diff --git "a/202601/qa_202601_all.jsonl" "b/202601/qa_202601_all.jsonl" new file mode 100644--- /dev/null +++ "b/202601/qa_202601_all.jsonl" @@ -0,0 +1,67 @@ +{"id": "2601.00782v2", "paper_link": "http://arxiv.org/abs/2601.00782v2", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm:main1}\n Let $P$ be a weakly ranked, \n finite, bounded poset of weak rank $n$. Then $h(P)$ forms a $\\operatorname{SI}$-sequence.", "start_pos": 7074, "end_pos": 7234, "label": "thm:main1"}, "ref_dict": {"thm:main2": "\\begin{theorem}\\label{thm:main2}\n Let $P$ be a ranked, finite, bounded poset of rank $n$. Then the sequence\n \\[\n \\left(h_0,h_1 - h_0, \\ldots,h_{\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor} - h_{\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor - 1}\\right)\n \\]\n is a pure $O$-sequence.\n\\end{theorem}", "def:SI": "\\begin{definition}\\label{def:SI}\n A \\emph{Stanley-Iarrobino sequence} (or \\emph{$\\SI$-sequence} for short) is a nonnegative, palindromic sequence $h$ that is a \\emph{differentially $O$-sequence}, i.e., $\\Delta h$ is an $O$-sequence.\n \\end{definition}", "thm:characterization SI sequence": "\\begin{theorem}[\\cite{gThmSufficiency,gThmNecessity, gThmNecessity2,Lefschetz}]\\label{thm:characterization SI sequence}\n For a sequence $$h=(1,h_1,\\ldots,h_e)$$ the following statements are equivalent:\n \\begin{itemize}\n \\item $h$ is an $SI$-sequence,\n \\item $h$ is the $h$-vector of a simplicial polytope,\n \\item $h$ is the Hilbert--Poincaré series of a Gorenstein algebra with the Weak Lefschetz property,\n \\item $h$ is the Hilbert--Poincaré series of a Gorenstein algebra with the Strong Lefschetz property,\n \\item $h$ is the Hilbert--Poincaré series of a Gorenstein algebra with the Kähler package.\n \\end{itemize}\n \\end{theorem}", "cor:main lefschetz": "\\begin{corollary}\\label{cor:main lefschetz}\n Let $P$ be a weakly ranked, finite, bounded poset of weak rank $n$. Then there exists a Gorenstein algebra with Kähler package whose Hilbert--Poincaré series coincides with the Chow polynomial of $P$.\n\\end{corollary}", "def:O": "\\begin{definition}\\label{def:O}\n We say that a sequence of nonnegative integers $$h=(h_0,h_1,h_2,\\ldots,h_e)$$ is a \\textit{(pure) $O$-sequence} if it is the $h$-vector of a (pure) monomial order ideal.\n \\end{definition}", "thm:main1": "\\begin{theorem}\\label{thm:main1}\n Let $P$ be a weakly ranked, \n finite, bounded poset of weak rank $n$. Then $h(P)$ forms a $\\SI$-sequence. \n\\end{theorem}"}, "pre_theorem_intro_text_len": 2443, "pre_theorem_intro_text": "A central theme in the study of enumerative invariants of posets is the extent to which integer sequences satisfy strong regularity properties, such as unimodality, palindromicity, log-concavity, or the existence of combinatorial or algebraic models realizing them as Hilbert--Poincaré series. A classical example of such a construction is due to Stanley \\cite{gThmNecessity} and independently McMullen \\cite{gThmNecessity2}, who construct for every simplicial polytope $P$ a graded ring whose Hilbert--Poincaré series coincides with the \\emph{$h$-vector} of $P$.\nChow polynomials of partially ordered sets, introduced recently by Ferroni, Matherne, and the second author, provide a unifying framework for encoding subtle enumerative and algebraic properties of posets \\cite{ferroni-matherne-vecchi}. The name comes from the theory of combinatorial algebraic geometry for matroids; when the poset is a geometric lattice, it coincides with the Hilbert--Poincaré series of the Chow ring of the associated matroid, as defined by Feichtner and Yuzvinsky \\cite{FY}. In this setting, Adiprasito, Huh and Katz proved that the Chow ring of a matroid satisfies a combinatorial analogue of the Kähler package \\cite{chowring}. This collection of properties includes Poincaré duality, strong Lefschetz property and the Hodge--Riemann relations, which imply the palindromicity and unimodality of the corresponding Chow polynomials. For general posets there is not such a ring-theoretic interpretation, however similar properties still hold. In particular, it is known that the coefficients of the characteristic Chow polynomial of any poset are non-negative, palindromic, and unimodal, and that they are also $\\gamma$-positive for Cohen--Macaulay posets \\cite{ferroni-matherne-vecchi}. \nAn outstanding open problem is whether a ring with Poincaré duality and the strong Lefschetz property can be constructed for any poset so that its Hilbert--Poincaré series coincides with the Chow polynomial \\cite[Section 4.6]{ferroni-matherne-vecchi}.