_n\\), and \\(\\RS_n\\); equivalently,\n\\[\n\\sum_{\\sigma\\in\\AndI_n}s^{\\ides(\\sigma)}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}\n=\n\\sum_{\\sigma\\in\\AndII_n}s^{\\ides(\\sigma)}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}\n=\n\\sum_{\\sigma\\in\\RS_n}s^{\\ides(\\sigma)}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}.\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "The joint distribution of \\((\\ides,\\des,\\maj)\\) is the same on \\(\\AndI_n\\) and \\(\\AndII_n\\), and after forgetting \\(\\ides\\) it is also the same on \\(\\RS_n\\); equivalently,\n\\[\n\\sum_{\\sigma\\in\\AndI_n}s^{\\ides(\\sigma)}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}\n=\n\\sum_{\\sigma\\in\\AndII_n}s^{\\ides(\\sigma)}t^{\\des(\\sigma)}q^{\\maj(\\sigma)},\n\\qquad\n\\sum_{\\sigma\\in\\AndI_n}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}\n=\n\\sum_{\\sigma\\in\\RS_n}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}.\n\\]" + }, + { + "label": "C", + "text": "The bivariate distribution of \\((\\des,\\maj)\\) is the same on \\(\\AndI_n\\), \\(\\AndII_n\\), and \\(\\RS_n\\); equivalently,\n\\[\n\\sum_{\\sigma\\in\\AndI_n}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}\n=\n\\sum_{\\sigma\\in\\AndII_n}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}\n=\n\\sum_{\\sigma\\in\\RS_n}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}.\n\\]" + }, + { + "label": "D", + "text": "For every unlabeled binary tree \\(T\\in\\URL_n\\), the joint distribution of \\((\\ides,\\des,\\maj)\\) is the same on the shape classes \\(\\AndI(T)\\), \\(\\AndII(T)\\), and \\(\\RS(T)\\); equivalently,\n\\[\n\\sum_{\\sigma\\in\\AndI(T)}s^{\\ides(\\sigma)}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}\n=\n\\sum_{\\sigma\\in\\AndII(T)}s^{\\ides(\\sigma)}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}\n=\n\\sum_{\\sigma\\in\\RS(T)}s^{\\ides(\\sigma)}t^{\\des(\\sigma)}q^{\\maj(\\sigma)}.\n\\]" + }, + { + "label": "E", + "text": "There exists a bijection \\(\\phi_n:\\AndI_n\\to\\AndII_n\\to\\RS_n\\) preserving each of \\(\\ides\\), \\(\\des\\), and \\(\\maj\\) individually; in particular,\n\\[\n\\ides(\\sigma)=\\ides(\\phi_n(\\sigma)),\\qquad \\des(\\sigma)=\\des(\\phi_n(\\sigma)),\\qquad \\maj(\\sigma)=\\maj(\\phi_n(\\sigma))\n\\]\nfor all \\(\\sigma\\), and hence the three generating functions in \\(s,t,q\\) are equal." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "B" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "shape-based reduction only gives full conclusion after proving ides equidistribution on simsun as well", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "dropped the inverse-descent variable \\(s^{\\ides}\\)", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "case_split", + "tampered_component": "per-shape theorem concerns only \\(\\ides\\), not full \\((\\ides,\\des,\\maj)\\) within each fixed tree shape", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "geometric_construction", + "tampered_component": "equidistribution proved via shape reduction, shuffle refinements, and a bijection only in one part, not by a single global statistic-preserving bijection", + "template_used": "property_confusion" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the statistics and classes but does not reveal the theorem-level conclusion. It asks the test-taker to distinguish among several nearby claims, so there is no explicit or trivial answer leakage." + }, + "TAS": { + "score": 2, + "justification": "This is not a bare restatement of a single theorem in the stem. The options include weaker, stronger, and differently scoped variants, so the task is to identify the precise valid conclusion rather than echo a stated fact." + }, + "GPS": { + "score": 2, + "justification": "The item creates strong generative pressure: the choices differ in subtle ways involving joint vs marginal distribution, global vs shape-wise statements, and equidistribution vs statistic-preserving bijection. Selecting the correct option requires real discrimination and theorem-level reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target common failure modes: accepting a weaker true statement, overextending to shape classes, or confusing equidistribution with existence of a bijection preserving each statistic. They are distinct and well-aligned with likely misconceptions." + }, + "total_score": 8, + "overall_assessment": "High-quality MCQ. It avoids answer leakage, is non-tautological, and uses strong, nuanced distractors that genuinely test precise mathematical understanding." + } + }, + { + "id": "2511.10299v1", + "paper_link": "http://arxiv.org/abs/2511.10299v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "theorem", + "content": "\\label{tt3}\n\tLet $(Z_{t}, t\\in \\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$. Let $m\\geq 1$ and $0=t_{0}0$ (depending possibly on $m$ and $n$) such that for any grid $0=t_{0}0$, for every $x=(x_{1},..., x_{m}) $ such that $ x_{j}\\geq 2(t_{j}-t_{j-1})^{H}$, $j=1,..., m.$ \n\\end{theorem}", + "tt3": "\\begin{theorem}\\label{tt3}\n\tLet $(Z_{t}, t\\in \\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$. Let $m\\geq 1$ and $0=t_{0}0$ (depending possibly on $m$ and $n$) such that for any grid $0=t_{0}0$, for every $x=(x_{1},..., x_{m}) $ such that $ x_{j}\\geq 2(t_{j}-t_{j-1})^{H}$, $j=1,..., m.$ \n\\end{theorem}\n\n\\begin{theorem}\n Let $(Z_{t}, t\\in \\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$. Let $m\\geq 2$ and $0=t_{0}0\\right)= 1.\n\\end{equation*}\nIn particular, the random vector $Z_{\\pi}$ admits a density with respect to the Lebesgue measure on $\\mathbb{R} ^{m}$. \n\\end{theorem}\n\n\\begin{lemma}\\label{ll2}\nLet $(Z_{t}, t\\in \\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$. Let $m\\geq 2$ and $0=t_{0}0$ (depending possibly on $m$ and $n$) such that for any grid $0=t_{0}0$, for every $x=(x_{1},..., x_{m}) $ such that $ x_{j}\\geq 2(t_{j}-t_{j-1})^{H}$, $j=1,..., m.$ \n\\end{theorem}\n\nThe proof of\nTheorem \\ref{tt2} is based on a Malliavin integration by parts formula expressing the partial derivatives of $p_{\\pi}$ (see relation (\\ref{22o-1}) in Proposition \\ref{pp2} below) and on a precise control of the resulting terms using Malliavin calculus and tail estimates for random variables in Wiener chaos. \n\nThe paper is organized as follows. Section 2 provides the necessary preliminaries on Malliavin calculus, Wiener chaos and the Rosenblatt process. Section 3 contains the proof of the smoothness of the density of Rosenblatt vectors. Section 4 is devoted to the analysis of the regularity of the partial derivatives of this density.", + "sketch": "The proof of Theorem \\ref{tt3} \"relies on the classical Malliavin calculus theorem asserting that nondegenerate random vectors possess densities in the Schwartz space.\"", + "expanded_sketch": "The proof of the main theorem relies on the classical Malliavin calculus theorem asserting that nondegenerate random vectors possess densities in the Schwartz space.", + "expanded_theorem": "\\label{tt3}\n\tLet $(Z_{t}, t\\in \\mathbb{R})$ be a Rosenblatt process with self-similarity index $H\\in \\left( \\frac{1}{2}, 1\\right)$. Let $m\\geq 1$ and $0=t_{0}b$.} \nMore precisely, whenever a block of zeros is followed by a block of ones, the length of the ones block must be strictly less than $q$ times the length of the preceding zeros block (see \\cite{Barc,arXivBKV,bkv gray code,egecioglu irsic}). Therefore, such words can start with arbitrarily many 1's, and end with arbitrarily many 0's. Let $\\mathcal{W}_n^q$ be the set of $q$-decreasing words of length $n$. For instance, we have $\\mathcal{W}_4^1=\\{0000,0001,0010,1000,1001,1100,1110,1111\\}$, $\\mathcal{W}_4^{\\frac{1}{2}}=\\{0000,0001,1000,1100,1110,1111\\}$ and $\\mathcal{W}_3^{\\frac{\\pi}{2}}=\\{000,001,010,100,101,110,111\\}$. It directly follows that if $qq}\\mathcal{W}_n^r$, or equivalently $\\mathcal{W}_n^{q^+}$ is the set of binary words of length $n$ such that \\\\\n\\centerline{\\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\\cdot a\\geq b$,} \nthen we have $\\mathcal{W}_n^{q}=\\mathcal{W}_n^{q^+}$ for any $n$ when $q$ is irrational, and $\\mathcal{W}_n^{q}\\varsubsetneq\\mathcal{W}_n^{q^+}$ when $q=c/d$, with $c/d$ an irreducible fraction, and $n$ sufficiently large ($n\\geq c+d$). For instance, we have $$\\mathcal{W}_4^{1^+}=\\{0000,0001,0010,0100,1000,1001,1010,1100,0101,0011,1110,1101,1111\\},$$ which contains strictly $\\mathcal{W}_4^1$.\n\nAny word $w\\in\\mathcal{W}_n^q$ can be written \n $$w=1^m0^{a_1}1^{b_1}\\ldots0^{a_k}1^{b_k}0^\\ell,$$\n with $m,\\ell\\geq 0$, and $q\\cdot a_i>b_i\\geq1$ for $1\\leq i\\leq k$ (with \n$k$ possibly equal to zero). The maximal factors of the form $0^a1^b$ with $q\\cdot a>b\\geq 1$ will be called {\\it prime factors}.\n\nThe notion of $q$-decreasing words has recently attracted significant attention in the literature. This family of words exhibits a striking combinatorial property whenever $q$ is a positive integer. Indeed, they are in one-to-one correspondence with binary strings that avoid the pattern $1^{q+1}$, i.e. binary strings without $q+1$ consecutive 1 (see \\cite{bkv gray code}). So, this implies that $q$-decreasing words of length $n$ (when $q$ is a positive integer) are enumerated by the $(q+1)$-generalized Fibonacci numbers $F_{n+1}^{q+1}$ where $F_n^q$ is defined by \n\n$$F_n^{q}=F_{n-1}^{q}+F_{n-1}^{q}+\\cdot +F_{n-q}^{q},$$ with initial conditions $F_n^q=0$ for $n<0$ and $F_0^q=1$ (see \\cite{Fein,knuth,Mile}). It is well known that the generating function of these numbers is $$F_q(x)=\\sum_{n\\geq 0}F_n^qx^n=\\frac{1}{1-x-x^2-\\cdots -x^q}.$$\n\nRecently, Barcucci, Bernini, Bilotta and Pinzani \\cite{Barc} extended this bijection to $q$-decreasing words for any positive rational number $q$, showing that $\\mathcal{W}^q$ is in one-to-one correspondence with binary words avoiding some patterns. \n\nThese words have also been studied from a generative prospective. Baril et al. \\cite{bkv gray code} provide efficient algorithms for the generation of all $q$-decreasing words whenever $q$ is a positive integer. In particular, they construct a $3$‑Gray code for general $q$, and notably a $1$‑Gray code for the case $q=1$, thus resolving a conjecture posed in the context of interconnection networks by \\cite{egecioglu irsic}. More recently, Wong et al. \\cite{Wong} present a two-stage algorithm for generating cyclic $2$-Gray codes for $q$-decreasing words.\n\nMore generally, for any $q>0$, the generating function $W_q(x)$ for the number of $q$-decreasing words with respect to the length $n$ is given by \n\\begin{equation}\\label{eq:w_q general}\nW_q(x)=\\frac{1}{(1-x)\\left(1-\\sum_{i=0}^{+\\infty}x^{1+i+\\left\\lfloor\\frac{i}{q}\\right\\rfloor}\\right)}, \n\\end{equation}\nsee \\cite{sergey2,Serg}.\nThis expression can be simplified as follows when $q$ is rational, i.e. $q=c/d$ where $c$ and $d$ are positive integers:\n\\begin{equation}\\label{eq:w_q rational}\nW_q(x)=\\frac{1-x^{c+d}}{(1-x)\\left(1-x^{c+d}-\\sum_{i=0}^{c-1}x^{1+i+\\left\\lfloor\\frac{i}{q}\\right\\rfloor}\\right)}. \n\\end{equation}\nNote that when $q$ is an integer, i.e. when we fix $d=1$ and $c=q$ in the previous formula, we obtain $W_q(x)=\\frac{F_{q+1}(x)-1}{x}$. Using (\\ref{eq:w_q general}) and (\\ref{eq:w_q rational}), Dovgal and Kirgizov \\cite{sergey2} proved that for all real $q>0$, $[x^n]W_q(x)\\underset{n\\to\\infty}{\\sim}C_q\\cdot\\Phi(q)^n$, for a positive constant $C_q$, and a function $\\Phi(q)$ that interpolates the $q$-bonacci numbers. In particular, when $q=c/d$ is a rational number, $\\Phi(q)^{-1}$ is the smallest root in modulus of the polynomial $x^{c+d}+\\sum_{i=0}^{c-1}x^{1+i+\\left\\lfloor\\frac{i}{q}\\right\\rfloor}-1$. See \\cite{sergey2} for additional properties of $\\Phi(q)$.\n\nTo conclude this set of definitions, we introduce the main order-theoretic concepts used throughout this paper. These notions are standard and can be found, for instance, in \\cite{Grat,stanley}. A {\\it poset} $\\mathcal{L}$ is a set endowed with a partial order\nrelation $\\leq$. Given two elements $P,Q\\in \\mathcal{L}$, a {\\it meet} (or\n{\\it greatest lower bound}) of $P$ and $Q$, denoted $P\\wedge Q$, is an\nelement $R$ such that $R\\leq P$, $R\\leq Q$, and for any $S$ such that\n$S\\leq P$ and $S\\leq Q$, then we have $S\\leq R$. Dually, a {\\it join} (or\n{\\it least upper bound}) of $P$ and $Q$, denoted $P\\vee Q$, is an\nelement $R$ such that $P\\leq R$, $Q\\leq R$, and for any $S$ such that\n$P\\leq S$ and $Q\\leq S$, then we have $R\\leq S$. Notice that join and meet\nelements do not necessarily exist in a poset. A {\\it lattice} is a\nposet where any pair of elements admits a meet and a join.\nAn element $P\\in\\mathcal{L}$ is {\\it join-irreducible} (resp. {\\it\n meet-irreducible}) if $P=R\\vee S$ (resp. $P=R\\wedge S$) implies\n$P=R$ or $P=S$. \n An {\\it interval} $I$ in a poset $\\mathcal{L}$ is a subset of\n $\\mathcal{L}$ such that there exist $P,Q\\in I$, $P\\leq Q$, such that $I=\\{R\\in\\mathcal{L}\\ | \\ P\\leq R \\mbox{ and } R\\leq Q\\}$. An element $w$ is said to {\\it cover} an element $v$ if $vb$.} \nMore precisely, whenever a block of zeros is followed by a block of ones, the length of the ones block must be strictly less than $q$ times the length of the preceding zeros block (see \\cite{Barc,arXivBKV,bkv gray code,egecioglu irsic}). Therefore, such words can start with arbitrarily many 1's, and end with arbitrarily many 0's. Let $\\mathcal{W}_n^q$ be the set of $q$-decreasing words of length $n$. For instance, we have $\\mathcal{W}_4^1=\\{0000,0001,0010,1000,1001,1100,1110,1111\\}$, $\\mathcal{W}_4^{\\frac{1}{2}}=\\{0000,0001,1000,1100,1110,1111\\}$ and $\\mathcal{W}_3^{\\frac{\\pi}{2}}=\\{000,001,010,100,101,110,111\\}$. It directly follows that if $qq}\\mathcal{W}_n^r$, or equivalently $\\mathcal{W}_n^{q^+}$ is the set of binary words of length $n$ such that \\\\\n\\centerline{\\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\\cdot a\\geq b$,} \nthen we have $\\mathcal{W}_n^{q}=\\mathcal{W}_n^{q^+}$ for any $n$ when $q$ is irrational, and $\\mathcal{W}_n^{q}\\varsubsetneq\\mathcal{W}_n^{q^+}$ when $q=c/d$, with $c/d$ an irreducible fraction, and $n$ sufficiently large ($n\\geq c+d$). For instance, we have $$\\mathcal{W}_4^{1^+}=\\{0000,0001,0010,0100,1000,1001,1010,1100,0101,0011,1110,1101,1111\\},$$ which contains strictly $\\mathcal{W}_4^1$.\n\nAny word $w\\in\\mathcal{W}_n^q$ can be written \n $$w=1^m0^{a_1}1^{b_1}\\ldots0^{a_k}1^{b_k}0^\\ell,$$\n with $m,\\ell\\geq 0$, and $q\\cdot a_i>b_i\\geq1$ for $1\\leq i\\leq k$ (with \n$k$ possibly equal to zero). The maximal factors of the form $0^a1^b$ with $q\\cdot a>b\\geq 1$ will be called {\\it prime factors}.\n\n$$F_n^{q}=F_{n-1}^{q}+F_{n-1}^{q}+\\cdot +F_{n-q}^{q},$$ with initial conditions $F_n^q=0$ for $n<0$ and $F_0^q=1$ (see \\cite{Fein,knuth,Mile}). It is well known that the generating function of these numbers is $$F_q(x)=\\sum_{n\\geq 0}F_n^qx^n=\\frac{1}{1-x-x^2-\\cdots -x^q}.$$\n\nTo conclude this set of definitions, we introduce the main order-theoretic concepts used throughout this paper. These notions are standard and can be found, for instance, in \\cite{Grat,stanley}. A {\\it poset} $\\mathcal{L}$ is a set endowed with a partial order\nrelation $\\leq$. Given two elements $P,Q\\in \\mathcal{L}$, a {\\it meet} (or\n{\\it greatest lower bound}) of $P$ and $Q$, denoted $P\\wedge Q$, is an\nelement $R$ such that $R\\leq P$, $R\\leq Q$, and for any $S$ such that\n$S\\leq P$ and $S\\leq Q$, then we have $S\\leq R$. Dually, a {\\it join} (or\n{\\it least upper bound}) of $P$ and $Q$, denoted $P\\vee Q$, is an\nelement $R$ such that $P\\leq R$, $Q\\leq R$, and for any $S$ such that\n$P\\leq S$ and $Q\\leq S$, then we have $R\\leq S$. Notice that join and meet\nelements do not necessarily exist in a poset. A {\\it lattice} is a\nposet where any pair of elements admits a meet and a join.\nAn element $P\\in\\mathcal{L}$ is {\\it join-irreducible} (resp. {\\it\n meet-irreducible}) if $P=R\\vee S$ (resp. $P=R\\wedge S$) implies\n$P=R$ or $P=S$. \n An {\\it interval} $I$ in a poset $\\mathcal{L}$ is a subset of\n $\\mathcal{L}$ such that there exist $P,Q\\in I$, $P\\leq Q$, such that $I=\\{R\\in\\mathcal{L}\\ | \\ P\\leq R \\mbox{ and } R\\leq Q\\}$. An element $w$ is said to {\\it cover} an element $v$ if $vb$.} \nMore precisely, whenever a block of zeros is followed by a block of ones, the length of the ones block must be strictly less than $q$ times the length of the preceding zeros block (see \\cite{Barc,arXivBKV,bkv gray code,egecioglu irsic}). Therefore, such words can start with arbitrarily many 1's, and end with arbitrarily many 0's. Let $\\mathcal{W}_n^q$ be the set of $q$-decreasing words of length $n$. For instance, we have $\\mathcal{W}_4^1=\\{0000,0001,0010,1000,1001,1100,1110,1111\\}$, $\\mathcal{W}_4^{\\frac{1}{2}}=\\{0000,0001,1000,1100,1110,1111\\}$ and $\\mathcal{W}_3^{\\frac{\\pi}{2}}=\\{000,001,010,100,101,110,111\\}$. It directly follows that if $qq}\\mathcal{W}_n^r$, or equivalently $\\mathcal{W}_n^{q^+}$ is the set of binary words of length $n$ such that \\\\\n\\centerline{\\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\\cdot a\\geq b$,} \nthen we have $\\mathcal{W}_n^{q}=\\mathcal{W}_n^{q^+}$ for any $n$ when $q$ is irrational, and $\\mathcal{W}_n^{q}\\varsubsetneq\\mathcal{W}_n^{q^+}$ when $q=c/d$, with $c/d$ an irreducible fraction, and $n$ sufficiently large ($n\\geq c+d$). For instance, we have $$\\mathcal{W}_4^{1^+}=\\{0000,0001,0010,0100,1000,1001,1010,1100,0101,0011,1110,1101,1111\\},$$ which contains strictly $\\mathcal{W}_4^1$.\n\nAny word $w\\in\\mathcal{W}_n^q$ can be written \n $$w=1^m0^{a_1}1^{b_1}\\ldots0^{a_k}1^{b_k}0^\\ell,$$\n with $m,\\ell\\geq 0$, and $q\\cdot a_i>b_i\\geq1$ for $1\\leq i\\leq k$ (with \n$k$ possibly equal to zero). The maximal factors of the form $0^a1^b$ with $q\\cdot a>b\\geq 1$ will be called {\\it prime factors}.\n\n$$F_n^{q}=F_{n-1}^{q}+F_{n-1}^{q}+\\cdot +F_{n-q}^{q},$$ with initial conditions $F_n^q=0$ for $n<0$ and $F_0^q=1$ (see \\cite{Fein,knuth,Mile}). It is well known that the generating function of these numbers is $$F_q(x)=\\sum_{n\\geq 0}F_n^qx^n=\\frac{1}{1-x-x^2-\\cdots -x^q}.$$\n\nTo conclude this set of definitions, we introduce the main order-theoretic concepts used throughout this paper. These notions are standard and can be found, for instance, in \\cite{Grat,stanley}. A {\\it poset} $\\mathcal{L}$ is a set endowed with a partial order\nrelation $\\leq$. Given two elements $P,Q\\in \\mathcal{L}$, a {\\it meet} (or\n{\\it greatest lower bound}) of $P$ and $Q$, denoted $P\\wedge Q$, is an\nelement $R$ such that $R\\leq P$, $R\\leq Q$, and for any $S$ such that\n$S\\leq P$ and $S\\leq Q$, then we have $S\\leq R$. Dually, a {\\it join} (or\n{\\it least upper bound}) of $P$ and $Q$, denoted $P\\vee Q$, is an\nelement $R$ such that $P\\leq R$, $Q\\leq R$, and for any $S$ such that\n$P\\leq S$ and $Q\\leq S$, then we have $R\\leq S$. Notice that join and meet\nelements do not necessarily exist in a poset. A {\\it lattice} is a\nposet where any pair of elements admits a meet and a join.\nAn element $P\\in\\mathcal{L}$ is {\\it join-irreducible} (resp. {\\it\n meet-irreducible}) if $P=R\\vee S$ (resp. $P=R\\wedge S$) implies\n$P=R$ or $P=S$. \n An {\\it interval} $I$ in a poset $\\mathcal{L}$ is a subset of\n $\\mathcal{L}$ such that there exist $P,Q\\in I$, $P\\leq Q$, such that $I=\\{R\\in\\mathcal{L}\\ | \\ P\\leq R \\mbox{ and } R\\leq Q\\}$. An element $w$ is said to {\\it cover} an element $v$ if $vb$.} \nMore precisely, whenever a block of zeros is followed by a block of ones, the length of the ones block must be strictly less than $q$ times the length of the preceding zeros block (see \\cite{Barc,arXivBKV,bkv gray code,egecioglu irsic}). Therefore, such words can start with arbitrarily many 1's, and end with arbitrarily many 0's. Let $\\mathcal{W}_n^q$ be the set of $q$-decreasing words of length $n$. For instance, we have $\\mathcal{W}_4^1=\\{0000,0001,0010,1000,1001,1100,1110,1111\\}$, $\\mathcal{W}_4^{\\frac{1}{2}}=\\{0000,0001,1000,1100,1110,1111\\}$ and $\\mathcal{W}_3^{\\frac{\\pi}{2}}=\\{000,001,010,100,101,110,111\\}$. It directly follows that if $qq}\\mathcal{W}_n^r$, or equivalently $\\mathcal{W}_n^{q^+}$ is the set of binary words of length $n$ such that \\\\\n\\centerline{\\it every maximal factor of the form $0^a1^b$ satisfies either $a=0$ or $q\\cdot a\\geq b$,} \nthen we have $\\mathcal{W}_n^{q}=\\mathcal{W}_n^{q^+}$ for any $n$ when $q$ is irrational, and $\\mathcal{W}_n^{q}\\varsubsetneq\\mathcal{W}_n^{q^+}$ when $q=c/d$, with $c/d$ an irreducible fraction, and $n$ sufficiently large ($n\\geq c+d$). For instance, we have $$\\mathcal{W}_4^{1^+}=\\{0000,0001,0010,0100,1000,1001,1010,1100,0101,0011,1110,1101,1111\\},$$ which contains strictly $\\mathcal{W}_4^1$.\n\nAny word $w\\in\\mathcal{W}_n^q$ can be written \n $$w=1^m0^{a_1}1^{b_1}\\ldots0^{a_k}1^{b_k}0^\\ell,$$\n with $m,\\ell\\geq 0$, and $q\\cdot a_i>b_i\\geq1$ for $1\\leq i\\leq k$ (with \n$k$ possibly equal to zero). The maximal factors of the form $0^a1^b$ with $q\\cdot a>b\\geq 1$ will be called {\\it prime factors}.\n\nTo conclude this set of definitions, we introduce the main order-theoretic concepts used throughout this paper. These notions are standard and can be found, for instance, in \\cite{Grat,stanley}. A {\\it poset} $\\mathcal{L}$ is a set endowed with a partial order\nrelation $\\leq$. Given two elements $P,Q\\in \\mathcal{L}$, a {\\it meet} (or\n{\\it greatest lower bound}) of $P$ and $Q$, denoted $P\\wedge Q$, is an\nelement $R$ such that $R\\leq P$, $R\\leq Q$, and for any $S$ such that\n$S\\leq P$ and $S\\leq Q$, then we have $S\\leq R$. Dually, a {\\it join} (or\n{\\it least upper bound}) of $P$ and $Q$, denoted $P\\vee Q$, is an\nelement $R$ such that $P\\leq R$, $Q\\leq R$, and for any $S$ such that\n$P\\leq S$ and $Q\\leq S$, then we have $R\\leq S$. Notice that join and meet\nelements do not necessarily exist in a poset. A {\\it lattice} is a\nposet where any pair of elements admits a meet and a join.\nAn element $P\\in\\mathcal{L}$ is {\\it join-irreducible} (resp. {\\it\n meet-irreducible}) if $P=R\\vee S$ (resp. $P=R\\wedge S$) implies\n$P=R$ or $P=S$. \n An {\\it interval} $I$ in a poset $\\mathcal{L}$ is a subset of\n $\\mathcal{L}$ such that there exist $P,Q\\in I$, $P\\leq Q$, such that $I=\\{R\\in\\mathcal{L}\\ | \\ P\\leq R \\mbox{ and } R\\leq Q\\}$. An element $w$ is said to {\\it cover} an element $v$ if $v 0 \\), we provide a closed-form expression for the generating function counting the number of coverings in \\( \\W^q_n \\). When \\( q \\) is irrational, we present a formula that enables efficient computation of the initial terms of the series expansion (e.g., using \\textsc{Maple}). We also prove that the asymptotic behavior of the number of coverings is connected to the function $\\Phi(q)$ defined in the introduction above. In Section~\\ref{sec:intervals}, we derive a closed-form expression for the generating function that enumerates the number of intervals in \\( \\W^q_n \\) for any rational number \\( q > 0 \\). Finally, Section~\\ref{sec:meet} presents the structure of meet-irreducible elements in $\\W_n^q$ for any positive rational number $q$. This structure is the same as the one of words over an alphabet of $2\\lceil q\\rceil+1$ letters avoiding $\\lceil q\\rceil^2+2\\lceil q\\rceil-1$ consecutive patterns of length 2. Taking advantage of this classical structure, we present a method to obtain the closed form of the generating function enumerating the number of meet-irreducible elements in $\\W_n^q$ for any rational number $q>0$.\n\nIn this section, we provide enumerative results for classical parameters of a lattice, namely the join-irreducible elements, as well as covering relations. The enumeration of meet-irreducible elements is a bit more intricate, so we treat it in Section \\ref{sec:meet}.\nWe first give the enumeration of join-irreducible elements for any $q>0$, and then we conclude by giving closed form for the generating functions of the covering for positive rational numbers $q$, and a method for computing arbitrarily many terms of the generating functions for positive irrational numbers $q$. \n\\subsection{Join-irreducible elements}\n\\begin{thm}\\label{join irreducible}\n For $q>0$ and $n\\geq1$, there are exactly $n$ join-irreducible elements in $\\W_n^q$.\n\\end{thm}\n\\begin{proof}\n In a finite lattice, an element is join-irreducible if and only if it covers exactly one element. We therefore count the words in $\\W_n^q$ that cover exactly one element. Note that a factor $0^a1^b$, with $qa>b\\geq 1$ covers only one element if and only if $b=1$ (otherwise it covers $0^a1^{b-1}0$ and $0^{a+1}1^{b-1}$). Now we investigate the elements covered by $1^m$. Given $i\\geq0$, there is at most one word covered by $1^m$ with suffix $01^i$. So the words covered by $1^m$ are exactly", + "post_theorem_intro_text_len": 3400, + "post_theorem_intro_text": "\\begin{proof} For any words $v$ and $w$ in $\\mathcal{W}_n^q$, we consider the binary word $a=a_1\\ldots a_n$ where $a_i=1$ if and only if $v_i=w_i=1$. It is straightforward to see that $a\\in \\mathcal{W}_n^q$, and then $a$ is the greatest lower bound of $v$ and $w$, which implies that $\\W_n^q$ is a meet-semilattice. Since $1^n$ is the maximum element of $\\W_n^q$, Proposition 3.3.1 in \\cite{stanley} implies that $\\W_n^q$ is a lattice. \n\\end{proof}\n\n\\begin{figure}[h]\n \\centering\n\\begin{tikzpicture}[scale=1.3]\n \\node (11111) at (0,7.5) {$11111$};\n \\node (11110) at (-1.5,6) {$11110$};\n \\node (11001) at (0,4.5) {$11001$};\n \\node (11100) at (-3,4.5) {$11100$};\n \\node (00011) at (3,3) {$00011$};\n \\node (10001) at (1,3) {$10001$};\n \\node (10010) at (-1,3) {$10010$};\n \\node (11000) at (-3,3) {$11000$};\n \\node (00001) at (3,1.5) {$00001$};\n \\node (00010) at (1,1.5) {$00010$};\n \\node (10000) at (-3,1.5) {$10000$};\n \\node (00100) at (-1,1.5) {$00100$};\n \\node (00000) at (0,0) {$00000$};\n \\draw (00000) -- (00100);\\draw (00000) -- (10000);\\draw (00000) -- (00001);\\draw (00000) -- (00010);\\draw (10000) -- (11000);\\draw (10000) -- (10010);\\draw (10000) -- (10001);\\draw (00010) -- (10010);\\draw (00010) -- (00011);\\draw (00001) -- (10001);\\draw (00001) -- (00011);\\draw (00100) -- (11100);\\draw (11000) -- (11001);\\draw (11000) -- (11100);\\draw (10010) -- (11110);\\draw (10001) -- (11001);\\draw (00011) -- (11111);\\draw (11100) -- (11110);\\draw (11001) -- (11111);\\draw (11110) -- (11111);\n\\end{tikzpicture}\n \\caption{The lattice $\\W_5^1$. It contains 20 coverings (edges), 7 meet-irreducible elements, 5 join-irreducible elements and 56 intervals. }\n \\label{fig:W_4^2}\n\\end{figure}\n\n\\noindent {\\bf Outline of the paper.} In Section~\\ref{sec:useful results}, we collect preliminary results that will be used throughout the paper. Many of these results are quite technical, due to the presence of floor and ceiling functions in generating functions related to $q$-decreasing words. Section~\\ref{sec:join and coverings} is devoted to enumerative results concerning classical lattice parameters, specifically join-irreducible elements, as well as covering relations. For any rational number \\( q > 0 \\), we provide a closed-form expression for the generating function counting the number of coverings in \\( \\mathbb{W}^q_n \\). When \\( q \\) is irrational, we present a formula that enables efficient computation of the initial terms of the series expansion (e.g., using \\textsc{Maple}). We also prove that the asymptotic behavior of the number of coverings is connected to the function $\\Phi(q)$ defined in the introduction above. In Section~\\ref{sec:intervals}, we derive a closed-form expression for the generating function that enumerates the number of intervals in \\( \\mathbb{W}^q_n \\) for any rational number \\( q > 0 \\). Finally, Section~\\ref{sec:meet} presents the structure of meet-irreducible elements in $\\W_n^q$ for any positive rational number $q$. This structure is the same as the one of words over an alphabet of $2\\lceil q\\rceil+1$ letters avoiding $\\lceil q\\rceil^2+2\\lceil q\\rceil-1$ consecutive patterns of length 2. Taking advantage of this classical structure, we present a method to obtain the closed form of the generating function enumerating the number of meet-irreducible elements in $\\W_n^q$ for any rational number $q>0$.", + "sketch": "For any words $v$ and $w$ in $\\mathcal{W}_n^q$, consider the binary word $a=a_1\\ldots a_n$ where $a_i=1$ iff $v_i=w_i=1$. It is \\\"straightforward to see\\\" that $a\\in \\mathcal{W}_n^q$, and then $a$ is the greatest lower bound of $v$ and $w$, so $\\W_n^q$ is a meet-semilattice. Since $1^n$ is the maximum element of $\\W_n^q$, Proposition 3.3.1 in \\cite{stanley} implies that $\\W_n^q$ is a lattice.", + "expanded_sketch": "For any words $v$ and $w$ in $\\mathcal{W}_n^q$, consider the binary word $a=a_1\\ldots a_n$ where $a_i=1$ iff $v_i=w_i=1$. It is \"straightforward to see\" that $a\\in \\mathcal{W}_n^q$, and then $a$ is the greatest lower bound of $v$ and $w$, so $\\W_n^q$ is a meet-semilattice. Since $1^n$ is the maximum element of $\\W_n^q$, Proposition 3.3.1 in \\cite{stanley} implies that $\\W_n^q$ is a lattice.,", + "expanded_theorem": "For $q\\geq 0$, the poset $\\W_n^q$ is a lattice for any $n\\geq 1$.", + "theorem_type": [ + "Universal" + ], + "mcq": { + "question": "Let \\(q\\ge 0\\) and \\(n\\ge 1\\). A binary word of length \\(n\\) is called \\(q\\)-decreasing if every maximal factor of the form \\(0^a1^b\\) satisfies either \\(a=0\\) or \\(q\\,a>b\\). Let \\(\\mathcal{W}_n^q\\) be the set of all such words, and equip it with the componentwise order: for \\(v=v_1\\cdots v_n\\) and \\(w=w_1\\cdots w_n\\), define \\(v\\le w\\) iff \\(v_i\\le w_i\\) for all \\(1\\le i\\le n\\). Write \\(\\W_n^q=(\\mathcal{W}_n^q,\\le)\\). Which statement holds for every choice of \\(q\\ge 0\\) and \\(n\\ge 1\\)?", + "correct_choice": { + "label": "A", + "text": "The poset \\(\\W_n^q\\) is a lattice; equivalently, every pair of elements of \\(\\mathcal{W}_n^q\\) has both a meet (greatest lower bound) and a join (least upper bound) in the componentwise order." + }, + "choices": [ + { + "label": "B", + "text": "The poset \\(\\W_n^q\\) is a meet-semilattice but, for some choices of \\(q\\ge 0\\) and \\(n\\ge 1\\), it is not a lattice; in general, componentwise meets of elements of \\(\\mathcal{W}_n^q\\) always exist, whereas componentwise joins need not belong to \\(\\mathcal{W}_n^q\\)." + }, + { + "label": "C", + "text": "For every \\(q\\ge 0\\) and \\(n\\ge 1\\), every pair of elements of \\(\\mathcal{W}_n^q\\) has a meet (greatest lower bound) in the componentwise order; equivalently, \\(\\W_n^q\\) is a meet-semilattice." + }, + { + "label": "D", + "text": "The poset \\(\\W_n^q\\) is a lattice for every irrational \\(q>0\\), but this can fail for rational values of \\(q\\) because the distinction between the strict condition \\(q\\,a>b\\) and the weak condition \\(q\\,a\\ge b\\) prevents componentwise joins from being well defined in general." + }, + { + "label": "E", + "text": "The poset \\(\\W_n^q\\) is a lattice precisely for \\(q>0\\); when \\(q=0\\), the condition on maximal factors forces a boundary case in which \\(\\mathcal{W}_n^q\\) is no longer closed under the meet and join operations in the componentwise order." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "finiteness", + "tampered_component": "use_of_top_element_to_upgrade_meet-semilattice_to_lattice", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped_existence_of_joins", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "characteristic", + "tampered_component": "uniformity_in_q_rational_vs_irrational", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "boundary_range", + "tampered_component": "inclusion_of_boundary_case_q=0", + "template_used": "boundary_range" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the objects and order relation but does not explicitly or implicitly reveal that the poset is a lattice. No wording strongly privileges choice A over the others." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-identification question: the correct option is essentially a clean theorem statement about \\(\\W_n^q\\). It is not a pure tautology because the alternatives offer nearby competing claims, but it still largely tests recognition of the main result." + }, + "GPS": { + "score": 1, + "justification": "There is some reasoning pressure because the student must distinguish lattice vs. meet-semilattice, universal vs. exceptional-in-\\(q\\) behavior, and boundary cases such as \\(q=0\\). However, the item mainly rewards theorem recall or result recognition rather than substantial derivation from first principles." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically meaningful: one is a weaker true statement, others exploit common failure modes involving joins, rational/irrational distinctions, and the boundary case \\(q=0\\). They are distinct and well targeted." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with good distractors and no answer leakage, but it leans more toward recall of the exact result than toward genuinely generative mathematical reasoning." + } + }, + { + "id": "2511.09176v1", + "paper_link": "http://arxiv.org/abs/2511.09176v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "lemma", + "content": "For two points $P,Q\\in k^n$ we have that $$\\dim_k\\ext^1_A(M_P,M_Q)=\\begin{cases}n,\\ P=Q\\\\0,\\ P\\neq Q.\\end{cases}$$", + "start_pos": 9024, + "end_pos": 9164, + "label": null + }, + "ref_dict": {}, + "pre_theorem_intro_text_len": 2190, + "pre_theorem_intro_text": "Real algebraic geometry can be thought of as a generalization of manifolds, where the continuous functions are replaced by polynomials with real coefficients. Application to physics also leads to a necessary generalization to associative algebraic geometry and a generalization of continuous Riemannian metrics, see the book of O.A. Laudal, \\cite{Laudal21}. One of the main problems with this, is that Riemannian metrics are defined over the reals, and the polynomial algebra over the reals, $\\mathbb R[x_1,\\dots,x_n],$ contains more simple modules than $\\mathbb R^n.$ Because the algebraic properties governing the simple modules are better controlled by an algebra over an algebraically closed field, the main result in this text is the construction of a $\\mathbb C$-algebra $A_{\\mathbb R}$ such that $\\simp(A_{\\mathbb R})\\cong\\mathbb R^n.$ Thus the points in $\\mathbb R^n$ is in one-to-one correspondence with the simple $A_{\\mathbb R}$-modules for which $\\aspec(A_{\\mathbb R})$ is a fine moduli, see the book \\cite{S23} or the preprint \\cite{S241}. For any real manifold $M$ we define an associative variety $(\\mathcal M,\\mathcal O^A_L)$ over $\\mathbb C$ such that the points in $M$ is in bijective correspondence with the closed points in $\\mathcal M,$ and such that the charts $U$ in $\\mathcal M$ corresponds to $\\mathcal O^A_L(U)=A_{\\mathbb R}.$\n\nIn short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$\n\nBecause of the discovery of localization in associative rings, \\cite{S252}, we include the general definition of associative schemes. Their purpose is to serve as moduli of algebraic objects which can be put in one-to-one correspondence with modules over associative schemes: That is, we need associative moduli to classify associative objects. This is most easily seen by the following.\nLet $k$ be a field and consider the polynomial algebra in $n\\geq 1$ variables $$A=k[x_1,\\dots,x_n]=k[\\underline x].$$ For a point $P=(p_1,\\dots,p_n)\\in k^n,$ we define the simple $A$-modules $$M(P)=A/(x_1-p_1,\\dots,x_n-p_n).$$", + "context": "Real algebraic geometry can be thought of as a generalization of manifolds, where the continuous functions are replaced by polynomials with real coefficients. Application to physics also leads to a necessary generalization to associative algebraic geometry and a generalization of continuous Riemannian metrics, see the book of O.A. Laudal, \\cite{Laudal21}. One of the main problems with this, is that Riemannian metrics are defined over the reals, and the polynomial algebra over the reals, $\\mathbb R[x_1,\\dots,x_n],$ contains more simple modules than $\\mathbb R^n.$ Because the algebraic properties governing the simple modules are better controlled by an algebra over an algebraically closed field, the main result in this text is the construction of a $\\mathbb C$-algebra $A_{\\mathbb R}$ such that $\\simp(A_{\\mathbb R})\\cong\\mathbb R^n.$ Thus the points in $\\mathbb R^n$ is in one-to-one correspondence with the simple $A_{\\mathbb R}$-modules for which $\\aspec(A_{\\mathbb R})$ is a fine moduli, see the book \\cite{S23} or the preprint \\cite{S241}. For any real manifold $M$ we define an associative variety $(\\mathcal M,\\mathcal O^A_L)$ over $\\mathbb C$ such that the points in $M$ is in bijective correspondence with the closed points in $\\mathcal M,$ and such that the charts $U$ in $\\mathcal M$ corresponds to $\\mathcal O^A_L(U)=A_{\\mathbb R}.$\n\nIn short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$\n\nBecause of the discovery of localization in associative rings, \\cite{S252}, we include the general definition of associative schemes. Their purpose is to serve as moduli of algebraic objects which can be put in one-to-one correspondence with modules over associative schemes: That is, we need associative moduli to classify associative objects. This is most easily seen by the following.\nLet $k$ be a field and consider the polynomial algebra in $n\\geq 1$ variables $$A=k[x_1,\\dots,x_n]=k[\\underline x].$$ For a point $P=(p_1,\\dots,p_n)\\in k^n,$ we define the simple $A$-modules $$M(P)=A/(x_1-p_1,\\dots,x_n-p_n).$$", + "full_context": "Real algebraic geometry can be thought of as a generalization of manifolds, where the continuous functions are replaced by polynomials with real coefficients. Application to physics also leads to a necessary generalization to associative algebraic geometry and a generalization of continuous Riemannian metrics, see the book of O.A. Laudal, \\cite{Laudal21}. One of the main problems with this, is that Riemannian metrics are defined over the reals, and the polynomial algebra over the reals, $\\mathbb R[x_1,\\dots,x_n],$ contains more simple modules than $\\mathbb R^n.$ Because the algebraic properties governing the simple modules are better controlled by an algebra over an algebraically closed field, the main result in this text is the construction of a $\\mathbb C$-algebra $A_{\\mathbb R}$ such that $\\simp(A_{\\mathbb R})\\cong\\mathbb R^n.$ Thus the points in $\\mathbb R^n$ is in one-to-one correspondence with the simple $A_{\\mathbb R}$-modules for which $\\aspec(A_{\\mathbb R})$ is a fine moduli, see the book \\cite{S23} or the preprint \\cite{S241}. For any real manifold $M$ we define an associative variety $(\\mathcal M,\\mathcal O^A_L)$ over $\\mathbb C$ such that the points in $M$ is in bijective correspondence with the closed points in $\\mathcal M,$ and such that the charts $U$ in $\\mathcal M$ corresponds to $\\mathcal O^A_L(U)=A_{\\mathbb R}.$\n\nIn short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$\n\nBecause of the discovery of localization in associative rings, \\cite{S252}, we include the general definition of associative schemes. Their purpose is to serve as moduli of algebraic objects which can be put in one-to-one correspondence with modules over associative schemes: That is, we need associative moduli to classify associative objects. This is most easily seen by the following.\nLet $k$ be a field and consider the polynomial algebra in $n\\geq 1$ variables $$A=k[x_1,\\dots,x_n]=k[\\underline x].$$ For a point $P=(p_1,\\dots,p_n)\\in k^n,$ we define the simple $A$-modules $$M(P)=A/(x_1-p_1,\\dots,x_n-p_n).$$\n\n\\begin{abstract}\nIn the preprint \\cite{S252} we proved that there exists a localizing ring $A_M$ for $A$ an associative ring with unit, and $M=\\oplus_{i=1}^rM_i$ a direct sum of $r\\geq 1$ simple right $A$-modules. For a homomorphism of associative rings $A\\rightarrow B$ we define the contraction of a simple $B$-module to $A.$\nThen we define the set of aprime right $A$-modules $\\aspec A$ to be the set of simple $A$-modules together with contractions of such. When $A$ is commutative, $\\aspec A=\\spec A,$ and we define a topology on $\\aspec A$ such that when $A$ is commutative, this is the Zariski topology. In the preprint \\cite{S251}, we proved that when we have a topology and a localizing subcategory, there exists a sheaf of associative rings $\\mathcal O_X$ on $\\aspec A,$ agreeing with the usual sheaf of rings on $\\spec A.$ In this text, we write out this construction, and we see that we can restrict the sheaf and topology to any subset $V\\subseteq\\aspec A.$ In particular, this \nproves that we can use complex varieties in real algebraic geometry, by restricting in accordance with $\\mathbb R\\subseteq\\mathbb C.$ Thus the theory of schemes over algebraically closed fields and its associative generalization can be applied to real (algebraic) geometry.\n\\end{abstract}\n\nIn short terms, if we want to generalize the study of manifolds by using algebraic scheme-theory, we need schemes of associative $\\mathbb C$-algebras for which the set of points is in bijective correspondence with $M.$\n\n\\begin{proof} From \\cite{S23} we know that $$\\ext^1_A(M_1,M_2)\\simeq\\der_k(A,\\hmm_k(M_1,M_2))/\\inner$$ where $\\hmm_k(M_1,M_2)$ is an $A-A$ bimodule by $a\\phi(m)=\\phi(am), \\phi a(m)=\\phi(m)a.$\n\nWhen $P\\neq Q$ we can consider $M_{\\underline 0}, M_P$ with $p_1\\neq 0.$ \nFor $$\\delta\\in\\der_k(A,\\hmm_k(M_{\\underline 0},M_P)$$ we have $$\\begin{aligned}\\delta(x_ix_1)&=\\delta(x_1x_i)\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)x_1+x_i\\delta(x_1)&=\\delta(x_1)x_i+x_1\\delta(x_i)\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)p_1&=\\delta(x_1)p_i\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)&=\\delta(x_i)\\frac{p_i}{p_1}.\n\\end{aligned}$$\nThis is also true for the inner derivations, proving that $\\dim_k\\inner=1$ so that $\\ext^1_A(M_P,M_Q)=0$ when $P\\neq Q.$\n\\end{proof}\n\n\\begin{proof} We have a homomorphism $f:A\\rightarrow A/\\mathfrak m$ such for every $s\\in A\\setminus\\mathfrak m,$ $f(s)$ is a unit. By the universal property of localization, there exists a unique homomorphism $\\phi:A_\\mathfrak m\\rightarrow A/\\mathfrak m$ such that $\\phi(\\frac{a}{s})=\\iota(a)\\iota(s)^{-1}.$ This homomorphism is clearly surjective, and its kernel is $\\mathfrak m A_\\mathfrak m$ giving the wanted isomorphism.\n\\end{proof}\n\n\\begin{proof} We send the prime ideal $\\mathfrak p\\subset A$ to $A_\\mathfrak p/\\mathfrak p A_\\mathfrak p$ which is $A$-prime by definition. On the other hand, let\n$M$ be an aprime $A$-module, defined by $\\iota_M: A\\rightarrow B$ such that $M$ is a simple $B$-module. Let $\\mathfrak m\\subset B$ be the maximal ideal defining $M$ as a simple $B$-module. Then $\\iota_M^{-1}(\\mathfrak m)$ is a prime ideal in $A.$ That these two operations are inverses to each other follows from the fact that for a maximal ideal in a ring $B$ we have $B/\\mathfrak m\\simeq B_{\\mathfrak m}/\\mathfrak m B_\\mathfrak m$ as proven i Lemma \\ref{anotherloclemma}.\n\\end{proof}\n\n\\begin{definition} Define the subset of $k$-points in $\\mathbb X$ by $$\\tilde{\\mathbb X}(k)=\\{x\\in X\\subseteq\\mathbb X|x\\text{ is simple}\\}\\subseteq\\mathbb X.$$ Then the induced associative subscheme $\\mathbb X(k)$ is called the associative subscheme of $k$-points. \n\\end{definition}\n\n\\begin{proof} Because $k\\subseteq\\Bbbk$ is a sub-algebra, it follows that if $\\phi\\otimes\\id$ is an isomorphism, then $\\dim_k V_1=\\dim_k V_2,$ and that choosing corresponding bases, $$0\\neq\\det(\\phi\\otimes\\id)=\\det\\phi.$$ \n\\end{proof}", + "post_theorem_intro_text_len": 1254, + "post_theorem_intro_text": "\\begin{proof} From \\cite{S23} we know that $$\\ext^1_A(M_1,M_2)\\simeq\\der_k(A,\\hmm_k(M_1,M_2))/\\inner$$ where $\\hmm_k(M_1,M_2)$ is an $A-A$ bimodule by $a\\phi(m)=\\phi(am), \\phi a(m)=\\phi(m)a.$\n\nWhen $P=Q$ we can assume $M_1=M_2=k[\\underline x]/(\\underline x).$ Then any inner derivation is on the form $\\ad_\\phi$ for which $\\ad_\\phi(x_i)=[\\phi,x_i]=0.$ Thus the inner derivations is of dimension zero and $\\ext^1_A(M_1,M_2)\\simeq\\der_k(A,k)$ where \n$\\operatorname d_i(x_i)=1,\\ i=1,\\dots,n$ gives a basis for the derivations.\n\nWhen $P\\neq Q$ we can consider $M_{\\underline 0}, M_P$ with $p_1\\neq 0.$ \nFor $$\\delta\\in\\der_k(A,\\hmm_k(M_{\\underline 0},M_P)$$ we have $$\\begin{aligned}\\delta(x_ix_1)&=\\delta(x_1x_i)\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)x_1+x_i\\delta(x_1)&=\\delta(x_1)x_i+x_1\\delta(x_i)\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)p_1&=\\delta(x_1)p_i\\\\\n&\\Updownarrow\\\\\n\\delta(x_i)&=\\delta(x_i)\\frac{p_i}{p_1}.\n\\end{aligned}$$\nThis is also true for the inner derivations, proving that $\\dim_k\\inner=1$ so that $\\ext^1_A(M_P,M_Q)=0$ when $P\\neq Q.$\n\\end{proof}\n\nIt follows from the lemma that not all finite dimensional simple modules over a noncommutative $k$-algebra can be classified by a finitely generated commutative algebra. See \\cite{S23} for a lot of examples.", + "sketch": "From \\cite{S23}: \\(\\ext^1_A(M_1,M_2)\\simeq \\der_k(A,\\hmm_k(M_1,M_2))/\\inner\\), with \\(\\hmm_k(M_1,M_2)\\) an \\(A\\)-\\(A\\) bimodule via \\(a\\phi(m)=\\phi(am)\\), \\(\\phi a(m)=\\phi(m)a\\).\n\n- Case \\(P=Q\\): assume \\(M_1=M_2=k[\\underline x]/(\\underline x)\\). Any inner derivation is \\(\\ad_\\phi\\) and \\(\\ad_\\phi(x_i)=[\\phi,x_i]=0\\), so \\(\\dim_k\\inner=0\\). Hence \\(\\ext^1_A(M_1,M_2)\\simeq \\der_k(A,k)\\), and the derivations \\(\\operatorname d_i\\) with \\(\\operatorname d_i(x_i)=1\\) for \\(i=1,\\dots,n\\) give a basis, so the dimension is \\(n\\).\n\n- Case \\(P\\neq Q\\): consider \\(M_{\\underline 0}, M_P\\) with \\(p_1\\neq 0\\). For \\(\\delta\\in\\der_k(A,\\hmm_k(M_{\\underline 0},M_P))\\), comparing \\(\\delta(x_ix_1)=\\delta(x_1x_i)\\) yields\n\\[\\delta(x_i)x_1+x_i\\delta(x_1)=\\delta(x_1)x_i+x_1\\delta(x_i)\\Rightarrow \\delta(x_i)p_1=\\delta(x_1)p_i\\Rightarrow \\delta(x_i)=\\delta(x_1)\\frac{p_i}{p_1}.\\]\nThis relation also holds for inner derivations, and it is concluded that \\(\\dim_k\\inner=1\\), so \\(\\ext^1_A(M_P,M_Q)=0\\) when \\(P\\neq Q\\).", + "expanded_sketch": "From \\cite{S23}: \\(\\ext^1_A(M_1,M_2)\\simeq \\der_k(A,\\hmm_k(M_1,M_2))/\\inner\\), with \\(\\hmm_k(M_1,M_2)\\) an \\(A\\)-\\(A\\) bimodule via \\(a\\phi(m)=\\phi(am)\\), \\(\\phi a(m)=\\phi(m)a\\).\n\n- Case \\(P=Q\\): assume \\(M_1=M_2=k[\\underline x]/(\\underline x)\\). Any inner derivation is \\(\\ad_\\phi\\) and \\(\\ad_\\phi(x_i)=[\\phi,x_i]=0\\), so \\(\\dim_k\\inner=0\\). Hence \\(\\ext^1_A(M_1,M_2)\\simeq \\der_k(A,k)\\), and the derivations \\(\\operatorname d_i\\) with \\(\\operatorname d_i(x_i)=1\\) for \\(i=1,\\dots,n\\) give a basis, so the dimension is \\(n\\).\n\n- Case \\(P\\neq Q\\): consider \\(M_{\\underline 0}, M_P\\) with \\(p_1\\neq 0\\). For \\(\\delta\\in\\der_k(A,\\hmm_k(M_{\\underline 0},M_P))\\), comparing \\(\\delta(x_ix_1)=\\delta(x_1x_i)\\) yields\n\\[\\delta(x_i)x_1+x_i\\delta(x_1)=\\delta(x_1)x_i+x_1\\delta(x_i)\\Rightarrow \\delta(x_i)p_1=\\delta(x_1)p_i\\Rightarrow \\delta(x_i)=\\delta(x_1)\\frac{p_i}{p_1}.\\]\nThis relation also holds for inner derivations, and it is concluded that \\(\\dim_k\\inner=1\\), so \\(\\ext^1_A(M_P,M_Q)=0\\) when \\(P\\neq Q\\).", + "expanded_theorem": "For two points $P,Q\\in k^n$ we have that $$\\dim_k\\ext^1_A(M_P,M_Q)=\\begin{cases}n,\\ P=Q\\\\0,\\ P\\neq Q.\\end{cases}$$,", + "theorem_type": [ + "Universal", + "Classification or Bijection" + ], + "mcq": { + "question": "Let $k$ be a field, let $A=k[x_1,\\dots,x_n]$ be the polynomial algebra in $n\\ge 1$ variables over $k$, and for each point $P=(p_1,\\dots,p_n)\\in k^n$ define the simple $A$-module\n$$M_P:=A/(x_1-p_1,\\dots,x_n-p_n).$$\nWhich statement holds for every pair of points $P,Q\\in k^n$ about the $k$-dimension of $\\operatorname{Ext}^1_A(M_P,M_Q)$?", + "correct_choice": { + "label": "A", + "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)=\\begin{cases}n,& P=Q,\\\\0,& P\\ne Q.\\end{cases}$$" + }, + "choices": [ + { + "label": "B", + "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)=\\begin{cases}n,& P=Q,\\\\1,& P\\ne Q.\\end{cases}$$" + }, + { + "label": "C", + "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_P)=n\\qquad\\text{for every }P\\in k^n.$$" + }, + { + "label": "D", + "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)=\\begin{cases}n,& P=Q,\\\\0,& P\\text{ and }Q\\text{ differ in every coordinate}.\\end{cases}$$" + }, + { + "label": "E", + "text": "$$\\dim_k\\operatorname{Ext}^1_A(M_P,M_Q)=\\begin{cases}n-1,& P=Q,\\\\0,& P\\ne Q.\\end{cases}$$" + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "quotient by inner derivations in the off-diagonal case", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "the vanishing statement for P\\ne Q", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "case_split", + "tampered_component": "use of a coordinate with nonzero difference after reducing to one point at the origin", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "dimension of derivations in the diagonal case", + "template_used": "boundary_range" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not state or strongly hint at the diagonal/off-diagonal Ext^1 dimensions; it only defines the modules and asks for the correct universal statement." + }, + "TAS": { + "score": 2, + "justification": "This is not a mere restatement of a theorem from the stem. The respondent must choose among several competing global descriptions, including a weaker true statement and subtly false variants." + }, + "GPS": { + "score": 2, + "justification": "Answering correctly requires genuine reasoning or recall of the Ext computation: one must identify both the diagonal value n and the off-diagonal vanishing, and also recognize that choice C is true but not the strongest statement asked for." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically targeted: B tests mistaken off-diagonal nonvanishing, C is a tempting weaker true statement, D reflects an over-specific coordinate-based vanishing claim, and E probes a dimension-count error." + }, + "total_score": 8, + "overall_assessment": "A strong MCQ: no answer leakage, non-tautological framing, real reasoning pressure, and high-quality distractors that reflect common mathematical missteps." + } + }, + { + "id": "2511.07607v2", + "paper_link": "http://arxiv.org/abs/2511.07607v2", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:main}\nLet $E_0\\in \\mathbb R$ be a fixed energy for which $L_d(\\omega,E_0)>0$.\nFor any $\\omega\\in \\mathrm{DC}$, the integrated density of states of $H_{\\omega,\\theta}$ satisfies\n \\begin{align}\n |\\mathcal{N}(\\omega,E)-\\mathcal{N}(\\omega,E')|\\leq |E-E'|^{\\beta},\n \\end{align}\n for any $E,E'\\in I_{E_0}$, a neighborhood of $E_0$, and any H\\\"older exponent $0<\\beta<1/(2\\kappa^d(\\omega,E_0))$.", + "start_pos": 12230, + "end_pos": 12668, + "label": "thm:main" + }, + "ref_dict": { + "def:fn": "\\begin{align}\\label{def:fn}\nf_n(\\theta,E):=\\det(H^p_{\\theta}|_{[0,nd-1]}-E).\n\\end{align}", + "def:symp": "\\begin{align}\\label{def:symp}\n M^*\\Omega M=\\Omega,\n\\end{align}", + "thm:main": "\\begin{theorem}\\label{thm:main}\nLet $E_0\\in \\R$ be a fixed energy for which $L_d(\\omega,E_0)>0$.\nFor any $\\omega\\in \\mathrm{DC}$, the integrated density of states of $H_{\\omega,\\theta}$ satisfies\n \\begin{align}\n |\\mathcal{N}(\\omega,E)-\\mathcal{N}(\\omega,E')|\\leq |E-E'|^{\\beta},\n \\end{align}\n for any $E,E'\\in I_{E_0}$, a neighborhood of $E_0$, and any H\\\"older exponent $0<\\beta<1/(2\\kappa^d(\\omega,E_0))$. \n\\end{theorem}", + "def:LE": "\\begin{align}\\label{def:LE}\nL_j(\\omega, A,n):=\\frac{1}{n}\\int_{\\T} \\log \\sigma_j(A_n(\\omega,\\theta))\\, \\mathrm{d}\\theta, \\text{ for } 1\\leq j\\leq k,\n\\end{align}", + "thm:acc=zeros": "\\begin{theorem}\\label{thm:acc=zeros}\nAssume that for some $E\\in \\R$, $L^1(\\omega,E)\\geq \\nu>0$.\nThen for some $\\gamma\\in (0,1)$ and for $n$ large enough,\n\\begin{align}\n\\left| \\frac{1}{2n} N_n (E,\\delta/2)-\\kappa^1(\\omega,E)\\right|\\leq \\delta^{-1} n^{-\\gamma}.\n\\end{align}\n\\end{theorem}", + "def:DC": "\\begin{aligned}\\label{def:DC}\n \\omega\\in \\mathrm{DC}:=&\\bigcup_{a>0, A>1} \\mathrm{DC}_{a,A}, \\text{ where}\\\\\n \\mathrm{DC}_{a,A}=&\\left\\{\\omega\\in \\T:\\, \\|k\\omega\\|_{\\T}\\geq \\frac{a}{|k|^A}\\, \\text{ for all } k\\in \\Z\\setminus \\{0\\}\\right\\}.\n\\end{aligned}", + "lem:numerator_diag": "\\begin{lemma}\\label{lem:numerator_diag}\n Let $\\omega\\in \\mathrm{DC}$ and $x=y\\in [2d,n-2d-1]$, \n Then for $\\gamma>0$ as in Lemma \\ref{lem:upperbd}, $n$ large, and $E'\\in \\C$ such that $|E'-E|\\leq n^{-2}e^{-n^{\\gamma}}$, uniformly in $\\theta\\in \\T$:\n \\begin{align}\n |\\mu^{E'}_{[0,nd-1],x,x}(\\theta)|\\leq e^{n(L^d(E)+\\langle \\log |\\det B|\\rangle+5n^{-\\gamma})}.\n \\end{align}\n\\end{lemma}", + "eq:acc": "\\begin{align}\\label{eq:acc}\n\\kappa^1_{\\varepsilon}(\\omega, A):=\\lim_{\\varepsilon'\\to 0^+} \\frac{L^1_{\\varepsilon+\\varepsilon'}(\\omega, A)-L^1_{\\varepsilon}(\\omega, A)}{2\\pi \\varepsilon'}.\n\\end{align}", + "eq:SDC": "\\begin{align}\\label{eq:SDC}\n \\mathrm{DC}_{\\mathrm{strong}}:= \\bigcup_{a>1, c>0} \\left\\{\\omega\\in \\T:\\, \\|k\\omega\\|_{\\T}\\geq \\frac{c}{|k|(\\log |k|)^a}\\, \\text{ for all } k\\in \\Z\\setminus \\{0,\\pm 1\\}\\right\\}.\n\\end{align}", + "lem:local_zero*": "\\begin{lemma}\\label{lem:local_zero*}\nLet $n$ be $\\kappa_0$-admissible and large. \nFor each ball $B(z_0,r_n)$, with $z_0\\in \\kreis^1$ and $r\\simeq r_n:=e^{-(\\log n)^{C_0}}$ with $C_0>1$, there exists $|k|<(1-\\epsilon)n/2$ such that $f_n(e^{2\\pi ik\\omega} z,E)$ has at most $2\\kappa^d$ zeros in $B(z_0,r)$.\n\\end{lemma}", + "eq:BVsys": "\\begin{equation}\\label{eq:BVsys}\n (H_{\\omega,\\theta} \\Phi)_n = B_{n+1}(\\theta)\\Phi_{n+1}+ B_n^{(*)}(\\theta) \\Phi_{n-1}+ V_n (\\theta)\\Phi_n\n\\end{equation}" + }, + "pre_theorem_intro_text_len": 4535, + "pre_theorem_intro_text": "In this paper we study Schr\\\"odinger cocycles on a strip with Hamiltonian\n\\begin{equation}\\label{eq:BVsys}\n (H_{\\omega,\\theta} \\Phi)_n = B_{n+1}(\\theta)\\Phi_{n+1}+ B_n^{(*)}(\\theta) \\Phi_{n-1}+ V_n (\\theta)\\Phi_n\n\\end{equation}\nwhere $F_n(\\theta):=F(\\theta+n\\omega)$ for any $d\\times d$-matrix valued function. We set $B^{(*)}(\\theta)=(B(\\theta))^*$ for $\\theta\\in \\mathbb T$, and require it to be the analytic extension of $(B(\\theta))^*$ off of the real torus. Here $\\omega\\in\\mathbb T$ and we assume that $\\omega$ is Diophantine, i.e.,\n\\begin{equation}\\begin{aligned}\\label{def:DC}\n \\omega\\in \\mathrm{DC}:=&\\bigcup_{a>0, A>1} \\mathrm{DC}_{a,A}, \\text{ where}\\\\\n \\mathrm{DC}_{a,A}=&\\left\\{\\omega\\in \\mathbb T:\\, \\|k\\omega\\|_{\\mathbb T}\\geq \\frac{a}{|k|^A}\\, \\text{ for all } k\\in \\mathbb Z\\setminus \\{0\\}\\right\\}.\n\\end{aligned} \n\\end{equation}\nWe further assume that $B,V\\in C^{\\omega}(\\T_{\\delta}, \\mathrm{Mat}(d,\\mathbb C))$ are analytic, where $$\\T_{\\delta}:=\\{\\theta+i\\varepsilon:\\, \\theta\\in \\mathbb T,\\, \\varepsilon\\in \\mathbb R, \\text{ and } |\\varepsilon|\\leq \\delta\\}$$with some positive $\\delta>0$. \nWe assume throughout the paper that $V$ is Hermitian, and that~$B$ is invertible ($\\det B(\\theta)\\neq 0$ for any $\\theta\\in \\T_{\\delta}$). \nThe difference equation $H_{\\omega,\\theta} \\Phi=E\\Phi$ is equivalent to the cocycle \n\\begin{align}\\label{def:M_E}\n \\mathcal{C} &\\:: (\\theta,\\Psi)\\in \\mathbb T\\times \\mathbb C^{2d}\\mapsto (\\theta+\\omega, M_E(\\theta)\\Psi), \\notag \\\\\n \\quad M_{E}(\\theta) &= \\left [ \\begin{matrix}\n ( E-V(\\theta))B(\\theta)^{-1} & -B^{(*)}(\\theta) \\\\\n B(\\theta)^{-1} & 0 \n\\end{matrix}\\right] \n\\end{align}\nin the sense that for $n\\ge1$, \n\\begin{equation}\\label{eq:MnE}\n\\mathcal{C}^n (\\theta,\\Psi)=(\\theta+n\\omega, M_{n,E}(\\theta)\\Psi), \\quad M_{n,E}(\\theta)=\\prod_{j=n-1}^0 M_E(\\theta+j\\omega), \\quad \n\\Psi_n:=\\binom{B_{n}\\Phi_n}{\\Phi_{n-1}}\n\\end{equation} \nsatisfies $\\Psi_n = M_{n,E} (\\theta) \\Psi_0$. The model~\\eqref{eq:BVsys} was studied systematically by S.~Klein~\\cite{SK_block}, and Duarte and Klein~\\cites{DK1, DK2} within a very general framework, including the identically singular case when $\\det B(\\theta)\\equiv 0$ on $\\mathbb T^k$, $k\\geq 1$. These models generalize the quasi-periodic Schr\\\"odinger Hamiltonians on a strip considered by Bourgain and Jitomirskaya~\\cite{BJ}, who proved localization perturbatively for large disorder. \n\nSince $M_E(\\theta)$ is (complex) symplectic, see \\eqref{def:symp}, for $\\theta\\in \\mathbb T$, the Lyapunov exponents $\\{L_j(\\omega,M_E)\\}_{j=1}^{2d}$, see definition in \\eqref{def:LE}, satisfy $L_{2d+1-j}=-L_j$ for $1\\le j\\le d$. \nLet $L^d(\\omega,M_E)$ be the sum of the top $d$ Lyapunov exponents, and let $\\kappa^d(\\omega,M_E)$ be the acceleration, defined to be the right-derivative of $L^d(\\omega,M_E(\\cdot+i\\varepsilon))$ in the imaginary direction $\\varepsilon$, see \\eqref{eq:acc}.\n\nThroughout the paper we shall fix an energy $E_0\\in \\mathbb R$ such that $L_d(\\omega,E_0)\\geq \\tau>0$. It is known, see \\cite{AJS}, that $L^d(\\omega,M_E(\\cdot+i\\varepsilon))$ and $L_d(\\omega,M_E)$ are continuous in $E,\\varepsilon$ and $\\kappa^d(\\omega,M_E)$ is upper semi-continuous in $E,\\varepsilon$, hence we can assume that in a neighborhood $I_{E_0}$ of $E_0$, the following holds uniformly in $E\\in I_{E_0}$ and $|\\varepsilon|\\leq \\delta$, $\\varepsilon\\in \\mathbb R$, for some $\\delta=\\delta(E_0)>0$:\n\\begin{align}\\label{def:IE0}\n\\begin{cases}\nL_d(\\omega,M_E(\\cdot+i\\varepsilon))\\geq \\tau/2,\\\\\n\\kappa^d(\\omega,M_E(\\cdot+i\\varepsilon))\\leq \\kappa^d(\\omega,M_{E_0})\\\\\nL^d(\\omega,M_E(\\cdot+i\\varepsilon))\\leq L^d(\\omega,M_E)+2\\pi \\kappa^d(\\omega,M_{E_0})\\varepsilon.\n\\end{cases}\n\\end{align}\nNote the third inequality is a corollary of the second one.\n\nThe density of states $\\mathcal{N}(\\omega,\\cdot)$ is the limiting cumulative distribution function of the finite-volume eigenvalues. \nLet $H_{\\omega,\\theta}|_{\\Lambda}$ be the restriction of $H_{\\omega,\\theta}$ to the interval $\\Lambda=[a,b]\\subset \\mathbb Z$.\nLet $\\{E_{\\Lambda,j}(\\omega,\\theta)\\}_{j=1}^{|\\Lambda|}$ be the eigenvalues of $H_{\\omega,\\theta}|_{\\Lambda}$.\nConsider\n\\begin{align}\n N_{\\Lambda}(\\omega,E,\\theta):=\\frac{1}{|\\Lambda|}\\sum_{j=1}^{|\\Lambda|}\\chi_{(-\\infty,E)}(E_{\\Lambda,j}(\\omega,\\theta)).\n\\end{align}\nIt is well-known that the weak-limit\n\\begin{align}\n \\lim_{a\\to-\\infty, b\\to \\infty}\\mathrm{d} N_{\\Lambda}(\\omega,E,\\theta)=:\\mathrm{d} \\mathcal{N}(\\omega,E).\n\\end{align}\nexists and is independent of $\\theta$. In this paper we prove the following result.", + "context": "In this paper we study Schr\\\"odinger cocycles on a strip with Hamiltonian\n\\begin{equation}\\label{eq:BVsys}\n (H_{\\omega,\\theta} \\Phi)_n = B_{n+1}(\\theta)\\Phi_{n+1}+ B_n^{(*)}(\\theta) \\Phi_{n-1}+ V_n (\\theta)\\Phi_n\n\\end{equation}\nwhere $F_n(\\theta):=F(\\theta+n\\omega)$ for any $d\\times d$-matrix valued function. We set $B^{(*)}(\\theta)=(B(\\theta))^*$ for $\\theta\\in \\mathbb T$, and require it to be the analytic extension of $(B(\\theta))^*$ off of the real torus. Here $\\omega\\in\\mathbb T$ and we assume that $\\omega$ is Diophantine, i.e.,\n\\begin{equation}\\begin{aligned}\\label{def:DC}\n \\omega\\in \\mathrm{DC}:=&\\bigcup_{a>0, A>1} \\mathrm{DC}_{a,A}, \\text{ where}\\\\\n \\mathrm{DC}_{a,A}=&\\left\\{\\omega\\in \\mathbb T:\\, \\|k\\omega\\|_{\\mathbb T}\\geq \\frac{a}{|k|^A}\\, \\text{ for all } k\\in \\mathbb Z\\setminus \\{0\\}\\right\\}.\n\\end{aligned} \n\\end{equation}\nWe further assume that $B,V\\in C^{\\omega}(\\T_{\\delta}, \\mathrm{Mat}(d,\\mathbb C))$ are analytic, where $$\\T_{\\delta}:=\\{\\theta+i\\varepsilon:\\, \\theta\\in \\mathbb T,\\, \\varepsilon\\in \\mathbb R, \\text{ and } |\\varepsilon|\\leq \\delta\\}$$with some positive $\\delta>0$. \nWe assume throughout the paper that $V$ is Hermitian, and that~$B$ is invertible ($\\det B(\\theta)\\neq 0$ for any $\\theta\\in \\T_{\\delta}$). \nThe difference equation $H_{\\omega,\\theta} \\Phi=E\\Phi$ is equivalent to the cocycle \n\\begin{align}\\label{def:M_E}\n \\mathcal{C} &\\:: (\\theta,\\Psi)\\in \\mathbb T\\times \\mathbb C^{2d}\\mapsto (\\theta+\\omega, M_E(\\theta)\\Psi), \\notag \\\\\n \\quad M_{E}(\\theta) &= \\left [ \\begin{matrix}\n ( E-V(\\theta))B(\\theta)^{-1} & -B^{(*)}(\\theta) \\\\\n B(\\theta)^{-1} & 0 \n\\end{matrix}\\right] \n\\end{align}\nin the sense that for $n\\ge1$, \n\\begin{equation}\\label{eq:MnE}\n\\mathcal{C}^n (\\theta,\\Psi)=(\\theta+n\\omega, M_{n,E}(\\theta)\\Psi), \\quad M_{n,E}(\\theta)=\\prod_{j=n-1}^0 M_E(\\theta+j\\omega), \\quad \n\\Psi_n:=\\binom{B_{n}\\Phi_n}{\\Phi_{n-1}}\n\\end{equation} \nsatisfies $\\Psi_n = M_{n,E} (\\theta) \\Psi_0$. The model~\\eqref{eq:BVsys} was studied systematically by S.~Klein~\\cite{SK_block}, and Duarte and Klein~\\cites{DK1, DK2} within a very general framework, including the identically singular case when $\\det B(\\theta)\\equiv 0$ on $\\mathbb T^k$, $k\\geq 1$. These models generalize the quasi-periodic Schr\\\"odinger Hamiltonians on a strip considered by Bourgain and Jitomirskaya~\\cite{BJ}, who proved localization perturbatively for large disorder.\n\nSince $M_E(\\theta)$ is (complex) symplectic, see \\eqref{def:symp}, for $\\theta\\in \\mathbb T$, the Lyapunov exponents $\\{L_j(\\omega,M_E)\\}_{j=1}^{2d}$, see definition in \\eqref{def:LE}, satisfy $L_{2d+1-j}=-L_j$ for $1\\le j\\le d$. \nLet $L^d(\\omega,M_E)$ be the sum of the top $d$ Lyapunov exponents, and let $\\kappa^d(\\omega,M_E)$ be the acceleration, defined to be the right-derivative of $L^d(\\omega,M_E(\\cdot+i\\varepsilon))$ in the imaginary direction $\\varepsilon$, see \\eqref{eq:acc}.\n\nThroughout the paper we shall fix an energy $E_0\\in \\mathbb R$ such that $L_d(\\omega,E_0)\\geq \\tau>0$. It is known, see \\cite{AJS}, that $L^d(\\omega,M_E(\\cdot+i\\varepsilon))$ and $L_d(\\omega,M_E)$ are continuous in $E,\\varepsilon$ and $\\kappa^d(\\omega,M_E)$ is upper semi-continuous in $E,\\varepsilon$, hence we can assume that in a neighborhood $I_{E_0}$ of $E_0$, the following holds uniformly in $E\\in I_{E_0}$ and $|\\varepsilon|\\leq \\delta$, $\\varepsilon\\in \\mathbb R$, for some $\\delta=\\delta(E_0)>0$:\n\\begin{align}\\label{def:IE0}\n\\begin{cases}\nL_d(\\omega,M_E(\\cdot+i\\varepsilon))\\geq \\tau/2,\\\\\n\\kappa^d(\\omega,M_E(\\cdot+i\\varepsilon))\\leq \\kappa^d(\\omega,M_{E_0})\\\\\nL^d(\\omega,M_E(\\cdot+i\\varepsilon))\\leq L^d(\\omega,M_E)+2\\pi \\kappa^d(\\omega,M_{E_0})\\varepsilon.\n\\end{cases}\n\\end{align}\nNote the third inequality is a corollary of the second one.\n\nThe density of states $\\mathcal{N}(\\omega,\\cdot)$ is the limiting cumulative distribution function of the finite-volume eigenvalues. \nLet $H_{\\omega,\\theta}|_{\\Lambda}$ be the restriction of $H_{\\omega,\\theta}$ to the interval $\\Lambda=[a,b]\\subset \\mathbb Z$.\nLet $\\{E_{\\Lambda,j}(\\omega,\\theta)\\}_{j=1}^{|\\Lambda|}$ be the eigenvalues of $H_{\\omega,\\theta}|_{\\Lambda}$.\nConsider\n\\begin{align}\n N_{\\Lambda}(\\omega,E,\\theta):=\\frac{1}{|\\Lambda|}\\sum_{j=1}^{|\\Lambda|}\\chi_{(-\\infty,E)}(E_{\\Lambda,j}(\\omega,\\theta)).\n\\end{align}\nIt is well-known that the weak-limit\n\\begin{align}\n \\lim_{a\\to-\\infty, b\\to \\infty}\\mathrm{d} N_{\\Lambda}(\\omega,E,\\theta)=:\\mathrm{d} \\mathcal{N}(\\omega,E).\n\\end{align}\nexists and is independent of $\\theta$. In this paper we prove the following result.\n\n\\begin{align}\\label{def:LE}\nL_j(\\omega, A,n):=\\frac{1}{n}\\int_{\\T} \\log \\sigma_j(A_n(\\omega,\\theta))\\, \\mathrm{d}\\theta, \\text{ for } 1\\leq j\\leq k,\n\\end{align}\n\n\\begin{align}\\label{def:symp}\n M^*\\Omega M=\\Omega,\n\\end{align}\n\n\\begin{equation}\\label{eq:BVsys}\n (H_{\\omega,\\theta} \\Phi)_n = B_{n+1}(\\theta)\\Phi_{n+1}+ B_n^{(*)}(\\theta) \\Phi_{n-1}+ V_n (\\theta)\\Phi_n\n\\end{equation}\n\n\\begin{align}\\label{eq:acc}\n\\kappa^1_{\\varepsilon}(\\omega, A):=\\lim_{\\varepsilon'\\to 0^+} \\frac{L^1_{\\varepsilon+\\varepsilon'}(\\omega, A)-L^1_{\\varepsilon}(\\omega, A)}{2\\pi \\varepsilon'}.\n\\end{align}", + "full_context": "In this paper we study Schr\\\"odinger cocycles on a strip with Hamiltonian\n\\begin{equation}\\label{eq:BVsys}\n (H_{\\omega,\\theta} \\Phi)_n = B_{n+1}(\\theta)\\Phi_{n+1}+ B_n^{(*)}(\\theta) \\Phi_{n-1}+ V_n (\\theta)\\Phi_n\n\\end{equation}\nwhere $F_n(\\theta):=F(\\theta+n\\omega)$ for any $d\\times d$-matrix valued function. We set $B^{(*)}(\\theta)=(B(\\theta))^*$ for $\\theta\\in \\mathbb T$, and require it to be the analytic extension of $(B(\\theta))^*$ off of the real torus. Here $\\omega\\in\\mathbb T$ and we assume that $\\omega$ is Diophantine, i.e.,\n\\begin{equation}\\begin{aligned}\\label{def:DC}\n \\omega\\in \\mathrm{DC}:=&\\bigcup_{a>0, A>1} \\mathrm{DC}_{a,A}, \\text{ where}\\\\\n \\mathrm{DC}_{a,A}=&\\left\\{\\omega\\in \\mathbb T:\\, \\|k\\omega\\|_{\\mathbb T}\\geq \\frac{a}{|k|^A}\\, \\text{ for all } k\\in \\mathbb Z\\setminus \\{0\\}\\right\\}.\n\\end{aligned} \n\\end{equation}\nWe further assume that $B,V\\in C^{\\omega}(\\T_{\\delta}, \\mathrm{Mat}(d,\\mathbb C))$ are analytic, where $$\\T_{\\delta}:=\\{\\theta+i\\varepsilon:\\, \\theta\\in \\mathbb T,\\, \\varepsilon\\in \\mathbb R, \\text{ and } |\\varepsilon|\\leq \\delta\\}$$with some positive $\\delta>0$. \nWe assume throughout the paper that $V$ is Hermitian, and that~$B$ is invertible ($\\det B(\\theta)\\neq 0$ for any $\\theta\\in \\T_{\\delta}$). \nThe difference equation $H_{\\omega,\\theta} \\Phi=E\\Phi$ is equivalent to the cocycle \n\\begin{align}\\label{def:M_E}\n \\mathcal{C} &\\:: (\\theta,\\Psi)\\in \\mathbb T\\times \\mathbb C^{2d}\\mapsto (\\theta+\\omega, M_E(\\theta)\\Psi), \\notag \\\\\n \\quad M_{E}(\\theta) &= \\left [ \\begin{matrix}\n ( E-V(\\theta))B(\\theta)^{-1} & -B^{(*)}(\\theta) \\\\\n B(\\theta)^{-1} & 0 \n\\end{matrix}\\right] \n\\end{align}\nin the sense that for $n\\ge1$, \n\\begin{equation}\\label{eq:MnE}\n\\mathcal{C}^n (\\theta,\\Psi)=(\\theta+n\\omega, M_{n,E}(\\theta)\\Psi), \\quad M_{n,E}(\\theta)=\\prod_{j=n-1}^0 M_E(\\theta+j\\omega), \\quad \n\\Psi_n:=\\binom{B_{n}\\Phi_n}{\\Phi_{n-1}}\n\\end{equation} \nsatisfies $\\Psi_n = M_{n,E} (\\theta) \\Psi_0$. The model~\\eqref{eq:BVsys} was studied systematically by S.~Klein~\\cite{SK_block}, and Duarte and Klein~\\cites{DK1, DK2} within a very general framework, including the identically singular case when $\\det B(\\theta)\\equiv 0$ on $\\mathbb T^k$, $k\\geq 1$. These models generalize the quasi-periodic Schr\\\"odinger Hamiltonians on a strip considered by Bourgain and Jitomirskaya~\\cite{BJ}, who proved localization perturbatively for large disorder.\n\nSince $M_E(\\theta)$ is (complex) symplectic, see \\eqref{def:symp}, for $\\theta\\in \\mathbb T$, the Lyapunov exponents $\\{L_j(\\omega,M_E)\\}_{j=1}^{2d}$, see definition in \\eqref{def:LE}, satisfy $L_{2d+1-j}=-L_j$ for $1\\le j\\le d$. \nLet $L^d(\\omega,M_E)$ be the sum of the top $d$ Lyapunov exponents, and let $\\kappa^d(\\omega,M_E)$ be the acceleration, defined to be the right-derivative of $L^d(\\omega,M_E(\\cdot+i\\varepsilon))$ in the imaginary direction $\\varepsilon$, see \\eqref{eq:acc}.\n\nThroughout the paper we shall fix an energy $E_0\\in \\mathbb R$ such that $L_d(\\omega,E_0)\\geq \\tau>0$. It is known, see \\cite{AJS}, that $L^d(\\omega,M_E(\\cdot+i\\varepsilon))$ and $L_d(\\omega,M_E)$ are continuous in $E,\\varepsilon$ and $\\kappa^d(\\omega,M_E)$ is upper semi-continuous in $E,\\varepsilon$, hence we can assume that in a neighborhood $I_{E_0}$ of $E_0$, the following holds uniformly in $E\\in I_{E_0}$ and $|\\varepsilon|\\leq \\delta$, $\\varepsilon\\in \\mathbb R$, for some $\\delta=\\delta(E_0)>0$:\n\\begin{align}\\label{def:IE0}\n\\begin{cases}\nL_d(\\omega,M_E(\\cdot+i\\varepsilon))\\geq \\tau/2,\\\\\n\\kappa^d(\\omega,M_E(\\cdot+i\\varepsilon))\\leq \\kappa^d(\\omega,M_{E_0})\\\\\nL^d(\\omega,M_E(\\cdot+i\\varepsilon))\\leq L^d(\\omega,M_E)+2\\pi \\kappa^d(\\omega,M_{E_0})\\varepsilon.\n\\end{cases}\n\\end{align}\nNote the third inequality is a corollary of the second one.\n\nThe density of states $\\mathcal{N}(\\omega,\\cdot)$ is the limiting cumulative distribution function of the finite-volume eigenvalues. \nLet $H_{\\omega,\\theta}|_{\\Lambda}$ be the restriction of $H_{\\omega,\\theta}$ to the interval $\\Lambda=[a,b]\\subset \\mathbb Z$.\nLet $\\{E_{\\Lambda,j}(\\omega,\\theta)\\}_{j=1}^{|\\Lambda|}$ be the eigenvalues of $H_{\\omega,\\theta}|_{\\Lambda}$.\nConsider\n\\begin{align}\n N_{\\Lambda}(\\omega,E,\\theta):=\\frac{1}{|\\Lambda|}\\sum_{j=1}^{|\\Lambda|}\\chi_{(-\\infty,E)}(E_{\\Lambda,j}(\\omega,\\theta)).\n\\end{align}\nIt is well-known that the weak-limit\n\\begin{align}\n \\lim_{a\\to-\\infty, b\\to \\infty}\\mathrm{d} N_{\\Lambda}(\\omega,E,\\theta)=:\\mathrm{d} \\mathcal{N}(\\omega,E).\n\\end{align}\nexists and is independent of $\\theta$. In this paper we prove the following result.\n\n\\begin{align}\\label{def:LE}\nL_j(\\omega, A,n):=\\frac{1}{n}\\int_{\\T} \\log \\sigma_j(A_n(\\omega,\\theta))\\, \\mathrm{d}\\theta, \\text{ for } 1\\leq j\\leq k,\n\\end{align}\n\n\\begin{align}\\label{def:symp}\n M^*\\Omega M=\\Omega,\n\\end{align}\n\n\\begin{equation}\\label{eq:BVsys}\n (H_{\\omega,\\theta} \\Phi)_n = B_{n+1}(\\theta)\\Phi_{n+1}+ B_n^{(*)}(\\theta) \\Phi_{n-1}+ V_n (\\theta)\\Phi_n\n\\end{equation}\n\n\\begin{align}\\label{eq:acc}\n\\kappa^1_{\\varepsilon}(\\omega, A):=\\lim_{\\varepsilon'\\to 0^+} \\frac{L^1_{\\varepsilon+\\varepsilon'}(\\omega, A)-L^1_{\\varepsilon}(\\omega, A)}{2\\pi \\varepsilon'}.\n\\end{align}\n\nIn \\cite{HS3}*{Lemma 2.8} we established the following large deviation theorem for $f_n$.\n\\begin{lemma}\\label{lem:deno}\nLet $\\omega\\in \\mathrm{DC}$, and $\\gamma>0$ be as in Lemma \\ref{lem:LDTsig}.\nAssume $L_d(\\omega,M_E)\\geq \\nu>0$.\nThere exist $\\gamma>0$, $N_0>1$ large and $0<\\kappa_0\\ll 1$ so that the {\\it $\\kappa_0$-admissible} sequence \n\\begin{align}\\label{def:admissible}\n\\mathcal{N}:=\\{n\\geq N_0: \\|n\\omega\\|_{\\T}\\leq \\kappa_0\\}\n\\end{align}\nhas the following property: \nfor any $|\\varepsilon|\\leq \\delta/2$, and all large $\\kappa_0$-admissible $n$, the following large deviation set \n\\begin{align}\\label{def:B_fEn}\n\\mathcal{B}_{f,E,n,\\varepsilon}:=\n\\big\\{\\theta\\in \\T: n^{-1}\\log |f_{n}(\\theta+i\\varepsilon,E)|<\\langle \\log |\\det B(\\cdot+i\\varepsilon)|\\rangle +{L}^d_{\\varepsilon}(\\omega,M_E)- n^{-\\gamma}\\big\\}\n\\end{align}\nsatisfies $\\mathrm{mes}(\\mathcal{B}_{f,E,n,\\varepsilon})0$ there exists an admissible $\\tilde n>0$ with $|n-\\tilde n|\\le C_*$ for some constant $C_*$.\n\\end{remark}\n\nRecall that $E\\in I_{E_0}$.\nOur goal is to analyze the following for an arbitrary fixed $\\theta$ as $N\\to\\infty$:\n\\begin{align}\\label{eq:goal}\n &d_{\\omega,N}(\\theta,E-\\eta,E+\\eta)=\\frac{1}{N}\\mathrm{tr}(P_{[E-\\eta, E+\\eta)}(H_{\\omega,\\theta}|_{[0,N-1]})),\n\\end{align}\nin which $H_{\\omega,\\theta}|_{[0,N-1]}$ is $H_{\\omega,\\theta}$ restricted to the interval $[0,N-1]$ with Dirichlet boundary condition and $P_{[E-\\eta,E+\\eta)}$ stands for the spectral projection.\nBounding \\eqref{eq:goal} through the trace of the resolvent yields\n\\begin{align}\\label{eq:dN\\exp(2m(L(E,2m)-(2m)^{-\\gamma}))$. A telescoping argument, as in the proof of Lemma \\ref{lem:un_lower_zeros_eta}, using that $\\gamma_1>1-\\gamma/2$, yields\n\\begin{align}\n |D_{2m}(e^{2\\pi i\\theta},E+i\\eta)|>\\frac{1}{2}|D_{2m}(e^{2\\pi i\\theta},E)|>\\frac{1}{2}\\exp(2m(L(E,2m)-(2m)^{-\\gamma})).\n\\end{align}\nThis implies $(\\mathcal{B}_{2m,1}^0)^c\\subset (\\mathcal{B}_{2m,2}^{\\eta})^c$,\nand hence \\eqref{eq:mes_B2m_2}, when combining with \\eqref{eq:mes_B2m_1}.\n\nThe proof is analogous to the scalar-valued case. \nOur goal is to estimate the following expression for arbitrary fixed $E\\in I_{E_0}$ and $\\theta\\in \\T$ as $N\\to\\infty$:\n\\begin{align}\\label{eq:goal*}\n &d_{\\omega,dN}(\\theta,E-\\eta,E+\\eta)=\\frac{1}{dN}\\mathrm{tr}(P_{[E-\\eta, E+\\eta)}(H_{\\omega,\\theta}|_{[0,dN-1]})),\n\\end{align}\ncf.\\ \\eqref{eq:goal}. As before, we \nbound~\\eqref{eq:goal*} through the trace of the Green's function: \n\\begin{align}\\label{eq:dN1-\\gamma/2$, we have\n\\begin{align}\n |f_{2m}(e^{2\\pi i\\theta},E+i\\eta)|\\geq \\frac{1}{2} |f_{2m}(e^{2\\pi i\\theta}, E)|\\geq \\exp(2m(L^d(E,2m)+\\langle \\log |\\det B(\\cdot) |\\rangle-2(2m)^{-\\gamma})).\n\\end{align}\nThis implies $(\\mathcal{B}_{2m,1}^0)^c\\subset (\\mathcal{B}_{2m,2}^{\\eta})^c$, hence the estimate \\eqref{eq:B2m_eta_block} follows from \\eqref{eq:B2m_0_block}.\nLet $\\{\\xi_j\\, e^{2\\pi i\\theta_j}\\}_{j=1}^{J_0}$, $\\xi_j>0$ and $\\theta_j\\in \\T$, be the zeros of $f_{2m}(z,E+i\\eta)$. \nCartan's estimate implies that \n\\begin{align}\\label{eq:B2m_theta_pm_rm*}\n\\mathcal{B}_{2m,m^{\\gamma/2}}^{\\eta}\\subset \\bigcup_{j=1}^{J_0}(\\theta_j-\\tilde{r}_m, \\theta_j+\\tilde{r}_m),\n\\end{align}\nwith $\\tilde{r}_m:=e^{-cm^{\\gamma/2}}$ for some constant $c>0$, and $J_0\\lesssim m$.\nApplying Lemma~\\ref{lem:un_lower_zeros_eta*} to $n=2m$ and $z_0=e^{2\\pi i\\theta_j}$, for each $1\\leq j\\leq J_0$, yields the existence of $|k_j|<(1-\\varepsilon)m$, $C_j\\in [1,2\\kappa^d+1]$, and $\\ell_j\\in [1,2\\kappa^d]$, such that $w_j(z):=f_{2m}(z e^{2\\pi ik_j\\omega},E+i\\eta)$\nhas $\\ell_j$ zeros in $B(e^{2\\pi i\\theta_j}, (4C_j+1)r_m)$, $r_m\\simeq \\exp(-(\\log m)^{C_0})\\gg \\tilde{r}_m$.\nDenoting the zeros of $f_{2m}(z e^{2\\pi ik_j\\omega},E+i\\eta)$ by $\\{\\xi_{j,\\ell}\\, e^{2\\pi i\\theta_{j,\\ell}}\\}_{\\ell=1}^{\\ell_j}$ by $\\xi_{j,\\ell}>0$, $\\theta_{j,\\ell}\\in \\T$, one has\n\\begin{align}\\label{eq:wj_lower_zeros*}\n \\log |w_j(z)|\\geq 2m(L^d(E,2m)+\\langle \\log |\\det B(\\cdot)|\\rangle-m^{-\\gamma/2})+\\sum_{\\ell=1}^{\\ell_j}\\log |z-\\xi_{j,\\ell}\\, e^{2\\pi i\\theta_{j,\\ell}}|,\n\\end{align}\nfor all $z\\in B(e^{2\\pi i\\theta_j},(4C_j+2)r_m)$.\nFor each $1\\leq j\\leq J_0$ and $1\\leq \\ell\\leq \\ell_j$, let \n $ \\tilde{I}_j, \n I_{j,\\ell}$ be defined as in~\\eqref{eq:Ijdef}. \nWith $\\tau$ as in \\eqref{eq:choose_tau} and $\\tau\\ll r_m$, we have for each $j,\\ell$ that $I_{j,\\ell}\\subset \\tilde{I}_j$.\nAs before, we let $\\tilde{\\mathcal{B}}_{2m}^{\\eta}:=\\bigcup_{j=1}^{J_0} \\tilde{I}_j$. \nClearly, by \\eqref{eq:B2m_theta_pm_rm*}, $\\mathcal{B}_{2m,m^{\\gamma/2}}^{\\eta}\\subset \\tilde{\\mathcal{B}}_{2m}^{\\eta}$.\nWe again divide the analysis of $G^{E+i\\eta}_{[0,dN-1]}(\\theta; k,k)$ into three cases.", + "post_theorem_intro_text_len": 6093, + "post_theorem_intro_text": "\\begin{remark}\n The exponent $2\\kappa^d(\\omega,E)$ comes from the local zero count of the finite-volume characteristic polynomials, which is derived from the global zero count. When $d\\geq 2$, this is the zero count of $f_n(\\theta,E)$, see \\eqref{def:fn}. If the blocks $B,V$ satisfy additional symmetries, then it is possible to deduce smaller local zero counts, and hence improved H\\\"older exponents. We refer interested readers to \\cite[Theorem 1.4 \\& Theorem 1.6]{HS3} where the effects of additional symmetries are studied. \n\\end{remark}\n\nAs for the history of this type of regularity result: $\\log$--H\\\"older continuity was established by Craig and Simon in \\cite{CS} for general ergodic potentials. The stronger H\\\"older continuity of the integrated density of states associated with~\\eqref{eq:BVsys} was first proved by Goldstein and the second author in \\cite{GS1} for the scalar case $d=1$.\nLater, \\cite{GS2}*{Theorem 1.1} established that for a trigonometric polynomial $f$ of degree $k_0\\ge1$ and assuming $\\omega\\in \\mathrm{DC}_{\\mathrm{strong}}$, see \\eqref{eq:SDC}, and \n$L(\\omega,E)>0$ the IDS is $\\beta$-H\\\"older continuous for any $\\beta<1/(2k_0)$. \nThis result, for the same set of frequencies, was improved to $\\beta<1/(2\\kappa^1(\\omega,E))$ in \\cite{HS1}. Note for trigonometric $f$ one has $\\kappa^1(\\omega,E)\\leq k_0$.\nTheorem~\\ref{thm:main}, when $d=1$, is an improvement of \\cite{HS1}*{Theorem 1.3} in the sense that it covers more general Diophantine frequencies: $\\mathrm{DC}_{\\mathrm{strong}}\\subset \\mathrm{DC}$. When $d\\geq 2$, Theorem~\\ref{thm:main} is completely new.\n\nFor the almost Mathieu operator and $\\alpha\\in \\mathrm{DC}$, the IDS was proved to be $1/2$-H\\\"older if $|\\lambda|\\neq 0,1$, see Avila, Jitomirskaya~\\cite{AJ}.\nThe $1/2$-H\\\"older exponent was also proved for Diophantine $\\alpha$ and $f=\\lambda\\, g$ with $g$ being analytic and the coupling constant $|\\lambda|$ being small, first in the perturbative regime (smallness depends on $\\alpha, g$) by Amor~\\cite{Amor}, and then in the non-perturbative regime \\cite{AJ} (with dependence on $\\alpha$ removed).\nFor quasi-periodic long-range operators with large trigonometric polynomial potentials and Diophantine frequencies, the H\\\"older exponent in \\cite{GS2} was improved recently in the perturbative regime~\\cite{GYZ}.\nThese results are proved using the reducibility method.\n\nFor the proof of Theorem~\\ref{thm:main} we first single out the Schr\\\"odinger operator case, where $d=1$, $B\\equiv 1$ and $V=v$ is a non-constant analytic function. Sections~\\ref{sec:sch1}, \\ref{sec:sch2} present this case in details. The argument is variant of the one in~\\cite{GS2}, but is simpler and more robust.\nThe difference lies in how we derive a favorable local zero count for a finite volume determinant relative to some interval $\\Lambda\\subset\\mathbb Z$. In~\\cite{GS2}, this is accomplished by slightly varying the size of~$\\Lambda$, but not the position. Here we keep the size fixed but change the position. The strip case ($d>1$ in~\\eqref{eq:BVsys}) follows the same outline but is more involved because it relies on zero counting techniques that apply to the Jacobi block case. Thankfully, these were developed in~\\cite{HS3}, so we can treat them as a black box, cf.~Theorem~\\ref{thm:acc=zeros}. \n\nWe also point out that the Aubry dual of a subcritical Schr\\\"odinger operator $H$, with trigonometric potential with degree $d$, is a special family of $d\\times d$ Jacobi block-valued matrix $\\hat{H}$ with uniformly positive $L_d(\\omega,E)>0$, see e.g.~\\cite[Corollary 1.12]{HS2}. An easy computation shows that $\\kappa^d(\\omega,E)\\leq d$, but one can show the local zero count is at most $2$, instead of $2d$. This is a small modification of Lemma \\ref{lem:local_zero*} with considering not only $f_n(e^{2\\pi ik\\omega}z,E)$ with $|k|<(1-\\varepsilon)n/2$, but also $f_n(e^{2\\pi ik\\omega/d}z,E)$ with $|k|<(1-\\varepsilon)nd/2$. This is a feature due to the special form of $\\hat{H}$, and has been explored in the proof of arithmetic Anderson localization of $\\hat{H}$ in \\cite[Theorem 1.7]{HS2} and \\cite[Corollary 1.3]{HS3}. The local zero count being at most $2$ then implies almost $1/2$ H\\\"older continuity of the integrated density of states of the original subcritical Schr\\\"odinger with arbitrary trigonometric potentials with Diophantine frequencies.\n\nFinally, we comment that the (non-)H\\\"older continuity of the integrated density of states for Liouville frequencies has also been studied \\cite{YZ,HZ,ALSZ,HS4} when $d=1$, \nand it would be natural to combine those considerations with the techniques developed here for $d\\geq 2$.\n\nOur proofs are essentially self-contained. The only ingredients we require are the large deviation estimates, developed in \\cites{BG,GS1,HS3} and global zero counts of finite-volume determinants in \\cites{HS1,HS3}. \nAs we mentioned, the Diophantine condition \\eqref{def:DC} is weaker than that in \\cites{GS2,HS1} where it was defined to be\n\\begin{align}\\label{eq:SDC}\n \\mathrm{DC}_{\\mathrm{strong}}:= \\bigcup_{a>1, c>0} \\left\\{\\omega\\in \\mathbb T:\\, \\|k\\omega\\|_{\\mathbb T}\\geq \\frac{c}{|k|(\\log |k|)^a}\\, \\text{ for all } k\\in \\mathbb Z\\setminus \\{0,\\pm 1\\}\\right\\}.