\n\nIn this paper, we will show that there exist algebras with these properties for any weakly ranked poset. This serves as further evidence for the possibility of finding a suitable definition for the generalization of Chow rings. \nLet us denote by $h = h(P) = (h_0,h_1, \\ldots,h_{n-1})$ the sequence of coefficients of the Chow polynomial of a given poset $P$ of weak rank $n$. \nOur first main result is the following.", "context": "A central theme in the study of enumerative invariants of posets is the extent to which integer sequences satisfy strong regularity properties, such as unimodality, palindromicity, log-concavity, or the existence of combinatorial or algebraic models realizing them as Hilbert--Poincaré series. A classical example of such a construction is due to Stanley \\cite{gThmNecessity} and independently McMullen \\cite{gThmNecessity2}, who construct for every simplicial polytope $P$ a graded ring whose Hilbert--Poincaré series coincides with the \\emph{$h$-vector} of $P$.\nChow polynomials of partially ordered sets, introduced recently by Ferroni, Matherne, and the second author, provide a unifying framework for encoding subtle enumerative and algebraic properties of posets \\cite{ferroni-matherne-vecchi}. The name comes from the theory of combinatorial algebraic geometry for matroids; when the poset is a geometric lattice, it coincides with the Hilbert--Poincaré series of the Chow ring of the associated matroid, as defined by Feichtner and Yuzvinsky \\cite{FY}. In this setting, Adiprasito, Huh and Katz proved that the Chow ring of a matroid satisfies a combinatorial analogue of the Kähler package \\cite{chowring}. This collection of properties includes Poincaré duality, strong Lefschetz property and the Hodge--Riemann relations, which imply the palindromicity and unimodality of the corresponding Chow polynomials. For general posets there is not such a ring-theoretic interpretation, however similar properties still hold. In particular, it is known that the coefficients of the characteristic Chow polynomial of any poset are non-negative, palindromic, and unimodal, and that they are also $\\gamma$-positive for Cohen--Macaulay posets \\cite{ferroni-matherne-vecchi}. \nAn outstanding open problem is whether a ring with Poincaré duality and the strong Lefschetz property can be constructed for any poset so that its Hilbert--Poincaré series coincides with the Chow polynomial \\cite[Section 4.6]{ferroni-matherne-vecchi}.\n\nIn this paper, we will show that there exist algebras with these properties for any weakly ranked poset. This serves as further evidence for the possibility of finding a suitable definition for the generalization of Chow rings. \nLet us denote by $h = h(P) = (h_0,h_1, \\ldots,h_{n-1})$ the sequence of coefficients of the Chow polynomial of a given poset $P$ of weak rank $n$. \nOur first main result is the following.", "full_context": "A central theme in the study of enumerative invariants of posets is the extent to which integer sequences satisfy strong regularity properties, such as unimodality, palindromicity, log-concavity, or the existence of combinatorial or algebraic models realizing them as Hilbert--Poincaré series. A classical example of such a construction is due to Stanley \\cite{gThmNecessity} and independently McMullen \\cite{gThmNecessity2}, who construct for every simplicial polytope $P$ a graded ring whose Hilbert--Poincaré series coincides with the \\emph{$h$-vector} of $P$.