\n\\end{align}\nThe Anderson localization result of \\cite{HS1} holds for the weaker Diophantine condition \\eqref{def:DC}, see \\cite{HS1}*{Remark 1.9}. However, the H\\\"older continuity result of \\cite{HS1} holds under \\eqref{eq:SDC} due to the need for a sharp large deviation estimate as in \\cite{GS2}. \n\nWe organize the paper as follows: Section~\\ref{sec:Pre} contains the preliminary results. Sections~\\ref{sec:sch1} and \\ref{sec:sch2} prove Theorem~\\ref{thm:main} in the Schr\\\"odinger case, more specifically, Sec.~\\ref{sec:sch1} presents the crucial local factorization of the finite volume characteristic polynomials which lays the foundation of the proof of Theorem~\\ref{thm:main} in Sec.~\\ref{sec:sch2}. The Jacobi block-valued case is proved in Sections~\\ref{sec:block1} and \\ref{sec:block2}. Finally, the proof of Lemma~\\ref{lem:numerator_diag} is contained in Sec.~\\ref{sec:numerator}.", + "sketch": "For the proof of Theorem~\\ref{thm:main} the authors “first single out the Schr\\\"odinger operator case, where $d=1$, $B\\equiv 1$ and $V=v$ is a non-constant analytic function.” Sections~\\ref{sec:sch1}, \\ref{sec:sch2} treat this case; “the argument is variant of the one in~\\cite{GS2}, but is simpler and more robust.” The key difference is “how we derive a favorable local zero count for a finite volume determinant relative to some interval $\\Lambda\\subset\\mathbb Z$”: in~\\cite{GS2} this is done “by slightly varying the size of~$\\Lambda$, but not the position,” whereas here “we keep the size fixed but change the position.”\n\nThe strip/block case (“$d>1$ in~\\eqref{eq:BVsys}”) “follows the same outline but is more involved because it relies on zero counting techniques that apply to the Jacobi block case”; these zero counting tools were developed in~\\cite{HS3} and are used “as a black box, cf.~Theorem~\\ref{thm:acc=zeros}.”\n\nStructurally, “Sec.~\\ref{sec:sch1} presents the crucial local factorization of the finite volume characteristic polynomials which lays the foundation of the proof of Theorem~\\ref{thm:main} in Sec.~\\ref{sec:sch2},” and “the Jacobi block-valued case is proved in Sections~\\ref{sec:block1} and \\ref{sec:block2}.” The underlying inputs are stated to be “large deviation estimates” (from \\cites{BG,GS1,HS3}) and “global zero counts of finite-volume determinants” (from \\cites{HS1,HS3}); the remark also explains that the exponent $2\\kappa^d(\\omega,E)$ “comes from the local zero count of the finite-volume characteristic polynomials, which is derived from the global zero count.”", + "expanded_sketch": "No expanded sketch found.", + "expanded_theorem": "\\label{thm:main}\nLet $E_0\\in \\mathbb R$ be a fixed energy for which $L_d(\\omega,E_0)>0$.\nFor any $\\omega\\in \\mathrm{DC}$, the integrated density of states of $H_{\\omega,\\theta}$ satisfies\n \\begin{align}\n |\\mathcal{N}(\\omega,E)-\\mathcal{N}(\\omega,E')|\\leq |E-E'|^{\\beta},\n \\end{align}\n for any $E,E'\\in I_{E_0}$, a neighborhood of $E_0$, and any H\\\"older exponent $0<\\beta<1/(2\\kappa^d(\\omega,E_0))$.", + "theorem_type": [ + "Inequality or Bound", + "Universal" + ], + "mcq": { + "question": "Consider the analytic quasi-periodic Schr\\\"odinger operator on a strip acting on sequences \\(\\Phi_n\\in\\mathbb C^d\\) by\n\\[\n(H_{\\omega,\\theta}\\Phi)_n=B(\\theta+(n+1)\\omega)\\Phi_{n+1}+B(\\theta+n\\omega)^*\\Phi_{n-1}+V(\\theta+n\\omega)\\Phi_n,\n\\]\nwhere \\(B,V\\in C^{\\omega}(\\mathbb T_{\\delta},\\mathrm{Mat}(d,\\mathbb C))\\), \\(V\\) is Hermitian, and \\(\\det B(\\theta)\\neq 0\\) on \\(\\mathbb T_{\\delta}\\). Assume the frequency \\(\\omega\\) is Diophantine, i.e.\n\\[\n\\omega\\in \\mathrm{DC}:=\\bigcup_{a>0,\\,A>1}\\left\\{\\omega\\in\\mathbb T:\\ \\|k\\omega\\|_{\\mathbb T}\\ge \\frac{a}{|k|^A}\\ \\text{for all }k\\in\\mathbb Z\\setminus\\{0\\}\\right\\}.\n\\]\nFor each energy \\(E\\in\\mathbb R\\), let the associated cocycle be\n\\[\nM_E(\\theta)=\\begin{pmatrix}\n(E-V(\\theta))B(\\theta)^{-1} & -B(\\theta)^*\\\\\nB(\\theta)^{-1} & 0\n\\end{pmatrix}.\n\\]\nLet \\(L_d(\\omega,E)\\) denote the \\(d\\)-th Lyapunov exponent of this cocycle, let \\(L^d\\) be the sum of its top \\(d\\) Lyapunov exponents, and let \\(\\kappa^d(\\omega,E)\\) be the right derivative at \\(\\varepsilon=0\\) of \\(L^d(\\omega,M_E(\\cdot+i\\varepsilon))\\) with respect to the imaginary shift \\(\\varepsilon\\). Let \\(\\mathcal N(\\omega,E)\\) be the integrated density of states, i.e. the phase-independent limiting cumulative distribution function of the finite-volume eigenvalues. Fix \\(E_0\\in\\mathbb R\\) such that \\(L_d(\\omega,E_0)>0\\). Which statement holds for every such Diophantine \\(\\omega\\)?", + "correct_choice": { + "label": "A", + "text": "There exists a neighborhood \\(I_{E_0}\\) of \\(E_0\\) such that for every \\(E,E'\\in I_{E_0}\\) and every H\\\"older exponent \\(0<\\beta<1/(2\\kappa^d(\\omega,E_0))\\), one has\n\\[\n|\\mathcal N(\\omega,E)-\\mathcal N(\\omega,E')|\\le |E-E'|^{\\beta}.\n\\]" + }, + "choices": [ + { + "label": "B", + "text": "There exists a neighborhood \\(I_{E_0}\\) of \\(E_0\\) such that for every \\(E,E'\\in I_{E_0}\\) one has\n\\[\n|\\mathcal N(\\omega,E)-\\mathcal N(\\omega,E')|\\le |E-E'|^{\\beta}\n\\]\nfor every H\\\"older exponent \\(0<\\beta<1/\\kappa^d(\\omega,E_0)\\)." + }, + { + "label": "C", + "text": "There exists a neighborhood \\(I_{E_0}\\) of \\(E_0\\) and some H\\\"older exponent \\(\\beta>0\\) such that for every \\(E,E'\\in I_{E_0}\\),\n\\[\n|\\mathcal N(\\omega,E)-\\mathcal N(\\omega,E')|\\le |E-E'|^{\\beta}.\n\\]" + }, + { + "label": "D", + "text": "For every H\\\"older exponent \\(0<\\beta<1/(2\\kappa^d(\\omega,E_0))\\), there exists a neighborhood \\(I_{E_0,\\beta}\\) of \\(E_0\\) such that for every \\(E,E'\\in I_{E_0,\\beta}\\),\n\\[\n|\\mathcal N(\\omega,E)-\\mathcal N(\\omega,E')|\\le |E-E'|^{\\beta},\n\\]\nand one may choose \\(I_{E_0,\\beta}\\) independently of the Diophantine frequency \\(\\omega\\in\\mathrm{DC}\\)." + }, + { + "label": "E", + "text": "There exists a neighborhood \\(I_{E_0}\\) of \\(E_0\\) such that for every \\(E,E'\\in I_{E_0}\\) and every H\\\"older exponent \\(0<\\beta<1/(2\\kappa^d(\\omega,E))\\), one has\n\\[\n|\\mathcal N(\\omega,E)-\\mathcal N(\\omega,E')|\\le |E-E'|^{\\beta}.\n\\]" + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "critical exponent bound factor 2", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "uniformity over all \\(0<\\beta<1/(2\\kappa^d(\\omega,E_0))\\) replaced by existence of some positive \\(\\beta\\)", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "dependence of the neighborhood on the fixed frequency \\(\\omega\\)", + "template_used": "uniformity_effectivity" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "acceleration evaluated at \\(E_0\\) replaced by acceleration at the variable energy \\(E\\)", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives hypotheses and definitions but does not reveal the conclusion. The correct choice is not explicitly or trivially implied by wording in the prompt." + }, + "TAS": { + "score": 1, + "justification": "The item is largely a theorem-recall question: under a full list of assumptions, the student must identify the exact stated conclusion. The answer choices introduce subtle quantifier and parameter changes, so it is not a pure verbatim restatement, but it remains close to the source theorem." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to compare the strength of the Hölder exponent bound, dependence on E versus E0, and neighborhood uniformity. However, the task mainly tests precise recall/discrimination of the theorem statement rather than deeper derivation or synthesis." + }, + "DQS": { + "score": 2, + "justification": "The distractors are mathematically plausible and target realistic failure modes: missing the factor 2, weakening to mere existence of some β, incorrectly claiming ω-uniform neighborhoods, and replacing κ^d(ω,E0) by κ^d(ω,E). They are distinct and well aligned with common misreadings." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it is still fairly close to a direct recall of the theorem rather than a genuinely generative reasoning task." + } + }, + { + "id": "2511.07817v1", + "paper_link": "http://arxiv.org/abs/2511.07817v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:main}\n $\\mathsf{Frob}$ is a Poisson embedding.", + "start_pos": 4950, + "end_pos": 5036, + "label": "thm:main" + }, + "ref_dict": { + "thm:main": "\\begin{theorem}\\label{thm:main}\n $\\mathsf{Frob}$ is a Poisson embedding.\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 1643, + "pre_theorem_intro_text": "Let $X,\\, Y$ be closed oriented surfaces and $f \\,:\\, Y\\,\\longrightarrow\\, X$ a degree $d$ branched\ncovering. Given a flat vector\nbundle $(V,\\, \\nabla)$ of rank $r$ on $Y$, by taking the direct image (pushforward) we get a flat bundle \n$(f_*V,\\, f_*\\nabla)$ of rank $dr$ on $X\\setminus B$ by Grauert's Theorem, where $B$ is the branch locus of $f$.\n\nDenote $G_m\\,=\\,\\mathrm{GL}(m,\\mathbb{C})$. The isomorphism class of the bundle $(V, \\,\\nabla)$ is determined by the\n$G_r$--conjugation orbit of its holonomy $\\rho\\,:\\,\\pi_1(Y)\\,\\longrightarrow\\, G_r$. Denote this orbit\n(conjugacy class) by $[\\rho]$. Consequently, the moduli space of such bundles is identified with the (affine)\nGeometric Invariant Theory (GIT) quotient $\\mathcal{M}_r(Y)\\,:=\\,\\mathrm{Hom}(\\pi_1(Y),\\,G_r)/\\!\\!/G_r$.\nWe note that the {\\it affine} GIT quotient is the affine variety whose coordinate ring is the ring of\n$\\mathrm{Inn}(G_r)$-invariant polynomials on $\\mathrm{Hom}(\\pi_1(Y),\\,G_r)$. The above GIT quotient\n$\\mathrm{Hom}(\\pi_1(Y),\\,G_r)/\\!\\!/G_r$ is also known as a {\\it character variety} or more precisely\nthe $G_r$--character variety of $\\pi_1(Y)$. We likewise consider the character variety\n$\\mathcal{M}_{dr}(X-B)\\,:=\\,\\mathrm{Hom}(\\pi_1(X-B),\\,G_{dr})/\\!\\!/G_{dr}$.\n\nBy \\cite{Go, La, BJ}, we know $\\mathcal{M}_r(Y)$ is (holomorphic) symplectic and $\\mathcal{M}_{dr}(X-B)$ is \n(holomorphic) Poisson on their smooth loci. By sending the holonomy of $(V,\\, \\nabla)$ to the holonomy of \n$(f_*V,\\, f_*\\nabla)$, we get a natural map $\\mathsf{Frob} \\,:\\, \\mathcal{M}_r(Y)\n\\,\\longrightarrow\\, \\mathcal{M}_{dr}(X-B)$ which we call the {\\it Frobenius reciprocity map}.", + "context": "Let $X,\\, Y$ be closed oriented surfaces and $f \\,:\\, Y\\,\\longrightarrow\\, X$ a degree $d$ branched\ncovering. Given a flat vector\nbundle $(V,\\, \\nabla)$ of rank $r$ on $Y$, by taking the direct image (pushforward) we get a flat bundle \n$(f_*V,\\, f_*\\nabla)$ of rank $dr$ on $X\\setminus B$ by Grauert's Theorem, where $B$ is the branch locus of $f$.\n\nDenote $G_m\\,=\\,\\mathrm{GL}(m,\\mathbb{C})$. The isomorphism class of the bundle $(V, \\,\\nabla)$ is determined by the\n$G_r$--conjugation orbit of its holonomy $\\rho\\,:\\,\\pi_1(Y)\\,\\longrightarrow\\, G_r$. Denote this orbit\n(conjugacy class) by $[\\rho]$. Consequently, the moduli space of such bundles is identified with the (affine)\nGeometric Invariant Theory (GIT) quotient $\\mathcal{M}_r(Y)\\,:=\\,\\mathrm{Hom}(\\pi_1(Y),\\,G_r)/\\!\\!/G_r$.\nWe note that the {\\it affine} GIT quotient is the affine variety whose coordinate ring is the ring of\n$\\mathrm{Inn}(G_r)$-invariant polynomials on $\\mathrm{Hom}(\\pi_1(Y),\\,G_r)$. The above GIT quotient\n$\\mathrm{Hom}(\\pi_1(Y),\\,G_r)/\\!\\!/G_r$ is also known as a {\\it character variety} or more precisely\nthe $G_r$--character variety of $\\pi_1(Y)$. We likewise consider the character variety\n$\\mathcal{M}_{dr}(X-B)\\,:=\\,\\mathrm{Hom}(\\pi_1(X-B),\\,G_{dr})/\\!\\!/G_{dr}$.\n\nBy \\cite{Go, La, BJ}, we know $\\mathcal{M}_r(Y)$ is (holomorphic) symplectic and $\\mathcal{M}_{dr}(X-B)$ is \n(holomorphic) Poisson on their smooth loci. By sending the holonomy of $(V,\\, \\nabla)$ to the holonomy of \n$(f_*V,\\, f_*\\nabla)$, we get a natural map $\\mathsf{Frob} \\,:\\, \\mathcal{M}_r(Y)\n\\,\\longrightarrow\\, \\mathcal{M}_{dr}(X-B)$ which we call the {\\it Frobenius reciprocity map}.", + "full_context": "Let $X,\\, Y$ be closed oriented surfaces and $f \\,:\\, Y\\,\\longrightarrow\\, X$ a degree $d$ branched\ncovering. Given a flat vector\nbundle $(V,\\, \\nabla)$ of rank $r$ on $Y$, by taking the direct image (pushforward) we get a flat bundle \n$(f_*V,\\, f_*\\nabla)$ of rank $dr$ on $X\\setminus B$ by Grauert's Theorem, where $B$ is the branch locus of $f$.\n\nDenote $G_m\\,=\\,\\mathrm{GL}(m,\\mathbb{C})$. The isomorphism class of the bundle $(V, \\,\\nabla)$ is determined by the\n$G_r$--conjugation orbit of its holonomy $\\rho\\,:\\,\\pi_1(Y)\\,\\longrightarrow\\, G_r$. Denote this orbit\n(conjugacy class) by $[\\rho]$. Consequently, the moduli space of such bundles is identified with the (affine)\nGeometric Invariant Theory (GIT) quotient $\\mathcal{M}_r(Y)\\,:=\\,\\mathrm{Hom}(\\pi_1(Y),\\,G_r)/\\!\\!/G_r$.\nWe note that the {\\it affine} GIT quotient is the affine variety whose coordinate ring is the ring of\n$\\mathrm{Inn}(G_r)$-invariant polynomials on $\\mathrm{Hom}(\\pi_1(Y),\\,G_r)$. The above GIT quotient\n$\\mathrm{Hom}(\\pi_1(Y),\\,G_r)/\\!\\!/G_r$ is also known as a {\\it character variety} or more precisely\nthe $G_r$--character variety of $\\pi_1(Y)$. We likewise consider the character variety\n$\\mathcal{M}_{dr}(X-B)\\,:=\\,\\mathrm{Hom}(\\pi_1(X-B),\\,G_{dr})/\\!\\!/G_{dr}$.\n\nBy \\cite{Go, La, BJ}, we know $\\mathcal{M}_r(Y)$ is (holomorphic) symplectic and $\\mathcal{M}_{dr}(X-B)$ is \n(holomorphic) Poisson on their smooth loci. By sending the holonomy of $(V,\\, \\nabla)$ to the holonomy of \n$(f_*V,\\, f_*\\nabla)$, we get a natural map $\\mathsf{Frob} \\,:\\, \\mathcal{M}_r(Y)\n\\,\\longrightarrow\\, \\mathcal{M}_{dr}(X-B)$ which we call the {\\it Frobenius reciprocity map}.\n\n\\begin{abstract}\nWe show that a Frobenius reciprocity map on character varieties of surfaces is a Poisson embedding.\n\\end{abstract}\n\nBy \\cite{Go, La, BJ}, we know $\\mathcal{M}_r(Y)$ is (holomorphic) symplectic and $\\mathcal{M}_{dr}(X-B)$ is \n(holomorphic) Poisson on their smooth loci. By sending the holonomy of $(V,\\, \\nabla)$ to the holonomy of \n$(f_*V,\\, f_*\\nabla)$, we get a natural map $\\mathsf{Frob} \\,:\\, \\mathcal{M}_r(Y)\n\\,\\longrightarrow\\, \\mathcal{M}_{dr}(X-B)$ which we call the {\\it Frobenius reciprocity map}.\n\nLet $Y_0\\,=\\,Y-f^{-1}(B)$ and $\\mathcal{M}_r(Y_0)\\,:=\\,\\mathrm{Hom}(\\pi_1(Y_0),\\,G_r)/\\!\\!/G_r$. Again,\n$\\mathcal{M}_r(Y_0)$ is (holomorphic) Poisson on its smooth locus. Since $Y_0\\,\\subset\\, Y$ there is a\nnatural map $\\iota\\,:\\,\\pi_1(Y_0)\\,\\longrightarrow\\, \\pi_1(Y)$. The map $\\iota$ is surjective (and hence has a\nright inverse) because any loop in $Y$ can be homotoped to avoid $f^{-1}(B)$ since $\\dim f^{-1}(B)\\,=\\,0$.\n\nLet $\\mathsf{Res}\\,:\\,\\mathcal{M}_r(Y)\\,\\longrightarrow\\, \\mathcal{M}_r(Y_0)$ be given by $\\mathsf{Res}([h])\n\\,=\\,[h\\circ \\iota]$, where $h\\,:\\,\\pi_1(Y)\\,\\longrightarrow\\, G_r$. If\n$\\mathsf{Res}([h_1])\\,=\\,\\mathsf{Res}([h_1])$, then \n$[h_1\\circ\\iota]\\,=\\,[h_2\\circ\\iota]$ and so $[h_1]=[h_2]$ since $\\iota$ has a right inverse. Therefore, \n$\\mathsf{Res}$ is an embedding. The inclusion map $Y_0\\,\\subset\\, Y$ preserves transversality of based loops \nand thus by \\cite[Theorem A]{BHJL} the ``restriction\" map $\\mathsf{Res}$ is Poisson.\n\nSince $f\\,:\\,Y_0\\,\\longrightarrow\\, X-B$ is an unramified covering map, $\\pi_1(Y_0)$ is isomorphic to a\nsubgroup of $\\pi_1(X-B)$. Therefore, there\nis a map $\\mathsf{Ind}\\,:\\,\\mathcal{M}_r(Y_0)\\,\\longrightarrow\\, \\mathcal{M}_{dr}(X-B)$ given by \n$\\mathsf{Ind}([h])\\,=\\,[\\mathsf{ind}(h)]$, where $\\mathsf{ind}(h)\\,:\\,\\pi_1(X-B)\\,\\longrightarrow\\, G_{dr}$\nis the representation induced by $h\\,:\\,\\pi_1(Y_0)\\,\\longrightarrow\\, G_r$. Since $f\\,:\\,Y_0\\,\\longrightarrow\\,\nX-B$ is a $d$--sheeted cover, the index $[\\pi_1(X-B):\\,\\pi_1(Y_0)]\\,=\\,d$ is finite and \n$\\pi_1(X-B)/\\pi_1(Y_0)\\,=\\, \\{\\gamma_1\\pi_1(Y_0),\\, \\cdots,\\,\\gamma_d\\pi_1(Y_0)\\}$. And\nso $\\mathsf{ind}(h)$ is defined \nby the natural action of $\\pi_1(X-B)$ on $\\oplus_{i=1}^d\\gamma_i\\C^r\\,\\cong\\,\\C^{dr}$ by $\\gamma\\cdot \n\\sum_{i=1}^d\\gamma_iv_i\\,:=\\,\\sum_{i=1}^d\\gamma_{j_i}\\pi(\\delta_i)v_i$, where \n$\\gamma\\gamma_i\\,=\\,\\gamma_{j_i}\\delta_i$ with $\\delta_i\\,\\in\\,\\pi_1(Y_0)$. Thus, $\\mathsf{ind}(h)$ takes\nvalues in $\\mathsf{Sym}_d\\times G_r$ inside $G_{dr}$ where $\\mathsf{Sym}_d$ is the symmetric group on $d$ symbols \n(realized by permutation matrices).\n\nBy construction of the direct image sheaf, we have $\\mathsf{Frob}\\,=\\,\\mathsf{Ind}\\circ \\mathsf{Res}$ (see \nSection \\ref{sec:directimage} for details). Consequently, $\\mathsf{Frob}$ is algebraic. By the Mackey \nFormula \\cite{Mackey} applied to this context, we see that $\\mathsf{Ind}$ is injective. Therefore, \n$\\mathsf{Frob}$ is an embedding. In Section \\ref{sec3}, we show that $\\mathsf{Ind}$ is Poisson. Having \nboth $\\mathsf{Res}$ and $\\mathsf{Ind}$ injective and Poisson, we have outlined the proof of Theorem \n\\ref{thm:main}.\n\nLet $\\text{Hom}(\\Delta,\\, \\text{GL}(r,{\\mathbb C}))^{\\rm ir}\\, \\subset\\, \\text{Hom}(\\Delta,\\, \\text{GL}\n(r,{\\mathbb C}))$ be the space of irreducible representations of $\\Delta$ in $\\text{GL}(r,{\\mathbb C})$.\nThe conjugation action of $\\text{GL}(r,{\\mathbb C})$ on itself produces an action of $\\text{GL}\n(r,{\\mathbb C})$ on $\\text{Hom}(\\Delta,\\, \\text{GL}(r,{\\mathbb C}))^{\\rm ir}$. The corresponding quotient \n\\begin{equation}\\label{e12}\n{\\mathcal C}_r(Y_0)\\ :=\\ \\text{Hom}(\\Delta,\\, \\text{GL}(r,{\\mathbb C}))^{\\rm ir}/\\text{GL}(r,{\\mathbb C})\n\\end{equation}\nis a complex manifold equipped with a holomorphic Poisson structure \\cite{La}, \\cite{BJ}, \\cite{Go}.\nSimilarly, $\\text{Hom}(\\Gamma,\\, \\text{GL}(rd,{\\mathbb C}))^{\\rm ir}\\, \\subset\\, \\text{Hom}(\\Gamma,\\, \\text{GL}\n(rd,{\\mathbb C}))$ is the space of irreducible representations of $\\Gamma$ in $\\text{GL}(rd,{\\mathbb C})$.\nAs before,\n\\begin{equation}\\label{e13}\n{\\mathcal C}_{rd}(X_0)\\ :=\\ \\text{Hom}(\\Gamma,\\, \\text{GL}(rd,{\\mathbb C}))^{\\rm ir}/\\text{GL}(rd,{\\mathbb C}) \n\\end{equation}\nis the quotient complex manifold equipped with a holomorphic Poisson structure. Sending\nany $(V,\\, \\nabla)\\, \\in\\, {\\mathcal C}_r(Y_0)$ to its direct image $(\\phi_* V,\\, \\phi_*\\nabla)\\, \\in\\,\n{\\mathcal C}_{rd}(X_0)$ (see \\eqref{e12} and \\eqref{e13}) we obtain a map\n\\begin{equation}\\label{e14}\n\\Psi\\ :\\ {\\mathcal C}_r(Y_0)\\ \\longrightarrow\\ {\\mathcal C}_{rd}(X_0).\n\\end{equation}\n\n\\begin{proof}\nThe holomorphic tangent (respectively, cotangent) bundle of ${\\mathcal C}_r(Y_0)$ will be\ndenoted by $T{\\mathcal C}_r(Y_0)$ (respectively, $T^*{\\mathcal C}_r(Y_0)$). The Poisson structure\non ${\\mathcal C}_r(Y_0)$ is given by a holomorphic homomorphism\n\\begin{equation}\\label{e15}\nP\\ :\\ T^*{\\mathcal C}_r(Y_0) \\ \\longrightarrow\\ T{\\mathcal C}_r(Y_0)\n\\end{equation}\nwhich is skew-symmetric and the Poisson bracket defined by $P$ satisfies the Jacobi identity. Similarly,\n\\begin{equation}\\label{e16}\nQ\\ :\\ T^*{\\mathcal C}_{rd}(X_0) \\ \\longrightarrow\\ T{\\mathcal C}_{rd}(X_0)\n\\end{equation}\nis the Poisson structure on ${\\mathcal C}_{rd}(X_0)$. The map $\\Psi$ in \\eqref{e14} is said to be\ncompatible with the Poisson structures on ${\\mathcal C}_r(Y_0)$ and ${\\mathcal C}_{rd}(X_0)$ if the\nfollowing diagram of homomorphisms of vector bundles over ${\\mathcal C}_r(Y_0)$ is commutative:\n\\begin{equation}\\label{e17}\n\\begin{matrix}\nT^*{\\mathcal C}_r(Y_0) & \\xrightarrow{\\,\\,\\, P\\,\\,\\,} & T{\\mathcal C}_r(Y_0)\\\\\n\\, \\,\\,\\,\\,\\, \\Big\\uparrow (d\\Psi)^* && \\,\\,\\,\\, \\Big\\downarrow d\\Psi\\\\\n\\Psi^*T^*{\\mathcal C}_{rd}(X_0) & \\xrightarrow{\\,\\,\\, \\Psi^*Q\\,\\,\\,} & \\Psi^* T{\\mathcal C}_{rd}(X_0)\n\\end{matrix}\n\\end{equation}\nwhere $P$ (respectively, $Q$) is constructed in \\eqref{e15} (respectively, \\eqref{e16})\nand $d\\Psi$ is the differential of the map $\\Psi$ in \\eqref{e14} while $(d\\Psi)^*$ is its dual.\n\n\\begin{theorem}\\label{thm:main}\n $\\mathsf{Frob}$ is a Poisson embedding.\n\\end{theorem}", + "post_theorem_intro_text_len": 2624, + "post_theorem_intro_text": "Let $Y_0\\,=\\,Y-f^{-1}(B)$ and $\\mathcal{M}_r(Y_0)\\,:=\\,\\mathrm{Hom}(\\pi_1(Y_0),\\,G_r)/\\!\\!/G_r$. Again,\n$\\mathcal{M}_r(Y_0)$ is (holomorphic) Poisson on its smooth locus. Since $Y_0\\,\\subset\\, Y$ there is a\nnatural map $\\iota\\,:\\,\\pi_1(Y_0)\\,\\longrightarrow\\, \\pi_1(Y)$. The map $\\iota$ is surjective (and hence has a\nright inverse) because any loop in $Y$ can be homotoped to avoid $f^{-1}(B)$ since $\\dim f^{-1}(B)\\,=\\,0$.\n\nLet $\\mathsf{Res}\\,:\\,\\mathcal{M}_r(Y)\\,\\longrightarrow\\, \\mathcal{M}_r(Y_0)$ be given by $\\mathsf{Res}([h])\n\\,=\\,[h\\circ \\iota]$, where $h\\,:\\,\\pi_1(Y)\\,\\longrightarrow\\, G_r$. If\n$\\mathsf{Res}([h_1])\\,=\\,\\mathsf{Res}([h_1])$, then \n$[h_1\\circ\\iota]\\,=\\,[h_2\\circ\\iota]$ and so $[h_1]=[h_2]$ since $\\iota$ has a right inverse. Therefore, \n$\\mathsf{Res}$ is an embedding. The inclusion map $Y_0\\,\\subset\\, Y$ preserves transversality of based loops \nand thus by \\cite[Theorem A]{BHJL} the ``restriction\" map $\\mathsf{Res}$ is Poisson.\n\nSince $f\\,:\\,Y_0\\,\\longrightarrow\\, X-B$ is an unramified covering map, $\\pi_1(Y_0)$ is isomorphic to a\nsubgroup of $\\pi_1(X-B)$. Therefore, there\nis a map $\\mathsf{Ind}\\,:\\,\\mathcal{M}_r(Y_0)\\,\\longrightarrow\\, \\mathcal{M}_{dr}(X-B)$ given by \n$\\mathsf{Ind}([h])\\,=\\,[\\mathsf{ind}(h)]$, where $\\mathsf{ind}(h)\\,:\\,\\pi_1(X-B)\\,\\longrightarrow\\, G_{dr}$\nis the representation induced by $h\\,:\\,\\pi_1(Y_0)\\,\\longrightarrow\\, G_r$. Since $f\\,:\\,Y_0\\,\\longrightarrow\\,\nX-B$ is a $d$--sheeted cover, the index $[\\pi_1(X-B):\\,\\pi_1(Y_0)]\\,=\\,d$ is finite and \n$\\pi_1(X-B)/\\pi_1(Y_0)\\,=\\, \\{\\gamma_1\\pi_1(Y_0),\\, \\cdots,\\,\\gamma_d\\pi_1(Y_0)\\}$. And\nso $\\mathsf{ind}(h)$ is defined \nby the natural action of $\\pi_1(X-B)$ on $\\oplus_{i=1}^d\\gamma_i\\mathbb{C}^r\\,\\cong\\,\\mathbb{C}^{dr}$ by $\\gamma\\cdot \n\\sum_{i=1}^d\\gamma_iv_i\\,:=\\,\\sum_{i=1}^d\\gamma_{j_i}\\pi(\\delta_i)v_i$, where \n$\\gamma\\gamma_i\\,=\\,\\gamma_{j_i}\\delta_i$ with $\\delta_i\\,\\in\\,\\pi_1(Y_0)$. Thus, $\\mathsf{ind}(h)$ takes\nvalues in $\\mathsf{Sym}_d\\times G_r$ inside $G_{dr}$ where $\\mathsf{Sym}_d$ is the symmetric group on $d$ symbols \n(realized by permutation matrices).\n\nBy construction of the direct image sheaf, we have $\\mathsf{Frob}\\,=\\,\\mathsf{Ind}\\circ \\mathsf{Res}$ (see \nSection \\ref{sec:directimage} for details). Consequently, $\\mathsf{Frob}$ is algebraic. By the Mackey \nFormula \\cite{Mackey} applied to this context, we see that $\\mathsf{Ind}$ is injective. Therefore, \n$\\mathsf{Frob}$ is an embedding. In Section \\ref{sec3}, we show that $\\mathsf{Ind}$ is Poisson. Having \nboth $\\mathsf{Res}$ and $\\mathsf{Ind}$ injective and Poisson, we have outlined the proof of Theorem \n\\ref{thm:main}.", + "sketch": "Work with $Y_0:=Y-f^{-1}(B)$ and the Poisson (on the smooth locus) character variety $\\mathcal{M}_r(Y_0)=\\mathrm{Hom}(\\pi_1(Y_0),G_r)/\\!\\!/G_r$. Since $Y_0\\subset Y$ there is a natural surjection $\\iota:\\pi_1(Y_0)\\to\\pi_1(Y)$ (any loop in $Y$ can be homotoped to avoid $f^{-1}(B)$ because $\\dim f^{-1}(B)=0$), hence $\\iota$ has a right inverse.\n\nDefine the “restriction” map $\\mathsf{Res}:\\mathcal{M}_r(Y)\\to\\mathcal{M}_r(Y_0)$ by $\\mathsf{Res}([h])=[h\\circ\\iota]$. If $\\mathsf{Res}([h_1])=\\mathsf{Res}([h_2])$ then $[h_1\\circ\\iota]=[h_2\\circ\\iota]$, and since $\\iota$ has a right inverse this implies $[h_1]=[h_2]$; thus $\\mathsf{Res}$ is an embedding. The inclusion $Y_0\\subset Y$ “preserves transversality of based loops” and hence by \\cite[Theorem A]{BHJL} $\\mathsf{Res}$ is Poisson.\n\nBecause $f:Y_0\\to X-B$ is an unramified $d$--sheeted cover, $\\pi_1(Y_0)$ is a subgroup of finite index $d$ in $\\pi_1(X-B)$. Define an induction map $\\mathsf{Ind}:\\mathcal{M}_r(Y_0)\\to\\mathcal{M}_{dr}(X-B)$ by $\\mathsf{Ind}([h])=[\\mathsf{ind}(h)]$, where $\\mathsf{ind}(h):\\pi_1(X-B)\\to G_{dr}$ is the induced representation via the action of $\\pi_1(X-B)$ on $\\oplus_{i=1}^d\\gamma_i\\mathbb{C}^r\\cong\\mathbb{C}^{dr}$; thus $\\mathsf{ind}(h)$ lands in $\\mathsf{Sym}_d\\times G_r\\subset G_{dr}$. By construction of the direct image sheaf, $\\mathsf{Frob}=\\mathsf{Ind}\\circ\\mathsf{Res}$.\n\nConsequently $\\mathsf{Frob}$ is algebraic. By the Mackey Formula \\cite{Mackey}, $\\mathsf{Ind}$ is injective, hence $\\mathsf{Frob}$ is an embedding. In Section \\ref{sec3} it is shown that $\\mathsf{Ind}$ is Poisson. With both $\\mathsf{Res}$ and $\\mathsf{Ind}$ injective and Poisson, this “outlines the proof of Theorem~\\ref{thm:main},” i.e. that $\\mathsf{Frob}$ is a Poisson embedding.", + "expanded_sketch": "Work with $Y_0:=Y-f^{-1}(B)$ and the Poisson (on the smooth locus) character variety $\\mathcal{M}_r(Y_0)=\\mathrm{Hom}(\\pi_1(Y_0),G_r)/\\!\\!/G_r$. Since $Y_0\\subset Y$ there is a natural surjection $\\iota:\\pi_1(Y_0)\\to\\pi_1(Y)$ (any loop in $Y$ can be homotoped to avoid $f^{-1}(B)$ because $\\dim f^{-1}(B)=0$), hence $\\iota$ has a right inverse.\n\nDefine the “restriction” map $\\mathsf{Res}:\\mathcal{M}_r(Y)\\to\\mathcal{M}_r(Y_0)$ by $\\mathsf{Res}([h])=[h\\circ\\iota]$. If $\\mathsf{Res}([h_1])=\\mathsf{Res}([h_2])$ then $[h_1\\circ\\iota]=[h_2\\circ\\iota]$, and since $\\iota$ has a right inverse this implies $[h_1]=[h_2]$; thus $\\mathsf{Res}$ is an embedding. The inclusion $Y_0\\subset Y$ “preserves transversality of based loops” and hence by \\cite[Theorem A]{BHJL} $\\mathsf{Res}$ is Poisson.\n\nBecause $f:Y_0\\to X-B$ is an unramified $d$--sheeted cover, $\\pi_1(Y_0)$ is a subgroup of finite index $d$ in $\\pi_1(X-B)$. Define an induction map $\\mathsf{Ind}:\\mathcal{M}_r(Y_0)\\to\\mathcal{M}_{dr}(X-B)$ by $\\mathsf{Ind}([h])=[\\mathsf{ind}(h)]$, where $\\mathsf{ind}(h):\\pi_1(X-B)\\to G_{dr}$ is the induced representation via the action of $\\pi_1(X-B)$ on $\\oplus_{i=1}^d\\gamma_i\\mathbb{C}^r\\cong\\mathbb{C}^{dr}$; thus $\\mathsf{ind}(h)$ lands in $\\mathsf{Sym}_d\\times G_r\\subset G_{dr}$. By construction of the direct image sheaf, $\\mathsf{Frob}=\\mathsf{Ind}\\circ\\mathsf{Res}$.\n\nConsequently $\\mathsf{Frob}$ is algebraic. By the Mackey Formula \\cite{Mackey}, $\\mathsf{Ind}$ is injective, hence $\\mathsf{Frob}$ is an embedding. In Section \\ref{sec3} it is shown that $\\mathsf{Ind}$ is Poisson. With both $\\mathsf{Res}$ and $\\mathsf{Ind}$ injective and Poisson, this outlines the proof of the main theorem, i.e. that $\\mathsf{Frob}$ is a Poisson embedding.", + "expanded_theorem": "\\label{thm:main}\n $\\mathsf{Frob}$ is a Poisson embedding.,", + "theorem_type": [ + "Universal" + ], + "mcq": { + "question": "Let $X$ and $Y$ be closed oriented surfaces, and let $f:Y\\to X$ be a degree $d$ branched covering with branch locus $B\\subset X$. For $m\\ge 1$, write $G_m=\\mathrm{GL}(m,\\mathbb C)$ and define the character varieties\n\\[\\mathcal M_r(Y):=\\mathrm{Hom}(\\pi_1(Y),G_r)/\\!/G_r,\\qquad \\mathcal M_{dr}(X\\setminus B):=\\mathrm{Hom}(\\pi_1(X\\setminus B),G_{dr})/\\!/G_{dr}.\\]\nIf $(V,\\nabla)$ is a rank-$r$ flat vector bundle on $Y$, its holonomy class gives a point of $\\mathcal M_r(Y)$, and the direct image flat bundle $(f_*V,f_*\\nabla)$ on $X\\setminus B$ gives a point of $\\mathcal M_{dr}(X\\setminus B)$. Let\n\\[\\mathsf{Frob}:\\mathcal M_r(Y)\\longrightarrow \\mathcal M_{dr}(X\\setminus B)\\]\nbe the resulting map on character varieties, called the Frobenius reciprocity map. Which statement holds for this map, with respect to the natural holomorphic Poisson structures on these character varieties?", + "correct_choice": { + "label": "A", + "text": "The map $\\mathsf{Frob}$ is a Poisson embedding; that is, it is an embedding of $\\mathcal M_r(Y)$ into $\\mathcal M_{dr}(X\\setminus B)$ and it is compatible with the natural holomorphic Poisson structures." + }, + "choices": [ + { + "label": "B", + "text": "The map $\\mathsf{Frob}$ is a Poisson immersion on the smooth locus of $\\mathcal M_r(Y)$, but it need not be injective as a map into $\\mathcal M_{dr}(X\\setminus B)$." + }, + { + "label": "C", + "text": "The map $\\mathsf{Frob}$ is Poisson; that is, it is compatible with the natural holomorphic Poisson structures on $\\mathcal M_r(Y)$ and $\\mathcal M_{dr}(X\\setminus B)$." + }, + { + "label": "D", + "text": "The map $\\mathsf{Frob}$ is an embedding of $\\mathcal M_r(Y)$ into $\\mathcal M_{dr}(X\\setminus B)$, but in general it is only algebraic and does not preserve the natural holomorphic Poisson structures." + }, + { + "label": "E", + "text": "The map $\\mathsf{Frob}$ is a holomorphic symplectic embedding onto its image; equivalently, its image is a symplectic submanifold of $\\mathcal M_{dr}(X\\setminus B)$ and $\\mathsf{Frob}$ preserves the symplectic forms." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "finiteness", + "tampered_component": "Mackey-based injectivity of induction", + "template_used": "quantifier_dependence" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "embedding/injectivity conclusion", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "Poisson compatibility of Res and Ind", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "Poisson target is not symplectic in general", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not explicitly state that the map is a Poisson embedding. It defines the construction and asks for the correct structural property, without giving away the conclusion." + }, + "TAS": { + "score": 1, + "justification": "This is close to a theorem-recall item: it essentially asks which formal statement about the Frobenius reciprocity map is true. The multiple choices introduce stronger, weaker, and distorted variants, so it is not pure tautology, but it remains a mild reformulation of a likely theorem statement." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish among 'Poisson', 'Poisson embedding', 'immersion', and 'symplectic embedding', especially since one distractor is a weaker true statement. Still, the item mainly tests recognition of the exact theorem rather than substantial derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong: one is weaker but true in part, others reflect common confusions between injective/immersive, algebraic/Poisson, and Poisson/symplectic. They are distinct and mathematically meaningful." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with no answer leakage and high-quality distractors, but it leans toward theorem recognition rather than deep generative reasoning." + } + }, + { + "id": "2511.07345v1", + "paper_link": "http://arxiv.org/abs/2511.07345v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "proposition", + "content": "\\label{Proposition:Phi:compact}\n The input-output operator $\\Psi:L^2(0,T;L^2(\\Omega))\\to L^2(\\Omega)$ is compact.", + "start_pos": 710022, + "end_pos": 710180, + "label": "Proposition:Phi:compact" + }, + "ref_dict": { + "eq:intro:01": "\\begin{align}\n \\label{eq:intro:01}\n \\begin{cases}\n \\pt y -\\text{div}((a(x)+ib(x))\\nabla y) + \\vec{r}(x)\\cdot \\nabla y +p(x)y=f(x,t)&\\text{ in }\\Omega\\times (0,T),\\\\\n y=0&\\text{ on }\\partial \\Omega\\times (0,T),\\\\\n y(\\cdot,0)=y_0&\\text{ in }\\Omega. \n \\end{cases}\n\\end{align}", + "def:functional:J": "\\begin{align}\n \\label{def:functional:J}\n \\mathcal{J}(f):=\\frac{1}{2}\\int_\\Omega |\\Psi(f)-u_T|^2\\,dx,\n\\end{align}", + "eq:Input-output:op": "\\begin{align}\n \\label{eq:Input-output:op}\n \\Psi(f)=u_T(x),\\quad x\\in \\Omega,\\quad f\\in \\mathcal{U}. \n\\end{align}", + "Proposition:Phi:compact": "\\begin{proposition}\n \\label{Proposition:Phi:compact}\n The input-output operator $\\Psi:L^2(0,T;L^2(\\Omega))\\to L^2(\\Omega)$ is compact.\n\\end{proposition}", + "proposition:existence:strong:solutions": "\\begin{proposition}\n \\label{proposition:existence:strong:solutions}\n Let $\\Omega\\subset \\mathbb{R}^N$ be a bounded domain with $C^2$ boundary and let $T>0$. Assume that the coefficients $\\alpha$ and $\\beta$ satisfy \n \\begin{align}\n \\label{assumption:prop:existence:02:alpha:beta}\n \\alpha,\\beta\\in W^{1,\\infty}(\\Omega;\\mathbb{R}),\\quad \\alpha(x)\\geq \\alpha_0 >0\\text{ almost everywhere in }\\Omega.\n \\end{align}\n Moreover, consider \\eqref{prop:existence:statement:02}, $v_0\\in H_0^1(\\Omega)$ and $h\\in L^2(0,T;L^2(\\Omega))$. Then, there exists a strong solution $v$ of \\eqref{problem:existence:uniqueness} satisfying\n \\begin{align*}\n v\\in H^1(0,T;L^2(\\Omega)) \\cap C^0([0,T];H_0^1(\\Omega)) \\cap L^2(0,T; H^2(\\Omega) \\cap H_0^1(\\Omega)).\n \\end{align*}\n\n Moreover, there exists a constant $C>0$ depending only on $\\alpha_0$, $\\|\\alpha\\|_{W^{1,\\infty}}$, $\\||\\vec{\\sigma}|\\|_{L^\\infty}$, $\\|p\\|_\\infty$, $\\Omega$ and $T$ such that \n \\begin{align*}\n \\begin{split} \n &\\|v\\|_{L^2(0,T;H^2(\\Omega))} + \\|v\\|_{C^0([0,T];H_0^1(\\Omega))} + \\|\\pt v\\|_{L^2(0,T;L^2(\\Omega))} \\\\\n \\leq &C\\left(\\|h\\|_{L^2(0,T;L^2(\\Omega))} + \\|v_0\\|_{H_0^1(\\Omega)} \\right).\n \\end{split}\n \\end{align*}\n\\end{proposition}" + }, + "pre_theorem_intro_text_len": 12018, + "pre_theorem_intro_text": "This section introduces the mathematical formulation of the inverse problem and specifies the functional framework adopted throughout the paper. We also describe the model equation, its physical motivation, and the regularity assumptions required to ensure well-posedness.\n\nLet $\\Omega\\subset \\mathbb{R}^N$ be a bounded domain ($N\\geqslant 1$) with boundary $\\partial \\Omega$ of class $C^2$ and $T>0$. Here and in the sequel, the function spaces refer to spaces of complex-valued functions unless otherwise specified. \n\nWe consider $a,b$, $\\vec{r}$ and $p$ with the following assumptions:\n\\begin{itemize}\n \\item[{\\bf (H1)}] $a,b\\in W^{1,\\infty}(\\Omega;\\mathbb{R})$ with\n \\begin{align*}\n a(x)\\geqslant a_\\star >0\\quad \\text{ almost everywhere in }\\Omega, \n \\end{align*}\n for some $a_\\star$.\n \\item[{\\bf (H2)}] $\\vec{r}=\\vec{r}(x)$, $p=p(x)$ and $y_0=y_0(x)$ are complex-valued functions such that \n \\begin{align*}\n \\vec{r}\\in [W^{1,\\infty}(\\Omega)]^N,\\,\n q\\in L^\\infty(\\Omega)\\text{ and }y_0\\in L^2(\\Omega).\n \\end{align*}\n\\end{itemize}\n\nNow, consider the following Ginzburg-Landau equation with Dirichlet boundary conditions:\n\\begin{align}\n \\label{eq:intro:01}\n \\begin{cases}\n \\partial_t y -\\text{div}((a(x)+ib(x))\\nabla y) + \\vec{r}(x)\\cdot \\nabla y +p(x)y=f(x,t)&\\text{ in }\\Omega\\times (0,T),\\\\\n y=0&\\text{ on }\\partial \\Omega\\times (0,T),\\\\\n y(\\cdot,0)=y_0&\\text{ in }\\Omega. \n \\end{cases}\n\\end{align}\n\nUnder the assumptions {\\bf (H1)}, {\\bf (H2)} and $f\\in L^2(0,T;L^2(\\Omega))$, there exists a unique weak solution of \\eqref{eq:intro:01} with the following regularity:\n\\begin{align*}\n y\\in C^0([0,T];L^2(\\Omega)) \\cap L^2(0,T;H_0^1(\\Omega)).\n\\end{align*}\n\nFor completeness and self-containment, we include in Appendix \\ref{section:appendix:existence:uniqueness:results} general statements, together with their proofs, concerning the existence of weak and strong solutions of the equation \\eqref{eq:intro:01}. \n\nIn this article, we will consider the following {\\bf Inverse Source Problem (ISP):} Identify the unknown spatial-temporal source term $f$ in the space $L^2(0,T;L^2(\\Omega))$ in \\eqref{eq:intro:01} from the following final time measured output:\n\\begin{align*}\n u_T(x):=y(x,T),\\quad x\\in \\Omega. \n\\end{align*}\n\nHere, $y(x,T)\\equiv y(x,t)\\big|_{t=T}$ is a trace appropriately defined of the weak solution $y(x,t)$ of \\eqref{eq:intro:01}. In addition, $u_T(x)$ represents the measured output containing a random noise. \n\nInverse source problems of this type are not only mathematically rich but also arise naturally in applied sciences. In many experimental settings, direct access to distributed sources is often impossible, and only partial or final-time measurements are available. For instance, in nonlinear optics one may measure the profile of a pulse at the end of a cavity, while the internal gain or loss mechanisms driving its formation remain hidden. Similarly, in chemical or biological pattern formation, one often observes the state of the system after some transient dynamics but seeks to infer the localized heterogeneities or forcing terms that generated it. This gap between accessible measurements and hidden dynamics motivates the study of reconstruction methods capable of identifying unknown excitations from indirect data.\n\nThe choice of the Ginzburg-Landau equation as the underlying model is particularly significant. It serves as a universal amplitude equation near the onset of instability in diverse physical systems, ranging from fluid dynamics and plasma physics to optics and chemical oscillations \\cite{Aranson2002,levermore1996complex,chen1998numerical,bartuccelli1990possibility,mielke2002ginzburg}. Consequently, advances in the understanding of inverse problems for this equation are not limited to a single discipline but carry implications across multiple fields. Identifying unknown sources in such a model not only enriches the theory of dissipative-dispersive PDEs but also provides a mathematical framework for interpreting experiments in which only end-point information is available.\n\nAt the mathematical level, the inverse source problem we consider is severely ill-posed. Regulation strategies and variational formulations thus become essential. Our work adopts this perspective, combining the theory of weak solutions, adjoint-based gradient formulas, and Tikhonov regularization to rigorously justify reconstruction algorithms and test their performance through numerical simulations.\n\n\\subsection{Literature}\n\nWe now review some representative contributions on the complex Ginzburg-Landau equation and its variants, emphasizing their role as canonical amplitude equations and discussing prior work related to inverse or identification problems.\n\nThe complex Ginzburg-Landau (GL) equation is the canonical amplitude equation near a Hopf or Turing-Hopf bifurcation. In its cubic form, it reads\n\\begin{align*}\n \\partial_t A=(\\mu+i\\omega)A + (1+c_1i)\\Delta A -(1+c_3 i)|A|^2 A,\n\\end{align*}\nwhile the cubic-quintic extension adds a saturating nonlinearity:\n\\begin{align*}\n \\partial_t A=(\\mu +\\omega i)A +(1+c_1i)\\Delta A -(1+c_3 i)|A|^2 A -(\\nu +c_5i)|A|^4A,\n\\end{align*}\nwith real parameters $\\mu,\\nu,c_j$ ($j=1,3,5$) that encode growth, dispersion/diffusion, and nonlinear gain/loss. The cubic model is the universal normal form near onset and captures phase diffusion, plane-wave selection and Benjamin-Feir/Newell instabilities. On the other hand, the quintic term become relevant farther from threshold to model saturation and complex dissipative structures (e.g. localized pulses). See the reviews \\cite{Cross1993} and \\cite{Aranson2002}, which also discuss higher-order variants and parameter scalings.\n\nThe cubic and cubic-quintic complex GL equations serve as canonical models of pattern formation and nonlinear wave dynamics in nonequilibrium systems. These models describe the slow complex envelope $A(x,t)$ of oscillatory modes near stability. In hydrodynamics and Rayleigh-B\\'enard convection, the cubic GL equation explains the emergence of roll patterns, travelling waves, and their stability domains \\cite{Newell1969} and \\cite{Cross1993}. In nonlinear optics, it governs mode-locked lasers, pulse propagation in fiber cavities, and the formation of dissipative solutions. In this case, the cubic-quintic equation extension accounts for gain saturation and quintic nonlinear effects that are essential for bounded amplitude states \\cite{SotoCrespo1997}, \\cite{Akhmediev2005}. Similarly, in chemical oscillations and biological excitable media, it models spiral waves, defect turbulence, and spatiotemporal chaos, highlighting its universality across disciplines \\cite{Aranson2002}.