\nChow polynomials of partially ordered sets, introduced recently by Ferroni, Matherne, and the second author, provide a unifying framework for encoding subtle enumerative and algebraic properties of posets \\cite{ferroni-matherne-vecchi}. The name comes from the theory of combinatorial algebraic geometry for matroids; when the poset is a geometric lattice, it coincides with the Hilbert--Poincaré series of the Chow ring of the associated matroid, as defined by Feichtner and Yuzvinsky \\cite{FY}. In this setting, Adiprasito, Huh and Katz proved that the Chow ring of a matroid satisfies a combinatorial analogue of the Kähler package \\cite{chowring}. This collection of properties includes Poincaré duality, strong Lefschetz property and the Hodge--Riemann relations, which imply the palindromicity and unimodality of the corresponding Chow polynomials. For general posets there is not such a ring-theoretic interpretation, however similar properties still hold. In particular, it is known that the coefficients of the characteristic Chow polynomial of any poset are non-negative, palindromic, and unimodal, and that they are also $\\gamma$-positive for Cohen--Macaulay posets \\cite{ferroni-matherne-vecchi}. \nAn outstanding open problem is whether a ring with Poincaré duality and the strong Lefschetz property can be constructed for any poset so that its Hilbert--Poincaré series coincides with the Chow polynomial \\cite[Section 4.6]{ferroni-matherne-vecchi}.\n\nIn this paper, we will show that there exist algebras with these properties for any weakly ranked poset. This serves as further evidence for the possibility of finding a suitable definition for the generalization of Chow rings. \nLet us denote by $h = h(P) = (h_0,h_1, \\ldots,h_{n-1})$ the sequence of coefficients of the Chow polynomial of a given poset $P$ of weak rank $n$. \nOur first main result is the following.\n\n\\section{Introduction}\nA central theme in the study of enumerative invariants of posets is the extent to which integer sequences satisfy strong regularity properties, such as unimodality, palindromicity, log-concavity, or the existence of combinatorial or algebraic models realizing them as Hilbert--Poincaré series. A classical example of such a construction is due to Stanley \\cite{gThmNecessity} and independently McMullen \\cite{gThmNecessity2}, who construct for every simplicial polytope $P$ a graded ring whose Hilbert--Poincaré series coincides with the \\emph{$h$-vector} of $P$.\nChow polynomials of partially ordered sets, introduced recently by Ferroni, Matherne, and the second author, provide a unifying framework for encoding subtle enumerative and algebraic properties of posets \\cite{ferroni-matherne-vecchi}. The name comes from the theory of combinatorial algebraic geometry for matroids; when the poset is a geometric lattice, it coincides with the Hilbert--Poincaré series of the Chow ring of the associated matroid, as defined by Feichtner and Yuzvinsky \\cite{FY}. In this setting, Adiprasito, Huh and Katz proved that the Chow ring of a matroid satisfies a combinatorial analogue of the Kähler package \\cite{chowring}. This collection of properties includes Poincaré duality, strong Lefschetz property and the Hodge--Riemann relations, which imply the palindromicity and unimodality of the corresponding Chow polynomials. For general posets there is not such a ring-theoretic interpretation, however similar properties still hold. In particular, it is known that the coefficients of the characteristic Chow polynomial of any poset are non-negative, palindromic, and unimodal, and that they are also $\\gamma$-positive for Cohen--Macaulay posets \\cite{ferroni-matherne-vecchi}. \nAn outstanding open problem is whether a ring with Poincaré duality and the strong Lefschetz property can be constructed for any poset so that its Hilbert--Poincaré series coincides with the Chow polynomial \\cite[Section 4.6]{ferroni-matherne-vecchi}.\n\n\\begin{corollary}\\label{cor:main lefschetz}\n Let $P$ be a weakly ranked, finite, bounded poset of weak rank $n$. Then there exists a Gorenstein algebra with Kähler package whose Hilbert--Poincaré series coincides with the Chow polynomial of $P$.\n\\end{corollary}\n\nFurthermore, we are able to show that if the poset is ranked, this monomial ideal is pure, i.e., all its maximal monomials are of the same degree. This is the second main result of our paper.\n\\begin{theorem}\\label{thm:main2}\n Let $P$ be a ranked, finite, bounded poset of rank $n$. Then the sequence\n \\[\n \\left(h_0,h_1 - h_0, \\ldots,h_{\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor} - h_{\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor - 1}\\right)\n \\]\n is a pure $O$-sequence.