\n\nFrom a mathematical viewpoint, the GL equations combine parabolic smoothing (via the Laplacian operator) with dispersive phase dynamics (via imaginary coefficients), which generates a delicate interplay between diffusion, dispersion, and nonlinearity \\cite{Akhmediev2005,Cross1993, levermore1996complex, mielke2002ginzburg}. This hybrid character produces a wide spectrum of phenomena: existence of plane-wave solutions, modulational (Benjamin-Feir/Newell) instabilities, turbulence, and complex coherent structures \\cite{Akhmediev2005, Cross1993, Newell1969, SotoCrespo1997}. The cubic equation alone already yields chaotic attractors, defect-mediated turbulence, and spiral wave breakup \\cite{Aranson2002,bartuccelli1990possibility, SotoCrespo1997}, while the quintic term stabilizes or destabilizes localized patterns depending on parameter regimes \\cite{Akhmediev2005,SotoCrespo1997}. The GL equations have been used to study attractors, bifurcations, and long-time dynamics of nonlinear PDEs, serving as a testbed for methods in dissipative systems, control, and inverse problems \\cite{chen1998numerical,mielke2002ginzburg,santos2019insensitizing,borzi2005analysis,junge2000synchronization}. \nRecovering sources and parameters in GL-type models is crucial in practice: in optics, it enables diagnosing distributed gain/loss or saturable absorbers in cavity models from end-of-pulse measurements; in chemical/biological pattern formation, it helps infer spatially localized forcing or heterogeneities driving wave patterns; in fluid and plasma contexts, it allows one to back-out effective forcing or feedback from limited snapshots near transitions to turbulence. Because the GL framework is a universal amplitude model, successful inversion translates across disciplines, providing interpretable maps of where and how energy is injected or dissipated. From a computational perspective, adjoint-based Tikhonov schemes scale to high-dimensional discretizations and accommodate realistic noise models \\cite{Cross1993}, \\cite{Aranson2002}, \\cite{MR2516528}.\n\nWithin this broad literature, inverse problems for GL-type models remain relatively less explored. Classical studies on inverse source problems for parabolic or Schr\\\"odinger-type equations \\cite{hasanouglu2021introduction,hasanov2007simultaneous,GarciaOssesTapia+2013+755+779,chorfi2025identification} provide methodological foundations, yet the adaptation to GL equations, which combine diffusion, dispersion, and nonlinearity, poses unique analytical and numerical challenges.\n\nWe also note that other GL-type models, such as those with dynamic boundary conditions, have recently attracted attention in the context of controllability \\cite{carreno2025local}. While these studies highlight the richness of the GL framework under more complex boundary interactions, the analysis of inverse problems in such settings remains largely open. Our contribution is thus complementary, as we address the identification of space-time dependent source terms for the standard GL equation, providing a first step toward broader inverse problem formulations in extended GL models.\n\nIt is worth emphasizing that the linear GL equation can also be expressed as a system of two coupled real-valued PDE equations, corresponding to the real and imaginary parts of the complex amplitude. In this formulation, the coupling terms are of second order and involve cross-derivative operators, which profoundly alter the analytical structure of the system. As a consequence, classical techniques from the theory of control and inverse problems for scalar parabolic equations (based on Carleman estimates and observability inequalities \\cite{fursikov1996carleman}) cannot be directly applied. In particular, the presence of complex coefficients and dispersive-type interactions requires adapted Carleman weights, refined energy estimates, and the development of new stability results tailored to mixed dissipative-dispersive operators. \n\nThe present work is intended as a first step toward a comprehensive theory of inverse problems for GL-type equations, focusing here on the standard setting with Dirichlet boundary conditions.\n\n\\subsection{The input-output operator}\nTo formalize the inverse problem, we introduce the input-output operator mapping the source term to the final-state of the system. This operator provides a compact representation of the forward model and allows for a variational reformulation of the inverse problem.\n\nLet $\\mathcal{U}\\subset L^2(0,T;L^2(\\Omega))$ be a non-empty set, which is supposed to be bounded, closed and convex. In addition, we define the operator $\\Psi: \\mathcal{U}\\to L^2(\\Omega)$ in the following way:\n\\begin{align*}\n \\Psi(f)=y(\\cdot,T)\\quad f\\in \\mathcal{U},\n\\end{align*}\nwhere $y=y(x,t)$ is the solution of \\eqref{eq:intro:01} associated to the source term $f$.\n\nWe point out that, the {\\bf (ISP)} can be formulated in terms of $\\Psi$ as the following functional equation:\n\\begin{align}\n \\label{eq:Input-output:op}\n \\Psi(f)=u_T(x),\\quad x\\in \\Omega,\\quad f\\in \\mathcal{U}. \n\\end{align}\n\nWe emphasize that, due to measurement errors in the output $u_T$, exact equality in the equation \\eqref{eq:Input-output:op} cannot be satisfied in general. Moreover, we have the following result", + "context": "Let $\\Omega\\subset \\mathbb{R}^N$ be a bounded domain ($N\\geqslant 1$) with boundary $\\partial \\Omega$ of class $C^2$ and $T>0$. Here and in the sequel, the function spaces refer to spaces of complex-valued functions unless otherwise specified.\n\nNow, consider the following Ginzburg-Landau equation with Dirichlet boundary conditions:\n\\begin{align}\n \\label{eq:intro:01}\n \\begin{cases}\n \\partial_t y -\\text{div}((a(x)+ib(x))\\nabla y) + \\vec{r}(x)\\cdot \\nabla y +p(x)y=f(x,t)&\\text{ in }\\Omega\\times (0,T),\\\\\n y=0&\\text{ on }\\partial \\Omega\\times (0,T),\\\\\n y(\\cdot,0)=y_0&\\text{ in }\\Omega. \n \\end{cases}\n\\end{align}\n\n\\subsection{The input-output operator}\nTo formalize the inverse problem, we introduce the input-output operator mapping the source term to the final-state of the system. This operator provides a compact representation of the forward model and allows for a variational reformulation of the inverse problem.\n\nLet $\\mathcal{U}\\subset L^2(0,T;L^2(\\Omega))$ be a non-empty set, which is supposed to be bounded, closed and convex. In addition, we define the operator $\\Psi: \\mathcal{U}\\to L^2(\\Omega)$ in the following way:\n\\begin{align*}\n \\Psi(f)=y(\\cdot,T)\\quad f\\in \\mathcal{U},\n\\end{align*}\nwhere $y=y(x,t)$ is the solution of \\eqref{eq:intro:01} associated to the source term $f$.\n\nWe point out that, the {\\bf (ISP)} can be formulated in terms of $\\Psi$ as the following functional equation:\n\\begin{align}\n \\label{eq:Input-output:op}\n \\Psi(f)=u_T(x),\\quad x\\in \\Omega,\\quad f\\in \\mathcal{U}. \n\\end{align}\n\nWe emphasize that, due to measurement errors in the output $u_T$, exact equality in the equation \\eqref{eq:Input-output:op} cannot be satisfied in general. Moreover, we have the following result\n\n\\begin{align}\n \\label{eq:Input-output:op}\n \\Psi(f)=u_T(x),\\quad x\\in \\Omega,\\quad f\\in \\mathcal{U}. \n\\end{align}\n\n\\begin{align}\n \\label{eq:intro:01}\n \\begin{cases}\n \\pt y -\\text{div}((a(x)+ib(x))\\nabla y) + \\vec{r}(x)\\cdot \\nabla y +p(x)y=f(x,t)&\\text{ in }\\Omega\\times (0,T),\\\\\n y=0&\\text{ on }\\partial \\Omega\\times (0,T),\\\\\n y(\\cdot,0)=y_0&\\text{ in }\\Omega. \n \\end{cases}\n\\end{align}", + "full_context": "Let $\\Omega\\subset \\mathbb{R}^N$ be a bounded domain ($N\\geqslant 1$) with boundary $\\partial \\Omega$ of class $C^2$ and $T>0$. Here and in the sequel, the function spaces refer to spaces of complex-valued functions unless otherwise specified.\n\nNow, consider the following Ginzburg-Landau equation with Dirichlet boundary conditions:\n\\begin{align}\n \\label{eq:intro:01}\n \\begin{cases}\n \\partial_t y -\\text{div}((a(x)+ib(x))\\nabla y) + \\vec{r}(x)\\cdot \\nabla y +p(x)y=f(x,t)&\\text{ in }\\Omega\\times (0,T),\\\\\n y=0&\\text{ on }\\partial \\Omega\\times (0,T),\\\\\n y(\\cdot,0)=y_0&\\text{ in }\\Omega. \n \\end{cases}\n\\end{align}\n\n\\subsection{The input-output operator}\nTo formalize the inverse problem, we introduce the input-output operator mapping the source term to the final-state of the system. This operator provides a compact representation of the forward model and allows for a variational reformulation of the inverse problem.\n\nLet $\\mathcal{U}\\subset L^2(0,T;L^2(\\Omega))$ be a non-empty set, which is supposed to be bounded, closed and convex. In addition, we define the operator $\\Psi: \\mathcal{U}\\to L^2(\\Omega)$ in the following way:\n\\begin{align*}\n \\Psi(f)=y(\\cdot,T)\\quad f\\in \\mathcal{U},\n\\end{align*}\nwhere $y=y(x,t)$ is the solution of \\eqref{eq:intro:01} associated to the source term $f$.\n\nWe point out that, the {\\bf (ISP)} can be formulated in terms of $\\Psi$ as the following functional equation:\n\\begin{align}\n \\label{eq:Input-output:op}\n \\Psi(f)=u_T(x),\\quad x\\in \\Omega,\\quad f\\in \\mathcal{U}. \n\\end{align}\n\nWe emphasize that, due to measurement errors in the output $u_T$, exact equality in the equation \\eqref{eq:Input-output:op} cannot be satisfied in general. Moreover, we have the following result\n\n\\begin{align}\n \\label{eq:Input-output:op}\n \\Psi(f)=u_T(x),\\quad x\\in \\Omega,\\quad f\\in \\mathcal{U}. \n\\end{align}\n\n\\begin{align}\n \\label{eq:intro:01}\n \\begin{cases}\n \\pt y -\\text{div}((a(x)+ib(x))\\nabla y) + \\vec{r}(x)\\cdot \\nabla y +p(x)y=f(x,t)&\\text{ in }\\Omega\\times (0,T),\\\\\n y=0&\\text{ on }\\partial \\Omega\\times (0,T),\\\\\n y(\\cdot,0)=y_0&\\text{ in }\\Omega. \n \\end{cases}\n\\end{align}\n\nLet $\\mathcal{U}\\subset L^2(0,T;L^2(\\Omega))$ be a non-empty set, which is supposed to be bounded, closed and convex. In addition, we define the operator $\\Psi: \\mathcal{U}\\to L^2(\\Omega)$ in the following way:\n\\begin{align*}\n \\Psi(f)=y(\\cdot,T)\\quad f\\in \\mathcal{U},\n\\end{align*}\nwhere $y=y(x,t)$ is the solution of \\eqref{eq:intro:01} associated to the source term $f$.\n\nWe point out that, the {\\bf (ISP)} can be formulated in terms of $\\Psi$ as the following functional equation:\n\\begin{align}\n \\label{eq:Input-output:op}\n \\Psi(f)=u_T(x),\\quad x\\in \\Omega,\\quad f\\in \\mathcal{U}. \n\\end{align}\n\n\\begin{proof}\n Let $(f_k)_{k\\in \\mathbb{N}}$ be a bounded sequence in $L^2(\\Omega\\times (0,T))$. Then, by Proposition \\ref{proposition:existence:strong:solutions}, the sequence of associated weak solutions $(y(\\cdot,,\\cdot,f_k))_{k\\in \\mathbb{N}}$ is bounded in $C^0([0,T];H_0^1(\\Omega))$. In particular, the sequence $(Y_k)_{k\\in \\mathbb{N}}$ given by $Y_k:=y(\\cdot,T,f_k)$ is bounded in $H_0^1(\\Omega)\\cap H^2(\\Omega)$. Thus, using the Sobolev-Gagliardo-Nirenberg compact embedding $H_0^1(\\Omega) \\hookrightarrow L^2(\\Omega)$, there exists a subsequence of $ (Y_k)_{k\\in \\mathbb{N}}$ which converges strongly in $L^2(\\Omega)$. This implies that the input-output operator $\\Psi$ is compact and the proof of Proposition \\ref{Proposition:Phi:compact} is finished. \n\\end{proof}\n\nIn view of the Proposition \\ref{Proposition:Phi:compact}, it is evident that the inverse problem {\\bf (ISP)} is ill-posed in the sense of Hadamard. For this reason, one needs to introduce the functional $\\mathcal{J}:\\mathcal{U}\\to \\mathbb{R}$ by\n\\begin{align}\n \\label{def:functional:J}\n \\mathcal{J}(f):=\\frac{1}{2}\\int_\\Omega |\\Psi(f)-u_T|^2\\,dx,\n\\end{align}\nand reformulate {\\bf (ISP)} in terms of the quasi-solution method, i.e., we minimize the following extremal problem\n\\begin{align*}\n \\mathcal{J}(f^\\star)=\\inf_{f\\in \\mathcal{U}} \\mathcal{J}(f).\n\\end{align*}\n\nSince the operator $\\Psi$ is compact, small perturbations in the data may cause large variations in the reconstructed source. Therefore, regularization becomes essential to obtain stable approximations. In this context, it is customary to consider a regularized Tikhonov version of the functional $J$ in \\eqref{def:functional:J}. For $\\epsilon>0$, we introduce the regularized functional\n\\begin{align}\\label{regfunct}\n \\mathcal{J}_\\epsilon (f):=\\frac{1}{2}\\int_\\Omega |\\Psi(f)-u_T|^2\\,dx + \\frac{\\epsilon}{2}\\int_0^T\\int_\\Omega |f|^2\\,dx\\,dt ,\\quad f\\in \\mathcal{U}. \n\\end{align}\n\n\\begin{proposition}\n \\label{proposition:Frechet:formula}\n Consider the assumptions {\\bf (H1)} and {\\bf (H2)}. Then, the functional $\\mathcal{J}:\\mathcal{U}\\to \\mathbb{R}$ is Fr\\'echet differentiable and its gradient at each $f\\in \\mathcal{U}$ is given by \n \\begin{align}\n \\label{eq:frechet:gradient:formula}\n \\mathcal{J}'(f)=\\phi,\\quad f\\in \\mathcal{U},\n \\end{align}\n where $\\phi$ is the unique weak solution of the following adjoint system \n \\begin{align}\n \\label{problem:Frechet:dif}\n \\begin{cases}\n -\\pt \\phi -\\text{div}((a(x)-ib(x))\\nabla \\phi) - \\overline{\\vec{r}(x)}\\cdot \\nabla \\phi +(\\overline{p(x)}- \\overline{\\text{div}(\\vec{r})(x))}\\phi =0&\\text{ in }\\Omega\\times (0,T),\\\\\n \\phi=0&\\text{ on }\\partial\\Omega \\times (0,T),\\\\\n \\phi(\\cdot,T)=y(\\cdot,T;f)-u_T&\\text{ in }\\Omega. \n \\end{cases}\n \\end{align}\n\\end{proposition}\n\nUsing the complex identity\n\\begin{align*}\n \\frac{1}{2}(|x-z|^2-|y-z|^2)=\\Re [(x-y)\\overline{(y-z)}] +\\frac{1}{2} |x-y|^2,\\quad \\forall x,y,z\\in \\mathbb{C},\n\\end{align*}\nwe have \n\\begin{align}\n \\nonumber \n \\delta \\mathcal{J}(f)=&\\Re \\int_\\Omega (y(\\cdot,T;f+\\delta f)-y(\\cdot,T;f)) \\overline{y(\\cdot,T;f)-u_T}\\,dx \\\\\n \\nonumber \n &+\\frac{1}{2}\\int_\\Omega |y(\\cdot,T;f+\\delta f)-y(\\cdot,T;f)|^2\\,dx \\\\\n \\label{eq:Frechet:for:01} \n =&-\\Re \\int_\\Omega \\delta y(\\cdot,T;f)\\overline{\\phi (\\cdot,T;f)}\\,dx +\\frac{1}{2}\\int_\\Omega |\\delta y(\\cdot,T;f)|^2\\,dx \n\\end{align}\nwhere $\\phi(\\cdot,\\cdot,f)$ is the weak solution of \\eqref{problem:Frechet:dif} (and thanks to Proposition \\ref{proposition:existence:weak:solutions}, $\\phi(\\cdot,T,f)=y(\\cdot,T;f)-u_T$ in $L^2(\\Omega)$) and $\\delta y$ is the solution of the following \\textit{sensitivity problem}\n\\begin{align}\n \\label{eq:sensitivity:problem}\n \\begin{cases}\n \\pt \\delta y -\\text{div}((a(x)+b(x)i)\\nabla \\delta y)+\\vec{r}(x)\\cdot \\nabla \\delta y + p(x)\\delta y=\\delta f(x,t)&\\text{ in }\\Omega\\times (0,T),\\\\\n \\delta y=0&\\text{ on }\\partial \\Omega\\times (0,T),\\\\\n \\delta y(\\cdot,0)=0&\\text{ in }\\Omega. \n \\end{cases}\n\\end{align}\n\n\\begin{proof}\n Let us fix $f,\\delta f\\in \\mathcal{U}$ such that $f+\\delta f\\in \\mathcal{U}$. Then, the function $\\delta \\phi=\\delta \\phi (\\cdot,\\cdot,f)$ is the solution of\n \\begin{align}\n \\label{eq:Lipschitz:01}\n \\begin{cases}\n -\\pt \\delta \\phi -\\text{div}((a(x)-ib(x))\\nabla \\phi) - \\overline{\\vec{r}(x)}\\cdot \\nabla \\phi +(\\overline{p(x)}-\\overline{\\text{div}(\\vec{r}(x))}\\phi =0&\\text{ in }\\Omega\\times (0,T),\\\\\n \\delta \\phi =0&\\text{ on }\\partial \\Omega \\times (0,T),\\\\\n \\delta \\phi(\\cdot,T)=\\delta y(\\cdot,T;f)&\\text{ in }\\Omega, \n \\end{cases}\n \\end{align}\n where $\\delta y$ is the solution of \\eqref{problem:Frechet:dif} associated to $\\delta f$. We remark that, by Proposition \\ref{proposition:existence:strong:solutions}, $\\delta \\phi$ is a strong solution of \\eqref{eq:Lipschitz:01}. Thus, by Proposition \\ref{proposition:Frechet:formula} we have, for some constant $C>0$:\n \\begin{align*}\n |\\mathcal{J}'(f+\\delta f)-\\mathcal{J}'(f)|= \\|\\delta \\phi\\|_{L^2(\\Omega\\times (0,T))}\\leq C\\|\\delta y(\\cdot,T;f)\\|_{L^2(\\Omega)}\\leq C\\|\\delta f\\|_{L^2(0,T;L^2(\\Omega))},\n \\end{align*}\n where we have used the continuity of the strong solutions applied to $\\delta y$ respect to the data. This proves the assertion of the Lemma \\ref{Lemma:Lipschitz:gradient}. \n\\end{proof}\n\n\\begin{proposition}\n \\label{Proposition:Phi:compact}\n The input-output operator $\\Psi:L^2(0,T;L^2(\\Omega))\\to L^2(\\Omega)$ is compact.\n\\end{proposition}\n\n\\begin{align}\n \\label{def:functional:J}\n \\mathcal{J}(f):=\\frac{1}{2}\\int_\\Omega |\\Psi(f)-u_T|^2\\,dx,\n\\end{align}\n\n\\begin{align}\n \\label{eq:intro:01}\n \\begin{cases}\n \\pt y -\\text{div}((a(x)+ib(x))\\nabla y) + \\vec{r}(x)\\cdot \\nabla y +p(x)y=f(x,t)&\\text{ in }\\Omega\\times (0,T),\\\\\n y=0&\\text{ on }\\partial \\Omega\\times (0,T),\\\\\n y(\\cdot,0)=y_0&\\text{ in }\\Omega. \n \\end{cases}\n\\end{align}\n\n\\begin{proposition}\n \\label{proposition:existence:strong:solutions}\n Let $\\Omega\\subset \\mathbb{R}^N$ be a bounded domain with $C^2$ boundary and let $T>0$. Assume that the coefficients $\\alpha$ and $\\beta$ satisfy \n \\begin{align}\n \\label{assumption:prop:existence:02:alpha:beta}\n \\alpha,\\beta\\in W^{1,\\infty}(\\Omega;\\mathbb{R}),\\quad \\alpha(x)\\geq \\alpha_0 >0\\text{ almost everywhere in }\\Omega.\n \\end{align}\n Moreover, consider \\eqref{prop:existence:statement:02}, $v_0\\in H_0^1(\\Omega)$ and $h\\in L^2(0,T;L^2(\\Omega))$. Then, there exists a strong solution $v$ of \\eqref{problem:existence:uniqueness} satisfying\n \\begin{align*}\n v\\in H^1(0,T;L^2(\\Omega)) \\cap C^0([0,T];H_0^1(\\Omega)) \\cap L^2(0,T; H^2(\\Omega) \\cap H_0^1(\\Omega)).\n \\end{align*}\n\n Moreover, there exists a constant $C>0$ depending only on $\\alpha_0$, $\\|\\alpha\\|_{W^{1,\\infty}}$, $\\||\\vec{\\sigma}|\\|_{L^\\infty}$, $\\|p\\|_\\infty$, $\\Omega$ and $T$ such that \n \\begin{align*}\n \\begin{split} \n &\\|v\\|_{L^2(0,T;H^2(\\Omega))} + \\|v\\|_{C^0([0,T];H_0^1(\\Omega))} + \\|\\pt v\\|_{L^2(0,T;L^2(\\Omega))} \\\\\n \\leq &C\\left(\\|h\\|_{L^2(0,T;L^2(\\Omega))} + \\|v_0\\|_{H_0^1(\\Omega)} \\right).\n \\end{split}\n \\end{align*}\n\\end{proposition}", + "post_theorem_intro_text_len": 2627, + "post_theorem_intro_text": "\\begin{proof}\n Let $(f_k)_{k\\in \\mathbb{N}}$ be a bounded sequence in $L^2(\\Omega\\times (0,T))$. Then, by Proposition \\ref{proposition:existence:strong:solutions}, the sequence of associated weak solutions $(y(\\cdot,,\\cdot,f_k))_{k\\in \\mathbb{N}}$ is bounded in $C^0([0,T];H_0^1(\\Omega))$. In particular, the sequence $(Y_k)_{k\\in \\mathbb{N}}$ given by $Y_k:=y(\\cdot,T,f_k)$ is bounded in $H_0^1(\\Omega)\\cap H^2(\\Omega)$. Thus, using the Sobolev-Gagliardo-Nirenberg compact embedding $H_0^1(\\Omega) \\hookrightarrow L^2(\\Omega)$, there exists a subsequence of $ (Y_k)_{k\\in \\mathbb{N}}$ which converges strongly in $L^2(\\Omega)$. This implies that the input-output operator $\\Psi$ is compact and the proof of Proposition \\ref{Proposition:Phi:compact} is finished. \n\\end{proof}\n\nIn view of the Proposition \\ref{Proposition:Phi:compact}, it is evident that the inverse problem {\\bf (ISP)} is ill-posed in the sense of Hadamard. For this reason, one needs to introduce the functional $\\mathcal{J}:\\mathcal{U}\\to \\mathbb{R}$ by\n\\begin{align}\n \\label{def:functional:J}\n \\mathcal{J}(f):=\\frac{1}{2}\\int_\\Omega |\\Psi(f)-u_T|^2\\,dx,\n\\end{align}\nand reformulate {\\bf (ISP)} in terms of the quasi-solution method, i.e., we minimize the following extremal problem\n\\begin{align*}\n \\mathcal{J}(f^\\star)=\\inf_{f\\in \\mathcal{U}} \\mathcal{J}(f).\n\\end{align*}\n\nSince the operator $\\Psi$ is compact, small perturbations in the data may cause large variations in the reconstructed source. Therefore, regularization becomes essential to obtain stable approximations. In this context, it is customary to consider a regularized Tikhonov version of the functional $J$ in \\eqref{def:functional:J}. For $\\epsilon>0$, we introduce the regularized functional\n\\begin{align}\\label{regfunct}\n \\mathcal{J}_\\epsilon (f):=\\frac{1}{2}\\int_\\Omega |\\Psi(f)-u_T|^2\\,dx + \\frac{\\epsilon}{2}\\int_0^T\\int_\\Omega |f|^2\\,dx\\,dt ,\\quad f\\in \\mathcal{U}. \n\\end{align}\n\n\\subsection{Outline}\nThe rest of the paper is as follows. In Section \\ref{section:Frechet:formula}, we focus on the properties of $J$, i.e., a detailed characterization of its Fr\\'echet derivative via a suitable adjoint system. In Section \\ref{section:Existence:ISP}, we give sufficient conditions for the existence and uniqueness of quasi-solutions to {\\bf (ISP)}. In Section \\ref{section:Numerical:experiments}, we validate our theoretical results by some numerical experiments to reconstruct source terms for 1-D and 2-D case. Finally, in Section \\ref{section:Summary:perspectives} we give additional comments concerning the theoretical and numerical results obtained in this article.", + "sketch": "Take a bounded sequence $(f_k)\\subset L^2(\\Omega\\times(0,T))$. By Proposition \\ref{proposition:existence:strong:solutions}, the associated weak solutions $y(\\cdot,\\cdot,f_k)$ are bounded in $C^0([0,T];H_0^1(\\Omega))$, hence the terminal states $Y_k:=y(\\cdot,T,f_k)$ form a bounded sequence in $H_0^1(\\Omega)\\cap H^2(\\Omega)$. Using the compact embedding $H_0^1(\\Omega)\\hookrightarrow L^2(\\Omega)$, extract a subsequence of $(Y_k)$ converging strongly in $L^2(\\Omega)$. Therefore $\\Psi(f_k)=Y_k$ has a strongly convergent subsequence in $L^2(\\Omega)$ for every bounded $(f_k)$, which implies $\\Psi$ is compact (Proposition~\\ref{Proposition:Phi:compact}).", + "expanded_sketch": "Take a bounded sequence $(f_k)\\subset L^2(\\Omega\\times(0,T))$. We first use the following result.\n\n\\begin{proposition}\n \\label{proposition:existence:strong:solutions}\n Let $\\Omega\\subset \\mathbb{R}^N$ be a bounded domain with $C^2$ boundary and let $T>0$. Assume that the coefficients $\\alpha$ and $\\beta$ satisfy \n \\begin{align}\n \\label{assumption:prop:existence:02:alpha:beta}\n \\alpha,\\beta\\in W^{1,\\infty}(\\Omega;\\mathbb{R}),\\quad \\alpha(x)\\geq \\alpha_0 >0\\text{ almost everywhere in }\\Omega.\n \\end{align}\n Moreover, consider \\eqref{prop:existence:statement:02}, $v_0\\in H_0^1(\\Omega)$ and $h\\in L^2(0,T;L^2(\\Omega))$. Then, there exists a strong solution $v$ of \\eqref{problem:existence:uniqueness} satisfying\n \\begin{align*}\n v\\in H^1(0,T;L^2(\\Omega)) \\cap C^0([0,T];H_0^1(\\Omega)) \\cap L^2(0,T; H^2(\\Omega) \\cap H_0^1(\\Omega)).\n \\end{align*}\n\n Moreover, there exists a constant $C>0$ depending only on $\\alpha_0$, $\\|\\alpha\\|_{W^{1,\\infty}}$, $\\||\\vec{\\sigma}|\\|_{L^\\infty}$, $\\|p\\|_\\infty$, $\\Omega$ and $T$ such that \n \\begin{align*}\n \\begin{split} \n &\\|v\\|_{L^2(0,T;H^2(\\Omega))} + \\|v\\|_{C^0([0,T];H_0^1(\\Omega))} + \\|\\pt v\\|_{L^2(0,T;L^2(\\Omega))} \\\\\n \\leq &C\\left(\\|h\\|_{L^2(0,T;L^2(\\Omega))} + \\|v_0\\|_{H_0^1(\\Omega)} \\right).\n \\end{split}\n \\end{align*}\n\\end{proposition}\n\nBy this proposition, the associated weak solutions $y(\\cdot,\\cdot,f_k)$ are bounded in $C^0([0,T];H_0^1(\\Omega))$, hence the terminal states $Y_k:=y(\\cdot,T,f_k)$ form a bounded sequence in $H_0^1(\\Omega)\\cap H^2(\\Omega)$. Using the compact embedding $H_0^1(\\Omega)\\hookrightarrow L^2(\\Omega)$, extract a subsequence of $(Y_k)$ converging strongly in $L^2(\\Omega)$. Therefore $\\Psi(f_k)=Y_k$ has a strongly convergent subsequence in $L^2(\\Omega)$ for every bounded $(f_k)$, which implies compactness. This completes the proof of the main theorem.", + "expanded_theorem": "\\label{Proposition:Phi:compact}\n The input-output operator $\\Psi:L^2(0,T;L^2(\\Omega))\\to L^2(\\Omega)$ is compact.,", + "theorem_type": [ + "Universal" + ], + "mcq": { + "question": "Let \\(\\Omega\\subset \\mathbb{R}^N\\) be a bounded domain with \\(C^2\\) boundary, let \\(T>0\\), and fix the coefficients and initial datum in the Dirichlet Ginzburg--Landau problem\n\\[\n\\begin{cases}\n\\partial_t y-\\operatorname{div}((a(x)+ib(x))\\nabla y)+\\vec r(x)\\cdot \\nabla y+p(x)y=f(x,t) & \\text{in } \\Omega\\times(0,T),\\\\\ny=0 & \\text{on } \\partial\\Omega\\times(0,T),\\\\\ny(\\cdot,0)=y_0 & \\text{in } \\Omega.\n\\end{cases}\n\\]\nFor each source term \\(f\\in L^2(0,T;L^2(\\Omega))\\), let \\(y\\) denote the corresponding solution and define the input-output operator\n\\[\n\\Psi:L^2(0,T;L^2(\\Omega))\\to L^2(\\Omega),\\qquad \\Psi(f)=y(\\cdot,T).\n\\]\nWhich statement holds for this operator?", + "correct_choice": { + "label": "A", + "text": "The operator \\(\\Psi:L^2(0,T;L^2(\\Omega))\\to L^2(\\Omega)\\) is compact; equivalently, it sends every bounded subset of \\(L^2(0,T;L^2(\\Omega))\\) into a relatively compact subset of \\(L^2(\\Omega)\\)." + }, + "choices": [ + { + "label": "B", + "text": "The operator \\(\\Psi:L^2(0,T;L^2(\\Omega))\\to L^2(\\Omega)\\) is bounded but not compact; in fact, for every bounded sequence \\((f_k)\\subset L^2(0,T;L^2(\\Omega))\\), the sequence \\((\\Psi(f_k))\\) is only guaranteed to be bounded in \\(L^2(\\Omega)\\), not to admit a strongly convergent subsequence there." + }, + { + "label": "C", + "text": "The operator \\(\\Psi:L^2(0,T;L^2(\\Omega))\\to L^2(\\Omega)\\) is continuous (in particular, bounded as a linear map)." + }, + { + "label": "D", + "text": "The operator \\(\\Psi:L^2(0,T;L^2(\\Omega))\\to H_0^1(\\Omega)\\) is compact; equivalently, it sends every bounded subset of \\(L^2(0,T;L^2(\\Omega))\\) into a relatively compact subset of \\(H_0^1(\\Omega)\\)." + }, + { + "label": "E", + "text": "The operator \\(\\Psi:L^2(0,T;L^2(\\Omega))\\to L^2(\\Omega)\\) is compact only when the source set is restricted to a bounded, closed, convex subset \\(\\mathcal U\\subset L^2(0,T;L^2(\\Omega))\\); on the whole space \\(L^2(0,T;L^2(\\Omega))\\) one can assert continuity but not compactness." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "B" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "upgrade from boundedness of terminal states to relative compactness via compact embedding", + "template_used": "wildcard" + }, + { + "label": "C", + "sketch_hook_type": "regularity", + "tampered_component": "relative compactness conclusion", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "regularity", + "tampered_component": "compactness is only obtained after embedding into L^2, not in H_0^1", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "finiteness", + "tampered_component": "domain of the operator: compactness proved on the whole bounded-sequence criterion in L^2, not only on a preset bounded closed convex subset", + "template_used": "quantifier_dependence" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem only defines the PDE and the terminal-state map; it does not explicitly state or strongly hint that compactness is the key conclusion." + }, + "TAS": { + "score": 2, + "justification": "This is not a bare restatement of a theorem in the stem. The solver must distinguish among continuity, compactness in L^2, and a stronger false compactness claim in H_0^1." + }, + "GPS": { + "score": 1, + "justification": "Identifying A requires some reasoning about parabolic regularization and compact embeddings, so it is not immediate. However, the presence of C, which is also true as a weaker statement, reduces the need to generate the uniquely strongest conclusion in a clean way." + }, + "DQS": { + "score": 1, + "justification": "B, D, and E are plausible and reflect common misconceptions about boundedness vs compactness and target-space regularity. But C is a weakly true statement rather than a genuine distractor, which weakens the single-best-answer design." + }, + "total_score": 6, + "overall_assessment": "A conceptually solid question with no answer leakage and good mathematical discrimination, but its quality is notably reduced by having a weaker true option (C), making the distractor set only partially effective." + } + }, + { + "id": "2511.07102v1", + "paper_link": "http://arxiv.org/abs/2511.07102v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:MAIN}\nFor any stratum $\\Omega\\mathcal M_{g}(\\mu)$ of holomorphic differentials, the projectivized stratum $\\mathbb P\\Omega\\mathcal M_{g}(\\mu)$ contains no positive-dimensional complete subvariety.", + "start_pos": 12717, + "end_pos": 12954, + "label": "thm:MAIN" + }, + "ref_dict": { + "thm:MAIN": "\\begin{theorem}\\label{thm:MAIN}\nFor any stratum $\\Omega\\mathcal M_{g}(\\mu)$ of holomorphic differentials, the projectivized stratum $\\mathbb P\\Omega\\mathcal M_{g}(\\mu)$ contains no positive-dimensional complete subvariety. \n\\end{theorem}", + "prop:RigidPair": "\\begin{proposition}\\label{prop:RigidPair}\nLet $M$ be a positive-dimensional complete subvariety contained in the projectivized stratum $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ of holomorphic differentials. Then, for any $(X,\\omega) \\in M$, every pair of cylinders in $(X,\\omega)$ is rigid.\n\\end{proposition}", + "prop:RigidCylinder": "\\begin{proposition}\\label{prop:RigidCylinder}\nLet $M$ be a positive-dimensional subvariety of the projectivized stratum $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ of holomorphic differentials.\nThen for every $(X,\\omega) \\in M$, any cylinder $\\mathcal{C}$ in $(X,\\omega)$ has constant conformal modulus on a neighborhood of $(X,\\omega)$ in $M$; equivalently, every cylinder in $(X,\\omega)$ is rigid.\n\\end{proposition}" + }, + "pre_theorem_intro_text_len": 2208, + "pre_theorem_intro_text": "Let $\\mu$ be a partition of $2g-2$. Denote by $\\Omega\\mathcal M_{g}(\\mu)$ the stratum of Abelian differentials (i.e., differential one-forms) on smooth, connected, genus-$g$ complex curves whose zero and pole orders are prescribed by~$\\mu$, and let $\\mathbb P\\Omega\\mathcal{M}_{g}(\\mu) = \\Omega\\mathcal{M}_{g}(\\mu)/\\mathbb C^{*}$ be the corresponding projectivized stratum. \n\nIt is a natural and meaningful question to study how the strata of differentials appear from the perspective of affine geometry—for instance, whether a given stratum can contain a positive-dimensional complete algebraic subvariety.\n\nFor a holomorphic signature~$\\mu$, that is, when all entries of~$\\mu$ are nonnegative, Gendron showed that the unprojectivized stratum $\\Omega\\mathcal M_{g}(\\mu)$ of holomorphic differentials contains no positive-dimensional complete subvariety, by applying the maximum modulus principle to shortest saddle connections; see~\\cite{G20}. In this case, an alternative proof was later given by the first-named author, using the positivity of certain divisor classes on the moduli space of curves; see~\\cite{C23}.\n\nWhen $\\mu$ is a signature of (strictly) meromorphic differentials, again by exploiting the positivity of divisor classes, the first-named author proved that both the projectivized and unprojectivized strata, $\\mathbb P\\Omega\\mathcal M_{g}(\\mu)$ and $\\Omega\\mathcal M_{g}(\\mu)$, of strictly meromorphic differentials contain no positive-dimensional complete subvarieties; see~\\cite{C19, C24}. Therefore, the remaining question is whether the projectivized strata of holomorphic differentials can contain a positive-dimensional complete subvariety.\n\nThis remaining problem is not only the most challenging case, but also significant from the viewpoint of the geometry of canonical divisors. For instance, for $\\mu = (2g-2)$, the minimal stratum $\\mathbb P\\Omega\\mathcal M_{g}(2g-2)$ parameterizes subcanonical points $z$, where $(2g-2)z$ is a canonical divisor. Harris asked whether there exist complete families of such subcanonical points (see~\\cite[Proof of Corollary~5]{harris84}), a question that has remained open for four decades.\n\nIn this paper, we resolve this problem as follows.", + "context": "Let $\\mu$ be a partition of $2g-2$. Denote by $\\Omega\\mathcal M_{g}(\\mu)$ the stratum of Abelian differentials (i.e., differential one-forms) on smooth, connected, genus-$g$ complex curves whose zero and pole orders are prescribed by~$\\mu$, and let $\\mathbb P\\Omega\\mathcal{M}_{g}(\\mu) = \\Omega\\mathcal{M}_{g}(\\mu)/\\mathbb C^{*}$ be the corresponding projectivized stratum.\n\nIt is a natural and meaningful question to study how the strata of differentials appear from the perspective of affine geometry—for instance, whether a given stratum can contain a positive-dimensional complete algebraic subvariety.\n\nFor a holomorphic signature~$\\mu$, that is, when all entries of~$\\mu$ are nonnegative, Gendron showed that the unprojectivized stratum $\\Omega\\mathcal M_{g}(\\mu)$ of holomorphic differentials contains no positive-dimensional complete subvariety, by applying the maximum modulus principle to shortest saddle connections; see~\\cite{G20}. In this case, an alternative proof was later given by the first-named author, using the positivity of certain divisor classes on the moduli space of curves; see~\\cite{C23}.\n\nWhen $\\mu$ is a signature of (strictly) meromorphic differentials, again by exploiting the positivity of divisor classes, the first-named author proved that both the projectivized and unprojectivized strata, $\\mathbb P\\Omega\\mathcal M_{g}(\\mu)$ and $\\Omega\\mathcal M_{g}(\\mu)$, of strictly meromorphic differentials contain no positive-dimensional complete subvarieties; see~\\cite{C19, C24}. Therefore, the remaining question is whether the projectivized strata of holomorphic differentials can contain a positive-dimensional complete subvariety.\n\nThis remaining problem is not only the most challenging case, but also significant from the viewpoint of the geometry of canonical divisors. For instance, for $\\mu = (2g-2)$, the minimal stratum $\\mathbb P\\Omega\\mathcal M_{g}(2g-2)$ parameterizes subcanonical points $z$, where $(2g-2)z$ is a canonical divisor. Harris asked whether there exist complete families of such subcanonical points (see~\\cite[Proof of Corollary~5]{harris84}), a question that has remained open for four decades.\n\nIn this paper, we resolve this problem as follows.", + "full_context": "Let $\\mu$ be a partition of $2g-2$. Denote by $\\Omega\\mathcal M_{g}(\\mu)$ the stratum of Abelian differentials (i.e., differential one-forms) on smooth, connected, genus-$g$ complex curves whose zero and pole orders are prescribed by~$\\mu$, and let $\\mathbb P\\Omega\\mathcal{M}_{g}(\\mu) = \\Omega\\mathcal{M}_{g}(\\mu)/\\mathbb C^{*}$ be the corresponding projectivized stratum.\n\nIt is a natural and meaningful question to study how the strata of differentials appear from the perspective of affine geometry—for instance, whether a given stratum can contain a positive-dimensional complete algebraic subvariety.\n\nFor a holomorphic signature~$\\mu$, that is, when all entries of~$\\mu$ are nonnegative, Gendron showed that the unprojectivized stratum $\\Omega\\mathcal M_{g}(\\mu)$ of holomorphic differentials contains no positive-dimensional complete subvariety, by applying the maximum modulus principle to shortest saddle connections; see~\\cite{G20}. In this case, an alternative proof was later given by the first-named author, using the positivity of certain divisor classes on the moduli space of curves; see~\\cite{C23}.\n\nWhen $\\mu$ is a signature of (strictly) meromorphic differentials, again by exploiting the positivity of divisor classes, the first-named author proved that both the projectivized and unprojectivized strata, $\\mathbb P\\Omega\\mathcal M_{g}(\\mu)$ and $\\Omega\\mathcal M_{g}(\\mu)$, of strictly meromorphic differentials contain no positive-dimensional complete subvarieties; see~\\cite{C19, C24}. Therefore, the remaining question is whether the projectivized strata of holomorphic differentials can contain a positive-dimensional complete subvariety.\n\nThis remaining problem is not only the most challenging case, but also significant from the viewpoint of the geometry of canonical divisors. For instance, for $\\mu = (2g-2)$, the minimal stratum $\\mathbb P\\Omega\\mathcal M_{g}(2g-2)$ parameterizes subcanonical points $z$, where $(2g-2)z$ is a canonical divisor. Harris asked whether there exist complete families of such subcanonical points (see~\\cite[Proof of Corollary~5]{harris84}), a question that has remained open for four decades.\n\nIn this paper, we resolve this problem as follows.\n\nIn this paper, we resolve this problem as follows.\n\n\\begin{lemma}\\label{lem:LocalHarmonic}\nLet $M$ be a positive-dimensional subvariety of the projectivized stratum $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ of holomorphic differentials.\nFor any $(X,\\omega) \\in M$ and any cylinder $\\mathcal{C}$ in $(X,\\omega)$, exactly one of the following holds:\n\\begin{enumerate}\n\\item There exists a neighborhood $U$ of $(X,\\omega)$ in $M$ such that the conformal modulus of $\\mathcal{C}$ remains constant on $U$ (\\textbf{rigid cylinder});\n\\item The conformal modulus of $\\mathcal{C}$ is a non-constant positive pluriharmonic function in a neighborhood of $(X,\\omega)$ in $M$ (\\textbf{flexible cylinder}).\n\\end{enumerate} \n\\end{lemma}\n\n\\begin{proposition}\\label{prop:RigidCylinder}\nLet $M$ be a positive-dimensional subvariety of the projectivized stratum $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ of holomorphic differentials.\nThen for every $(X,\\omega) \\in M$, any cylinder $\\mathcal{C}$ in $(X,\\omega)$ has constant conformal modulus on a neighborhood of $(X,\\omega)$ in $M$; equivalently, every cylinder in $(X,\\omega)$ is rigid.\n\\end{proposition}\n\n\\begin{lemma}\\label{lem:DichotomyPeriods}\nLet $M$ be a positive-dimensional subvariety of the projectivized stratum $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ of holomorphic differentials.\nFor any $(X,\\omega) \\in M$, and for any pair of cylinders $\\mathcal{C}_{1},\\mathcal{C}_{2}$ in $(X,\\omega)$ with waist curves $\\alpha_{1}$ and $\\alpha_{2}$, exactly one of the following holds:\n\\begin{enumerate}\n\\item There exists a neighborhood $U$ of $(X,\\omega)$ in $M$ such that the ratio $ \\frac{\\int_{\\alpha_{1}}\\omega}{\\int_{\\alpha_{2}}\\omega}$ is constant on $U$ (\\textbf{rigid pair of cylinders});\n\\item The function $ \\log\\left|\\frac{\\int_{\\alpha_{1}}\\omega}{\\int_{\\alpha_{2}}\\omega}\\right|$ is a non-constant pluriharmonic function in a neighborhood of $(X,\\omega)$ in $M$ (\\textbf{flexible pair of cylinders}).\n\\end{enumerate} \n\\end{lemma}\n\n\\begin{proposition}\\label{prop:RigidPair}\nLet $M$ be a positive-dimensional complete subvariety contained in the projectivized stratum $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ of holomorphic differentials. Then, for any $(X,\\omega) \\in M$, every pair of cylinders in $(X,\\omega)$ is rigid.\n\\end{proposition}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:MAIN}]\nWe first consider the case of the projectivized stratum of flat tori $\\mathbb{P}\\Omega\\mathcal{M}_{1}(\\emptyset)$, which is one-dimensional. Any holomorphic curve contained in $\\mathbb{P}\\Omega\\mathcal{M}_{1}(\\emptyset)$ must coincide with it, and hence is not compact (since flat tori degenerate to a nodal sphere). In the remainder of the proof, we will therefore assume that $\\mu$ is nonempty, so that the results of Section~\\ref{sec:Cylinders} apply. \n\\par\nWe assume, for the sake of contradiction, that there exists a positive-dimensional complete subvariety $M$ in $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$. In the definition of the normalized length, we use the set $MaxCyl(X,\\omega)$ of the cylinders of largest width among those having the largest conformal modulus. According to Propositions~\\ref{prop:RigidCylinder} and~\\ref{prop:RigidPair}, these cylinders remain the ones of largest width among those with the largest conformal modulus as $(X,\\omega)$ is deformed in $M$. Deformation along a closed path in $M$ can, however, permute the elements of $MaxCyl(X,\\omega)$. \n\\par\nA saddle connection $\\alpha$ is called \\textbf{rigid} if its normalized length $l(\\alpha)$ remains constant in a neighborhood of $(X,\\omega)$ in $M$, and \\textbf{flexible} otherwise. Observe that for any rigid (resp. flexible) saddle connection $\\alpha$ of $(X,\\omega)$, there exists a neighborhood of $(X,\\omega)$ in $M$ where $\\alpha$ persists as a saddle connection and remains rigid (resp. flexible).\nThis follows from the fact that the ratio $\\frac{\\int_{\\alpha} \\omega}{\\int_{\\beta} \\omega}$ is holomorphic in the local coordinates of $M$.