\n\\end{theorem}\n\n\\subsection{Chow polynomials}\nIn this section, we briefly recall the definition of our main object of interest, i.e., the \\emph{Chow polynomial} of a \\emph{partially ordered set} (or poset for short). Recall that an interval of a poset is a subposet of $P$ of the form $[x,y]= \\{z \\in P : x\\leq z \\leq y\\}$, where $x,y\\in P$. \nA poset $P$ is \\emph{locally finite} if each of its intervals is finite. We also say that a poset is \\emph{bounded} if it has unique least and largest elements, which we denote by $\\zero$ and $\\one$, respectively.\nGiven a poset $P$, a \\emph{weak rank function} is a function $\\rho: P\\times P \\to \\bN$ such that \n\\begin{itemize}\n \\item $\\rho(x,y) = \\rho_{x,y} > 0$ if and only if $x < y$, and\n \\item $\\rho_{x,y} = \\rho_{x,z} + \\rho_{z,y}$, for all $x\\leq z \\leq y$ in $P$.\n\\end{itemize}\nA \\emph{weakly ranked} poset consists of a pair $(P,\\rho)$, where $\\rho$ is a weak rank function for $P$. By slight abuse of notation, we say that $P$ is a weakly ranked poset, when this does not create confusion. \nIf the poset has a least element $\\zero$, we write $\\rho(x):= \\rho_{\\zero,x}$ for an element $x \\in P$ and call this the \\emph{weak rank} of $x$ in $P$. If $\\rho_{x,y} =1$ whenever $y$ covers $x$, then we say that $P$ is \\emph{ranked}. Moreover, if $P$ is bounded, then we say that $P$ has weak rank $\\rho(\\one)$. Notice that if a poset is bounded and locally finite, then in particular it is finite. All the posets considered in this paper are finite, bounded and weakly ranked.\nLastly, we will denote by $W(P) = (W_0,W_1,\\ldots W_{\\rho(\\one)})$, where $W_i=|\\{p\\in P | \\rho(p)=i\\}|,$ the sequence of \\emph{Whitney numbers of the first kind of $P$}.\n\nWhile the following is not the standard definition of Chow polynomials, it is the one which is going to be more convenient for us. \n\\begin{definition}[{\\cite[Theorem~4.1]{ferroni-matherne-vecchi}}]\\label{def:chow polynomial}\n Let $P$ be a weakly ranked poset. Then its Chow polynomial is\n \\[\n H_P(t) = \\sum_{s \\geq 0}\\sum_{\\zero=p_0 < p_1 < \\cdots < p_s \\leq \\one} \\prod_{i=1}^{s}\\frac{t\\left(t^{\\rho(p_i)-\\rho(p_{i-1})-1}-1 \\right)}{t-1} .\n \\]\n\\end{definition}\n\\begin{remark}\nThe polynomial $H_P$ is usually called \\emph{characteristic} Chow polynomial, as it is proven to be the inverse of the negative reduced characteristic function inside the incidence algebra of the poset $P$. Indeed, the original construction in \\cite{ferroni-matherne-vecchi} associates a Chow polynomial to any poset with a choice of extra data called \\emph{$P$-kernel}. We also observe that this definition would associate a polynomial to every interval in a poset, that would not need to be finite (only locally finite) nor bounded. However, since the characteristic Chow polynomial is invariant under isomorphism of posets, we can reduce ourselves to studying bounded, finite posets. Since in this article we are only concerned with the characteristic Chow polynomial, we choose to ease the notation and only refer to it as \\emph{Chow polynomial}.\n\\end{remark}\n\nThe following observation will help us construct symmetric chain decompositions.\n\\begin{remark}\\label{rem:subdiv-decomp}\n Let $P$ be a ranked poset of rank $n$. If we have a partition $P = \\bigsqcup_i P_i$ into ranked subposets, each with a sequence of Whitney numbers of the first kind palindromic with center of symmetry $n/2$, then the union of any symmetric chain decomposition of $P_i$ constitutes a symmetric chain decomposition of $P$. \n\\end{remark}\n\n\\begin{theorem}\\label{thm:description SFY}\n For any weakly ranked poset $P$ there exists a symmetric chain decomposition $S$ of $\\FY$ such that the set of initial elements, which we denote by $\\SFY$, coincides with the family of FY monomials of the form\n \\[\n m = \\prod_{k=1}^s x_{p_k}^{\\ell_k},\n \\]\n where $\\zero = p_0 < p_1 < \\cdots < p_s < \\one$ is a chain of the poset $P$ and \n \\[\n 1 \\leq \\ell_k \\leq \\min\\left(\\rho(p_k) - \\rho(p_{k-1}) - 1,\\ \\rho(P) - \\rho(p_k) - 2\\sum_{i= k+1}^{s}\\ell_i\\right).\n \\]\n\\end{theorem}\n\\begin{proof}\n Let us build a symmetric chain decomposition of $\\FY$.\n\n\\section{Proof of Theorem \\ref{thm:main1}}\\label{sec:main1}\n We are now ready to prove Theorem \\ref{thm:main1}.\n According to Theorem \\ref{thm:properties of characteristic Chow}, we only need to prove that $h$ forms a differential $O$-sequence. We will do so by constructing directly a monomial order ideal. The proof of the theorem will follow immediately from the following result.\n \\begin{theorem}\\label{thm:SFY monomial order ideal}\n Let $P$ be a poset.\n The set of monomials $\\SFY$ is a monomial order ideal.\n \\end{theorem}\n \\begin{proof}\n To prove the statement, it is sufficient to show that for any monomial\n $$m = \\prod_{k=1}^s x_{p_k}^{\\ell_k}\\in \\SFY,$$ any monomial $m'=\\prod_{k=1}^s x_{p_k}^{\\ell'_k}$ that divides $m$ is also in $\\SFY$.\n By the definition of $\\SFY$ we know that \\begin{equation}\\label{eq:mon-1}\n \\ell_k\\leq\\rho(p_k)-\\rho(p_{k-1})-1\n \\end{equation} and that \\begin{equation}\\label{eq:mon-2}\n \\ell_k\\leq \\rho(\\one)-\\rho(p_k)-2\\sum_{i=k+1}^{s} \\ell_i.\n \\end{equation}", "post_theorem_intro_text_len": 2275, "post_theorem_intro_text": "See Definitions \\ref{def:O} and \\ref{def:SI} for the definition of $\\operatorname{SI}$-sequence. This statement is equivalent to saying that $h(P)$ is the $h$-vector of a simplicial polytope, see also Theorem \\ref{thm:characterization SI sequence} below. The proof proceeds by giving an explicit combinatorial construction of a monomial order ideal whose graded pieces realize the successive differences of the Chow coefficients. A key ingredient is an analysis of Feichtner–-Yuzvinsky monomials, combined with symmetric chain decompositions of products of chains. \nTheorem \\ref{thm:main1} allows us to state the following corollary immediately.\n\n\\begin{corollary}\\label{cor:main lefschetz}\n Let $P$ be a weakly ranked, finite, bounded poset of weak rank $n$. Then there exists a Gorenstein algebra with Kähler package whose Hilbert--Poincaré series coincides with the Chow polynomial of $P$.\n\\end{corollary}\n\nFurthermore, we are able to show that if the poset is ranked, this monomial ideal is pure, i.e., all its maximal monomials are of the same degree. This is the second main result of our paper.\n\\begin{theorem}\\label{thm:main2}\n Let $P$ be a ranked, finite, bounded poset of rank $n$. Then the sequence\n \\[\n \\left(h_0,h_1 - h_0, \\ldots,h_{\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor} - h_{\\left\\lfloor\\frac{n-1}{2}\\right\\rfloor - 1}\\right)\n \\]\n is a pure $O$-sequence.\n\\end{theorem}\n\nAs an application, we study the log-concavity of Chow polynomials. Using general properties of $O$-sequences, we show that Chow polynomials are log-concave for all posets of weak rank at most six. We also construct explicit counterexamples in every higher weak rank.\n\nThe paper is organized as follows. In Section \\ref{sec:background} we recall the necessary background on standard graded algebras, Chow polynomials, monomial order ideals and $O$-sequences and $\\operatorname{SI}$-sequences. In Section \\ref{sec:symmetric chain decomp} we recall and prove relevant results about symmetric chain decompositions. Section \\ref{sec:main1} contains the proof of Theorem \\ref{thm:main1} and hence Corollary \\ref{cor:main lefschetz}. Section \\ref{sec:main2} proves Theorem \\ref{thm:main2}. Lastly, Section \\ref{sec:log-concavity} discusses the log-concavity of Chow polynomials.", "sketch": "The proof of Theorem~\\ref{thm:main1} “proceeds by giving an explicit combinatorial construction of a monomial order ideal whose graded pieces realize the successive differences of the Chow coefficients.” A “key ingredient is an analysis of Feichtner–-Yuzvinsky monomials, combined with symmetric chain decompositions of products of chains.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm:main1}\n Let $P$ be a weakly ranked, \n finite, bounded poset of weak rank $n$. Then $h(P)$ forms a $\\operatorname{SI}$-sequence.,", "theorem_type": ["Universal"], "mcq": {"question": "Let $P$ be a weakly ranked, finite, bounded poset of weak rank $n$, and let $h(P)=(h_0,h_1,\\ldots,h_{n-1})$ denote the sequence of coefficients of the Chow polynomial of $P$. Which statement holds for every such poset $P$?", "correct_choice": {"label": "A", "text": "The coefficient sequence $h(P)$ forms an $\\operatorname{SI}$-sequence."}, "choices": [{"label": "B", "text": "The coefficient sequence $h(P)$ forms a $\\gamma$-positive sequence."}, {"label": "C", "text": "The coefficient sequence $h(P)$ is unimodal."