\n\\par\nBy hypothesis, any translation surface in $M$ can be deformed nontrivially within $M$, so every $(X,\\omega)$ in $M$ contains at least one flexible saddle connection. We define the function \n$$L\\colon M \\to \\mathbb R^{\\geq 0}$$ \nby assigning to $(X,\\omega)$ the normalized length $l(\\alpha)$, where $\\alpha$ is one of the shortest flexible saddle connections in $(X,\\omega)$.\nSince a unit-area translation surface contains only finitely many saddle connections of length at most $K$ for any $K>0$, the function $L$ is well defined.\nWe first show that $L$ is a continuous function on $M$. \n\\par\nWe consider a sequence of surfaces $(X_{n},\\omega_{n})_{n \\in \\mathbb{N}}$ in $M$ converging to a surface $(X_{\\infty},\\omega_{\\infty})$ in $M$.\nIn each $(X_{n},\\omega_{n})$, let $\\alpha_{n}$ be a flexible saddle connection such that $L(X_{n},\\omega_{n}) = l(\\alpha_{n})$.\nDenote by $\\beta_{n}$ the waist curve of a cylinder in $MaxCyl(X_{n},\\omega_{n})$.\nUp to passing to a subsequence of $(X_{n},\\omega_{n})_{n \\in \\mathbb{N}}$, we may assume that \n$$l(\\alpha_{n})_{n \\in \\mathbb{N}}=\\frac{|\\int_{\\alpha_{n}} \\omega_n|}{|\\int_{\\beta_{n}} \\omega_n|}$$ \nconverges to some limit $\\lambda$.\nLet $\\alpha_{\\infty}$ be a flexible saddle connection such that $L(X_{\\infty},\\omega_{\\infty}) = l(\\alpha_{\\infty})$.\nSince $\\alpha_{\\infty}$ persists as a flexible saddle connection in a sufficiently small neighborhood of $(X_{\\infty},\\omega_{\\infty})$ in $M$, we deduce that $\\lambda \\leq l(\\alpha_{\\infty})$.\nIt remains to prove that $\\lambda \\geq l(\\alpha_{\\infty})$. \n\\par\nUp to passing to a subsequence again, we may assume that the ratios\n$$\\frac{\\int_{\\alpha_{n}} \\omega_n}{\\int_{\\beta_n} \\omega_n}$$ \nconverge to some limit value $\\nu$ satisfying $0 \\leq |\\nu| = \\lambda$. Lemma~\\ref{lem:NormalizedLength} already establishes that the normalized length of any saddle connection is bounded below by some positive constant. We therefore deduce that $|\\nu|>0$. \n\\par\nSince this convergence takes place in an arbitrarily small contractible neighborhood of $(X_{\\infty},\\omega_{\\infty})$, we can mark the zeros of $\\omega_{\\infty}$ and, up to taking a further subsequence, require that all the saddle connections $(\\alpha_{n})_{n \\in \\mathbb{N}}$ join the same oriented pair of zeros. The cylinders of $MaxCyl(X_{n},\\omega_{n})$ can be marked within this neighborhood, so we may also assume that all the waist curves $\\beta_{n}$ belong to the same cylinder. We denote by $\\beta$ a waist curve of this cylinder in $(X_{\\infty},\\omega_{\\infty})$. \n\\par \nA subsequence of saddle connections $(\\alpha_{\\phi(n)})_{n \\in \\mathbb{N}}$ accumulates on an arc $\\gamma$ satisfying $$\\frac{\\int_{\\gamma} \\omega_{\\infty}}{\\int_{\\beta} \\omega_{\\infty}}=\\nu,$$ \nwhich is formed by one or several oriented saddle connections of the same oriented slope. For $n$ sufficiently large, the relative homology classes $[\\alpha_{\\phi(n)}]$ and $[\\gamma]$ coincide, so at least one saddle connection forming $\\gamma$ must be flexible, which we denote by $\\delta$. Then this saddle connection $\\delta$ satisfies\n$l(\\delta) \\leq |\\nu| = \\lambda$,\nand hence $\\lambda \\geq l(\\alpha_{\\infty})$.\nConsequently, we obtain $L(X_{n},\\omega_{n}) \\rightarrow L(X_{\\infty},\\omega_{\\infty})$ as $n \\to \\infty$, thereby establishing the continuity of $L$. \n\\par\nSince $L$ is continuous on $M$, which is compact, it attains a global minimum realized by some flexible saddle connection $\\alpha$ on a translation surface $(X,\\omega)$ in $M$. Because the ratio $\\frac{\\int_{\\alpha} \\omega}{\\int_{\\beta} \\omega}$ is holomorphic in the local coordinates of $M$, its modulus can decrease again along certain local deformation paths, leading to a contradiction. Therefore, no flexible saddle connection can exist on any translation surface in $M$, which implies that no nontrivial deformation exists for any translation surface in $M$. In conclusion, we deduce that there is no positive-dimensional complete subvariety in $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$. \n\\end{proof}", + "post_theorem_intro_text_len": 2445, + "post_theorem_intro_text": "Besides Abelian differentials, one can also study the strata of $k$-differentials with prescribed zero and pole orders, where a $k$-differential is a section of the $k$-th power of the canonical bundle. Note that a $k$-differential with pole orders at most $k-1$ (i.e., when the corresponding $\\frac{1}{k}$-translation surface has finite area) can be lifted via the canonical cyclic covering construction to the $k$-th power of a holomorphic one-form; see~\\cite[Section 2]{BCGGM-k}. Theorem~\\ref{thm:MAIN} therefore implies the following corollary, which settles the remaining case concerning complete subvarieties in the strata of $k$-differentials; see~\\cite[Remark]{C23}. \n\n\\begin{corollary}\\label{cor:k-diff}\n For any stratum $\\Omega^k\\mathcal M_{g}(\\mu)$ of $k$-differentials whose pole orders are bounded by $k-1$, the projectivized stratum $\\mathbb P\\Omega^k\\mathcal M_{g}(\\mu)$ contains no positive-dimensional complete subvarieties. \n\\end{corollary}\n\nTo prove Theorem~\\ref{thm:MAIN}, we refine Gendron’s argument from~\\cite{G20}. By applying the maximum principle to suitably chosen plurisubharmonic functions, we show that, in any complete subvariety contained in a projectivized stratum, the following quantities are rigid:\n\\begin{itemize}\n\\item the conformal moduli of cylinders (see Proposition~\\ref{prop:RigidCylinder});\n\\item the periods of closed geodesics, up to a global scaling (see Proposition~\\ref{prop:RigidPair});\n\\item suitably normalized lengths of saddle connections (see Section~\\ref{sec:Main}).\n\\end{itemize} \n\nFinally, we remark that since $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ is a quasi-projective variety, its complete algebraic subvarieties and complex analytic subvarieties coincide by the GAGA principle. Moreover, if a complete subvariety $M$ is singular, we may work with a resolution of singularities of $M$ and pull back the family of differentials accordingly. Alternatively, one can intersect a higher-dimensional complete subvariety with ample hypersurfaces until obtaining a complete algebraic curve. Therefore, Theorem~\\ref{thm:MAIN} is equivalent to showing that $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ contains no complete algebraic curves, in which case we may work with the normalization of such a curve, which is smooth. Therefore, in these senses, even when $M$ is singular, we can still speak of and make use of holomorphic local coordinates on $M$. We shall do so without further comment.", + "sketch": "To prove Theorem~\\ref{thm:MAIN}, the authors “refine Gendron’s argument from~\\cite{G20}.” The strategy is to “apply[] the maximum principle to suitably chosen plurisubharmonic functions” and deduce that on any complete subvariety contained in a projectivized stratum, several geometric quantities are rigid: (1) “the conformal moduli of cylinders” (Proposition~\\ref{prop:RigidCylinder}); (2) “the periods of closed geodesics, up to a global scaling” (Proposition~\\ref{prop:RigidPair}); and (3) “suitably normalized lengths of saddle connections” (Section~\\ref{sec:Main}). They also reduce to the curve case: since $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ is quasi-projective, complete algebraic and complex analytic subvarieties coincide by GAGA; if needed one passes to a resolution/normalization, and by intersecting with ample hypersurfaces, Theorem~\\ref{thm:MAIN} is “equivalent to showing that $\\mathbb{P}\\Omega\\mathcal{M}_{g}(\\mu)$ contains no complete algebraic curves.”", + "expanded_sketch": "No expanded sketch found.", + "expanded_theorem": "\\label{thm:MAIN}\nFor any stratum $\\Omega\\mathcal M_{g}(\\mu)$ of holomorphic differentials, the projectivized stratum $\\mathbb P\\Omega\\mathcal M_{g}(\\mu)$ contains no positive-dimensional complete subvariety.,", + "theorem_type": [ + "Nonexistence", + "Universal" + ], + "mcq": { + "question": "Let $g$ be a genus and let $\\mu$ be any holomorphic signature with total order $2g-2$ (so $\\Omega\\mathcal M_g(\\mu)$ is the stratum of holomorphic Abelian differentials $\\omega$ on smooth, connected, genus-$g$ complex curves $X$ whose zero orders are prescribed by $\\mu$). Let\n$$\\mathbb P\\Omega\\mathcal M_g(\\mu)=\\Omega\\mathcal M_g(\\mu)/\\mathbb C^*$$\nbe the projectivized stratum, obtained by identifying differentials that differ by nonzero scalar multiplication. Which statement holds for every such projectivized stratum?", + "correct_choice": { + "label": "A", + "text": "The projectivized stratum $\\mathbb P\\Omega\\mathcal M_g(\\mu)$ contains no positive-dimensional complete subvariety." + }, + "choices": [ + { + "label": "B", + "text": "The projectivized stratum $\\mathbb P\\Omega\\mathcal M_g(\\mu)$ contains no complete subvariety of codimension one." + }, + { + "label": "C", + "text": "The projectivized stratum $\\mathbb P\\Omega\\mathcal M_g(\\mu)$ is not itself a positive-dimensional complete variety." + }, + { + "label": "D", + "text": "The projectivized stratum $\\mathbb P\\Omega\\mathcal M_g(\\mu)$ contains no positive-dimensional complete subvariety whenever $g\\ge 2$." + }, + { + "label": "E", + "text": "The unprojectivized stratum $\\Omega\\mathcal M_g(\\mu)$ contains no positive-dimensional complete subvariety." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "finiteness", + "tampered_component": "forbidden_dimension_of_complete_subvariety", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "finiteness", + "tampered_component": "absence_of_all_positive_dimensional_complete_subvarieties weakened to non-completeness of the whole stratum", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "case_split", + "tampered_component": "genus_1_exception_recast_as_hypothesis", + "template_used": "boundary_range" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "projectivized_vs_unprojectivized_conclusion", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives definitions and asks for a universal property, but it does not state or strongly hint at the correct conclusion. There is no explicit answer leakage." + }, + "TAS": { + "score": 1, + "justification": "The correct option is essentially a theorem-style statement about projectivized strata, so the item is close to direct recall. However, the presence of nearby alternatives prevents it from being a pure restatement." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish the strongest true statement from weaker or altered variants, especially versus the weaker true option and the projectivized/unprojectivized confusion. Still, this is more theorem recognition than deep generative reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are well designed: one is a weaker true statement, others reflect common confusions about codimension, genus restrictions, and projectivized versus unprojectivized strata. They are plausible and mathematically distinct." + }, + "total_score": 6, + "overall_assessment": "A solid MCQ with no answer leakage and strong distractors, but it mainly tests precise theorem recall rather than substantial generative mathematical reasoning." + } + }, + { + "id": "2511.06517v1", + "paper_link": "http://arxiv.org/abs/2511.06517v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "theorem", + "content": "\\label{th_homo_image} The relation $\\{\\langle A,B \\rangle \\colon A \\leftarrow\\mathrel{\\mkern-14mu}\\leftarrow B\\}$ on the Borel space of countable groups is a complete analytic quasi-order. Thus, the relation of bi-epimorphism on the same Borel space is a complete analytic equivalence relation.", + "start_pos": 8697, + "end_pos": 9014, + "label": "th_homo_image" + }, + "ref_dict": {}, + "pre_theorem_intro_text_len": 2039, + "pre_theorem_intro_text": "The set of groups with domain $\\omega$ admits a natural topology which makes it into a Borel space. Using the tools of invariant descriptive set theory (see~\\cite{gao}), one can attempt to determine the (Borel) complexity of equivalence relations and quasi-orders of interest on this space. Friedman and Stanley~\\cite{friedman_and_stanley} proved that the isomorphism relation on countable groups is Borel bireducible with the isomorphism relation between countable graphs on $\\omega$, using a construction of Mekler~\\cite{mekler}.\n\t In fact they show that also isomorphism relation on the class of $2$-nilpotent group of exponent a fixed odd prime is Borel bi-reducible with graph isomorphism Recently, \n\t the fourth author of the present paper and S.\\ Shelah~\\cite{1205} proved that the same applies to countable torsion-free abelian groups. The isomorphism relation between finitely generated groups is a universal essentially countable Borel equivalence relation~\\cite{veli}. In another direction, Calderoni and Thomas~\\cite{calderoni} show that the embeddability relation between countable torsion free abelian groups is a complete analytic quasi-order (and hence bi-embeddability is a complete analytic equivalence relation). Our aim is to determine the Borel complexity of a quasi-order on groups dual to that of embeddability: being an epimorphic image.\n\n\t\\smallskip\n\n\tThroughout, when we say a structure is countable we mean that it has $\\omega$ as its domain. Classes of structures we are interested in will always form a standard Borel space. For countable groups $A,B$, we write $A \\leftarrow\\mathrel{\\mkern-14mu}\\leftarrow B$ if there is a surjective homomorphism from $B$ onto $A$; equivalently, $A$ is isomorphic to a quotient of $B$. This relation is easily shown to be analytic. Our main result is that it has the maximum possible complexity. Camerlo et al. \\cite{camerlo2} showed such a result for several classes of structures. Notably, the case of countable groups was left open.", + "context": "The set of groups with domain $\\omega$ admits a natural topology which makes it into a Borel space. Using the tools of invariant descriptive set theory (see~\\cite{gao}), one can attempt to determine the (Borel) complexity of equivalence relations and quasi-orders of interest on this space. Friedman and Stanley~\\cite{friedman_and_stanley} proved that the isomorphism relation on countable groups is Borel bireducible with the isomorphism relation between countable graphs on $\\omega$, using a construction of Mekler~\\cite{mekler}.\n In fact they show that also isomorphism relation on the class of $2$-nilpotent group of exponent a fixed odd prime is Borel bi-reducible with graph isomorphism Recently, \n the fourth author of the present paper and S.\\ Shelah~\\cite{1205} proved that the same applies to countable torsion-free abelian groups. The isomorphism relation between finitely generated groups is a universal essentially countable Borel equivalence relation~\\cite{veli}. In another direction, Calderoni and Thomas~\\cite{calderoni} show that the embeddability relation between countable torsion free abelian groups is a complete analytic quasi-order (and hence bi-embeddability is a complete analytic equivalence relation). Our aim is to determine the Borel complexity of a quasi-order on groups dual to that of embeddability: being an epimorphic image.\n\n\\smallskip\n\nThroughout, when we say a structure is countable we mean that it has $\\omega$ as its domain. Classes of structures we are interested in will always form a standard Borel space. For countable groups $A,B$, we write $A \\leftarrow\\mathrel{\\mkern-14mu}\\leftarrow B$ if there is a surjective homomorphism from $B$ onto $A$; equivalently, $A$ is isomorphic to a quotient of $B$. This relation is easily shown to be analytic. Our main result is that it has the maximum possible complexity. Camerlo et al. \\cite{camerlo2} showed such a result for several classes of structures. Notably, the case of countable groups was left open.", + "full_context": "The set of groups with domain $\\omega$ admits a natural topology which makes it into a Borel space. Using the tools of invariant descriptive set theory (see~\\cite{gao}), one can attempt to determine the (Borel) complexity of equivalence relations and quasi-orders of interest on this space. Friedman and Stanley~\\cite{friedman_and_stanley} proved that the isomorphism relation on countable groups is Borel bireducible with the isomorphism relation between countable graphs on $\\omega$, using a construction of Mekler~\\cite{mekler}.\n In fact they show that also isomorphism relation on the class of $2$-nilpotent group of exponent a fixed odd prime is Borel bi-reducible with graph isomorphism Recently, \n the fourth author of the present paper and S.\\ Shelah~\\cite{1205} proved that the same applies to countable torsion-free abelian groups. The isomorphism relation between finitely generated groups is a universal essentially countable Borel equivalence relation~\\cite{veli}. In another direction, Calderoni and Thomas~\\cite{calderoni} show that the embeddability relation between countable torsion free abelian groups is a complete analytic quasi-order (and hence bi-embeddability is a complete analytic equivalence relation). Our aim is to determine the Borel complexity of a quasi-order on groups dual to that of embeddability: being an epimorphic image.\n\n\\smallskip\n\nThroughout, when we say a structure is countable we mean that it has $\\omega$ as its domain. Classes of structures we are interested in will always form a standard Borel space. For countable groups $A,B$, we write $A \\leftarrow\\mathrel{\\mkern-14mu}\\leftarrow B$ if there is a surjective homomorphism from $B$ onto $A$; equivalently, $A$ is isomorphic to a quotient of $B$. This relation is easily shown to be analytic. Our main result is that it has the maximum possible complexity. Camerlo et al. \\cite{camerlo2} showed such a result for several classes of structures. Notably, the case of countable groups was left open.\n\n\\begin{abstract} \n We prove that the epimorphism relation is a complete analytic quasi-order on the space of countable groups. \n In the process we obtain the result of indepent interest showing that the epimorphism relation on pointed reflexive graph is complete.\n\\end{abstract}\n\nThe set of groups with domain $\\omega$ admits a natural topology which makes it into a Borel space. Using the tools of invariant descriptive set theory (see~\\cite{gao}), one can attempt to determine the (Borel) complexity of equivalence relations and quasi-orders of interest on this space. Friedman and Stanley~\\cite{friedman_and_stanley} proved that the isomorphism relation on countable groups is Borel bireducible with the isomorphism relation between countable graphs on $\\omega$, using a construction of Mekler~\\cite{mekler}.\n In fact they show that also isomorphism relation on the class of $2$-nilpotent group of exponent a fixed odd prime is Borel bi-reducible with graph isomorphism Recently, \n the fourth author of the present paper and S.\\ Shelah~\\cite{1205} proved that the same applies to countable torsion-free abelian groups. The isomorphism relation between finitely generated groups is a universal essentially countable Borel equivalence relation~\\cite{veli}. In another direction, Calderoni and Thomas~\\cite{calderoni} show that the embeddability relation between countable torsion free abelian groups is a complete analytic quasi-order (and hence bi-embeddability is a complete analytic equivalence relation). Our aim is to determine the Borel complexity of a quasi-order on groups dual to that of embeddability: being an epimorphic image.\n\n\\smallskip\n\nOur reduction proceeds in two steps. In the first step, we use a result of Louveau and Rosendal \\cite{LR} to show that the epimorphism relation on countable pointed reflexive graphs is analytic complete; here a pointed reflexive graph is one that has a distinguished vertex that is adajacent to each vertex. In the second step, we Borel reduce this quasi-order to the epimorphism relation on countable groups, using \n a new group theoretic construction based on certain countably generated Coxeter groups.\n\n\\begin{question} Is the epimorphism relation a complete-analytic quasi-order on the Borel space of countable {\\em abelian} groups?\n\\end{question}\n\n\\begin{remark} \\label{LRrem}\nLouveau and Rosendal \\cite[Theorem 3.5]{LR} proved that the homomorphism relation between countable graphs is a complete analytic quasi-order. They give a Borel reduction $T \\to G_T$ of their complete quasi-order $\\leq_{max} $ on normal trees (see \\cite[Definition 2.3 and above]{LR}) to it. One observes from the proof that for each normal tree $T$, the constructed graph $G_T$ has not isolated vertex. As they remark at the end of their proof, if $S \\le_{max} T$ then $G_S$ is in fact isomorphic to an induced subgraph of $G_T$, and so in particular $G_S\\preceqc_1 G_T$. Thus, their proof shows that \\emph{the relation $\\preceqc_1$ on countable graphs with no isolated vertices is a complete analytic quasi-order.}\n\\end{remark}\n\n\\begin{proposition}\\label{reduction}\n There is a Borel map $F$ from the class of countable graphs without isolated vertices to the class of countable pointed reflexive graphs, such that for any countable connected graphs $\\Gamma$ and $\\Delta$, we have\n $$\\Gamma \\preceqc_1 \\Delta \\iff F(\\Gamma) \\leftarrowdbl F(\\Delta). $$\n Thus, the relation $\\leftarrowdbl$ between countable pointed reflexive graphs is a complete analytic quasi-order using Remark~\\ref{LRrem}.\n\\end{proposition}\n\n\\begin{fact}[{\\cite[3.1.7]{krammer}}]\\label{krammer_fact} Let $(G, S)$ be a Coxeter system and let $I, J \\subseteq S$. Then $G_I = \\langle I \\rangle_G$ and $G_J = \\langle J \\rangle_G$ are conjugate in $G$ if and only if\n$I$ and $J$ are in the same connected component of $\\mathcal{K}_S$ (cf. \\ref{def_K}).\n\\end{fact}", + "post_theorem_intro_text_len": 714, + "post_theorem_intro_text": "Our reduction proceeds in two steps. In the first step, we use a result of Louveau and Rosendal \\cite{LR} to show that the epimorphism relation on countable pointed reflexive graphs is analytic complete; here a pointed reflexive graph is one that has a distinguished vertex that is adajacent to each vertex. In the second step, we Borel reduce this quasi-order to the epimorphism relation on countable groups, using \n\ta new group theoretic construction based on certain countably generated Coxeter groups. \n\nWe leave open the following question.\n\n\t\\begin{question} Is the epimorphism relation a complete-analytic quasi-order on the Borel space of countable {\\em abelian} groups?\n\\end{question}", + "sketch": "Our reduction proceeds in two steps. In the first step, we use a result of Louveau and Rosendal \\cite{LR} to show that the epimorphism relation on countable pointed reflexive graphs is analytic complete (where a pointed reflexive graph has a distinguished vertex adjacent to each vertex). In the second step, we Borel reduce this quasi-order to the epimorphism relation on countable groups, using a new group theoretic construction based on certain countably generated Coxeter groups.", + "expanded_sketch": "Our reduction proceeds in two steps. In the first step, we use a result of Louveau and Rosendal \\cite{LR} to show that the epimorphism relation on countable pointed reflexive graphs is analytic complete (where a pointed reflexive graph has a distinguished vertex adjacent to each vertex). In the second step, we Borel reduce this quasi-order to the epimorphism relation on countable groups, using a new group theoretic construction based on certain countably generated Coxeter groups.", + "expanded_theorem": "\\label{th_homo_image} The relation $\\{\\langle A,B \\rangle \\colon A \\leftarrow\\mathrel{\\mkern-14mu}\\leftarrow B\\}$ on the Borel space of countable groups is a complete analytic quasi-order. Thus, the relation of bi-epimorphism on the same Borel space is a complete analytic equivalence relation.", + "theorem_type": [ + "Classification or Bijection", + "Universal" + ], + "mcq": { + "question": "Let a countable group mean a group with domain \\(\\omega\\), viewed in the standard Borel space of all countable groups. For countable groups \\(A\\) and \\(B\\), write \\(A \\leftarrow\\!\\!\\leftarrow B\\) if there exists a surjective homomorphism from \\(B\\) onto \\(A\\) (equivalently, \\(A\\) is isomorphic to a quotient of \\(B\\)). Let bi-epimorphism mean the symmetric relation \\(A \\leftarrow\\!\\!\\leftarrow B\\) and \\(B \\leftarrow\\!\\!\\leftarrow A\\). Which statement holds for these relations on the Borel space of countable groups? Here, “complete analytic quasi-order” means an analytic quasi-order to which every analytic quasi-order on a standard Borel space Borel reduces, and “complete analytic equivalence relation” is defined analogously among analytic equivalence relations.", + "correct_choice": { + "label": "A", + "text": "The relation \\(\\{\\langle A,B\\rangle : A \\leftarrow\\!\\!\\leftarrow B\\}\\) on countable groups is a complete analytic quasi-order. Consequently, the bi-epimorphism relation on the same Borel space is a complete analytic equivalence relation." + }, + "choices": [ + { + "label": "B", + "text": "The relation \\(\\{\\langle A,B\\rangle : A \\leftarrow\\!\\!\\leftarrow B\\}\\) on countable groups is an analytic quasi-order, but it is not complete analytic. Consequently, the bi-epimorphism relation on the same Borel space is analytic, though not a complete analytic equivalence relation." + }, + { + "label": "C", + "text": "The relation \\(\\{\\langle A,B\\rangle : A \\leftarrow\\!\\!\\leftarrow B\\}\\) on countable groups is an analytic quasi-order. Consequently, the bi-epimorphism relation on the same Borel space is an analytic equivalence relation." + }, + { + "label": "D", + "text": "The relation \\(\\{\\langle A,B\\rangle : A \\leftarrow\\!\\!\\leftarrow B\\}\\) on countable groups is a complete analytic equivalence relation. Consequently, the bi-epimorphism relation on the same Borel space is also a complete analytic equivalence relation." + }, + { + "label": "E", + "text": "The relation \\(\\{\\langle A,B\\rangle : A \\leftarrow\\!\\!\\leftarrow B\\}\\) on countable groups is a complete analytic quasi-order, and moreover there is a direct Borel reduction from every analytic quasi-order to epimorphism on countable groups without passing through pointed reflexive graphs." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "analytic-completeness obtained via the two-step Borel reduction", + "template_used": "stronger_trap" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped completeness in both the quasi-order and equivalence-relation conclusions", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "quasi-order versus equivalence relation", + "template_used": "property_confusion" + }, + { + "label": "E", + "sketch_hook_type": "geometric_construction", + "tampered_component": "necessity of the intermediate pointed reflexive graph reduction step", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem defines the epimorphism and bi-epimorphism relations and asks for the correct descriptive-set-theoretic classification, but it does not explicitly or implicitly reveal that the relation is complete analytic." + }, + "TAS": { + "score": 2, + "justification": "The question is not a direct restatement of a theorem from the stem itself; the student must choose among materially different claims about analyticity, completeness, and quasi-order versus equivalence-relation status." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to compare the strength and type of the options, but the item mainly tests recall of a known theorem rather than derivation. Generative pressure is also weakened because one distractor states a weaker claim that would still be true if the correct choice is true." + }, + "DQS": { + "score": 1, + "justification": "Several distractors are plausible and target real confusions (analytic vs complete analytic, quasi-order vs equivalence relation, overclaiming a direct reduction). However, choice C is a weaker true statement implied by A, so it is not a clean distractor and introduces ambiguity." + }, + "total_score": 6, + "overall_assessment": "Good on avoiding leakage and tautology, but only moderate as a reasoning item. Its main flaw is ambiguity: option C appears true as a weaker consequence of A, so the MCQ does not cleanly enforce a unique correct answer." + } + }, + { + "id": "2511.06071v1", + "paper_link": "http://arxiv.org/abs/2511.06071v1", + "theorems_cnt": 2, + "theorem": { + "env_name": "thm", + "content": "\\label{thm:symmetry}\n\tLet $(u,v) \\in \\mathcal{D}^{1,p}(\\mathbb{R}^n) \\times \\mathcal{D}^{1,p} (\\mathbb{R}^n)$ be a solution to \\eqref{P}. Then $(u,v)$ is radially symmetric and radially decreasing around the origin.", + "start_pos": 19963, + "end_pos": 20199, + "label": "thm:symmetry" + }, + "ref_dict": { + "AubinTalenti": "\\begin{equation}\\label{AubinTalenti}\n\\mathcal{V}_{\\lambda,x_0}(x) := \\left[\\frac{\\lambda^{\\frac{1}{p-1}}\\sqrt[p]{n\\left(\\frac{n-p}{p-1}\\right)^{p-1}}}{\\lambda^{\\frac{p}{p-1}} + |x - x_0|^{\\frac{p}{p-1}}}\\right]^{\\frac{n-p}{p}}\n\\end{equation}", + "eq:Sobolev": "\\begin{equation}\\label{eq:Sobolev}\n\t\\begin{cases}\n\t\t\\displaystyle -\\Delta_p u\\,= u^{p^*-1} & \\text{in}\\quad\\R^n\\\\\n\t\tu > 0 & \\text{in}\\quad\\R^n.\n\t\\end{cases}\n\\end{equation}", + "eq:pBSsystem": "\\begin{equation}\\label{eq:pBSsystem}\n\t\\begin{cases}\n\t\t-\\Delta u + V(x) u = \\mu u^{2q-1} + \\nu u^{q-1} v^{q} & \\text{in } \\mathbb{R}^n \\\\\n\t\t-\\Delta v + V(x) v = \\mu v^{2q-1} + \\nu u^{q}v^{q-1} & \\text{in } \\mathbb{R}^n,\n\t\\end{cases}\n\\end{equation}", + "P": "\\begin{equation}\\tag{$\\mathcal{S}^*$}\\label{P}\n\\begin{cases}\n\\displaystyle -\\Delta_p u\\,=\\gamma \\frac{u^{p-1}}{|x|^p} + u^{p^*-1}+ \\nu \\alpha u^{\\alpha-1} v^\\beta & \\text{in}\\quad\\R^n \\vspace{0.2cm} \\\\\n\\displaystyle -\\Delta_p v\\,=\\gamma \\frac{v^{p-1}}{|x|^p} + v^{p^*-1}+ \\nu \\beta u^\\alpha v^{\\beta-1} & \\text{in}\\quad\\R^n \\vspace{0.2cm} \\\\\nu,v > 0 & \\text{in}\\quad\\R^n \\setminus \\{0\\}\\\\\nu,v \\in \\mathcal D^{1,p}(\\R^n),\n\\end{cases}\n\\end{equation}", + "eq:terraciniane": "\\begin{equation}\\label{eq:terraciniane}\n\t\\mathcal{U}_\\lambda(x)=\\lambda^{\\frac{2-n}{2}}\\mathcal{U}\\left(\\frac{x}{\\lambda}\\right) \\qquad \\mbox{ with } \\qquad \\mathcal{U}(x)= \\dfrac{A(n,\\gamma)}{|x|^{\\mu}\\left(1+|x|^{2-\\frac{4\\mu}{n-2}}\\right)^{\\frac{n-2}{2}}},\n\\end{equation}", + "eq:HardySobolev": "\\begin{equation}\\label{eq:HardySobolev}\n\\begin{cases}\n-\\Delta_p u = \\gamma \\frac{u^{p-1}}{|x|^p} + u^{p^*-1} & \\text{in } \\mathbb{R}^n \\\\\nu > 0 & \\text{in } \\mathbb{R}^n \\setminus \\{0\\}.\n\\end{cases} \n\\end{equation}" + }, + "pre_theorem_intro_text_len": 12259, + "pre_theorem_intro_text": "In this paper, we investigate qualitative properties of solutions to the following doubly critical system involving the $p$-Laplace operator:\n\\begin{equation}\\tag{$\\mathcal{S}^*$}\\label{P}\n\\begin{cases}\n\\displaystyle -\\Delta_p u\\,=\\gamma \\frac{u^{p-1}}{|x|^p} + u^{p^*-1}+ \\nu \\alpha u^{\\alpha-1} v^\\beta & \\text{in}\\quad\\R^n \\vspace{0.2cm} \\\\\n\\displaystyle -\\Delta_p v\\,=\\gamma \\frac{v^{p-1}}{|x|^p} + v^{p^*-1}+ \\nu \\beta u^\\alpha v^{\\beta-1} & \\text{in}\\quad\\R^n \\vspace{0.2cm} \\\\\nu,v > 0 & \\text{in}\\quad\\R^n \\setminus \\{0\\}\\\\\nu,v \\in \\mathcal D^{1,p}(\\R^n),\n\\end{cases}\n\\end{equation}\nwhere $\\gamma \\in [0, \\Lambda_{n,p})$ with $\\Lambda_{n,p} = \\left[(n-p)/p\\right]^p$ representing the optimal constant in Hardy's inequality for $n > p$. We note that this system is well posed once one assumes that both\n$\\alpha, \\beta > 1$ are real parameters that satisfy\n\\begin{equation*}\n\\alpha + \\beta = p^*. \n\\end{equation*}\nHere, $p^* = np/(n-p)$ denotes the critical Sobolev exponent, $\\nu > 0$ is the coupling parameter. By the previous homogeneity condition we deduce that $p^*> 2$. Thus, here and in all the paper we will assume that\n$$\n\\frac{2n}{n+2} < p< n.\n$$\nWe point out that $\\mathcal D^{1, p}(\\mathbb R^n)$ is the completion of $\\mathcal C^\\infty_c(\\mathbb R^n)$, the space of smooth functions with compact support, with respect to the norm \n$$\\|u\\|:=\\left(\\int_{\\mathbb R^n}|\\nabla u|^p \\, dx \\right)^{\\frac 1p}.$$\nIt is well known, by standard regularity theory (see e.g.~\\cite{Di,T}), it follows that solutions to \\eqref{P} are of class $\\mathcal C^{1,\\alpha}$ far from the origin.\\\\\n\nOur main objective is to prove radial symmetry of solutions to system \\eqref{P}. When $\\gamma = \\nu = 0$, the system reduces to the standard $p$-Sobolev critical equation\n\\begin{equation}\\label{eq:Sobolev}\n\t\\begin{cases}\n\t\t\\displaystyle -\\Delta_p u\\,= u^{p^*-1} & \\text{in}\\quad\\R^n\\\\\n\t\tu > 0 & \\text{in}\\quad\\R^n.\n\t\\end{cases}\n\\end{equation}\nIt is well-known that the Aubin-Talenti bubbles, defined as\n\\begin{equation}\\label{AubinTalenti}\n\\mathcal{V}_{\\lambda,x_0}(x) := \\left[\\frac{\\lambda^{\\frac{1}{p-1}}\\sqrt[p]{n\\left(\\frac{n-p}{p-1}\\right)^{p-1}}}{\\lambda^{\\frac{p}{p-1}} + |x - x_0|^{\\frac{p}{p-1}}}\\right]^{\\frac{n-p}{p}}\n\\end{equation}\nis a family of positive radial solutions to \\eqref{eq:Sobolev}, with $\\lambda > 0$, and $x_0 \\in \\mathbb{R}^n$. These functions achieve equality in the Sobolev inequality in $\\mathbb{R}^n$. The situation in case $p=2$ is fully understood. In fact, in \\cite{GNN2}, it has been shown that any positive solution $u$ to \\eqref{eq:Sobolev} that satisfies the decay assumption $u(x) = \\mathcal{O}(1/|x|^m)$ at infinity for $m > 0$ is radially symmetric and decreasing around some point $x_0 \\in \\mathbb{R}^n$, i.e.~$u(x)=\\mathcal{V}_{\\lambda,x_0}(x)$ (with $p=2$). The authors used a refined version of the moving planes technique taking into account the lack of compactness in the whole space $\\R^n$, developed by themselves in bounded domains in \\cite{GNN}. The problem was completely solved in the seminal paper by Caffarelli, Gidas and Spruck \\cite{CGS}, where the authors classified all the solutions to \\eqref{eq:Sobolev} with $p=2$ and without an apriori finite energy assumption. Exploiting the Kelvin transformation, they showed that the moving planes procedure can start. Finally, they proved that any solution $u \\in H^1_{loc}(\\R^n)$ to \\eqref{eq:Sobolev} is of the form \\eqref{AubinTalenti}, hence solutions are unique up to rescaling and translations.\n\nThe situation in the case $p \\neq 2$ is much more involved, since the approach with Kelvin transformation is not applicable for this equation. Under the finite energy assumption, i.e.~$u \\in \\mathcal{D}^{1,p}(\\R^n)$, the classification of all the positive solutions to \\eqref{eq:Sobolev} has been in fact an open and challenging problem recently solved in a series of papers by Damascelli, Merch\\'an, Montoro and Sciunzi~\\cite{DMMS} ($2n/(n+2)p_n$ for some number $p_n\\in (n/3,(n+1)/3)$ such that $p_n\\sim n/3+ 1/n$.\n\nNow, we focus our attention on the case $\\gamma \\neq 0$ and $\\nu = 0$; in this setting, system \\eqref{P} reduces to the Hardy-Sobolev doubly critical equation\n\\begin{equation}\\label{eq:HardySobolev}\n\\begin{cases}\n-\\Delta_p u = \\gamma \\frac{u^{p-1}}{|x|^p} + u^{p^*-1} & \\text{in } \\mathbb{R}^n \\\\\nu > 0 & \\text{in } \\mathbb{R}^n \\setminus \\{0\\}.\n\\end{cases} \n\\end{equation}\nLet us now turn to the case $p=2$ so that $\\Lambda_{n,2}$ is the best constant in the Hardy-Sobolev inequality for $p=2$. The main contribution in the semilinear case is due to Terracini in \\cite{terracini}, where the author, by means of variational arguments and the concentration compactness principle, showed the existence of solutions to a more general version of the equation \\eqref{eq:HardySobolev}. Making use of the Kelvin transformation and of the moving plane technique, she was able to prove that any positive solution with finite energy to \\eqref{eq:HardySobolev} is radially symmetric about the origin. Finally, thanks to a detailed ODE's analysis, she gave a complete classification of the solutions to \\eqref{eq:HardySobolev}, given by\n\\begin{equation}\\label{eq:terraciniane}\n\t\\mathcal{U}_\\lambda(x)=\\lambda^{\\frac{2-n}{2}}\\mathcal{U}\\left(\\frac{x}{\\lambda}\\right) \\qquad \\mbox{ with } \\qquad \\mathcal{U}(x)= \\dfrac{A(n,\\gamma)}{|x|^{\\mu}\\left(1+|x|^{2-\\frac{4\\mu}{n-2}}\\right)^{\\frac{n-2}{2}}},\n\\end{equation}\nwhere \n\\begin{equation*}\n\t\\mu:=\\frac{n-2}{2}-\\sqrt{\\left( \\frac{n-2}{2}\\right)^2-\\gamma}, \\qquad A(n,\\gamma):=\\left(\\frac{n(n-2-2\\mu)^2}{n-2}\\right)^{\\frac{n-2}{4}},\n\\end{equation*}\nand $\\lambda>0$ is a scaling factor. Obviously, $\\mathcal{U}_\\lambda=\\mathcal{V}_{\\lambda,0}$ if $\\gamma=0$.\n\nThe case of $p$-Lapalce operator was firstly treated in \\cite{abdellaoui}, where the authors studied the existence of solutions to \\eqref{eq:HardySobolev} and a very fine ODE's analysis was carried out. The radial symmetry of the solutions there was an assumption. In \\cite{OSV} the authors proved that each solution to \\eqref{eq:HardySobolev} with finite energy is radially symmetric (and radially decreasing) about the origin. Having in force the radial symmetry of the solution it is easy to derive the ordinary differential equation associated to the PDE and fulfilled by the solution $u = u(r)$ and, hence, to apply the results in \\cite{abdellaoui}. In particular, they proved also uniqueness up to scaling of the \\emph{radial} solutions as well as the asymptotic behavior are proved showing in particular that, a \\emph{radial} solution \\eqref{eq:HardySobolev}, satisfy the following properties:\n\\begin{equation*}\n \\begin{split}\n &\\mathcal{U}_{p,\\lambda}(x)= \\lambda^\\frac{ p-n}{p} \\mathcal{U}_p\\left(\\frac{x}{\\lambda}\\right) \\quad \\text{ with } \\quad \\mathcal U_p(x)= u(r)\\\\\n &\\lim_{r\\to0^+} r^{\\mu_1} \\mathcal U_p(r)=C_1,\\qquad \\lim_{r\\to+\\infty}r^{\\mu_2} \\mathcal U_p(r)=C_2,\\\\\n &\\lim_{r\\to0^+} r^{\\mu_1+1} |\\mathcal U_p'(r)|=C_1\\mu_1,\\qquad \\lim_{r\\to+\\infty}r^{\\mu_2+1}|\\mathcal{U}_p'(r)|=C_2\\mu_2,\n \\end{split}\n\\end{equation*}\nfor some positive constants $C_1, C_2$. {Obviously, $\\mathcal{U}_{p,\\lambda}=\\mathcal{U}_{\\lambda}$ and $\\mathcal{U}_{p}=\\mathcal{U}$ if $p=2$.} Here and in the sequel $\\mu_1, \\mu_2\\in [0, +\\infty)$, $\\mu_1<\\mu_2$ are two roots of the equation\n\\begin{equation*}\n\\mu^{p-2}\\left[(p-1)\\mu^2-(n-p)\\mu\\right]+\\gamma=0.\n\\end{equation*}\nWe remark (for later use) that\n\\begin{equation*}\\label{relgammaj}\n0\\leqslant \\mu_1<\\frac{n-p}{p}<\\mu_2\\leqslant \\frac{n-p}{p-1}.\n\\end{equation*}\nNote that when $p=2$ then $\\mu_{1}=\\tau$ and $\\mu_2=n-2-\\tau$. Instead, when $p\\neq 2$ but $\\gamma=0$ then $\\mu_1=0$ and $\\mu_2=\\frac{n-p}{p-1}$.\\\\\n\nWe now turn our attention to the study of \\eqref{P}. Nonlinear Schrödinger problems, particularly those of Gross--Pitaevskii type, seems to have strong connections to various physical phenomena. These kind of systems appear naturally in the Hartree--Fock theory for double condensates specifically in binary mixtures of Bose Einstein condensates occupying different hyperfine states while spatially overlapping (see \\cite{Esry} and \\cite{Frantz} for comprehensive details). \nThe solitary-wave solutions to coupled Gross--Pitaevski equations satisfy the following system:\n\\begin{equation}\\label{eq:pBSsystem}\n\t\\begin{cases}\n\t\t-\\Delta u + V(x) u = \\mu u^{2q-1} + \\nu u^{q-1} v^{q} & \\text{in } \\mathbb{R}^n \\\\\n\t\t-\\Delta v + V(x) v = \\mu v^{2q-1} + \\nu u^{q}v^{q-1} & \\text{in } \\mathbb{R}^n,\n\t\\end{cases}\n\\end{equation}\nwhere $V$ represents the potential of the system and $1 < q \\leq \\frac{2^*}{2}$. This formulation is commonly referred to as the Bose--Einstein condensate system. For subcritical regimes, we refer to \\cite{ambrosetti}, \\cite{bartsch2}, \\cite{lin1}, \\cite{sirakov}, and \\cite{nicola2} for existence and multiplicity results under various assumptions on $V$ and $\\nu$. \n\nIn the critical case where $q = \\frac{2^*}{2}$, if $V$ is a non-zero constant, system \\eqref{eq:pBSsystem} admits only the trivial solution $(0,0)$, that is a consequence of the Pohozaev identity. When $V=0$, \\cite{Wang} demonstrated the uniqueness of ground states under specific parameter conditions for generalized systems. Conversely, \\cite{pistoia} examined the competitive setting ($\\nu < 0$), establishing the existence of infinitely many fully nontrivial solutions that are not conformally equivalent. \n\nIn \\cite{ELS}, the authors treated the case of Hardy-type potential $V = -\\frac{\\gamma}{|x|^2}$. With this choice of $V$, the authors were able to study problem \\eqref{P} when $p=2$. What they have proved in \\cite{ELS} is that any $(u,v) \\in \\mathcal{D}^{1,2}(\\mathbb{R}^n) \\times \\mathcal{D}^{1,2}(\\mathbb{R}^n)$ solution to \\eqref{P} is of synchronized type:\n\\begin{equation*}\n(u,v)=\\left(c_1 \\, \\mathcal{U}_{\\lambda_0}, c_2 \\, \\mathcal{U}_{\\lambda_0}\\right),\n\\end{equation*}\nwhere $\\mathcal{U}_\\lambda$ is given in \\eqref{eq:terraciniane}, $\\lambda_0>0$ and $c_1,c_2$ are any positive constants satisfying the system\n\\begin{equation*}\n\t\t\\left\\{\\begin{array}{ll}\n\t\t\t\\displaystyle{c_1^{2^*-2}+\\nu \\alpha c_1^{\\alpha-2}c_2^\\beta}=1 & \\vspace{.3cm}\\\\\n\t\t\t\\displaystyle{c_2^{2^*-2}+\\nu \\beta c_1^{\\alpha}c_2^{\\beta-2}=1}. &\n\t\t\\end{array}\\right.