}, {"label": "D", "text": "For every weakly ranked, finite, bounded poset $P$ of weak rank $n$, the coefficient sequence $h(P)$ forms an $\\operatorname{O}$-sequence."}, {"label": "E", "text": "The coefficient sequence $h(P)$ forms a palindromic $\\operatorname{SI}$-sequence."}], "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": "replacing SI with gamma-positivity for all weakly ranked posets", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the SI-condition and kept only the unimodality consequence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "substituting O-sequence for SI-sequence", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "adding palindromicity as an extra universal conclusion", "template_used": "stronger_trap"}]}} +{"id": "2601.01001v1", "paper_link": "http://arxiv.org/abs/2601.01001v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{Teo2}\n\\begin{enumerate}\n\\item[i)] {($\\Gamma$-$\\liminf$ inequality)}\n\nLet \\( \\{( \\boldsymbol{u}^{(j)}, \\alpha^{(j)})\\}_{j \\in \\mathbb{N}} \\) be a sequence \nin~\\( \\mathcal{A} \\). \nAssume that for some $\\left( \\widehat{\\boldsymbol u}, \\widehat{\\alpha} \\right) \\in \\mathcal{A}$,\nthe sequence $\\Big ( \\overline{u}_3^{(j)} \\Big )_{j\\in\\mathbb{N}}$ defined by \\eqref{eq:horizontal_averages_u3}\nconverges weakly in $L^2(\\Omega)$ to $\\widehat{u}_3$, and that $\\alpha^{(j)}$ converges strongly in $L^2(\\Omega)$ to $\\widehat{\\alpha}$.\nThen \n\\begin{align*}\n\t E_\\infty[\\widehat {\\boldsymbol u}, \\widehat \\alpha]\n\t \\leq \\liminf_{j\\to\\infty} E_j[\\boldsymbol u^{(j)}, \\alpha^{(j)}].\n\\end{align*}\n\n\\item[ii)] {(\\, $\\Gamma$-$\\limsup$ inequality)}\n\nGiven any $(\\boldsymbol u,\\alpha)\\in \\mathcal{S}$, it is possible to construct a sequence $(\\boldsymbol{u}^{(j)},\\alpha^{(j)})$ in $\\mathcal{A}$ such that\n $$ \\big (u_3^{(j)}, \\alpha^{(j)}\\big) \\quad \\text{converges strongly in } L^{2}(\\Omega)\\times L^2(\\Omega)\n \\text{ to } \\big (u_3, \\alpha\\big ), $$\n and\n \\[\n \\lim_{j \\to \\infty} E_j[{\\boldsymbol u}^{(j)}, \\alpha^{(j)}] =\\int_0^1 a_\\eta\\big(\\alpha(z)\\big) \\cdot \\frac{1}{2} E u_3'(z)^2 + w\\big(\\alpha(z)\\big) + \\frac{w_1 \\ell^2}{2L^2} |\\alpha'(z)|^2 \\, \\mathrm{d}z.\n \\]\n\\end{enumerate}", "start_pos": 11098, "end_pos": 12374, "label": "Teo2"}, "ref_dict": {"Teo2": "\\begin{theorem}\\label{Teo2}\n\\begin{enumerate}\n\\item[i)] {($\\Gamma$-$\\liminf$ inequality)}\n\nLet \\( \\{( \\boldsymbol{u}^{(j)}, \\alpha^{(j)})\\}_{j \\in \\mathbb{N}} \\) be a sequence \nin~\\( \\mathcal{A} \\). \nAssume that for some $\\left( \\widehat{\\vec u}, \\widehat{\\alpha} \\right) \\in \\mathcal{A}$,\nthe sequence $\\Big ( \\overline{u}_3^{(j)} \\Big )_{j\\in\\N}$ defined by \\eqref{eq:horizontal_averages_u3}\nconverges weakly in $L^2(\\Omega)$ to $\\widehat{u}_3$, and that $\\alpha^{(j)}$ converges strongly in $L^2(\\Omega)$ to $\\widehat{\\alpha}$.\nThen \n\\begin{align*}\n\t E_\\infty[\\widehat {\\vec u}, \\widehat \\alpha]\n\t \\leq \\liminf_{j\\to\\infty} E_j[\\vec u^{(j)}, \\alpha^{(j)}].\n\\end{align*}\n\n\\item[ii)] {(\\, $\\Gamma$-$\\limsup$ inequality)}\n\nGiven any $(\\vec u,\\alpha)\\in \\mathcal{S}$, it is possible to construct a sequence $(\\boldsymbol{u}^{(j)},\\alpha^{(j)})$ in $\\mathcal{A}$ such that\n $$ \\big (u_3^{(j)}, \\alpha^{(j)}\\big) \\quad \\text{converges strongly in } L^{2}(\\Omega)\\times L^2(\\Omega)\n \\text{ to } \\big (u_3, \\alpha\\big ), $$\n and\n \\[\n \\lim_{j \\to \\infty} E_j[{\\vec u}^{(j)}, \\alpha^{(j)}] =\\int_0^1 a_\\eta\\big(\\alpha(z)\\big) \\cdot \\frac{1}{2} E u_3'(z)^2 + w\\big(\\alpha(z)\\big) + \\frac{w_1 \\ell^2}{2L^2} |\\alpha'(z)|^2 \\, \\mathrm{d}z.\n \\]\n\\end{enumerate}\n\\end{theorem}", "eq:uniaxial_strain": "\\begin{align}\n \\label{eq:uniaxial_strain}\n \\widehat{\\varepsilon}_{11}=\\widehat{\\varepsilon}_{22}= -\\nu \\widehat{u}'(z).\n\\end{align}", "se:preliminaries": "\\begin{itemize}\n\\item[1.]\n Theorem~\\ref{Teo2} is a $\\Gamma$-convergence result, but for a non-standard topology. Its proof is based\n on Theorem~3 below, a compactness result, which characterizes this `dimension reduction' topology.\n The latter Theorem indeed describes the structure of weak limits of admissible $(\\vec{u}^{(j)}, \\alpha^{(j)})$'s\nwhen only some of the components of the displacement fields or their derivatives can be expected to be controlled, \nwhen the energies $E(\\vec{u}^{(j)}, \n\\alpha^{(j)})$ are uniformly bounded.