\n\\end{equation*}\nThis result was new also in the case $\\gamma=0$. Under that assumption the explicit solutions in \\eqref{eq:terraciniane} reduce to those of \\eqref{AubinTalenti}.\\\\ \n\nDoubly critical problems have received significant attention in recent years. The pioneering work of \\cite{AFP} investigated general Hardy-Sobolev type systems making use of variational techniques. In cooperative regimes ($\\nu > 0$), they established the existence of ground and bound states contingent on parameters $\\alpha$, $\\beta$, dimension $n$, and a potential function $h$ in the coupling term. We also refer to \\cite{ChenZou3, EduRafaAle} for further existence results in the critical regime.\n\nThe primary objective of this paper is to show that any positive solutions, with finite energy, to problem \\eqref{P} is radially symmetric and radially decreasing around the origin, in the same spirit of \\cite{ELS}.\n\nThus our main result is given by the following:", + "context": "Our main objective is to prove radial symmetry of solutions to system \\eqref{P}. When $\\gamma = \\nu = 0$, the system reduces to the standard $p$-Sobolev critical equation\n\\begin{equation}\\label{eq:Sobolev}\n \\begin{cases}\n \\displaystyle -\\Delta_p u\\,= u^{p^*-1} & \\text{in}\\quad\\R^n\\\\\n u > 0 & \\text{in}\\quad\\R^n.\n \\end{cases}\n\\end{equation}\nIt is well-known that the Aubin-Talenti bubbles, defined as\n\\begin{equation}\\label{AubinTalenti}\n\\mathcal{V}_{\\lambda,x_0}(x) := \\left[\\frac{\\lambda^{\\frac{1}{p-1}}\\sqrt[p]{n\\left(\\frac{n-p}{p-1}\\right)^{p-1}}}{\\lambda^{\\frac{p}{p-1}} + |x - x_0|^{\\frac{p}{p-1}}}\\right]^{\\frac{n-p}{p}}\n\\end{equation}\nis a family of positive radial solutions to \\eqref{eq:Sobolev}, with $\\lambda > 0$, and $x_0 \\in \\mathbb{R}^n$. These functions achieve equality in the Sobolev inequality in $\\mathbb{R}^n$. The situation in case $p=2$ is fully understood. In fact, in \\cite{GNN2}, it has been shown that any positive solution $u$ to \\eqref{eq:Sobolev} that satisfies the decay assumption $u(x) = \\mathcal{O}(1/|x|^m)$ at infinity for $m > 0$ is radially symmetric and decreasing around some point $x_0 \\in \\mathbb{R}^n$, i.e.~$u(x)=\\mathcal{V}_{\\lambda,x_0}(x)$ (with $p=2$). The authors used a refined version of the moving planes technique taking into account the lack of compactness in the whole space $\\R^n$, developed by themselves in bounded domains in \\cite{GNN}. The problem was completely solved in the seminal paper by Caffarelli, Gidas and Spruck \\cite{CGS}, where the authors classified all the solutions to \\eqref{eq:Sobolev} with $p=2$ and without an apriori finite energy assumption. Exploiting the Kelvin transformation, they showed that the moving planes procedure can start. Finally, they proved that any solution $u \\in H^1_{loc}(\\R^n)$ to \\eqref{eq:Sobolev} is of the form \\eqref{AubinTalenti}, hence solutions are unique up to rescaling and translations.\n\nNow, we focus our attention on the case $\\gamma \\neq 0$ and $\\nu = 0$; in this setting, system \\eqref{P} reduces to the Hardy-Sobolev doubly critical equation\n\\begin{equation}\\label{eq:HardySobolev}\n\\begin{cases}\n-\\Delta_p u = \\gamma \\frac{u^{p-1}}{|x|^p} + u^{p^*-1} & \\text{in } \\mathbb{R}^n \\\\\nu > 0 & \\text{in } \\mathbb{R}^n \\setminus \\{0\\}.\n\\end{cases} \n\\end{equation}\nLet us now turn to the case $p=2$ so that $\\Lambda_{n,2}$ is the best constant in the Hardy-Sobolev inequality for $p=2$. The main contribution in the semilinear case is due to Terracini in \\cite{terracini}, where the author, by means of variational arguments and the concentration compactness principle, showed the existence of solutions to a more general version of the equation \\eqref{eq:HardySobolev}. Making use of the Kelvin transformation and of the moving plane technique, she was able to prove that any positive solution with finite energy to \\eqref{eq:HardySobolev} is radially symmetric about the origin. Finally, thanks to a detailed ODE's analysis, she gave a complete classification of the solutions to \\eqref{eq:HardySobolev}, given by\n\\begin{equation}\\label{eq:terraciniane}\n \\mathcal{U}_\\lambda(x)=\\lambda^{\\frac{2-n}{2}}\\mathcal{U}\\left(\\frac{x}{\\lambda}\\right) \\qquad \\mbox{ with } \\qquad \\mathcal{U}(x)= \\dfrac{A(n,\\gamma)}{|x|^{\\mu}\\left(1+|x|^{2-\\frac{4\\mu}{n-2}}\\right)^{\\frac{n-2}{2}}},\n\\end{equation}\nwhere \n\\begin{equation*}\n \\mu:=\\frac{n-2}{2}-\\sqrt{\\left( \\frac{n-2}{2}\\right)^2-\\gamma}, \\qquad A(n,\\gamma):=\\left(\\frac{n(n-2-2\\mu)^2}{n-2}\\right)^{\\frac{n-2}{4}},\n\\end{equation*}\nand $\\lambda>0$ is a scaling factor. Obviously, $\\mathcal{U}_\\lambda=\\mathcal{V}_{\\lambda,0}$ if $\\gamma=0$.\n\nThe case of $p$-Lapalce operator was firstly treated in \\cite{abdellaoui}, where the authors studied the existence of solutions to \\eqref{eq:HardySobolev} and a very fine ODE's analysis was carried out. The radial symmetry of the solutions there was an assumption. In \\cite{OSV} the authors proved that each solution to \\eqref{eq:HardySobolev} with finite energy is radially symmetric (and radially decreasing) about the origin. Having in force the radial symmetry of the solution it is easy to derive the ordinary differential equation associated to the PDE and fulfilled by the solution $u = u(r)$ and, hence, to apply the results in \\cite{abdellaoui}. In particular, they proved also uniqueness up to scaling of the \\emph{radial} solutions as well as the asymptotic behavior are proved showing in particular that, a \\emph{radial} solution \\eqref{eq:HardySobolev}, satisfy the following properties:\n\\begin{equation*}\n \\begin{split}\n &\\mathcal{U}_{p,\\lambda}(x)= \\lambda^\\frac{ p-n}{p} \\mathcal{U}_p\\left(\\frac{x}{\\lambda}\\right) \\quad \\text{ with } \\quad \\mathcal U_p(x)= u(r)\\\\\n &\\lim_{r\\to0^+} r^{\\mu_1} \\mathcal U_p(r)=C_1,\\qquad \\lim_{r\\to+\\infty}r^{\\mu_2} \\mathcal U_p(r)=C_2,\\\\\n &\\lim_{r\\to0^+} r^{\\mu_1+1} |\\mathcal U_p'(r)|=C_1\\mu_1,\\qquad \\lim_{r\\to+\\infty}r^{\\mu_2+1}|\\mathcal{U}_p'(r)|=C_2\\mu_2,\n \\end{split}\n\\end{equation*}\nfor some positive constants $C_1, C_2$. {Obviously, $\\mathcal{U}_{p,\\lambda}=\\mathcal{U}_{\\lambda}$ and $\\mathcal{U}_{p}=\\mathcal{U}$ if $p=2$.} Here and in the sequel $\\mu_1, \\mu_2\\in [0, +\\infty)$, $\\mu_1<\\mu_2$ are two roots of the equation\n\\begin{equation*}\n\\mu^{p-2}\\left[(p-1)\\mu^2-(n-p)\\mu\\right]+\\gamma=0.\n\\end{equation*}\nWe remark (for later use) that\n\\begin{equation*}\\label{relgammaj}\n0\\leqslant \\mu_1<\\frac{n-p}{p}<\\mu_2\\leqslant \\frac{n-p}{p-1}.\n\\end{equation*}\nNote that when $p=2$ then $\\mu_{1}=\\tau$ and $\\mu_2=n-2-\\tau$. Instead, when $p\\neq 2$ but $\\gamma=0$ then $\\mu_1=0$ and $\\mu_2=\\frac{n-p}{p-1}$.\\\\\n\nThe primary objective of this paper is to show that any positive solutions, with finite energy, to problem \\eqref{P} is radially symmetric and radially decreasing around the origin, in the same spirit of \\cite{ELS}.\n\nThus our main result is given by the following:", + "full_context": "Our main objective is to prove radial symmetry of solutions to system \\eqref{P}. When $\\gamma = \\nu = 0$, the system reduces to the standard $p$-Sobolev critical equation\n\\begin{equation}\\label{eq:Sobolev}\n \\begin{cases}\n \\displaystyle -\\Delta_p u\\,= u^{p^*-1} & \\text{in}\\quad\\R^n\\\\\n u > 0 & \\text{in}\\quad\\R^n.\n \\end{cases}\n\\end{equation}\nIt is well-known that the Aubin-Talenti bubbles, defined as\n\\begin{equation}\\label{AubinTalenti}\n\\mathcal{V}_{\\lambda,x_0}(x) := \\left[\\frac{\\lambda^{\\frac{1}{p-1}}\\sqrt[p]{n\\left(\\frac{n-p}{p-1}\\right)^{p-1}}}{\\lambda^{\\frac{p}{p-1}} + |x - x_0|^{\\frac{p}{p-1}}}\\right]^{\\frac{n-p}{p}}\n\\end{equation}\nis a family of positive radial solutions to \\eqref{eq:Sobolev}, with $\\lambda > 0$, and $x_0 \\in \\mathbb{R}^n$. These functions achieve equality in the Sobolev inequality in $\\mathbb{R}^n$. The situation in case $p=2$ is fully understood. In fact, in \\cite{GNN2}, it has been shown that any positive solution $u$ to \\eqref{eq:Sobolev} that satisfies the decay assumption $u(x) = \\mathcal{O}(1/|x|^m)$ at infinity for $m > 0$ is radially symmetric and decreasing around some point $x_0 \\in \\mathbb{R}^n$, i.e.~$u(x)=\\mathcal{V}_{\\lambda,x_0}(x)$ (with $p=2$). The authors used a refined version of the moving planes technique taking into account the lack of compactness in the whole space $\\R^n$, developed by themselves in bounded domains in \\cite{GNN}. The problem was completely solved in the seminal paper by Caffarelli, Gidas and Spruck \\cite{CGS}, where the authors classified all the solutions to \\eqref{eq:Sobolev} with $p=2$ and without an apriori finite energy assumption. Exploiting the Kelvin transformation, they showed that the moving planes procedure can start. Finally, they proved that any solution $u \\in H^1_{loc}(\\R^n)$ to \\eqref{eq:Sobolev} is of the form \\eqref{AubinTalenti}, hence solutions are unique up to rescaling and translations.\n\nNow, we focus our attention on the case $\\gamma \\neq 0$ and $\\nu = 0$; in this setting, system \\eqref{P} reduces to the Hardy-Sobolev doubly critical equation\n\\begin{equation}\\label{eq:HardySobolev}\n\\begin{cases}\n-\\Delta_p u = \\gamma \\frac{u^{p-1}}{|x|^p} + u^{p^*-1} & \\text{in } \\mathbb{R}^n \\\\\nu > 0 & \\text{in } \\mathbb{R}^n \\setminus \\{0\\}.\n\\end{cases} \n\\end{equation}\nLet us now turn to the case $p=2$ so that $\\Lambda_{n,2}$ is the best constant in the Hardy-Sobolev inequality for $p=2$. The main contribution in the semilinear case is due to Terracini in \\cite{terracini}, where the author, by means of variational arguments and the concentration compactness principle, showed the existence of solutions to a more general version of the equation \\eqref{eq:HardySobolev}. Making use of the Kelvin transformation and of the moving plane technique, she was able to prove that any positive solution with finite energy to \\eqref{eq:HardySobolev} is radially symmetric about the origin. Finally, thanks to a detailed ODE's analysis, she gave a complete classification of the solutions to \\eqref{eq:HardySobolev}, given by\n\\begin{equation}\\label{eq:terraciniane}\n \\mathcal{U}_\\lambda(x)=\\lambda^{\\frac{2-n}{2}}\\mathcal{U}\\left(\\frac{x}{\\lambda}\\right) \\qquad \\mbox{ with } \\qquad \\mathcal{U}(x)= \\dfrac{A(n,\\gamma)}{|x|^{\\mu}\\left(1+|x|^{2-\\frac{4\\mu}{n-2}}\\right)^{\\frac{n-2}{2}}},\n\\end{equation}\nwhere \n\\begin{equation*}\n \\mu:=\\frac{n-2}{2}-\\sqrt{\\left( \\frac{n-2}{2}\\right)^2-\\gamma}, \\qquad A(n,\\gamma):=\\left(\\frac{n(n-2-2\\mu)^2}{n-2}\\right)^{\\frac{n-2}{4}},\n\\end{equation*}\nand $\\lambda>0$ is a scaling factor. Obviously, $\\mathcal{U}_\\lambda=\\mathcal{V}_{\\lambda,0}$ if $\\gamma=0$.\n\nThe case of $p$-Lapalce operator was firstly treated in \\cite{abdellaoui}, where the authors studied the existence of solutions to \\eqref{eq:HardySobolev} and a very fine ODE's analysis was carried out. The radial symmetry of the solutions there was an assumption. In \\cite{OSV} the authors proved that each solution to \\eqref{eq:HardySobolev} with finite energy is radially symmetric (and radially decreasing) about the origin. Having in force the radial symmetry of the solution it is easy to derive the ordinary differential equation associated to the PDE and fulfilled by the solution $u = u(r)$ and, hence, to apply the results in \\cite{abdellaoui}. In particular, they proved also uniqueness up to scaling of the \\emph{radial} solutions as well as the asymptotic behavior are proved showing in particular that, a \\emph{radial} solution \\eqref{eq:HardySobolev}, satisfy the following properties:\n\\begin{equation*}\n \\begin{split}\n &\\mathcal{U}_{p,\\lambda}(x)= \\lambda^\\frac{ p-n}{p} \\mathcal{U}_p\\left(\\frac{x}{\\lambda}\\right) \\quad \\text{ with } \\quad \\mathcal U_p(x)= u(r)\\\\\n &\\lim_{r\\to0^+} r^{\\mu_1} \\mathcal U_p(r)=C_1,\\qquad \\lim_{r\\to+\\infty}r^{\\mu_2} \\mathcal U_p(r)=C_2,\\\\\n &\\lim_{r\\to0^+} r^{\\mu_1+1} |\\mathcal U_p'(r)|=C_1\\mu_1,\\qquad \\lim_{r\\to+\\infty}r^{\\mu_2+1}|\\mathcal{U}_p'(r)|=C_2\\mu_2,\n \\end{split}\n\\end{equation*}\nfor some positive constants $C_1, C_2$. {Obviously, $\\mathcal{U}_{p,\\lambda}=\\mathcal{U}_{\\lambda}$ and $\\mathcal{U}_{p}=\\mathcal{U}$ if $p=2$.} Here and in the sequel $\\mu_1, \\mu_2\\in [0, +\\infty)$, $\\mu_1<\\mu_2$ are two roots of the equation\n\\begin{equation*}\n\\mu^{p-2}\\left[(p-1)\\mu^2-(n-p)\\mu\\right]+\\gamma=0.\n\\end{equation*}\nWe remark (for later use) that\n\\begin{equation*}\\label{relgammaj}\n0\\leqslant \\mu_1<\\frac{n-p}{p}<\\mu_2\\leqslant \\frac{n-p}{p-1}.\n\\end{equation*}\nNote that when $p=2$ then $\\mu_{1}=\\tau$ and $\\mu_2=n-2-\\tau$. Instead, when $p\\neq 2$ but $\\gamma=0$ then $\\mu_1=0$ and $\\mu_2=\\frac{n-p}{p-1}$.\\\\\n\nThe primary objective of this paper is to show that any positive solutions, with finite energy, to problem \\eqref{P} is radially symmetric and radially decreasing around the origin, in the same spirit of \\cite{ELS}.\n\nThus our main result is given by the following:\n\nNow, we focus our attention on the case $\\gamma \\neq 0$ and $\\nu = 0$; in this setting, system \\eqref{P} reduces to the Hardy-Sobolev doubly critical equation\n\\begin{equation}\\label{eq:HardySobolev}\n\\begin{cases}\n-\\Delta_p u = \\gamma \\frac{u^{p-1}}{|x|^p} + u^{p^*-1} & \\text{in } \\mathbb{R}^n \\\\\nu > 0 & \\text{in } \\mathbb{R}^n \\setminus \\{0\\}.\n\\end{cases} \n\\end{equation}\nLet us now turn to the case $p=2$ so that $\\Lambda_{n,2}$ is the best constant in the Hardy-Sobolev inequality for $p=2$. The main contribution in the semilinear case is due to Terracini in \\cite{terracini}, where the author, by means of variational arguments and the concentration compactness principle, showed the existence of solutions to a more general version of the equation \\eqref{eq:HardySobolev}. Making use of the Kelvin transformation and of the moving plane technique, she was able to prove that any positive solution with finite energy to \\eqref{eq:HardySobolev} is radially symmetric about the origin. Finally, thanks to a detailed ODE's analysis, she gave a complete classification of the solutions to \\eqref{eq:HardySobolev}, given by\n\\begin{equation}\\label{eq:terraciniane}\n \\mathcal{U}_\\lambda(x)=\\lambda^{\\frac{2-n}{2}}\\mathcal{U}\\left(\\frac{x}{\\lambda}\\right) \\qquad \\mbox{ with } \\qquad \\mathcal{U}(x)= \\dfrac{A(n,\\gamma)}{|x|^{\\mu}\\left(1+|x|^{2-\\frac{4\\mu}{n-2}}\\right)^{\\frac{n-2}{2}}},\n\\end{equation}\nwhere \n\\begin{equation*}\n \\mu:=\\frac{n-2}{2}-\\sqrt{\\left( \\frac{n-2}{2}\\right)^2-\\gamma}, \\qquad A(n,\\gamma):=\\left(\\frac{n(n-2-2\\mu)^2}{n-2}\\right)^{\\frac{n-2}{4}},\n\\end{equation*}\nand $\\lambda>0$ is a scaling factor. Obviously, $\\mathcal{U}_\\lambda=\\mathcal{V}_{\\lambda,0}$ if $\\gamma=0$.\n\nThe case of $p$-Lapalce operator was firstly treated in \\cite{abdellaoui}, where the authors studied the existence of solutions to \\eqref{eq:HardySobolev} and a very fine ODE's analysis was carried out. The radial symmetry of the solutions there was an assumption. In \\cite{OSV} the authors proved that each solution to \\eqref{eq:HardySobolev} with finite energy is radially symmetric (and radially decreasing) about the origin. Having in force the radial symmetry of the solution it is easy to derive the ordinary differential equation associated to the PDE and fulfilled by the solution $u = u(r)$ and, hence, to apply the results in \\cite{abdellaoui}. In particular, they proved also uniqueness up to scaling of the \\emph{radial} solutions as well as the asymptotic behavior are proved showing in particular that, a \\emph{radial} solution \\eqref{eq:HardySobolev}, satisfy the following properties:\n\\begin{equation*}\n \\begin{split}\n &\\mathcal{U}_{p,\\lambda}(x)= \\lambda^\\frac{ p-n}{p} \\mathcal{U}_p\\left(\\frac{x}{\\lambda}\\right) \\quad \\text{ with } \\quad \\mathcal U_p(x)= u(r)\\\\\n &\\lim_{r\\to0^+} r^{\\mu_1} \\mathcal U_p(r)=C_1,\\qquad \\lim_{r\\to+\\infty}r^{\\mu_2} \\mathcal U_p(r)=C_2,\\\\\n &\\lim_{r\\to0^+} r^{\\mu_1+1} |\\mathcal U_p'(r)|=C_1\\mu_1,\\qquad \\lim_{r\\to+\\infty}r^{\\mu_2+1}|\\mathcal{U}_p'(r)|=C_2\\mu_2,\n \\end{split}\n\\end{equation*}\nfor some positive constants $C_1, C_2$. {Obviously, $\\mathcal{U}_{p,\\lambda}=\\mathcal{U}_{\\lambda}$ and $\\mathcal{U}_{p}=\\mathcal{U}$ if $p=2$.} Here and in the sequel $\\mu_1, \\mu_2\\in [0, +\\infty)$, $\\mu_1<\\mu_2$ are two roots of the equation\n\\begin{equation*}\n\\mu^{p-2}\\left[(p-1)\\mu^2-(n-p)\\mu\\right]+\\gamma=0.\n\\end{equation*}\nWe remark (for later use) that\n\\begin{equation*}\\label{relgammaj}\n0\\le \\mu_1<\\frac{n-p}{p}<\\mu_2\\le \\frac{n-p}{p-1}.\n\\end{equation*}\nNote that when $p=2$ then $\\mu_{1}=\\tau$ and $\\mu_2=n-2-\\tau$. Instead, when $p\\neq 2$ but $\\gamma=0$ then $\\mu_1=0$ and $\\mu_2=\\frac{n-p}{p-1}$.\\\\\n\nThe primary objective of this paper is to show that any positive solutions, with finite energy, to problem \\eqref{P} is radially symmetric and radially decreasing around the origin, in the same spirit of \\cite{ELS}.\n\n\\begin{rem}\nThis result holds (and is new) also in the case $\\gamma=0$. \n\\end{rem}\n\nNow, we state the upper and lower estimates of the solutions and the upper one of their gradient, contained in \\cite{KL20}, in our case.\n\\begin{thm} \\label{thm:asymptEst}$($Theorem $1.1$, \\cite{KL20}$)$\n Let $(u,v) \\in \\mathcal{D}^{1,p}(\\R^n)\\times \\mathcal{D}^{1,p}(\\R^n)$ be a weak solution of \\eqref{P}\n with $1 0 & \\text{in}\\quad\\R^n \\setminus \\{0\\}\\\\\nu,v \\in \\mathcal D^{1,p}(\\R^n),\n\\end{cases}\n\\end{equation}\nThen $(u,v)$ is radially symmetric and radially decreasing around the origin.,", + "theorem_type": [ + "Implication", + "Universal" + ], + "mcq": { + "question": "Let $10 & \\text{in }\\mathbb R^n\\setminus\\{0\\}.\n\\end{cases}\n\\]\nWhich statement holds for every such solution pair $(u,v)$?", + "correct_choice": { + "label": "A", + "text": "Both components are radially symmetric about the origin and radially decreasing; equivalently, there exist functions $U,V:(0,\\infty)\\to(0,\\infty)$ such that $u(x)=U(|x|)$ and $v(x)=V(|x|)$ for all $x\\neq 0$, and $U$ and $V$ decrease as $|x|$ increases." + }, + "choices": [ + { + "label": "B", + "text": "Both components are radially symmetric and radially decreasing with respect to the same center $x_0\\in\\mathbb R^n$; equivalently, there exist $x_0\\in\\mathbb R^n$ and functions $U,V:(0,\\infty)\\to(0,\\infty)$ such that $u(x)=U(|x-x_0|)$ and $v(x)=V(|x-x_0|)$ for all $x\\neq x_0$, and $U$ and $V$ decrease as $|x-x_0|$ increases." + }, + { + "label": "C", + "text": "Both components are radially symmetric about the origin; equivalently, there exist functions $U,V:(0,\\infty)\\to(0,\\infty)$ such that $u(x)=U(|x|)$ and $v(x)=V(|x|)$ for all $x\\neq 0$." + }, + { + "label": "D", + "text": "There exist positive constants $R_0<10$, depending only on $n,p,\\gamma,\\nu,\\alpha,\\beta$, such that $|\\nabla u(x)|+|\\nabla v(x)|\\le C\\big(|x|^{-\\mu_1-1}+|x|^{-\\mu_2-1}\\big)$ for every $x\\in\\mathbb R^n\\setminus\\{0\\}$; in particular, both are radially decreasing." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "geometric_construction", + "tampered_component": "forced center at the origin due to the Hardy singularity", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "geometric_construction", + "tampered_component": "dropped radial monotonicity while keeping symmetry about the origin", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "geometric_construction", + "tampered_component": "global moving-plane conclusion replaced by only local symmetry near 0 and infinity", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "solution-dependent asymptotic/gradient constants made uniform and global", + "template_used": "uniformity_effectivity" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not state the symmetry/monotonicity conclusion, nor does it directly signal which option is correct. The Hardy singularity at the origin is relevant context, but it is not an explicit giveaway." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question: it asks for the exact qualitative conclusion satisfied by every solution pair. However, it is not a pure restatement, since the options vary in center of symmetry, monotonicity, locality vs. globality, and extra quantitative estimates." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is required to distinguish the strongest valid conclusion from nearby alternatives, especially between origin-centered symmetry, arbitrary-center symmetry, and symmetry without monotonicity. Still, for someone who knows the result, the task is mainly recognition rather than deep derivation." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically targeted: one weakens the true claim, one shifts the symmetry center, one reduces global to local symmetry, and one adds an unjustified uniform gradient bound. These align with realistic failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-understanding MCQ with strong distractors and little answer leakage, though it leans more toward theorem recognition than genuinely generative reasoning." + } + }, + { + "id": "2511.05384v1", + "paper_link": "http://arxiv.org/abs/2511.05384v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm_main}(Unique determination)\nLet $\\Omega\\subset\\mathbb{R}^n$ be a bounded open set with smooth \nboundary $\\partial\\Omega$. Let $s\\in \\mathbb{R}^+\\setminus \\mathbb{Z}$, $\\lfloor s \\rfloor>\\max\\{m,\\,n/2\\}$.\nSuppose that $q_j\\in L^\\infty(\\Omega)$ satisfies $q_j\\geq 0$ in $\\Omega$ for $j=1,\\,2$.\nLet $\\mathbf{P}_1$ and $\\mathbf{P}_2$ be nonlinearities of the form \\eqref{DEF:nonlinear P} with coefficients $a_{\\sigma,k}^{(1)},\\, a_{\\sigma,k}^{(2)}(x) \\in C (\\overline\\Omega)$. Let $W_1,\\, W_2\\subset\\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\\Lambda_{\\mathbf{P}_j}: \\mathcal{X}_{\\varepsilon_0}(W_1)\\rightarrow (H^{-s}(\\Omega_e))^*$ satisfy\n\\[\n\\Lambda_{\\mathbf{P}_1}f|_{W_2} = \\Lambda_{\\mathbf{P}_2}f|_{W_2} \\quad \\mbox{ for all } f \\in \\mathcal{X}_{\\varepsilon_0}(W_1), \n\\]\nthen for all multi-indices $\\sigma$ such that $0\\le |\\sigma|\\le m$ and for all $k=1,\\ldots, K-1$ it holds\n$$\t \nq_1=q_2\\quad\\hbox{and}\\quad\ta_{\\sigma,k}^{(1)} = a_{\\sigma,k}^{(2)} \\quad \\hbox{ in }\\Omega,\n$$ \nand thus the nonlinearities $\\mathbf P_1$ and $\\mathbf P_2$ coincide.", + "start_pos": 374900, + "end_pos": 375985, + "label": "thm_main" + }, + "ref_dict": { + "eq_1_IBVP": "\\begin{equation}\\label{eq_1_IBVP}\n\\begin{cases}\n(-\\Delta)^su + q(x)u +\\mathbf{P}(u)= 0 &\\hbox{ in } \\Omega,\\\\\nu = f &\\hbox{ in } \\Omega_e,\n\\end{cases}\n\\end{equation}", + "DEF:nonlinear P": "\\begin{align}\\label{DEF:nonlinear P}\n\\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\\ldots+u^{K-1}P_{K-1}(x,D)u,\n\\end{align}", + "thm_main": "\\begin{theorem}\\label{thm_main}(Unique determination)\nLet $\\Omega\\subset\\R^n$ be a bounded open set with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$, $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q_j\\in L^\\infty(\\Omega)$ satisfies $q_j\\geq 0$ in $\\Omega$ for $j=1,\\,2$.\nLet $\\mathbf{P}_1$ and $\\mathbf{P}_2$ be nonlinearities of the form \\eqref{DEF:nonlinear P} with coefficients $a_{\\sigma,k}^{(1)},\\, a_{\\sigma,k}^{(2)}(x) \\in C (\\overline\\Omega)$. Let $W_1,\\, W_2\\subset\\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\\Lambda_{\\mathbf{P}_j}: \\mathcal{X}_{\\varepsilon_0}(W_1)\\rightarrow (H^{-s}(\\Omega_e))^*$ satisfy\n\\[\n\\Lambda_{\\mathbf{P}_1}f|_{W_2} = \\Lambda_{\\mathbf{P}_2}f|_{W_2} \\quad \\mbox{ for all } f \\in \\mathcal{X}_{\\varepsilon_0}(W_1), \n\\]\nthen for all multi-indices $\\sigma$ such that $0\\le |\\sigma|\\le m$ and for all $k=1,\\ldots, K-1$ it holds\n$$\t \nq_1=q_2\\quad\\hbox{and}\\quad\ta_{\\sigma,k}^{(1)} = a_{\\sigma,k}^{(2)} \\quad \\hbox{ in }\\Omega,\n$$ \nand thus the nonlinearities $\\mathbf P_1$ and $\\mathbf P_2$ coincide.\n\\end{theorem}", + "thm_wellposdness": "\\begin{theorem}[Well-posedness for the nonlinear equation]\n\\label{thm_wellposdness}\nLet $\\Omega\\subset\\R^n$ be a bounded open domain with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$ and $m\\in \\mathbb{N}$ satisfy $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q\\in L^\\infty(\\Omega)$ satisfies $q\\geq 0$ in $\\Omega$, and the coefficients in $\\mathbf{P}(u)$ satisfy $a_{\\sigma,k}\\in C (\\overline\\Omega)$. \nThen there exists a sufficiently small $\\varepsilon_0>0$ such that for any $f\\in \\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$,\nthe Dirichlet problem \\eqref{eq_1_IBVP} has a unique solution $u\\in H^s(\\R^n)\\cap C^s(\\R^n)$. Moreover, there exists a constant $C>0$, independent of $u$ and $f$, such that \n\\[\n\\|u\\|_{C^{s}(\\R^n)}+ \\|u\\|_{H^{s}(\\R^n)}\\le C\\|f\\|_{C^\\infty_c(\\Omega_e)}.\n\\]\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 2434, + "pre_theorem_intro_text": "Let $\\Omega$ be a bounded open set in $\\mathbb{R}^n$, $n\\geq 1$, with smooth boundary $\\partial\\Omega$, and \n$\\Omega_e := \\mathbb{R}^n\\setminus \\overline\\Omega$ be the exterior of $\\Omega$. \nLet $m\\geq 0$ be an integer and $s\\in \\mathbb{R}^+\\setminus \\mathbb{Z}$, where $\\mathbb{R}^+:=(0,\\infty)$ and $\\mathbb{Z}$ denotes the set of all integers in $\\mathbb{R}$. \nWe consider inverse problems for the nonlinear fractional Schr\\\"odinger equation (NLFSE):\n\\begin{equation}\\label{eq_1_IBVP}\n\\begin{cases}\n(-\\Delta)^su + q(x)u +\\mathbf{P}(u)= 0 &\\hbox{ in } \\Omega,\\\\\nu = f &\\hbox{ in } \\Omega_e,\n\\end{cases}\n\\end{equation}\nwhere the fractional Laplacian $(-\\Delta)^s$ is the nonlocal pseudo-differential operator defined by $(-\\Delta)^su: = \\mathcal{F}^{-1}(|\\cdot|^{2s}\\mathcal{F}(u))$. Here $\\mathcal{F} (u)$ represents the Fourier transform of $u$. \nThe local nonlinear perturbation $\\mathbf{P} (u)$ is defined as \n\\begin{align}\\label{DEF:nonlinear P}\n\\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\\ldots+u^{K-1}P_{K-1}(x,D)u,\n\\end{align}\nwhere $K\\geq 2$ is an integer, and the linear differential operators $P_k(x,D)$ of order $m$ are of the form\n$$\nP_k(x,D):=\\sum_{\\sigma: \\,|\\sigma|\\leq m}a_{\\sigma,k}(x)D^\\sigma,\\quad k=1,\\ldots,K-1,\n$$\nwith scalar coefficients $a_{\\sigma,k}$ depending only on $x$. Here $\\sigma=(\\sigma_1,\\ldots,\\sigma_n)$ is multi-index of length $|\\sigma|=\\sigma_1+\\ldots+\\sigma_n=\\ell$ for $\\ell\\leq m$, and therefore $D^\\sigma u := \\frac{\\partial^{\\sigma_1}}{\\partial x_1^{\\sigma_1}}\\ldots\\frac{\\partial^{\\sigma_n}}{\\partial x_n^{\\sigma_n}} u$. \\\\\n\nWe show in Theorem \\ref{thm_wellposdness} that \\eqref{eq_1_IBVP} is well-posed in $H^s(\\mathbb{R}^n)$ for any data $f$ in a small ball $\\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$ in $C^\\infty_c(\\Omega_e)$, where for any open set $O\\subset \\mathbb R^n$ we define \n\\[\n\\mathcal{X}_{\\varepsilon_0}(O):=\\{f\\in C^\\infty_c(O):\\,\\|f\\|_{C^\\infty_c(O)}\\le\\varepsilon_0\\}.\n\\]\nThis well-posedness result for \\eqref{eq_1_IBVP} allows us to define the corresponding Dirichlet-to-Neumann (DN) map \n$$\n\\Lambda_{\\mathbf{P}}: H^{s}(\\Omega_e)\\rightarrow (H^{-s}(\\Omega_e))^*,\\quad f\\mapsto (-\\Delta)^s u_f|_{\\Omega_e}.\n$$\n\nIn this framework, we ask the following inverse problem: Does the map $\\Lambda_{\\mathbf{P}}$ uniquely determine the nonlinearity $\\mathbf P$? The answer is affirmative, and we give a full description of the results in the following Theorem~\\ref{thm_main}:", + "context": "Let $\\Omega$ be a bounded open set in $\\mathbb{R}^n$, $n\\geq 1$, with smooth boundary $\\partial\\Omega$, and \n$\\Omega_e := \\mathbb{R}^n\\setminus \\overline\\Omega$ be the exterior of $\\Omega$. \nLet $m\\geq 0$ be an integer and $s\\in \\mathbb{R}^+\\setminus \\mathbb{Z}$, where $\\mathbb{R}^+:=(0,\\infty)$ and $\\mathbb{Z}$ denotes the set of all integers in $\\mathbb{R}$. \nWe consider inverse problems for the nonlinear fractional Schr\\\"odinger equation (NLFSE):\n\\begin{equation}\\label{eq_1_IBVP}\n\\begin{cases}\n(-\\Delta)^su + q(x)u +\\mathbf{P}(u)= 0 &\\hbox{ in } \\Omega,\\\\\nu = f &\\hbox{ in } \\Omega_e,\n\\end{cases}\n\\end{equation}\nwhere the fractional Laplacian $(-\\Delta)^s$ is the nonlocal pseudo-differential operator defined by $(-\\Delta)^su: = \\mathcal{F}^{-1}(|\\cdot|^{2s}\\mathcal{F}(u))$. Here $\\mathcal{F} (u)$ represents the Fourier transform of $u$. \nThe local nonlinear perturbation $\\mathbf{P} (u)$ is defined as \n\\begin{align}\\label{DEF:nonlinear P}\n\\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\\ldots+u^{K-1}P_{K-1}(x,D)u,\n\\end{align}\nwhere $K\\geq 2$ is an integer, and the linear differential operators $P_k(x,D)$ of order $m$ are of the form\n$$\nP_k(x,D):=\\sum_{\\sigma: \\,|\\sigma|\\leq m}a_{\\sigma,k}(x)D^\\sigma,\\quad k=1,\\ldots,K-1,\n$$\nwith scalar coefficients $a_{\\sigma,k}$ depending only on $x$. Here $\\sigma=(\\sigma_1,\\ldots,\\sigma_n)$ is multi-index of length $|\\sigma|=\\sigma_1+\\ldots+\\sigma_n=\\ell$ for $\\ell\\leq m$, and therefore $D^\\sigma u := \\frac{\\partial^{\\sigma_1}}{\\partial x_1^{\\sigma_1}}\\ldots\\frac{\\partial^{\\sigma_n}}{\\partial x_n^{\\sigma_n}} u$. \\\\\n\nWe show in Theorem \\ref{thm_wellposdness} that \\eqref{eq_1_IBVP} is well-posed in $H^s(\\mathbb{R}^n)$ for any data $f$ in a small ball $\\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$ in $C^\\infty_c(\\Omega_e)$, where for any open set $O\\subset \\mathbb R^n$ we define \n\\[\n\\mathcal{X}_{\\varepsilon_0}(O):=\\{f\\in C^\\infty_c(O):\\,\\|f\\|_{C^\\infty_c(O)}\\le\\varepsilon_0\\}.\n\\]\nThis well-posedness result for \\eqref{eq_1_IBVP} allows us to define the corresponding Dirichlet-to-Neumann (DN) map \n$$\n\\Lambda_{\\mathbf{P}}: H^{s}(\\Omega_e)\\rightarrow (H^{-s}(\\Omega_e))^*,\\quad f\\mapsto (-\\Delta)^s u_f|_{\\Omega_e}.\n$$\n\nIn this framework, we ask the following inverse problem: Does the map $\\Lambda_{\\mathbf{P}}$ uniquely determine the nonlinearity $\\mathbf P$? The answer is affirmative, and we give a full description of the results in the following Theorem~\\ref{thm_main}:\n\n\\begin{theorem}\\label{thm_main}(Unique determination)\nLet $\\Omega\\subset\\R^n$ be a bounded open set with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$, $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q_j\\in L^\\infty(\\Omega)$ satisfies $q_j\\geq 0$ in $\\Omega$ for $j=1,\\,2$.\nLet $\\mathbf{P}_1$ and $\\mathbf{P}_2$ be nonlinearities of the form \\eqref{DEF:nonlinear P} with coefficients $a_{\\sigma,k}^{(1)},\\, a_{\\sigma,k}^{(2)}(x) \\in C (\\overline\\Omega)$. Let $W_1,\\, W_2\\subset\\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\\Lambda_{\\mathbf{P}_j}: \\mathcal{X}_{\\varepsilon_0}(W_1)\\rightarrow (H^{-s}(\\Omega_e))^*$ satisfy\n\\[\n\\Lambda_{\\mathbf{P}_1}f|_{W_2} = \\Lambda_{\\mathbf{P}_2}f|_{W_2} \\quad \\mbox{ for all } f \\in \\mathcal{X}_{\\varepsilon_0}(W_1), \n\\]\nthen for all multi-indices $\\sigma$ such that $0\\le |\\sigma|\\le m$ and for all $k=1,\\ldots, K-1$ it holds\n$$\t \nq_1=q_2\\quad\\hbox{and}\\quad\ta_{\\sigma,k}^{(1)} = a_{\\sigma,k}^{(2)} \\quad \\hbox{ in }\\Omega,\n$$ \nand thus the nonlinearities $\\mathbf P_1$ and $\\mathbf P_2$ coincide.\n\\end{theorem}\n\n\\begin{theorem}[Well-posedness for the nonlinear equation]\n\\label{thm_wellposdness}\nLet $\\Omega\\subset\\R^n$ be a bounded open domain with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$ and $m\\in \\mathbb{N}$ satisfy $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q\\in L^\\infty(\\Omega)$ satisfies $q\\geq 0$ in $\\Omega$, and the coefficients in $\\mathbf{P}(u)$ satisfy $a_{\\sigma,k}\\in C (\\overline\\Omega)$. \nThen there exists a sufficiently small $\\varepsilon_0>0$ such that for any $f\\in \\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$,\nthe Dirichlet problem \\eqref{eq_1_IBVP} has a unique solution $u\\in H^s(\\R^n)\\cap C^s(\\R^n)$. Moreover, there exists a constant $C>0$, independent of $u$ and $f$, such that \n\\[\n\\|u\\|_{C^{s}(\\R^n)}+ \\|u\\|_{H^{s}(\\R^n)}\\le C\\|f\\|_{C^\\infty_c(\\Omega_e)}.\n\\]\n\\end{theorem}", + "full_context": "Let $\\Omega$ be a bounded open set in $\\mathbb{R}^n$, $n\\geq 1$, with smooth boundary $\\partial\\Omega$, and \n$\\Omega_e := \\mathbb{R}^n\\setminus \\overline\\Omega$ be the exterior of $\\Omega$. \nLet $m\\geq 0$ be an integer and $s\\in \\mathbb{R}^+\\setminus \\mathbb{Z}$, where $\\mathbb{R}^+:=(0,\\infty)$ and $\\mathbb{Z}$ denotes the set of all integers in $\\mathbb{R}$. \nWe consider inverse problems for the nonlinear fractional Schr\\\"odinger equation (NLFSE):\n\\begin{equation}\\label{eq_1_IBVP}\n\\begin{cases}\n(-\\Delta)^su + q(x)u +\\mathbf{P}(u)= 0 &\\hbox{ in } \\Omega,\\\\\nu = f &\\hbox{ in } \\Omega_e,\n\\end{cases}\n\\end{equation}\nwhere the fractional Laplacian $(-\\Delta)^s$ is the nonlocal pseudo-differential operator defined by $(-\\Delta)^su: = \\mathcal{F}^{-1}(|\\cdot|^{2s}\\mathcal{F}(u))$. Here $\\mathcal{F} (u)$ represents the Fourier transform of $u$. \nThe local nonlinear perturbation $\\mathbf{P} (u)$ is defined as \n\\begin{align}\\label{DEF:nonlinear P}\n\\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\\ldots+u^{K-1}P_{K-1}(x,D)u,\n\\end{align}\nwhere $K\\geq 2$ is an integer, and the linear differential operators $P_k(x,D)$ of order $m$ are of the form\n$$\nP_k(x,D):=\\sum_{\\sigma: \\,|\\sigma|\\leq m}a_{\\sigma,k}(x)D^\\sigma,\\quad k=1,\\ldots,K-1,\n$$\nwith scalar coefficients $a_{\\sigma,k}$ depending only on $x$. Here $\\sigma=(\\sigma_1,\\ldots,\\sigma_n)$ is multi-index of length $|\\sigma|=\\sigma_1+\\ldots+\\sigma_n=\\ell$ for $\\ell\\leq m$, and therefore $D^\\sigma u := \\frac{\\partial^{\\sigma_1}}{\\partial x_1^{\\sigma_1}}\\ldots\\frac{\\partial^{\\sigma_n}}{\\partial x_n^{\\sigma_n}} u$. \\\\\n\nWe show in Theorem \\ref{thm_wellposdness} that \\eqref{eq_1_IBVP} is well-posed in $H^s(\\mathbb{R}^n)$ for any data $f$ in a small ball $\\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$ in $C^\\infty_c(\\Omega_e)$, where for any open set $O\\subset \\mathbb R^n$ we define \n\\[\n\\mathcal{X}_{\\varepsilon_0}(O):=\\{f\\in C^\\infty_c(O):\\,\\|f\\|_{C^\\infty_c(O)}\\le\\varepsilon_0\\}.\n\\]\nThis well-posedness result for \\eqref{eq_1_IBVP} allows us to define the corresponding Dirichlet-to-Neumann (DN) map \n$$\n\\Lambda_{\\mathbf{P}}: H^{s}(\\Omega_e)\\rightarrow (H^{-s}(\\Omega_e))^*,\\quad f\\mapsto (-\\Delta)^s u_f|_{\\Omega_e}.\n$$\n\nIn this framework, we ask the following inverse problem: Does the map $\\Lambda_{\\mathbf{P}}$ uniquely determine the nonlinearity $\\mathbf P$? The answer is affirmative, and we give a full description of the results in the following Theorem~\\ref{thm_main}:\n\n\\begin{theorem}\\label{thm_main}(Unique determination)\nLet $\\Omega\\subset\\R^n$ be a bounded open set with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$, $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q_j\\in L^\\infty(\\Omega)$ satisfies $q_j\\geq 0$ in $\\Omega$ for $j=1,\\,2$.\nLet $\\mathbf{P}_1$ and $\\mathbf{P}_2$ be nonlinearities of the form \\eqref{DEF:nonlinear P} with coefficients $a_{\\sigma,k}^{(1)},\\, a_{\\sigma,k}^{(2)}(x) \\in C (\\overline\\Omega)$. Let $W_1,\\, W_2\\subset\\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\\Lambda_{\\mathbf{P}_j}: \\mathcal{X}_{\\varepsilon_0}(W_1)\\rightarrow (H^{-s}(\\Omega_e))^*$ satisfy\n\\[\n\\Lambda_{\\mathbf{P}_1}f|_{W_2} = \\Lambda_{\\mathbf{P}_2}f|_{W_2} \\quad \\mbox{ for all } f \\in \\mathcal{X}_{\\varepsilon_0}(W_1), \n\\]\nthen for all multi-indices $\\sigma$ such that $0\\le |\\sigma|\\le m$ and for all $k=1,\\ldots, K-1$ it holds\n$$\t \nq_1=q_2\\quad\\hbox{and}\\quad\ta_{\\sigma,k}^{(1)} = a_{\\sigma,k}^{(2)} \\quad \\hbox{ in }\\Omega,\n$$ \nand thus the nonlinearities $\\mathbf P_1$ and $\\mathbf P_2$ coincide.\n\\end{theorem}\n\n\\begin{theorem}[Well-posedness for the nonlinear equation]\n\\label{thm_wellposdness}\nLet $\\Omega\\subset\\R^n$ be a bounded open domain with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$ and $m\\in \\mathbb{N}$ satisfy $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q\\in L^\\infty(\\Omega)$ satisfies $q\\geq 0$ in $\\Omega$, and the coefficients in $\\mathbf{P}(u)$ satisfy $a_{\\sigma,k}\\in C (\\overline\\Omega)$. \nThen there exists a sufficiently small $\\varepsilon_0>0$ such that for any $f\\in \\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$,\nthe Dirichlet problem \\eqref{eq_1_IBVP} has a unique solution $u\\in H^s(\\R^n)\\cap C^s(\\R^n)$. Moreover, there exists a constant $C>0$, independent of $u$ and $f$, such that \n\\[\n\\|u\\|_{C^{s}(\\R^n)}+ \\|u\\|_{H^{s}(\\R^n)}\\le C\\|f\\|_{C^\\infty_c(\\Omega_e)}.\n\\]\n\\end{theorem}\n\n\\subsection{Motivation and connection to the literature}\nThis work addresses a natural generalization of the inverse problems for linear fractional elliptic equations, such as the fractional Schr\\\"odinger equation $(-\\Delta)^s+q$ for $s\\in (0,1)$. Our unique determination result of Theorem \\ref{thm_main} particularly finds its place within the context of the very active field of inverse problems of fractional Calder\\'on problem, known as the fractional analogue of Calder\\'on problem. First introduced in the seminal work \\cite{GSU20}, the fractional Calder\\'on problem asks to uniquely determine a scalar potential $q$ in the fractional Schr\\\"odinger equation from the related DN-map. It corresponds to our problem \\eqref{eq_1_IBVP} in the case $\\mathbf P \\equiv 0$ for $s\\in (0,1)$. Already in the first work \\cite{GSU20}, the authors were able to provide a uniqueness result from partial data for $L^\\infty$ potentials.\n\nBecause the DN-map acts on a quotient space, we have to make sure that it is well-defined as indicated above. We also need to show that it is indeed a bounded linear operator. In what follows, we will use $f$ rather than $[f]$ in order to simplify the notation.\n\\begin{prop}\nLet $\\Omega\\subset\\R^n$ be a bounded open domain with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$ and $m\\in \\mathbb{N}$ satisfy $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q\\in L^\\infty(\\Omega)$ satisfies $q\\geq 0$ in $\\Omega$, and the coefficients in $\\mathbf{P}(u)$ satisfy $a_{\\sigma,k}\\in C (\\overline\\Omega)$. \nThen the DN-map $\\Lambda_\\mathbf{P}$ is well-defined and bounded.\n\\end{prop}\n\\begin{proof}\nWe first show that $\\Lambda_\\mathbf{P}$ only depends on the equivalence classes. Let $\\phi,\\,\\psi\\in \\widetilde{H}^s(\\Omega)$. Since $u_f$ and $u_{f+\\phi}$ both solve problem \\eqref{eq_1_IBVP} with the same exterior data, the well-posedness of \\eqref{eq_1_IBVP} implies $u_f = u_{f+\\phi}$.\nBy the linearity of $B_\\mathbf{P}$ in the second component, we have\n\\[\nB_\\mathbf{P}(u_{f+\\phi},v+\\psi) = B_\\mathbf{P}(u_f, v+\\psi) = B_\\mathbf{P}(u_f, v)+ B_\\mathbf{P}(u_f, \\psi).\n\\]\nUsing the fact that $\\psi = 0$ in $\\Omega_e$ and $u_f$ solves \\eqref{eq_1_IBVP}, we get $B_\\mathbf{P}(u_f,\\psi) = 0$. This proves that $\\LA \\Lambda_\\mathbf{P}(f+\\phi), v+\\psi \\RA =\\LA \\Lambda_\\mathbf{P}f,v\\RA $, and thus $\\Lambda_\\mathbf{P}$ is well-defined.\n\n\\subsection{Linearization}\\label{sec_linearization}\nFor $f=(f_1,\\ldots,f_K)$ with $f_\\ell\\in C^\\infty_c(\\Omega_e)$, $\\|f_\\ell\\|_{C_c^\\infty(\\Omega_e)}\\le 1/K$, Theorem \\ref{thm_wellposdness} yields that, for sufficiently small $\\varepsilon = (\\varepsilon_1,\\ldots,\\varepsilon_K)$ with $|\\varepsilon|<\\varepsilon_0$, there exists a unique solution $u_\\varepsilon \\in H^s(\\R^n)\\cap C^s(\\R^n)$ of the following problem \n\\begin{equation}\\label{eq_3_linearization}\n\\begin{cases}\n[(-\\Delta)^s + q(x)]u_\\varepsilon =- \\mathbf{P}(u_\\varepsilon) &\\hbox{ in } \\Omega,\\\\\nu_\\varepsilon = \\sum_{\\ell=1}^K \\varepsilon_\\ell f_\\ell=:\\varepsilon\\cdot f &\\hbox{ in } \\Omega_e.\n\\end{cases}\n\\end{equation}\nMoreover, we have \n\\begin{align}\\label{eq_3_linear_u_est}\n\\|u_\\varepsilon\\|_{H^{s}(\\R^n)}+\n\\|u_\\varepsilon\\|_{C^{s}(\\R^n)}\\le C\\|\\varepsilon\\cdot f\\|_{C_c^\\infty(\\Omega_e)}\\le C|\\varepsilon|\\sum_{\\ell=1}^K\\|f_\\ell\\|_{C_c^\\infty(\\Omega_e)}\\le C|\\varepsilon|,\n\\end{align}\nwhere the constant $C > 0$ is independent of $u_\\varepsilon$ and $f_\\ell$. We will expand the solution $u_\\varepsilon$ in terms of the small parameter $\\varepsilon$. Let $\\alpha=(\\ell_1,\\ldots,\\ell_K)\\in \\mathbb{N}^K_0 $ be a $K$-dimensional multi-index of nonnegative integers. Combining Propositions \\ref{prop_wellposedness_linear} and \\ref{prop_Holder_linear}, we know that there exists a unique solution $w_{\\alpha}\\in H^s(\\R^n)\\cap C^{s}(\\R^n)$ to the linear fractional Schr\\\"odinger equation \n\\begin{equation}\n\\label{eq_3_linear_w_3}\n\\begin{cases}\n[(-\\Delta)^s + q(x)]w_{\\alpha} =- \\mathcal{T}_{\\alpha} &\\hbox{ in } \\Omega,\\\\\nw_{\\alpha} = D^\\alpha_\\varepsilon (\\varepsilon\\cdot f)|_{\\varepsilon=0} &\\hbox{ in } \\Omega_e,\n\\end{cases}\n\\end{equation}\nwhere the inhomogeneous term is \n\\begin{equation}\\label{eq_3_def_T}\n\\begin{aligned}\n\\mathcal{T}_{\\alpha} :=\n\\sum^{|\\alpha|-1}_{\\ell=1} \\sum_{\\substack{ \\beta_1,\\ldots, \\beta_\\ell\\in \\mathbb{N}^K_0\\setminus \\{0,\\cdots,0\\} : \\\\ \\beta:= \\sum_{j=1}^\\ell\\beta_j < \\alpha }}\\binom{\\alpha}{\\beta} \\binom{\\beta}{\\beta_1,\\cdots, \\beta_\\ell}\\bigg( \\prod_{j=1}^\\ell w_{\\beta_j } \\bigg) P_\\ell(x,D)w_{\\alpha-\\beta}.\n\\end{aligned} \n\\end{equation}\nNote here $\\mathcal{T}_{\\alpha} = 0$ when $|\\alpha| = 0,\\, 1$.\nFurthermore, the solution satisfies the stability estimate \n\\begin{equation}\\label{eq_3_linear_w3_est}\n\\begin{aligned}\n\\|w_\\alpha\\|_{C^{s}(\\R^n)}+ \\|w_\\alpha\\|_{H^{s}(\\R^n)}\n\\le \n\\begin{cases}\nC\\|D^\\alpha_\\varepsilon (\\varepsilon\\cdot f)|_{\\varepsilon=0}\\|_{C^\\infty_c(\\Omega_e)}&|\\alpha|=1,\\\\\nC\\|\\mathcal{T}_\\alpha\\|_{L^\\infty(\\Omega)},&|\\alpha|\\geq 2.\n\\end{cases}\n\\end{aligned}\n\\end{equation}\nHere we denote\n$$\n\\binom{\\alpha}{\\beta} = \\frac{\\alpha !}{\\beta !(\\alpha-\\beta)!} \\quad \\hbox{and} \\quad \n\\binom{\\beta}{\\beta_1, \\cdots, \\beta_k}= \\frac{\\beta!}{\\beta_1! \\cdots\\beta_k!}.\n$$\nObserve that the definition of $\\mathcal T_\\alpha$ only requires knowledge of those $w_\\beta$ with $\\beta<\\alpha$, and that in turn $w_\\alpha$ can be computed from $\\mathcal T_\\alpha$ as the solution of \\eqref{eq_3_linear_w_3}. Thus the definitions of $\\mathcal T_\\alpha$ and $w_\\alpha$ are not circular.\n\nBy using the derivatives given above, and the fact $u_\\varepsilon=0$ when $\\varepsilon=0$, we obtain the following lemma through direct computation. \n\\begin{lemma}\\label{lamma_DN_higher}\nLet $f=(f_1,\\ldots,f_K)$, where $f_\\ell\\in C^\\infty_c(\\Omega_e)$ satisfies $\\|f_\\ell\\|_{C_c^\\infty(\\Omega_e)}\\le 1/K$. For sufficiently small $\\varepsilon = (\\varepsilon_1,\\ldots,\\varepsilon_K)$ such that $|\\varepsilon|<\\varepsilon_0$, we have for $|\\alpha|\\ge 1$\nthat\n\\begin{align*}\n\\LA \\frac{\\p^{|\\alpha|}}{\\p \\varepsilon^\\alpha} \\Big|_{\\varepsilon=0}\\Lambda_\\mathbf{P}(\\varepsilon\\cdot f),g \\RA = \\int_{\\R^n} (-\\Delta)^{\\frac{s}{2}}w_\\alpha (-\\Delta)^{\\frac{s}{2}}g \\dx + \\int_\\Omega q(x)w_\\alpha\\, g \\dx+\n\\int_{\\Omega}\\mathcal{T}_{\\alpha}g \\dx, \\quad\\hbox{for }g\\in H^s(\\R^n).\n\\end{align*}\nNotice that $\\mathcal T_\\alpha \\equiv 0$ when $|\\alpha|=1$. \n\\end{lemma}\n\\begin{proof}\nWe first show the first-order derivative of the DN-map. \nFor any $g\\in H^s(\\R^n)$, by the definition of $\\Lambda_\\mathbf{P}$, taking the derivatives on the bilinear form yields that \n\\begin{align*}\n\\LA \\p _{\\varepsilon_\\ell}\\big|_{\\varepsilon=0}\\Lambda_\\mathbf{P}(\\varepsilon\\cdot f),g\\RA =\\p_{\\varepsilon_{\\ell}} \\big|_{\\varepsilon=0}B_\\mathbf{P}(u_\\varepsilon,g) = \\int_{\\R^n}(-\\Delta)^{\\frac{s}{2}}v_\\ell (-\\Delta)^{\\frac{s}{2}}g \\dx + \\int_\\Omega q(x)v_\\ell\\, g \\dx.\n\\end{align*}\nNote that all nonlinear terms vanish as $u_\\varepsilon=0$ when $\\varepsilon=0$. \nSimilarly, let $\\alpha=e_{\\ell_1}+e_{\\ell_2}$, for second-order partial derivatives of $\\Lambda_\\mathbf{P}$, since only linear terms and the quadratic nonlinearity in $B_\\mathbf{P}$ remain after taking derivatives, we get\n\\begin{align*}\n\\LA \\p_{\\varepsilon_{\\ell_1}} \\p_{\\varepsilon_{\\ell_2}} \\big|_{\\varepsilon=0}\\Lambda_\\mathbf{P}(\\varepsilon\\cdot f),g \\RA &=\\p_{\\varepsilon_{\\ell_1}} \\p_{\\varepsilon_{\\ell_2}} \\big|_{\\varepsilon=0}B_\\mathbf{P}(u_\\varepsilon,g)\\\\\n&=\\int_{\\R^n} (-\\Delta)^{\\frac{s}{2}}w_{\\alpha} (-\\Delta)^{\\frac{s}{2}}g \\dx + \\int_\\Omega q(x) w_{\\alpha}\\, g \\dx\\\\\n&\\quad\n+\\int_{\\Omega} \\LC v_{\\ell_1}P_1(x,D)v_{\\ell_2} + v_{\\ell_2}P_1(x,D)v_{\\ell_1} \\RC g \\dx.\n\\end{align*}\nFollowing a similar argument, we can also derive the integral representation of \n$\\frac{\\p^{|\\alpha|}}{\\p \\varepsilon^\\alpha}\n\\Big|_{\\varepsilon=0}\\Lambda_\\mathbf{P}(\\varepsilon\\cdot f)$ for $|\\alpha|\\geq 3$.\n\\end{proof}", + "post_theorem_intro_text_len": 7797, + "post_theorem_intro_text": "\\subsection{Motivation and connection to the literature}\nThis work addresses a natural generalization of the inverse problems for linear fractional elliptic equations, such as the fractional Schr\\\"odinger equation $(-\\Delta)^s+q$ for $s\\in (0,1)$. Our unique determination result of Theorem \\ref{thm_main} particularly finds its place within the context of the very active field of inverse problems of fractional Calder\\'on problem, known as the fractional analogue of Calder\\'on problem. First introduced in the seminal work \\cite{GSU20}, the fractional Calder\\'on problem asks to uniquely determine a scalar potential $q$ in the fractional Schr\\\"odinger equation from the related DN-map. It corresponds to our problem \\eqref{eq_1_IBVP} in the case $\\mathbf P \\equiv 0$ for $s\\in (0,1)$. Already in the first work \\cite{GSU20}, the authors were able to provide a uniqueness result from partial data for $L^\\infty$ potentials. \n\nBesides the mathematical interest, the fractional Calder\\'on problem arises naturally in the sciences, due to the close connection between the fractional Laplace operator and anomalous diffusion \\cite{CL24,V09}. A novel application is that of the recent paper \\cite{LNOS25}, which considers an inverse problem of fractional Calder\\'on type related to quantum field theory. Many aspects of the fractional Calder\\'on problem have been studied in recent years: among these, we recall uniqueness for low regularity partial data \\cite{RS20a} and anisotropic background metrics \\cite{GLX17}, reconstruction for a single measurement \\cite{GRSU20} and by monotonicity methods \\cite{ HL20, HL19}, the stability and instability results of \\cite{ BCR25,RS18, RS20b}, general anisotropic metrics \\cite{Fei24, FGKRSU25, FGKU21},\nand boundary reconstruction \\cite{CR25}. Moreover, the entanglement principles are studied for fractional poly-elliptic \\cite{FKU24, FL24} and poly-parabolic operators \\cite{LLYan25} to decouple the entangled effect of nonlocal perturbations. We refer to the survey \\cite{S17} and to the recent book \\cite{LL25book} for many more related results on the topic of nonlocal inverse problems. Incidentally, the fractional Calder\\'on problem has been studied also in its conductivity formulation, as shown in \\cite{Cov20a} and the subsequent works \\cite{Cov20b, Cov22, CMR21, CMRU22, CRZ22}. We refer to \\cite{Cov25} for a recent survey on the topic. Other equations of fractional and, more generally, nonlocal nature have also been the object of intensive studies. Among these, we recall inverse problems for fractional elasticity \\cite{CdHS22}, the fractional Helmholtz equation via geometrical optics solutions \\cite{CdHS25}, and the fractional wave equation \\cite{KLW22, LZ25}, among many others \\cite{DGM25, Li23, LW25, Lin23}. The relations between the fractional and classical Calder\\'on problems have also attracted substantial attention \\cite{BCR25, BR25, CGRU23, R23}. \n\nThe two distinctive features of our equation \\eqref{eq_1_IBVP} include the \\textit{higher-order nonlocal leading operator} and \\textit{general local nonlinear perturbations} of the fractional Schr\\\"odinger equation. It is a local nonlinear perturbation of the higher-order fractional Schr\\\"odinger equation. As such, it differs substantially from the perturbations considered in \\cite{BGU21, CLR20, Cov21,Li21}, which are linear (and sometimes nonlocal), and from those of \\cite{CMRU22}, where the most general local linear perturbations of the fractional Schr\\\"odinger equation were studied. \n\nThe consideration of the higher-order fractional Laplace operator $(-\\Delta)^s$ with $s\\in \\mathbb R^+\\setminus\\mathbb Z$ in this paper is inspired by the work \\cite{CMRU22}, which investigated the unique determination of a general linear term. \nIndeed, the fractional Laplace operator with power $s>1$ has been object of study in several recent publications: we recall \\cite{CMRU22, DHP23} for higher-order inverse problems of fractional Schr\\\"odinger type, and \\cite{AJ24, CSZ25, DPS25, FF20, GR19, RS15} for theoretical results concerning the unique continuation property and existence of solutions for the higher-order fractional Laplacian. \n\nMeanwhile, the latter feature falls into a rapidly developing research direction in inverse problems, dealing with reconstruction of nonlinear terms in partial differential equations (PDEs), which are highly motivated by natural phenomena, such as nonlinear optics. To study inverse problems for nonlinear PDEs, a classical linearization technique first introduced in \\cite{Isakov93} is to perform first-order linearization of the DN-map. \nLately, the higher-order linearization method \\cite{KLU18} was launched to treat inverse problems for various kind of PDEs. In particular, it was observed that the presence of nonlinearity in an equation turns out to be a great benefit to the study of inverse problems, despite the potential difficulty nonlinearity introduced to solve the forward problems. Specifically, since the nonlinear interactions generate more fruitful information in the reconstruction process, many inverse problems for the nonlinear equations can be resolved, while the underlying inverse problems in the linear setting are still open.\nThe interested reader is referred to the following relevant literature and the reference therein. The nonlinear Schr\\\"odinger equation has been studied in \\cite{lai_partial_2023, LUY24, Lai_inverse_2021,lassas_inverse_2024}, while a nonlinear version of the wave equation and related acoustic problems were the object of \\cite{HTT25, Kal25b, Kal25,LLPT25}. Many more nonlinear-type inverse problems, including for the Boltzmann equation, were considered in \\cite{KK25, lai_reconstruction_2021, lai_stable_2023, LZ24,LL25, NS25}. We also recall a recent result about corrosion detection by identification of a nonlinear Robin boundary condition \\cite{J25}. The fractional Schr\\\"odinger \\cite{KMS23, KMSS25, lai_global_2018}, wave \\cite{LTZ24} and several other related equations \\cite{JNS23, JNS25, lai_inverse_2022, lai_recovery_2022} were studied also in the semilinear case. \n\nThere is a major difference between the cases $s\\in (0,1)$ and $s\\in \\mathbb{R}^+\\setminus\\mathbb{Z}$ in $(-\\Delta)^s$. For the nonlinear Schr\\\"odinger equation with $s\\in (0,1)$, the availability of the maximum principle helps to ensure the $L^\\infty$ bounded solution to the linear Schr\\\"odinger equation. This can be used to deduce the $C^s$ regularity solution for the nonlinear one, see \\cite{lai_global_2018}, which studied the unique determination of the nonlinear potential in $(-\\Delta)^s u +q(x,u)=0$. For our case, however, it is necessary to require the condition $\\lfloor s \\rfloor>\\max\\{m,\\,n/2\\}$ to guarantee the existence of such bounded solution so that we can further derive the $C^s$ regularity, see Section~\\ref{sec:pre} for details. With this, the contraction mapping principle is applied to establish the well-posedness for \\eqref{eq_1_IBVP}.\n\n\\subsection{Outline of the article}\n\nThe remaining part of the paper is organized as follows. In Section~\\ref{sec:pre} we survey and extend some existing results for the linear Schr\\\"odinger equation to suit our current setting, and we establish the well-posedness Theorem \\ref{thm_wellposdness}. In Section~\\ref{sec_linearization} we show that the solution to the nonlinear Schr\\\"odinger equation \\eqref{eq_1_IBVP} can be decomposed with respect to the order of a small parameter. This enables us to translate the inverse problem for \\eqref{eq_1_IBVP} into the study of recovery of the nonhomogeneous term in the linearized equation in Section~\\ref{sec:proof of theorem}, where we prove our main result, Theorem \\ref{thm_main}. After the acknowledgments of Section~\\ref{sec:ack}, we included an Appendix dedicated to computations.", + "sketch": "To prove Theorem~\\ref{thm_main}, the paper proceeds by first establishing well-posedness: in Section~\\ref{sec:pre} the authors “survey and extend some existing results for the linear Schr\\\"odinger equation to suit our current setting” and “establish the well-posedness Theorem \\ref{thm_wellposdness}.” A key point for the forward problem is that, since for $s\\in\\mathbb{R}^+\\setminus\\mathbb{Z}$ one cannot rely on the maximum principle as in the case $s\\in(0,1)$, they “require the condition $\\lfloor s \\rfloor>\\max\\{m,\\,n/2\\}$ to guarantee the existence of such bounded solution so that we can further derive the $C^s$ regularity,” and “with this, the contraction mapping principle is applied to establish the well-posedness for \\eqref{eq_1_IBVP}.”\n\nFor the inverse problem, in Section~\\ref{sec_linearization} they “show that the solution to the nonlinear Schr\\\"odinger equation \\eqref{eq_1_IBVP} can be decomposed with respect to the order of a small parameter,” which “enables us to translate the inverse problem for \\eqref{eq_1_IBVP} into the study of recovery of the nonhomogeneous term in the linearized equation.” Finally, in Section~\\ref{sec:proof of theorem} they “prove our main result, Theorem~\\ref{thm_main},” i.e. uniqueness of the coefficients via this linearization-based reduction.", + "expanded_sketch": "To prove the main theorem, the paper proceeds by first establishing well-posedness: next the authors survey and extend some existing results for the linear Schr\\\"odinger equation to suit the current setting, and establish the following theorem.\n\n\\begin{theorem}[Well-posedness for the nonlinear equation]\n\\label{thm_wellposdness}\nLet $\\Omega\\subset\\R^n$ be a bounded open domain with smooth \nboundary $\\p\\Omega$. Let $s\\in \\R^+\\setminus \\mathbb{Z}$ and $m\\in \\mathbb{N}$ satisfy $\\floor{s}>\\max\\{m,\\,n/2\\}$.\nSuppose that $q\\in L^\\infty(\\Omega)$ satisfies $q\\geq 0$ in $\\Omega$, and the coefficients in $\\mathbf{P}(u)$ satisfy $a_{\\sigma,k}\\in C (\\overline\\Omega)$. \nThen there exists a sufficiently small $\\varepsilon_0>0$ such that for any $f\\in \\mathcal{X}_{\\varepsilon_0}(\\Omega_e)$,\nthe Dirichlet problem \\eqref{eq_1_IBVP} has a unique solution $u\\in H^s(\\R^n)\\cap C^s(\\R^n)$. Moreover, there exists a constant $C>0$, independent of $u$ and $f$, such that \n\\[\n\\|u\\|_{C^{s}(\\R^n)}+ \\|u\\|_{H^{s}(\\R^n)}\\le C\\|f\\|_{C^\\infty_c(\\Omega_e)}.\n\\]\n\\end{theorem}\n\nA key point for the forward problem is that, since for $s\\in\\mathbb{R}^+\\setminus\\mathbb{Z}$ one cannot rely on the maximum principle as in the case $s\\in(0,1)$, they require the condition $\\lfloor s \\rfloor>\\max\\{m,\\,n/2\\}$ to guarantee the existence of such bounded solution so that one can further derive the $C^s$ regularity; with this, the contraction mapping principle is applied to establish well-posedness for\n\\begin{equation}\\label{eq_1_IBVP}\n\\begin{cases}\n(-\\Delta)^su + q(x)u +\\mathbf{P}(u)= 0 &\\hbox{ in } \\Omega,\\\\\\nu = f &\\hbox{ in } \\Omega_e,\n\\end{cases}\n\\end{equation}\n\nFor the inverse problem, they then show that the solution to the nonlinear Schr\\\"odinger equation given by the equation above can be decomposed with respect to the order of a small parameter, which enables one to translate the inverse problem for the same equation into the study of recovery of the nonhomogeneous term in the linearized equation. Finally, they complete the argument by proving the main theorem, i.e. uniqueness of the coefficients via this linearization-based reduction.", + "expanded_theorem": "\\label{thm_main}(Unique determination)\nLet $\\Omega\\subset\\mathbb{R}^n$ be a bounded open set with smooth \nboundary $\\partial\\Omega$. Let $s\\in \\mathbb{R}^+\\setminus \\mathbb{Z}$, $\\lfloor s \\rfloor>\\max\\{m,\\,n/2\\}$.\nSuppose that $q_j\\in L^\\infty(\\Omega)$ satisfies $q_j\\geq 0$ in $\\Omega$ for $j=1,\\,2$.\nLet $\\mathbf{P}_1$ and $\\mathbf{P}_2$ be nonlinearities of the form\n\\begin{align}\\label{DEF:nonlinear P}\n\\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\\ldots+u^{K-1}P_{K-1}(x,D)u,\n\\end{align}\nwith coefficients $a_{\\sigma,k}^{(1)},\\, a_{\\sigma,k}^{(2)}(x) \\in C (\\overline\\Omega)$. Let $W_1,\\, W_2\\subset\\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\\Lambda_{\\mathbf{P}_j}: \\mathcal{X}_{\\varepsilon_0}(W_1)\\rightarrow (H^{-s}(\\Omega_e))^*$ satisfy\n\\[\n\\Lambda_{\\mathbf{P}_1}f|_{W_2} = \\Lambda_{\\mathbf{P}_2}f|_{W_2} \\quad \\mbox{ for all } f \\in \\mathcal{X}_{\\varepsilon_0}(W_1), \n\\]\nthen for all multi-indices $\\sigma$ such that $0\\le |\\sigma|\\le m$ and for all $k=1,\\ldots, K-1$ it holds\n$$\t \nq_1=q_2\\quad\\hbox{and}\\quad\ta_{\\sigma,k}^{(1)} = a_{\\sigma,k}^{(2)} \\quad \\hbox{ in }\\Omega,\n$$ \nand thus the nonlinearities $\\mathbf P_1$ and $\\mathbf P_2$ coincide.,", + "theorem_type": [ + "Implication", + "Universal" + ], + "mcq": { + "question": "Let $\\Omega\\subset \\mathbb{R}^n$ be a bounded open set with smooth boundary, and let $\\Omega_e:=\\mathbb{R}^n\\setminus \\overline{\\Omega}$. Fix an integer $m\\ge 0$, an integer $K\\ge 2$, and $s\\in \\mathbb{R}^+\\setminus \\mathbb{Z}$ such that $\\lfloor s\\rfloor>\\max\\{m,n/2\\}$. For $j=1,2$, assume $q_j\\in L^\\infty(\\Omega)$ with $q_j\\ge 0$ in $\\Omega$, and let $\\mathbf P_j$ be nonlinearities of the form\n\\[\n\\mathbf P_j(u)=uP^{(j)}_1(x,D)u+u^2P^{(j)}_2(x,D)u+\\cdots+u^{K-1}P^{(j)}_{K-1}(x,D)u,\n\\]\nwhere\n\\[\nP^{(j)}_k(x,D)=\\sum_{|\\sigma|\\le m} a^{(j)}_{\\sigma,k}(x)D^\\sigma,\n\\qquad a^{(j)}_{\\sigma,k}\\in C(\\overline\\Omega),\n\\]\nfor every $k=1,\\dots,K-1$. Let $W_1,W_2\\subset \\Omega_e$ be arbitrary nonempty bounded open sets, and define\n\\[\n\\mathcal X_{\\varepsilon_0}(W_1):=\\{f\\in C_c^\\infty(W_1):\\ \\|f\\|_{C_c^\\infty(W_1)}\\le \\varepsilon_0\\}.\n\\]\nFor each $j$, let $\\Lambda_{\\mathbf P_j}$ be the Dirichlet-to-Neumann map for the exterior value problem\n\\[\n(-\\Delta)^s u+q_j(x)u+\\mathbf P_j(u)=0\\ \\text{in }\\Omega,\\qquad u=f\\ \\text{in }\\Omega_e,\n\\]\nnamely $\\Lambda_{\\mathbf P_j}(f)=(-\\Delta)^s u_f|_{\\Omega_e}$. Suppose that the partial DN data agree in the sense that\n\\[\n\\Lambda_{\\mathbf P_1}f|_{W_2}=\\Lambda_{\\mathbf P_2}f|_{W_2}\\qquad \\text{for all }f\\in \\mathcal X_{\\varepsilon_0}(W_1).\n\\]\nWhich statement is guaranteed for every such pair of nonlinear equations?", + "correct_choice": { + "label": "A", + "text": "One must have $q_1=q_2$ in $\\Omega$, and for every multi-index $\\sigma$ with $0\\le |\\sigma|\\le m$ and every $k=1,\\dots,K-1$, the coefficients satisfy $a^{(1)}_{\\sigma,k}=a^{(2)}_{\\sigma,k}$ in $\\Omega$; equivalently, the nonlinearities $\\mathbf P_1$ and $\\mathbf P_2$ coincide in $\\Omega$." + }, + "choices": [ + { + "label": "B", + "text": "One must have $q_1=q_2$ in $\\Omega$, and for every multi-index $\\sigma$ with $0\\le |\\sigma|< m$ and every $k=1,\\dots,K-1$, the coefficients satisfy $a^{(1)}_{\\sigma,k}=a^{(2)}_{\\sigma,k}$ in $\\Omega$; hence the lower-order parts of the operators $P_k^{(j)}(x,D)$ agree, though the coefficients of order exactly $m$ need not be determined." + }, + { + "label": "C", + "text": "One must have $q_1=q_2$ in $\\Omega$." + }, + { + "label": "D", + "text": "One must have $q_1=q_2$ in $\\Omega$, and for every multi-index $\\sigma$ with $0\\le |\\sigma|\\le m$ there exists an index $k=k(\\sigma)\\in\\{1,\\dots,K-1\\}$ such that $a^{(1)}_{\\sigma,k}=a^{(2)}_{\\sigma,k}$ in $\\Omega$; in particular, each differential order is represented by some matching coefficient, although not necessarily for every $k$." + }, + { + "label": "E", + "text": "There exists a sufficiently small $\\varepsilon_*>0$, depending only on $\\Omega$, $n$, $m$, $s$, and the a priori bounds for the coefficients, such that if the partial DN maps agree on $W_2$ for all $f\\in \\mathcal X_{\\varepsilon_*}(W_1)$, then one must have $q_1=q_2$ in $\\Omega$ and $a^{(1)}_{\\sigma,k}=a^{(2)}_{\\sigma,k}$ in $\\Omega$ for all $0\\le |\\sigma|\\le m$ and all $k=1,\\dots,K-1$." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "D" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "regularity", + "tampered_component": "full recovery up to top differential order $m$", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped identification of all nonlinear coefficients $a_{\\sigma,k}$", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "quantifier structure over $(\\sigma,k)$ in coefficient recovery", + "template_used": "wildcard" + }, + { + "label": "E", + "sketch_hook_type": "regularity", + "tampered_component": "dependence on the given small-data radius $\\varepsilon_0$ versus a newly asserted uniform threshold", + "template_used": "uniformity_effectivity" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem does not state the conclusion explicitly and does not reveal the correct option by wording. It gives the full hypotheses, but the answer is not leaked directly." + }, + "TAS": { + "score": 1, + "justification": "This is largely a theorem-recall item: the stem reproduces a full technical hypothesis set and asks for the guaranteed conclusion. The alternatives introduce quantifier/order variations, so it is not a pure verbatim restatement, but it is still close to one." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is needed to distinguish full recovery from weaker or tampered conclusions, especially among B, C, D, and E. However, the item mainly tests recognition of the exact theorem conclusion rather than substantial generative mathematical reasoning." + }, + "DQS": { + "score": 2, + "justification": "The distractors are strong and mathematically plausible: one weakens the conclusion, one drops top-order recovery, one alters the quantifier structure, and one changes the smallness/uniform-threshold claim. These reflect realistic failure modes." + }, + "total_score": 6, + "overall_assessment": "A solid theorem-recognition MCQ with high-quality distractors and no major answer leakage, but it is still fairly close to a restatement of a specific uniqueness theorem rather than a deeply generative reasoning task." + } + }, + { + "id": "2511.03979v1", + "paper_link": "http://arxiv.org/abs/2511.03979v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "\\label{main theorem}\nFor $n>0$, we have\n\\begin{align}\\label{main theorem eqn}\nA(n)=B(n)=C(n+1)=\\frac{1}{2}D(n+1),\n\\end{align}\nwhere $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.", + "start_pos": 6707, + "end_pos": 7145, + "label": "main theorem" + }, + "ref_dict": { + "bijective proofs": "\\label{bijective proofs}\n\n\\subsection{A Bijective proof of $A(n-1)=\\frac{1}{2}D(n)$}\nWe can first remove the non-negative part condition for $D(n)$ as follows:\n\\begin{quote}\nFor $n\\ge 2$, $D(n)$ count", + "bc": "\\label{bc}\n\nWe note that\n\\begin{align}\\label{bn}\n\\sum_{n=1}^\\infty B(n)q^{n+1}&=q\\sum_{n=1}^\\infty\\frac{q^{2n-1}}{(q;q^2)_n} \\notag \\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n},\n\\end{align}\nhere, an", + "cd": "\\begin{align}\n\\sum_{n=1}^\\infty C(n)q^{n}&=\\sum_{n=1}^\\infty\\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\\nonumber\\\\\n&=\\sum_{n=1}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \\label{c1n}\\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n}.\\label{cn}\n\\end{align}\n\nComparing \\eqref{bn} and \\eqref{cn}, we see that $B(n)=C(n+1)$.\n\n\\section{Proof that $C(n)=\\frac{1}{2}D(n)$}\\label{cd}\n\nLet us fix $C(0)=1$. From \\eqref{c1n}, we have\n\\begin{align}\\label{before p19}\n\\sum_{n=0}^\\infty C(n)q^{n}&=\\sum_{n=0}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{n=0}^\\infty\\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\\infty}.\n\\end{align}", + "main theorem": "\\begin{theorem}\\label{main theorem}\nFor $n>0$, we have\n\\begin{align}\\label{main theorem eqn}\nA(n)=B(n)=C(n+1)=\\frac{1}{2}D(n+1),\n\\end{align}\nwhere $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 802, + "pre_theorem_intro_text": "The grandfather of all partition identities is Euler's theorem \\cite{andrews book}, namely,\n\\begin{align*}\nA(n)=B(n),\n\\end{align*}\nwhere $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.\n\nThis theorem contains all the elements that would suggest generalizations, and, over the centuries, generalizations have been found in profusion. The Rogers-Ramanujan identities \\cite[p.~104]{andrews book} and Schur's 1926 theorem \\cite[p.~116]{andrews book} kicked off the twentieth century's contributions, consult Henry Alder's survey \\cite{alder} for an account of some of the results in the late 20th century. \n\nHowever, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.", + "context": "The grandfather of all partition identities is Euler's theorem \\cite{andrews book}, namely,\n\\begin{align*}\nA(n)=B(n),\n\\end{align*}\nwhere $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.\n\nThis theorem contains all the elements that would suggest generalizations, and, over the centuries, generalizations have been found in profusion. The Rogers-Ramanujan identities \\cite[p.~104]{andrews book} and Schur's 1926 theorem \\cite[p.~116]{andrews book} kicked off the twentieth century's contributions, consult Henry Alder's survey \\cite{alder} for an account of some of the results in the late 20th century.\n\nHowever, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.", + "full_context": "The grandfather of all partition identities is Euler's theorem \\cite{andrews book}, namely,\n\\begin{align*}\nA(n)=B(n),\n\\end{align*}\nwhere $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.\n\nThis theorem contains all the elements that would suggest generalizations, and, over the centuries, generalizations have been found in profusion. The Rogers-Ramanujan identities \\cite[p.~104]{andrews book} and Schur's 1926 theorem \\cite[p.~116]{andrews book} kicked off the twentieth century's contributions, consult Henry Alder's survey \\cite{alder} for an account of some of the results in the late 20th century.\n\nHowever, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.\n\n\\subjclass[2020]{Primary 11P84, 05A17}\n \\keywords{Euler's theorem, Partitions, Glaisher's bijection}\n\\maketitle\n\\pagenumbering{arabic}\n\\pagestyle{headings}\n\\begin{abstract}\nEuler's theorem asserts that $A(n)=B(n)$ where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts. In this paper, it is proved that for $n>0$,\n\\begin{align*}\nA(n)=B(n)=C(n+1)=\\frac{1}{2}D(n+1),\n\\end{align*}\nwhere $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.\n\nThe grandfather of all partition identities is Euler's theorem \\cite{andrews book}, namely,\n\\begin{align*}\nA(n)=B(n),\n\\end{align*}\nwhere $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.\n\nHowever, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.\n\nFor example, if $n=6$, the four sets of partitions are following:\n\\begin{center}\n\\begin{tabular}{ p{2cm} p{3.5cm} p{3cm} p{4.1cm} p{4.1cm}}\n\nOn the other hand, we have\n\\begin{align}\n\\sum_{n=1}^\\infty C(n)q^{n}&=\\sum_{n=1}^\\infty\\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\\nonumber\\\\\n&=\\sum_{n=1}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \\label{c1n}\\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n}.\\label{cn}\n\\end{align}\n\nLet us fix $C(0)=1$. From \\eqref{c1n}, we have\n\\begin{align}\\label{before p19}\n\\sum_{n=0}^\\infty C(n)q^{n}&=\\sum_{n=0}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{n=0}^\\infty\\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\\infty}.\n\\end{align}\nWe now employ the following Euler's identity \\cite[p.~19, (2.2.5)]{andrews book}\n\\begin{align}\\label{euler identity}\n\\frac{1}{(t;q)_\\infty}=\\sum_{m=0}^\\infty\\frac{t^m}{(q,q)_m},\n\\end{align}\n(with replacing $q$ by $q^2$ and then letting $t=q^{2n+2}$) in \\eqref{before p19} so as to obtain\n\\begin{align}\\label{series}\n\\sum_{n=0}^\\infty C(n)q^{n}&=(q^2;q^2)_\\infty\\sum_{n=0}^\\infty\\frac{q^{2n}}{(q;q)_{2n}}\\sum_{m=0}^\\infty\\frac{q^{2nm+2m}}{(q^2;q^2)_m}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{m,n=0}^\\infty\\frac{q^{2n+2nm+2m}}{(q;q)_{2n}(q^2;q^2)_m}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{m,n=0}^\\infty\\frac{1}{2}\\left(1+(-1)^n\\right)\\frac{q^{n+nm+2m}}{(q;q)_{n}(q^2;q^2)_m}\\nonumber\\\\\n&=\\frac{1}{2}(q^2;q^2)_\\infty\\sum_{m=0}^\\infty\\frac{q^{2m}}{(q^2;q^2)_m}\\left\\{\\sum_{n=0}^\\infty\\frac{q^{n(m+1)}}{(q;q)_n}+\\sum_{n=0}^\\infty\\frac{(-1)^nq^{n(m+1)}}{(q;q)_n}\\right\\}.\n\\end{align}\nUpon invoking \\eqref{euler identity} twice, once with letting $t=q^{m+1}$ and once with letting $t=-q^{m+1}$, and then substituting both resulting expressions in \\eqref{series}, we conclude that\n\\begin{align}\\label{3.4}\n\\sum_{n=0}^\\infty C(n)q^{n}&=\\frac{1}{2}(q^2;q^2)_\\infty\\sum_{m=0}^\\infty\\frac{q^{2m}}{(q^2;q^2)_m}\\left\\{\\frac{1}{(q^{m+1};q)_\\infty}+\\frac{1}{(-q^{m+1};q)_\\infty}\\right\\}\\nonumber\\\\\n&=\\frac{1}{2}(-q;q)_\\infty\\sum_{m=0}^\\infty\\frac{q^{2m}}{(-q;q)_m}+\\frac{1}{2}(q;q)_\\infty\\sum_{m=0}^\\infty\\frac{q^{2m}}{(q;q)_m}\\nonumber\\\\\n&=\\frac{1}{2}\\sum_{m=0}^\\infty q^{2m}(-q^{m+1};q)_\\infty+\\frac{1}{2}(1-q),\n\\end{align}\nwhere the last step follows upon again using \\eqref{euler identity} with letting $t=q^2$.\n\n\\subsection{A Bijective proof of $A(n-1)=\\frac{1}{2}D(n)$}\nWe can first remove the non-negative part condition for $D(n)$ as follows:\n\\begin{quote}\nFor $n\\ge 2$, $D(n)$ counts the number of partitions of $n$ where only the smallest part can repeat at most twice and all other parts are distinct. \n\\end{quote}\nIn other words, a partition $\\lambda=(\\lambda_1,\\ldots, \\lambda_{\\ell})$ counted by $D(n)$ satisfies\n$$\n\\lambda_1>\\lambda_2> \\cdots > \\lambda_{\\ell-1}\\ge \\lambda_{\\ell} \\text{ if $\\ell>1$} \n$$\nor \n$\\lambda=(\\lambda_1)$ has only one part.\n\nLet $M \\le 2N$ be an even number, which can be written as\n$$\nM=2^k a \\text{ for some odd integer $a$ and some $ k\\ge 1$}.\n$$\nWe apply Glaisher's bijection $\\phi$ to $M$ and obtain $2^k$ parts of size $a$. Note that since\n$\n2^k a \\le 2N,\n$\n$$\na \\le N.\n$$\nAlso, if $M\\le N$, then $M$ can appear only once, so we get all distinct positive powers $2^k$ such that\n\\begin{equation} \n2^k\\le N/a. \\label{power1}\n\\end{equation}\nOn the other hand, if $N< M\\le 2N$, then\n\\begin{equation} \n2^k> N/a, \\label{power2}\n\\end{equation}\nand $M$ can repeat. \nSuppose $M$ appears $f$ many times. Upon applying Glaisher's bijection $\\phi$ to $f$ copies of $M$, we obtain\n$$\n f 2^{k} \\text{ copies of $a$}. \n$$\nBy writing $f$ as a binary expansion, \n$$\nf 2^k =(f_0 \\,2^{0}+f_1 \\, 2^{1} +\\cdots ) 2^k, \n$$\nwhere $f_j$ is either $0$ or $1$ for $j\\ge 0$. By \\eqref{power2}, we see that each summand in the above expression represents a distinct power of $2$ greater than $N/a$, i.e., \n\\begin{equation}\nf_j \\, 2^{j+k} >N/a. \\label{power3}\n\\end{equation}\nIt follows from \\eqref{power1} and \\eqref{power3} that $a$ can appear with any multiplicity greater than $1$. This proves that the resulting partition is counted by $B(n)$.", + "post_theorem_intro_text_len": 1257, + "post_theorem_intro_text": "For example, if $n=6$, the four sets of partitions are following:\n\\begin{center}\n\\begin{tabular}{ p{2cm} p{3.5cm} p{3cm} p{4.1cm} p{4.1cm}}\n\n \\vspace{2mm}$A(6)$\\newline & \\vspace{2mm} $B(6)$ & \\vspace{2mm} $C(7)$& \\vspace{2mm} $D(7)$ \\\\\n\n $6$ \\newline 5+1 \\newline 4+2\\newline 3+2+1 & 5+1 \\newline 3+3 \\newline 3+1+1+1 \\newline 1+1+1+1+1+1 & 6+1\\newline 4+3 \\newline 4+2+1 \\newline 2+2+2+1 & 0+0+7\\newline 0+0+6+1\\newline 0+0+5+2\\newline 0+0+4+3\\newline 0+0+4+2+1\\newline 1+1+5\\newline 1+1+2+3\\newline 2+2+3\n\\end{tabular}\n\\end{center}\n\nWe conclude the introduction with the following remark.\n\\begin{remark}\nIn their recent paper \\cite{ba}, M. El Bachraoui and the first author considered partitions with multiple appearances by the first part. All parts were assumed to be positive. It would be a simple matter to extend the results of that paper to the case of non-negative parts in that this would add $(-q;q)_\\infty$ to the generating functions in question.\n\\end{remark}\n\nThis paper is organised as follows. In section \\ref{bc}, we provide the brief proof that $B(n)=C(n+1)$. In Section \\ref{cd}, we prove that $C(n)=\\frac{1}{2}D(n)$. We also provide bijective proofs our assertions in Theorem \\ref{main theorem} in Section \\ref{bijective proofs}.", + "sketch": "The post-theorem introduction does not give a proof sketch beyond an outline of the paper: in Section~\\ref{bc} they \"provide the brief proof that $B(n)=C(n+1)$\"; in Section~\\ref{cd} they \"prove that $C(n)=\\frac{1}{2}D(n)$\"; and in Section~\\ref{bijective proofs} they \"provide bijective proofs\" of the assertions in Theorem~\\ref{main theorem}.", + "expanded_sketch": "The post-theorem introduction does not give a proof sketch beyond an outline of the paper: next they note that\n\\begin{align}\\label{bn}\n\\sum_{n=1}^\\infty B(n)q^{n+1}&=q\\sum_{n=1}^\\infty\\frac{q^{2n-1}}{(q;q^2)_n} \\notag \\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n},\n\\end{align}\nhere, an\nand also that\n\\begin{align}\n\\sum_{n=1}^\\infty C(n)q^{n}&=\\sum_{n=1}^\\infty\\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\\nonumber\\\\\n&=\\sum_{n=1}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \\label{c1n}\\\\\n&=\\sum_{n=1}^\\infty\\frac{q^{2n}}{(q;q^2)_n}.\\label{cn}\n\\end{align}\n\nComparing \\eqref{bn} and \\eqref{cn}, we see that $B(n)=C(n+1)$.\n\n\\section{Proof that $C(n)=\\frac{1}{2}D(n)$}\\label{cd}\n\nLet us fix $C(0)=1$. From \\eqref{c1n}, we have\n\\begin{align}\\label{before p19}\n\\sum_{n=0}^\\infty C(n)q^{n}&=\\sum_{n=0}^\\infty\\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\\nonumber\\\\\n&=(q^2;q^2)_\\infty\\sum_{n=0}^\\infty\\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\\infty}.\n\\end{align}\nThey then prove that $C(n)=\\frac{1}{2}D(n)$, and finally they provide bijective proofs as follows:\n\\label{bijective proofs}\n\n\\subsection{A Bijective proof of $A(n-1)=\\frac{1}{2}D(n)$}\nWe can first remove the non-negative part condition for $D(n)$ as follows:\n\\begin{quote}\nFor $n\\ge 2$, $D(n)$ count", + "expanded_theorem": "\\label{main theorem}\nFor $n>0$, we have\n\\begin{align}\\label{main theorem eqn}\nA(n)=B(n)=C(n+1)=\\frac{1}{2}D(n+1),\n\\end{align}\nwhere $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.,", + "theorem_type": [ + "Universal", + "Biconditional or Equivalence" + ], + "mcq": { + "question": "For each positive integer n, let A(n) denote the number of partitions of n into distinct parts, let B(n) denote the number of partitions of n into odd parts, let C(n) denote the number of partitions of n whose largest part is even and for which every part not exceeding half of the largest part occurs at most once, and let D(n) denote the number of partitions of n into nonnegative parts in which the smallest part appears exactly twice and no other part is repeated. Which statement holds for every positive integer n?", + "correct_choice": { + "label": "A", + "text": "A(n)=B(n)=C(n+1)=\\frac{1}{2}D(n+1)." + }, + "choices": [ + { + "label": "B", + "text": "A(n)=B(n)=C(n)=\\frac{1}{2}D(n)." + }, + { + "label": "C", + "text": "A(n)=B(n)=C(n+1)." + }, + { + "label": "D", + "text": "A(n)=B(n)=C(n+1)=D(n+1)." + }, + { + "label": "E", + "text": "A(n)=B(n)=C(n)+1=\\frac{1}{2}D(n+1)." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "other", + "tampered_component": "index_shift_in_C_and_D", + "template_used": "boundary_range" + }, + { + "label": "C", + "sketch_hook_type": "other", + "tampered_component": "dropped_half_D_term", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "factor_one_half_for_D", + "template_used": "stronger_trap" + }, + { + "label": "E", + "sketch_hook_type": "other", + "tampered_component": "exact_shift_relation_between_B_and_C", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem only defines the four partition-counting functions and asks which identity holds universally. It does not reveal the correct shift by n+1 or the factor of 1/2 in D(n+1), so there is no meaningful answer leakage." + }, + "TAS": { + "score": 2, + "justification": "This is not a direct restatement of a theorem from the stem. The question requires comparing several closely related candidate identities involving shifts and scaling factors, rather than recognizing a verbatim claim." + }, + "GPS": { + "score": 2, + "justification": "Identifying the correct statement requires substantial reasoning or prior knowledge of nontrivial partition identities. The options differ in subtle but important ways (index shifts, omission of the D-term factor, stronger vs. weaker equalities), creating real generative pressure." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically structured: they reflect common failure modes such as missing an index shift, dropping a factor of 1/2, or selecting a weaker true statement instead of the strongest correct one. They are distinct and well-aligned with likely reasoning errors." + }, + "total_score": 8, + "overall_assessment": "High-quality MCQ. It avoids answer leakage and tautology, uses strong distractors, and genuinely tests mathematical reasoning about subtle partition identities." + } + }, + { + "id": "2511.02829v1", + "paper_link": "http://arxiv.org/abs/2511.02829v1", + "theorems_cnt": 1, + "theorem": { + "env_name": "theorem", + "content": "\\label{thm:main}\n The properad $\\mathscr{Y}^{(n)}_{prop}$ is Koszul contractible, and therefore it is a Koszul properad.", + "start_pos": 35133, + "end_pos": 35285, + "label": "thm:main" + }, + "ref_dict": { + "thm:main": "\\begin{theorem}\\label{thm:main}\n The properad $\\mathscr{Y}^{(n)}_{prop}$ is Koszul contractible, and therefore it is a Koszul properad.