\n\\item[2.] \nConcerning Theorem~2, its proof combines Theorem~3\nwith a precise lower bound on the energies, when the convergences \nin Theorem~3 hold. This lower bound is derived in Proposition~1.\n\\item[3.] In addition, under the hypotheses of Theorem~2,\nwe show strong convergence of the minimizers \n(or, more precisely, on the components $u_3^{(j)}$, on the strain tensors $\\varepsilon^{(j)}$, and on the damage variables $\\alpha^{(j)}$ and their gradients),\nanother particular feature of the underlying `dimension reduction' topology.\n\\item[4.]\nAs another feature of this topology,\none can choose $u_3^{(j)}(x,y,z) = \\widehat{u}_3(z)$ to construct a recovery sequence\nin the proof of Theorem~1, i.e., a function which is independent of $j$ (and of $x,y$). The horizontal components $u_1^{(j)}$, $u_2^{(j)}$ do depend on $j$ in the recovery sequence, even with gradients that grow unbounded as $j\\to\\infty$, but their contribution to the energy becomes negligible in the slender rod limit because of the prefactor $\\delta_j$ in the rescaled shear strains in \\eqref{eq:strainL2converges}, \\eqref{eq:EjInExtense}. The system is dominated by the behaviour of the axial displacement $u_3$, to which the horizontal components are able to adjust, guided by the energy-minimality criterion of reducing, as much as possible, the shear strains (see Equations \\eqref{eq:aspiration_UB}, \\eqref{eq:horizontal_adapt}, \\eqref{eq:test_function}, and \\eqref{eq:uniaxial_strain}).\nThis is consistent with the way in which the limit functional $E_\\infty$ depends on the vectorial displacement field $\\vec u$: through the axial component $u_3$ only.\n\\item[5.] In this work, the internal characteristic length~$\\ell$ and\nthe regularization parameter $\\eta$ are fixed positive quantities,\nso that for fixed damage $\\alpha$, the associated elastic displacement\nis the solution to an elliptic PDE. It would be interesting to study\nunder which regimes the dimension reduction could be performed, when\nthe parameters $\\ell_j$, $\\eta_j$ are allowed to tend to 0. \n\\end{itemize}\n\\end{remark*}\n\\medskip\n\nAn extensive literature is available for the rigorous derivation of reduced models \nobtained from three-dimensional elasticity (see, e.g., \\cite[Chapter 16]{antman2005} \nfor linear theories of rods; \\cite[Chapters A.1 and B.5]{ciarlet2021volII} for Kirchhoff-Love and von K\\'arm\\'an equations in the theory of plates,\n\\cite{FJM2006} and~\\cite{neukammRichter2024} for models for plates and rods in the context \nof nonlinear elasticity).\nThe techniques we use here are inspired by the rigorous derivation of the model for fracture \nand delamination of a thin plate on an elastic foundation, proposed in~\\cite{baldelli2014},\nand by the dimension reduction analysis of a brittle Kirchhoff-Love plate in the SBD setting\nderived~in \\cite{babadjianHenao2016}.\n\\medskip\n\nThe structure of the article is as follows: \nSection 2 presents preliminary definitions, notation, and the assumptions under\nwhich our analysis is carried out.\nIn Section 3, we detail the non-dimensionalization and rescaling process.\nIn Section~4, we proceed with a discussion on compactness results, while\nSection 5 is dedicated to deriving the lower and upper bounds for the energy \nfunctionals $E_j$.\nFinally, the proof of Theorem~\\ref{Main} is assembled in Section~\\ref{se:minimizers}.\n\n\\section{Preliminaries}\n\\label{se:preliminaries}\n\nIn this section we introduce the definitions, notation, and assumptions \non which we base the reduction of the three-dimensional gradient damage model \nto a one-dimensional model.\n\n\\subsubsection*{The domain and admissible displacements}\n\nFor a slender rod with \\( R \\ll L \\), we define the domain:\n\\[\n\\tilde{\\Omega} := \\{ \\boldsymbol{X} = (X, Y, Z) \\in \\mathbb{R}^3 \\mid X^2 + Y^2 < R^2, \\ 0 < Z < L \\}\n\\]\n(tildes are used for sets and functions in the physical domain, \nand will be dropped after non-dimensionalization; \nvector quantities are represented in boldface).