\n\\end{theorem}" + }, + "pre_theorem_intro_text_len": 8242, + "pre_theorem_intro_text": "In this note we work over a field $\\kk$ of characteristic zero, and everything is in $\\kk$-vector spaces. A \\emph{properad} is an object that encodes a type of algebra whose operations have multiple inputs and outputs; one allows composition along any connected graph, and imposes relations that involve compositions given by graphs of any genera. In a \\emph{dioperad}, one only records the data of compositions along trees, and is constrained to impose relations that only involve compositions along trees. A \\emph{planar dioperad} is an even more restrictive notion, where one requires these trees to be embedded in the plane in a coherent manner. There are functors\n\\[ \\PDiop \\xrightarrow{\\Phi} \\Diop \\xrightarrow{F} \\mathsf{Properads}\\]\nthat, starting from a planar dioperad, freely adjoin compositions along any connected graphs, quotienting out by planar genus zero relations.\n\nIf $\\mathscr{Q}$ is a quadratic dioperad that is Koszul (as a dioperad), it is not true in general that $F\\mathscr{Q}$ will be Koszul as a properad; this is notably the case with non-commutative Frobenius algebras \\cite{merkulov2009deformation} and $\\scrV^{(n)}$-algebras \\cite{poirier2019koszuality}. There is, however, a class of dioperads for which dioperadic Koszulity implies properadic Koszulity; this happens for a quadratic dioperad $\\mathscr{Q}$ when the Koszul dual coproperad of $F\\mathscr{Q}$ is in fact a codioperad, that is, its decomposition map does not generate any higher genus terms; in this case, one says that the properad is \\emph{contractible}.\n\nThe properad $\\scrY^{(n)}$ was defined in \\cite{emprin2025properadic}, and one of its uses is to encode the data of an orientation on a space $X$ into the chains on the based loop space. This properad is defined by planar genus zero relations, that is, it satisfies\n\\[ \\mathscr{Y}^{(n)}_{prop} = F \\mathscr{Y}^{(n)} = F\\Phi(\\mathscr{Y}^{(n)}_{pl})\\]\nfor a certain planar dioperad $\\mathscr{Y}^{(n)}_{pl}$. An algebra $A$ over this planar dioperad has\n\\begin{itemize}\n \\item A product $\\mu \\colon A \\otimes A \\to A$ of degree zero,\n \\item A coproduct $\\alpha \\colon A \\to A \\otimes A$ of homological degree $(1-n)$,\n \\item A cyclically symmetric rank three tensor $\\beta \\in A \\otimes A \\otimes A$ of homological degree $(2-2n)$,\n\\end{itemize} \nsatisfying the following relations.\n\n\\small\\begin{align*}\n &\\tikzfig{\n \\node [vertex] (bot) at (0,-0.2) {$\\mu$};\n \\node [vertex] (top) at (-0.5,0.5) {$\\mu$};\n \\draw [->-] (-1,1) to (top);\n \\draw [->-] (0,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (bot);\n \\draw [->-] (bot) to (0,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (bot) at (0,-0.2) {$\\mu$};\n \\node [vertex] (top) at (0.5,0.5) {$\\mu$};\n \\draw [->-] (-1,1) to (bot);\n \\draw [->-] (0,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (top);\n \\draw [->-] (bot) to (0,-1);\n } ~ \\quad ; \\quad \\tikzfig{\n \\node [vertex] (top) at (0,0.5) {$\\mu$};\n \\node [vertex] (bot) at (0,-0.5) {$\\alpha$};\n \\draw [->-] (-1,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (top);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (top) at (0.5,0) {$\\alpha$};\n \\node [vertex] (bot) at (-0.5,0) {$\\mu$};\n \\draw [->-] (-1,1) to (bot);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (top);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (top) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (top) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (bot) at (0.5,0) {$\\mu$};\n \\draw [->-] (-1,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (bot);\n \\draw [->-] (top) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~;\\\\\n &\\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (right) at (0.5,0) {$\\mu$};\n \\draw [->-] (0,1) to (left);\n \\draw [->-] (0,-1) to (right);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (left) to (right);\n \\draw [->-] (right) to (1.2,0);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\mu$};\n \\node [vertex] (right) at (0.5,0) {$\\alpha$};\n \\draw [->-] (0,1) to (right);\n \\draw [->-] (0,-1) to (left);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (right) to (left);\n \\draw [->-] (right) to (1.2,0);\n } ~ + (-1)^n \\left(\\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (right) at (0.5,0) {$\\mu$};\n \\draw [->-] (0,1) to (right);\n \\draw [->-] (0,-1) to (left);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (left) to (right);\n \\draw [->-] (right) to (1.2,0);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\mu$};\n \\node [vertex] (right) at (0.5,0) {$\\alpha$};\n \\draw [->-] (0,1) to (left);\n \\draw [->-] (0,-1) to (right);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (right) to (left);\n \\draw [->-] (right) to (1.2,0);\n }\\right) ~;\n\\end{align*}\n\\begin{align*}\n &\\tikzfig{\n \\node [vertex] (top) at (0,0.4) {$\\mu$};\n \\node [vertex] (bot) at (0,-0.6) {$\\beta$};\n \\draw [->-] (bot) to (top);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (bot) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (top);\n \\draw [->-] (top) to (0,1);\n } ~ - \\qquad \\tikzfig{\n \\node [vertex] (top) at (0,0.1) {$\\beta$};\n \\node [vertex] (bot) at (-0.7,-0.5) {$\\mu$};\n \\draw [->-] (top) to (0,1);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (top) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (bot);\n \\draw [->-] (top) to (bot);\n } ~ + ~ \\tikzfig{\n \\node [vertex] (top) at (0,0.4) {$\\alpha$};\n \\node [vertex] (bot) at (0,-0.6) {$\\alpha$};\n \\draw [->-] (top) to (bot);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (bot) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (top);\n \\draw [->-] (top) to (0,1);\n } ~ - (-1)^{n-1} ~ \\tikzfig{\n \\node [vertex] (top) at (0,0.1) {$\\alpha$};\n \\node [vertex] (bot) at (-0.7,-0.5) {$\\alpha$};\n \\draw [->-] (top) to (0,1);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (top) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (bot);\n \\draw [->-] (bot) to (top);\n } ~; \\\\ \\\\\n &\\tikzfig{\n \\node [vertex] (top) at (0,0.5) {$\\alpha$};\n \\node [vertex] (bot) at (0,-0.5) {$\\beta$};\n \\draw [->-] (bot) to (top);\n \\draw [->-] (top) to (-1,1);\n \\draw [->-] (top) to (1,1);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (right) at (0.5,0) {$\\beta$};\n \\draw [->-] (right) to (left);\n \\draw [->-] (left) to (-1,1);\n \\draw [->-] (left) to (-1,-1);\n \\draw [->-] (right) to (1,-1);\n \\draw [->-] (right) to (1,1);\n } ~ + (-1)^n \\left( ~ \\tikzfig{\n \\node [vertex] (top) at (0,0.5) {$\\beta$};\n \\node [vertex] (bot) at (0,-0.5) {$\\alpha$};\n \\draw [->-] (top) to (bot);\n \\draw [->-] (top) to (-1,1);\n \\draw [->-] (top) to (1,1);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\beta$};\n \\node [vertex] (right) at (0.5,0) {$\\alpha$};\n \\draw [->-] (right) to (left);\n \\draw [->-] (left) to (-1,1);\n \\draw [->-] (left) to (-1,-1);\n \\draw [->-] (right) to (1,-1);\n \\draw [->-] (right) to (1,1);\n } ~ \\right)\n\\end{align*}\\normalsize\n\nNote that all of these relations, and the cyclicity condition on $\\beta$, can be given by linear combinations of diagrams embedded in a disc with a consistent labeling of inputs and outputs, using only the cyclic action, which comes from the rotational symmetry of the disc; this is why this dioperad is planar, see \\cref{sec:planar}. In this note we establish the following theorem.", + "context": "The properad $\\scrY^{(n)}$ was defined in \\cite{emprin2025properadic}, and one of its uses is to encode the data of an orientation on a space $X$ into the chains on the based loop space. This properad is defined by planar genus zero relations, that is, it satisfies\n\\[ \\mathscr{Y}^{(n)}_{prop} = F \\mathscr{Y}^{(n)} = F\\Phi(\\mathscr{Y}^{(n)}_{pl})\\]\nfor a certain planar dioperad $\\mathscr{Y}^{(n)}_{pl}$. An algebra $A$ over this planar dioperad has\n\\begin{itemize}\n \\item A product $\\mu \\colon A \\otimes A \\to A$ of degree zero,\n \\item A coproduct $\\alpha \\colon A \\to A \\otimes A$ of homological degree $(1-n)$,\n \\item A cyclically symmetric rank three tensor $\\beta \\in A \\otimes A \\otimes A$ of homological degree $(2-2n)$,\n\\end{itemize} \nsatisfying the following relations.\n\n\\small\\begin{align*}\n &\\tikzfig{\n \\node [vertex] (bot) at (0,-0.2) {$\\mu$};\n \\node [vertex] (top) at (-0.5,0.5) {$\\mu$};\n \\draw [->-] (-1,1) to (top);\n \\draw [->-] (0,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (bot);\n \\draw [->-] (bot) to (0,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (bot) at (0,-0.2) {$\\mu$};\n \\node [vertex] (top) at (0.5,0.5) {$\\mu$};\n \\draw [->-] (-1,1) to (bot);\n \\draw [->-] (0,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (top);\n \\draw [->-] (bot) to (0,-1);\n } ~ \\quad ; \\quad \\tikzfig{\n \\node [vertex] (top) at (0,0.5) {$\\mu$};\n \\node [vertex] (bot) at (0,-0.5) {$\\alpha$};\n \\draw [->-] (-1,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (top);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (top) at (0.5,0) {$\\alpha$};\n \\node [vertex] (bot) at (-0.5,0) {$\\mu$};\n \\draw [->-] (-1,1) to (bot);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (top);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (top) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (top) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (bot) at (0.5,0) {$\\mu$};\n \\draw [->-] (-1,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (bot);\n \\draw [->-] (top) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~;\\\\\n &\\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (right) at (0.5,0) {$\\mu$};\n \\draw [->-] (0,1) to (left);\n \\draw [->-] (0,-1) to (right);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (left) to (right);\n \\draw [->-] (right) to (1.2,0);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\mu$};\n \\node [vertex] (right) at (0.5,0) {$\\alpha$};\n \\draw [->-] (0,1) to (right);\n \\draw [->-] (0,-1) to (left);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (right) to (left);\n \\draw [->-] (right) to (1.2,0);\n } ~ + (-1)^n \\left(\\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (right) at (0.5,0) {$\\mu$};\n \\draw [->-] (0,1) to (right);\n \\draw [->-] (0,-1) to (left);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (left) to (right);\n \\draw [->-] (right) to (1.2,0);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\mu$};\n \\node [vertex] (right) at (0.5,0) {$\\alpha$};\n \\draw [->-] (0,1) to (left);\n \\draw [->-] (0,-1) to (right);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (right) to (left);\n \\draw [->-] (right) to (1.2,0);\n }\\right) ~;\n\\end{align*}\n\\begin{align*}\n &\\tikzfig{\n \\node [vertex] (top) at (0,0.4) {$\\mu$};\n \\node [vertex] (bot) at (0,-0.6) {$\\beta$};\n \\draw [->-] (bot) to (top);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (bot) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (top);\n \\draw [->-] (top) to (0,1);\n } ~ - \\qquad \\tikzfig{\n \\node [vertex] (top) at (0,0.1) {$\\beta$};\n \\node [vertex] (bot) at (-0.7,-0.5) {$\\mu$};\n \\draw [->-] (top) to (0,1);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (top) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (bot);\n \\draw [->-] (top) to (bot);\n } ~ + ~ \\tikzfig{\n \\node [vertex] (top) at (0,0.4) {$\\alpha$};\n \\node [vertex] (bot) at (0,-0.6) {$\\alpha$};\n \\draw [->-] (top) to (bot);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (bot) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (top);\n \\draw [->-] (top) to (0,1);\n } ~ - (-1)^{n-1} ~ \\tikzfig{\n \\node [vertex] (top) at (0,0.1) {$\\alpha$};\n \\node [vertex] (bot) at (-0.7,-0.5) {$\\alpha$};\n \\draw [->-] (top) to (0,1);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (top) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (bot);\n \\draw [->-] (bot) to (top);\n } ~; \\\\ \\\\\n &\\tikzfig{\n \\node [vertex] (top) at (0,0.5) {$\\alpha$};\n \\node [vertex] (bot) at (0,-0.5) {$\\beta$};\n \\draw [->-] (bot) to (top);\n \\draw [->-] (top) to (-1,1);\n \\draw [->-] (top) to (1,1);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (right) at (0.5,0) {$\\beta$};\n \\draw [->-] (right) to (left);\n \\draw [->-] (left) to (-1,1);\n \\draw [->-] (left) to (-1,-1);\n \\draw [->-] (right) to (1,-1);\n \\draw [->-] (right) to (1,1);\n } ~ + (-1)^n \\left( ~ \\tikzfig{\n \\node [vertex] (top) at (0,0.5) {$\\beta$};\n \\node [vertex] (bot) at (0,-0.5) {$\\alpha$};\n \\draw [->-] (top) to (bot);\n \\draw [->-] (top) to (-1,1);\n \\draw [->-] (top) to (1,1);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\beta$};\n \\node [vertex] (right) at (0.5,0) {$\\alpha$};\n \\draw [->-] (right) to (left);\n \\draw [->-] (left) to (-1,1);\n \\draw [->-] (left) to (-1,-1);\n \\draw [->-] (right) to (1,-1);\n \\draw [->-] (right) to (1,1);\n } ~ \\right)\n\\end{align*}\\normalsize\n\nNote that all of these relations, and the cyclicity condition on $\\beta$, can be given by linear combinations of diagrams embedded in a disc with a consistent labeling of inputs and outputs, using only the cyclic action, which comes from the rotational symmetry of the disc; this is why this dioperad is planar, see \\cref{sec:planar}. In this note we establish the following theorem.", + "full_context": "The properad $\\scrY^{(n)}$ was defined in \\cite{emprin2025properadic}, and one of its uses is to encode the data of an orientation on a space $X$ into the chains on the based loop space. This properad is defined by planar genus zero relations, that is, it satisfies\n\\[ \\mathscr{Y}^{(n)}_{prop} = F \\mathscr{Y}^{(n)} = F\\Phi(\\mathscr{Y}^{(n)}_{pl})\\]\nfor a certain planar dioperad $\\mathscr{Y}^{(n)}_{pl}$. An algebra $A$ over this planar dioperad has\n\\begin{itemize}\n \\item A product $\\mu \\colon A \\otimes A \\to A$ of degree zero,\n \\item A coproduct $\\alpha \\colon A \\to A \\otimes A$ of homological degree $(1-n)$,\n \\item A cyclically symmetric rank three tensor $\\beta \\in A \\otimes A \\otimes A$ of homological degree $(2-2n)$,\n\\end{itemize} \nsatisfying the following relations.\n\n\\small\\begin{align*}\n &\\tikzfig{\n \\node [vertex] (bot) at (0,-0.2) {$\\mu$};\n \\node [vertex] (top) at (-0.5,0.5) {$\\mu$};\n \\draw [->-] (-1,1) to (top);\n \\draw [->-] (0,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (bot);\n \\draw [->-] (bot) to (0,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (bot) at (0,-0.2) {$\\mu$};\n \\node [vertex] (top) at (0.5,0.5) {$\\mu$};\n \\draw [->-] (-1,1) to (bot);\n \\draw [->-] (0,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (top);\n \\draw [->-] (bot) to (0,-1);\n } ~ \\quad ; \\quad \\tikzfig{\n \\node [vertex] (top) at (0,0.5) {$\\mu$};\n \\node [vertex] (bot) at (0,-0.5) {$\\alpha$};\n \\draw [->-] (-1,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (top);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (top) at (0.5,0) {$\\alpha$};\n \\node [vertex] (bot) at (-0.5,0) {$\\mu$};\n \\draw [->-] (-1,1) to (bot);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (top);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (top) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (top) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (bot) at (0.5,0) {$\\mu$};\n \\draw [->-] (-1,1) to (top);\n \\draw [->-] (top) to (bot);\n \\draw [->-] (1,1) to (bot);\n \\draw [->-] (top) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~;\\\\\n &\\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (right) at (0.5,0) {$\\mu$};\n \\draw [->-] (0,1) to (left);\n \\draw [->-] (0,-1) to (right);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (left) to (right);\n \\draw [->-] (right) to (1.2,0);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\mu$};\n \\node [vertex] (right) at (0.5,0) {$\\alpha$};\n \\draw [->-] (0,1) to (right);\n \\draw [->-] (0,-1) to (left);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (right) to (left);\n \\draw [->-] (right) to (1.2,0);\n } ~ + (-1)^n \\left(\\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (right) at (0.5,0) {$\\mu$};\n \\draw [->-] (0,1) to (right);\n \\draw [->-] (0,-1) to (left);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (left) to (right);\n \\draw [->-] (right) to (1.2,0);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\mu$};\n \\node [vertex] (right) at (0.5,0) {$\\alpha$};\n \\draw [->-] (0,1) to (left);\n \\draw [->-] (0,-1) to (right);\n \\draw [->-] (left) to (-1.2,0);\n \\draw [->-] (right) to (left);\n \\draw [->-] (right) to (1.2,0);\n }\\right) ~;\n\\end{align*}\n\\begin{align*}\n &\\tikzfig{\n \\node [vertex] (top) at (0,0.4) {$\\mu$};\n \\node [vertex] (bot) at (0,-0.6) {$\\beta$};\n \\draw [->-] (bot) to (top);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (bot) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (top);\n \\draw [->-] (top) to (0,1);\n } ~ - \\qquad \\tikzfig{\n \\node [vertex] (top) at (0,0.1) {$\\beta$};\n \\node [vertex] (bot) at (-0.7,-0.5) {$\\mu$};\n \\draw [->-] (top) to (0,1);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (top) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (bot);\n \\draw [->-] (top) to (bot);\n } ~ + ~ \\tikzfig{\n \\node [vertex] (top) at (0,0.4) {$\\alpha$};\n \\node [vertex] (bot) at (0,-0.6) {$\\alpha$};\n \\draw [->-] (top) to (bot);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (bot) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (top);\n \\draw [->-] (top) to (0,1);\n } ~ - (-1)^{n-1} ~ \\tikzfig{\n \\node [vertex] (top) at (0,0.1) {$\\alpha$};\n \\node [vertex] (bot) at (-0.7,-0.5) {$\\alpha$};\n \\draw [->-] (top) to (0,1);\n \\draw [->-] (bot) to (-1.2,-1);\n \\draw [->-] (top) to (1.2,-1);\n \\draw [->-] (-1.2,0.4) to (bot);\n \\draw [->-] (bot) to (top);\n } ~; \\\\ \\\\\n &\\tikzfig{\n \\node [vertex] (top) at (0,0.5) {$\\alpha$};\n \\node [vertex] (bot) at (0,-0.5) {$\\beta$};\n \\draw [->-] (bot) to (top);\n \\draw [->-] (top) to (-1,1);\n \\draw [->-] (top) to (1,1);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\alpha$};\n \\node [vertex] (right) at (0.5,0) {$\\beta$};\n \\draw [->-] (right) to (left);\n \\draw [->-] (left) to (-1,1);\n \\draw [->-] (left) to (-1,-1);\n \\draw [->-] (right) to (1,-1);\n \\draw [->-] (right) to (1,1);\n } ~ + (-1)^n \\left( ~ \\tikzfig{\n \\node [vertex] (top) at (0,0.5) {$\\beta$};\n \\node [vertex] (bot) at (0,-0.5) {$\\alpha$};\n \\draw [->-] (top) to (bot);\n \\draw [->-] (top) to (-1,1);\n \\draw [->-] (top) to (1,1);\n \\draw [->-] (bot) to (-1,-1);\n \\draw [->-] (bot) to (1,-1);\n } ~ - ~ \\tikzfig{\n \\node [vertex] (left) at (-0.5,0) {$\\beta$};\n \\node [vertex] (right) at (0.5,0) {$\\alpha$};\n \\draw [->-] (right) to (left);\n \\draw [->-] (left) to (-1,1);\n \\draw [->-] (left) to (-1,-1);\n \\draw [->-] (right) to (1,-1);\n \\draw [->-] (right) to (1,1);\n } ~ \\right)\n\\end{align*}\\normalsize\n\nNote that all of these relations, and the cyclicity condition on $\\beta$, can be given by linear combinations of diagrams embedded in a disc with a consistent labeling of inputs and outputs, using only the cyclic action, which comes from the rotational symmetry of the disc; this is why this dioperad is planar, see \\cref{sec:planar}. In this note we establish the following theorem.\n\nIf $\\mathscr{Q}$ is a quadratic dioperad that is Koszul (as a dioperad), it is not true in general that $F\\mathscr{Q}$ will be Koszul as a properad; this is notably the case with non-commutative Frobenius algebras \\cite{merkulov2009deformation} and $\\scrV^{(n)}$-algebras \\cite{poirier2019koszuality}. There is, however, a class of dioperads for which dioperadic Koszulity implies properadic Koszulity; this happens for a quadratic dioperad $\\mathscr{Q}$ when the Koszul dual coproperad of $F\\mathscr{Q}$ is in fact a codioperad, that is, its decomposition map does not generate any higher genus terms; in this case, one says that the properad is \\emph{contractible}.\n\nIt was proven in \\cite{emprin2025properadic} that the dioperad $\\mathscr{Y}^{(n)}$ is contractible; the other component that allows us to establish Koszulity of the properad $\\scrY^{(n)}_{prop}$ is a geometric interpretation of the bar complexes associated to (the Koszul dual of) the planar dioperad $\\scrY^{(n)}_{pl}$. This interpretation is in the same spirit as the description of the ``PROP of open-closed marked surfaces'' of \\cite{kontsevich2025pre} using moduli spaces of meromorphic Strebel differentials. In \\cref{sec:planar} we will recall this theory, in the specific case of differentials on $\\mathbb{CP}^1$ with a higher-order pole at infinity, or equivalently, polynomial quadratic differentials on $\\CC$. The moduli spaces of such objects has a natural non-compact cell complex structure that is dual to the cell complex structure of the \\emph{assocoipahedra} of \\cite{poirier2018combinatorics}, and gives an alternative proof of their compatibility, established in that reference; by the results of \\cite{poirier2019koszuality} this implies Koszulity of the dioperad $\\mathscr{V}^{(n)}$.\n\nAs a consequence of \\cref{thm:main}, the map \n\\[ \\scrY^{(n)}_\\infty = \\Omega (\\scrY^{(n)}_{prop})^{\\antish} \\to \\scrY^{(n)}_{prop} \\]\nis a quasi-isomorphism of dg properads and therefore gives a cofibrant resolution of the properad $\\scrY^{(n)}_{prop}$; the results on formality in \\cite{emprin2025properadic} can then be seen as completely analogous to the formality of $A_\\infty$-algebras, but in the properadic setting of $\\scrY^{(n)}_{prop}$-algebras. Together with the explicit formulas for homotopy transfer of \\cite{hoffbeck2021properadic}, we have as a consequence of our main theorem.\n\\begin{corollary}\n The homotopy transferred structures for $\\scrY^{(n)}_{prop}$-algebras can be computed by formulas only involving sums of planar trees.\n\\end{corollary}\n\nTherefore, for any $\\mathbf{C}$ as above, the subspace\n\\[ \\mathrm{ClovQ}_\\mathbf{C} \\subset Q^\\mathrm{Str}_\\mathrm{reg}(k;i_1,\\dots,i_k) \\]\nof regularized Strebel differentials that are cloven by $\\mathbf{C}$ is a union of cells, each of which labeled by a directed tree that has at least $r$ bivalent vertices. This union of cells is coclosed, in the sense that the if $a$ is in $\\mathrm{ClovQ}_\\mathbf{C}$ and is on the boundary of $b$, then so is $b$. So we can write\n\\[ \\mathrm{Clov}_\\mathbf{C} \\subset Z_{(k;i_1,\\dots,i_k)} \\]\nfor the corresponding union of cells in the assocoipahedron, which will be a cell subcomplex. We now give the main proposition of this section.\n\\begin{proposition}\\label{prop:contractible}\n For any collection of pairwise distinct equivalence classes \n \\[ \\mathbf{C} = \\{C_1,\\dots,C_r\\}, \\] \n the space $\\mathrm{Clov}_\\mathbf{C}$ is contractible if and only if there are $r$ cuts $\\gamma_1,\\dots,\\gamma_r$, one in each equivalence class, that are pairwise disjoint and divide the complex plane into regions such that each contains at least one of the $k$ output directions. In every other case, $\\mathrm{Clov}_\\mathbf{C}$ is empty.\n\\end{proposition}\n\\begin{proof}\n Let us start by proving the contrapositive of the last statement. If $\\mathrm{Clov}_\\mathbf{C}$ is not empty, $\\mathrm{ClovQ}_\\mathbf{C}$ must contain at least one point; taking the $r$ saddle cuts gives us the desired representatives $\\gamma_1,\\dots,\\gamma_r$.\n\n\\begin{proposition}\\label{prop:barcomplexKoszul}\n The complex $(\\mathbf{B}^*_{pl} (\\scrY^{(n)}_{pl})^!)_{(k;i_1,\\dots,i_k)}$ has cohomology concentrated in degree zero.\n\\end{proposition}\n\\begin{proof}\n For ease of notation, let us write\n \\[ Y^* = (\\mathbf{B}^*_{pl} (\\mathscr{Y}^{(n)}_{pl})^!)_{(k;i_1,\\dots,i_k)}, \\quad Z^* = (\\mathbf{B}^*_{pl} (\\mathscr{V}^{(n)}_{pl})^!)_{(k;i_1,\\dots,i_k)} \\cong C^*(Z_{(k;i_1,\\dots,i_k)},\\ZZ) \\]\n By definition, we have an exact sequence\n \\[ Y^{*-k+1} \\hookrightarrow Z^* \\twoheadrightarrow R^* \\]\n and the result then follows from the induced long exact sequence in cohomology, together with the vanishing results above, and contractibility of $Z^*$.\n\\end{proof}\n\n\\subsection{Dioperadic and properadic Koszulity}\nThe contractibility of these planar bar complexes immediately implies dioperadic Koszulity.\n\\begin{corollary}\\label{cor:diopKoszul}\n The dioperad $\\mathscr{Y}^{(n)} = \\Phi(\\scrY^{(n)}_{pl})$ is a Koszul dioperad.\n\\end{corollary}\n\\begin{proof}\n From \\cref{prop:KoszulDiop,prop:barcomplexKoszul} it follows that $(\\scrY^{(n)})^! \\cong \\Phi((\\scrY^{(n)}_{pl})^!)$ is a Koszul dioperad; since $\\scrY^{(n)}$ is finitely generated, we conclude that $\\scrY^{(n)} \\cong (\\scrY^{(n)})^{!!}$ is also a Koszul dioperad.\n\\end{proof}\n\nThe functor $F$ \\emph{does not} intertwine dioperadic and properadic Koszul duality. That is, for an arbitrary finitely-generated quadratic dioperad $\\mathscr{Q}$, the properads\n\\[ F(\\mathscr{Q}^!) \\quad \\text{and} \\quad (F \\mathscr{Q})^! \\]\nneed not be isomorphic, even if $\\mathscr{Q}$ is Koszul; in other words, the Koszul dual properad $(F \\mathscr{Q})^!$ might have nontrivial higher-genus relations, or dually, the decomposition maps of the Koszul dual coproperad $(F \\mathscr{Q})^{\\antish}$ is not a codioperad, that is, if its decomposition map generates higher-genus graphs from genus-zero ones.\n\\begin{definition}\n A quadratic dioperad $\\mathscr{Q}$ is called \\emph{contractible} when $(F \\mathscr{Q})^{\\antish}$ is a codioperad. \n\\end{definition}\nIf $\\mathscr{Q}$ is a finitely-generated quadratic dioperad, then it is contractible if and only if the canonical map $F(\\mathscr{Q}^!) \\to (F \\mathscr{Q})^!$ is an isomorphism of properads. Contractibility means that one can deduce Koszulity of the properad from Koszulity of the dioperad.\n\\begin{proposition}\n If a quadratic dioperad $\\mathscr{Q}$ is Koszul (as a dioperad) and contractible, then the properad $F\\mathscr{Q}$ is Koszul (as a properad).\n\\end{proposition}\nIn the case the proposition above holds, one says that the properad $F\\mathscr{Q}$ is \\emph{Koszul contractible}. Together with \\cite[Thm.1.23]{emprin2025properadic}, which shows that $\\scrY^{(n)}$ is a contractible dioperad, and \\cref{cor:diopKoszul}, we deduce the main theorem of this paper.\n\\begin{theorem*}(\\cref{thm:main})\n The properad $\\scrY^{(n)}_{\\mathrm{prop}} = F(\\scrY^{(n)})$ is Koszul contractible, and therefore a Koszul properad.\n\\end{theorem*}", + "post_theorem_intro_text_len": 6425, + "post_theorem_intro_text": "It was proven in \\cite{emprin2025properadic} that the dioperad $\\mathscr{Y}^{(n)}$ is contractible; the other component that allows us to establish Koszulity of the properad $\\scrY^{(n)}_{prop}$ is a geometric interpretation of the bar complexes associated to (the Koszul dual of) the planar dioperad $\\scrY^{(n)}_{pl}$. This interpretation is in the same spirit as the description of the ``PROP of open-closed marked surfaces'' of \\cite{kontsevich2025pre} using moduli spaces of meromorphic Strebel differentials. In \\cref{sec:planar} we will recall this theory, in the specific case of differentials on $\\mathbb{CP}^1$ with a higher-order pole at infinity, or equivalently, polynomial quadratic differentials on $\\CC$. The moduli spaces of such objects has a natural non-compact cell complex structure that is dual to the cell complex structure of the \\emph{assocoipahedra} of \\cite{poirier2018combinatorics}, and gives an alternative proof of their compatibility, established in that reference; by the results of \\cite{poirier2019koszuality} this implies Koszulity of the dioperad $\\mathscr{V}^{(n)}$.\n\nIn the $\\mathscr{Y}^{(n)}$ case, one needs to remove a certain subcomplex of \\emph{cloven Strebel differentials}; these are quadratic differentials that split the complex plane into regions in a specific way. Using geometric arguments, we can control the topology of this subcomplex, concluding that it is always homologous to a bouquet of $(k-2)$-spheres, where $k$ is the number of outputs we are looking at. Together with a comparison between the bar complexes associated to $\\mathscr{V}^{(n)}$ and $\\mathscr{Y}^{(n)}$, vanishing of this homology outside of that specific degree implies vanishing of higher cohomology of the relevant bar complex, establishing our main theorem.\n\nAs a consequence of \\cref{thm:main}, the map \n\\[ \\scrY^{(n)}_\\infty = \\Omega (\\scrY^{(n)}_{prop})^{\\antish} \\to \\scrY^{(n)}_{prop} \\]\nis a quasi-isomorphism of dg properads and therefore gives a cofibrant resolution of the properad $\\scrY^{(n)}_{prop}$; the results on formality in \\cite{emprin2025properadic} can then be seen as completely analogous to the formality of $A_\\infty$-algebras, but in the properadic setting of $\\scrY^{(n)}_{prop}$-algebras. Together with the explicit formulas for homotopy transfer of \\cite{hoffbeck2021properadic}, we have as a consequence of our main theorem.\n\\begin{corollary}\n The homotopy transferred structures for $\\scrY^{(n)}_{prop}$-algebras can be computed by formulas only involving sums of planar trees.\n\\end{corollary}\n\n\\subsection{Relation to other known results}\nWe would like to point out that there are three other closely related dioperads for which Koszulity has been studied. The first one is the dioperad $\\scrV^{(n)}$ of Poirier and Tradler, following earlier work of Tradler and Zeinalian \\cite{tradler2007infinity}; algebras over this dioperad are pre-Calabi--Yau algebras, in the formulation of \\cite{kontsevich2025pre}. We cite results of \\cite{poirier2019koszuality}.\n\\begin{theorem*}\n The dioperad $\\scrV^{(n)}$ is Koszul (as a dioperad) but not contractible.\n\\end{theorem*}\n\nAnother one is the dioperad of balanced infinitesimal bialgebras $\\mathrm{BIB}^\\lambda$ used by Quesney in \\cite{quesney2024balanced}; the algebras that this dioperad encodes are associative analogues of Lie bialgebras, and have a coproduct of degree $\\lambda$. It was proved in that paper that this dioperad is contractible, but using deformation theory techniques, Merkulov showed in \\cite{merkulov2025complex} the following result.\n\\begin{theorem*}\n The properad $F(\\mathrm{BIB}^\\lambda)$ is not Koszul, so as a consequence the dioperad $\\mathrm{BIB}^\\lambda$ cannot be Koszul.\n\\end{theorem*}\n\nFinally, the third dioperad that I would like to mention is the dioperad $\\mathrm{DPois}$ encoding the double Poisson algebras of van den Bergh; this dioperad was studied by Leray in \\cite{leray2019protoperadsI,leray2020protoperadsII}, who showed the following result, see also \\cite{leray2025precalabiyau} for a more recent discussion.\n\\begin{theorem*}\n The dioperad $\\mathrm{DPois}$ is Koszul contractible.\n\\end{theorem*}\n\nThe relationship between $\\scrY^(n)$ and these three other dioperads was established in \\cite{leray2025precalabiyau,quesney2024balanced,emprin2025properadic}. Their Koszul dual dioperads are related by inclusions\n\\[ \\mathrm{DPois}^! \\overset{n=2}\\hookrightarrow (\\mathrm{BIB}^{1-n})^! \\hookrightarrow (\\scrY^{(n)})^! \\hookrightarrow (\\scrV^{(n)})^! \\]\nwhere the superscript on the first arrow means that this map only exists for that value of $n$.\\footnote{There are shifted versions of double Poisson algebras, but the results we cite are only phrased for the unshifted case.} We note that the first and third maps do not respect weight-grading, and do not come from taking Koszul dual maps.\n\nDually, we have quotients of codioperads\n\\[ \\mathrm{DPois}^{\\antish} \\overset{n=2}\\twoheadleftarrow (\\mathrm{BIB}^{1-n})^{\\antish} \\twoheadleftarrow (\\scrY^{(n)})^{\\antish} \\twoheadleftarrow (\\scrV^{(n)})^{\\antish} \\]\nand taking cobar, quotients of dg codioperads\n\\[ \\mathrm{DPois}_\\infty \\overset{n=2}\\twoheadleftarrow \\Omega (\\mathrm{BIB}^{1-n})^{\\antish} \\twoheadleftarrow \\scrY^{(n)}_\\infty \\twoheadleftarrow \\scrV^{(n)}_\\infty = \\mathrm{pCY}, \\]\nso the notion of a $\\scrY^{(n)}_\\infty$-algebra is extremely close to the notion of a $\\scrV^{(n)}_\\infty$ or pre-CY algebra; in fact, it is just a pre-CY algebra with vanishing copairing. When $n \\ge 1$, every pre-CY algebra that is concentrated in non-negative homological degree must have vanishing copairing by degree reasons, and will be a $\\scrV^{(n)}_\\infty$-algebra. We summarize the results cited above in a table for convenience.\n\\begin{center}\n \\begin{tabular}{|c||c|c|c|c|}\n \\hline\n & $\\mathrm{DPois}$ & $\\mathrm{BIB}^{1-n}$ & $\\scrY^{(n)}$ & $\\scrV^{(n)}$ \\\\\n \\hline\\hline\n Dioperad is Koszul? & Yes & No & Yes & Yes \\\\\n \\hline\n Properad is Koszul? & Yes & No & Yes & Unknown \\\\\n \\hline\n \\end{tabular}\n\\end{center}\n\n\\noindent \\textbf{Acknowledgments:} I would like to thank C.~Emprin, S.~Merkulov and B.~Vallette for helpful discussions, and Uppsala University for the great working environment. This work was supported by the Knut and Alice Wallenberg foundation.", + "sketch": "It is recalled that \\cite{emprin2025properadic} proves the dioperad $\\mathscr{Y}^{(n)}$ is contractible; to establish Koszulity (and hence Koszul contractibility) for the properad $\\scrY^{(n)}_{prop}$, the other ingredient is “a geometric interpretation of the bar complexes associated to (the Koszul dual of) the planar dioperad $\\scrY^{(n)}_{pl}$.” The introduction outlines that one studies moduli spaces of meromorphic Strebel differentials (here specialized to $\\mathbb{CP}^1$ with a higher-order pole at infinity / polynomial quadratic differentials on $\\CC$), which carry “a natural non-compact cell complex structure” related to assocoipahedra; by \\cite{poirier2019koszuality} this type of input yields Koszulity in the $\\mathscr{V}^{(n)}$ setting.\n\nFor $\\mathscr{Y}^{(n)}$, the sketch says one must “remove a certain subcomplex of \\emph{cloven Strebel differentials}.” One then uses “geometric arguments” to control its topology, concluding this subcomplex “is always homologous to a bouquet of $(k-2)$-spheres, where $k$ is the number of outputs.” Finally, “together with a comparison between the bar complexes associated to $\\mathscr{V}^{(n)}$ and $\\mathscr{Y}^{(n)}$,” the fact that this homology vanishes outside that degree implies “vanishing of higher cohomology of the relevant bar complex,” which is said to “establish[] our main theorem” (i.e. Theorem~\\ref{thm:main}).", + "expanded_sketch": "It is recalled that \\cite{emprin2025properadic} proves the dioperad $\\mathscr{Y}^{(n)}$ is contractible; to establish Koszulity (and hence Koszul contractibility) for the properad $\\scrY^{(n)}_{prop}$, the other ingredient is “a geometric interpretation of the bar complexes associated to (the Koszul dual of) the planar dioperad $\\scrY^{(n)}_{pl}$.” The introduction outlines that one studies moduli spaces of meromorphic Strebel differentials (here specialized to $\\mathbb{CP}^1$ with a higher-order pole at infinity / polynomial quadratic differentials on $\\CC$), which carry “a natural non-compact cell complex structure” related to assocoipahedra; by \\cite{poirier2019koszuality} this type of input yields Koszulity in the $\\mathscr{V}^{(n)}$ setting.\n\nFor $\\mathscr{Y}^{(n)}$, the sketch says one must “remove a certain subcomplex of \\emph{cloven Strebel differentials}.” One then uses “geometric arguments” to control its topology, concluding this subcomplex “is always homologous to a bouquet of $(k-2)$-spheres, where $k$ is the number of outputs.” Finally, “together with a comparison between the bar complexes associated to $\\mathscr{V}^{(n)}$ and $\\mathscr{Y}^{(n)}$,” the fact that this homology vanishes outside that degree implies “vanishing of higher cohomology of the relevant bar complex,” which is said to establish the main theorem.", + "expanded_theorem": "\\label{thm:main}\n The properad $\\mathscr{Y}^{(n)}_{prop}$ is Koszul contractible, and therefore it is a Koszul properad.", + "theorem_type": [ + "Universal", + "Implication" + ], + "mcq": { + "question": "Let \\(\\mathscr{Y}^{(n)}\\) be the quadratic dioperad associated to the planar dioperad \\(\\mathscr{Y}^{(n)}_{pl}\\), and let \\(\\mathscr{Y}^{(n)}_{\\mathrm{prop}} = F(\\mathscr{Y}^{(n)})\\) be the properad obtained from it by the functor \\(F\\). Here, a properad of the form \\(F\\mathscr{Q}\\) is called \\emph{Koszul contractible} when the quadratic dioperad \\(\\mathscr{Q}\\) is both Koszul and contractible; for finitely generated quadratic dioperads, contractibility is equivalent to the canonical map \\(F(\\mathscr{Q}^!) \\to (F\\mathscr{Q})^!\\) being an isomorphism of properads. Which statement holds for \\(\\mathscr{Y}^{(n)}_{\\mathrm{prop}}\\)?", + "correct_choice": { + "label": "A", + "text": "The properad \\(\\mathscr{Y}^{(n)}_{\\mathrm{prop}}\\) is Koszul contractible, and therefore it is a Koszul properad." + }, + "choices": [ + { + "label": "B", + "text": "The properad \\(\\mathscr{Y}^{(n)}_{\\mathrm{prop}}\\) is a Koszul properad, but it is not Koszul contractible because the canonical map \\(F((\\mathscr{Y}^{(n)})^!) \\to (F\\mathscr{Y}^{(n)})^!\\) need not be an isomorphism of properads." + }, + { + "label": "C", + "text": "The properad \\(\\mathscr{Y}^{(n)}_{\\mathrm{prop}}\\) is a Koszul properad." + }, + { + "label": "D", + "text": "The properad \\(\\mathscr{Y}^{(n)}_{\\mathrm{prop}}\\) is contractible in the sense that the canonical map \\(F((\\mathscr{Y}^{(n)})^!) \\to (F\\mathscr{Y}^{(n)})^!\\) is an isomorphism of properads, but it is not Koszul contractible because Koszulity of \\(\\mathscr{Y}^{(n)}\\) as a dioperad does not imply Koszulity of \\(F(\\mathscr{Y}^{(n)})\\) as a properad." + }, + { + "label": "E", + "text": "The properad \\(\\mathscr{Y}^{(n)}_{\\mathrm{prop}}\\) is Koszul contractible only after restricting to the genus-zero part; equivalently, the canonical map \\(F((\\mathscr{Y}^{(n)})^!) \\to (F\\mathscr{Y}^{(n)})^!\\) is an isomorphism merely as a dioperad, not as a properad." + } + ], + "meta": { + "weaker_true_label": "C", + "false_labels": [ + "B", + "D", + "E" + ], + "wildcard_false_label": "E" + }, + "sketch_usage_meta": [ + { + "label": "B", + "sketch_hook_type": "geometric_construction", + "tampered_component": "comparison-plus-vanishing needed to upgrade contractibility to properadic Koszul contractibility", + "template_used": "property_confusion" + }, + { + "label": "C", + "sketch_hook_type": "geometric_construction", + "tampered_component": "drops the contractibility/Koszul-contractible assertion while keeping the properadic Koszul conclusion", + "template_used": "weaker_true" + }, + { + "label": "D", + "sketch_hook_type": "other", + "tampered_component": "proposition that Koszul + contractible dioperad implies \\(F\\mathscr Q\\) is Koszul as a properad", + "template_used": "quantifier_dependence" + }, + { + "label": "E", + "sketch_hook_type": "geometric_construction", + "tampered_component": "full properadic isomorphism versus genus-zero/dioperadic-only control", + "template_used": "wildcard" + } + ] + }, + "qa quality eval": { + "ALS": { + "score": 2, + "justification": "The stem gives definitions and a criterion for Koszul contractibility, but it does not explicitly state that \\(\\mathscr{Y}^{(n)}\\) or \\(\\mathscr{Y}^{(n)}_{\\mathrm{prop}}\\) has the required properties. The correct answer is not directly leaked." + }, + "TAS": { + "score": 1, + "justification": "The item is close to a theorem-recall question: it asks which global statement holds for a named object, and the correct option largely packages the standard conclusion. Still, the presence of a weaker true option and several subtle false variants prevents it from being a pure restatement." + }, + "GPS": { + "score": 1, + "justification": "Some reasoning is required to distinguish the strongest valid conclusion from the merely weaker true statement (C) and from distractors that confuse contractibility with properadic Koszulity. However, for a student who knows the result, the answer is fairly immediate." + }, + "DQS": { + "score": 2, + "justification": "The distractors are plausible and mathematically meaningful: they target common confusions about whether contractibility upgrades to properadic Koszulity, whether the canonical map is an isomorphism, and whether only a weaker conclusion is justified." + }, + "total_score": 6, + "overall_assessment": "A solid but theorem-proximal MCQ. It avoids direct answer leakage and uses strong distractors, but it leans more toward recognition of a known result than deep generative reasoning." + } + } +]