\nWe denote by $\\tilde{u} \\in H(\\tilde{\\Omega}; \\mathbb{R}^3)$\n\\begin{eqnarray*}\n \\tilde{u}:&& \\tilde{\\Omega} \\to \\mathbb{R}^3, \\quad \\\\\n && \\boldsymbol{X} \\longmapsto\\tilde{u}(\\boldsymbol{X}) = (\\tilde{u}_1(\\boldsymbol{X}), \\tilde{u}_2(\\boldsymbol{X}), \\tilde{u}_3(\\boldsymbol{X} ))\n\\end{eqnarray*}\nthe displacement of each material point relative to its initial position,\nwhile $\\tilde{\\alpha}\\in H^1(\\tilde(\\Omega))$ denotes the damage.\nWe assume that the rod $\\Omega$ is subject to the following mixed boundary conditions\n\\begin{eqnarray} \\label{bc's}\n\\tilde{u}_3 (X, Y, L) = -\\tilde{t}, \\quad \\tilde{u}_3 (X, Y, 0) = 0, \\quad \\forall X, Y \\text{ such that } X^2 + Y^2 < R^2.\n\\end{eqnarray}\nOur analysis is motivated by the modeling of uniaxial compressive tests ($\\tilde t>0$), \nhowever, it remains valid for a rod in tension ($\\tilde t<0$).\n\\medskip\n\nThe equilibrium state under the action of $\\tilde{t}$ is defined as the\nminimizer of the following 3D gradient damage energy, among all fields \n$(\\tilde{\\bf u},\\alpha) \\in H(\\tilde{\\Omega}; \\mathbb{R}^3) \\times H^1(\\tilde{\\Omega})$\nthat statisfy the boundary conditions~\\eqref{bc's}.\n\\[\n\\int_{\\tilde{\\Omega}} \\frac{1}{2} a_\\eta(\\tilde \\alpha(\\boldsymbol{X})) A \\varepsilon : \\varepsilon + w(\\tilde{\\alpha}(\\boldsymbol{X})) + \\frac{1}{2} w_1 \\ell^2 |\\nabla \\tilde{\\alpha}(\\boldsymbol{X})|^2 \\, d\\boldsymbol{X}\n\\]\nEach term is explained below.\n\n\\subsubsection*{The strain tensor $\\varepsilon$}\n\nThe strain tensor $\\varepsilon$, associated to the deformation of the body, \nis the symmetric rank-2 tensor defined by\n\\begin{equation*}\n \\varepsilon = \\frac{1}{2} \\left( \\nabla \\tilde{u} + \\nabla \\tilde{u}^{T} \\right)\n =\n \\begin{bmatrix}\n \\varepsilon_{11} & \\varepsilon_{12} & \\varepsilon_{13} \\\\\n \\varepsilon_{21} & \\varepsilon_{22} & \\varepsilon_{23} \\\\\n \\varepsilon_{31} & \\varepsilon_{32} & \\varepsilon_{33}\n \\end{bmatrix},\n \\qquad\n \\nabla \\tilde{u} =\n \\begin{bmatrix}\n \\displaystyle \\frac{\\partial \\tilde{u}_1}{\\partial X} & \\displaystyle \\frac{\\partial \\tilde{u}_1}{\\partial Y} & \\displaystyle \\frac{\\partial \\tilde{u}_1}{\\partial Z} \\\\\\\\\n \\displaystyle \\frac{\\partial \\tilde{u}_2}{\\partial X} & \\displaystyle \\frac{\\partial \\tilde{u}_2}{\\partial Y} & \\displaystyle \\frac{\\partial \\tilde{u}_2}{\\partial Z} \\\\\\\\\n \\displaystyle \\frac{\\partial \\tilde{u}_3}{\\partial X} & \\displaystyle \\frac{\\partial \\tilde{u}_3}{\\partial Y} & \\displaystyle \\frac{\\partial \\tilde{u}_3}{\\partial Z}\n \\end{bmatrix}.\n\\end{equation*}\n\n\\subsubsection*{The elastic constants}\n\nBefore damage, the material is governed by an isotropic linear elastic\nstress-strain relation, of the form\n\\begin{eqnarray*}\n\\sigma &=& A \\varepsilon \\;:=\\; 2\\mu \\varepsilon + \\lambda (\\text{tr} \\varepsilon), I\n\\end{eqnarray*}\nwhere the Lam\\'e coefficients \\( \\lambda \\) and \\( \\mu\\) respectively measure \nthe material's volumetric (compressive) response, and the material's rigidity. \nThe Young's modulus \\( E \\) and Poisson's ratio \\( \\nu \\) are given by\n\\begin{eqnarray}\\label{Young}\n E \\;=\\; \\frac{\\mu(3\\lambda + 2\\mu)}{\\lambda + \\mu} &\\quad\\text{and}\\quad& \n \\nu \\;=\\; \\frac{\\lambda}{2(\\lambda + \\mu)}.\n\\end{eqnarray}\n\n\\subsubsection*{The damage variable $\\tilde{\\alpha}$}\n\nThe scalar field $\\tilde{\\alpha} \\in H^1(\\tilde{\\Omega})$\n\\begin{eqnarray*}\n\\tilde{\\alpha}:&&\\tilde{\\Omega} \\to [0,1],\\\\\n &&\\boldsymbol{X} \\longmapsto \\tilde{\\alpha}(\\boldsymbol{X}) = \\tilde{\\alpha}(X, Y, Z).\n\\end{eqnarray*}\ndescribes the distribution of material damage in the physical domain $\\tilde{\\Omega}$.\nIt can be interpreted as the impact on the macroscopic stiffness of the medium\nof the presence of micro-cracks, of loss of internal cohesion, or of localized \nstructural degradation. In our context,\n\\begin{itemize}\n \\item $\\tilde{\\alpha}(\\boldsymbol{X}) = 0$ indicates that the material is intact at point $\\boldsymbol{X}$.\n \\item $0 < \\tilde{\\alpha}(\\boldsymbol{X}) < 1$ represents partial damage, which means that some mechanical properties, such as stiffness, have been reduced but are not completely lost. \n\\item $\\tilde{\\alpha}(\\boldsymbol{X}) = 1$ indicates that the material is completely damaged.\n\\end{itemize}", "eq:aspiration_UB": "\\begin{align} \\label{eq:aspiration_UB}\n\\varepsilon_{11}^{(j)} = \\varepsilon_{22}^{(j)} = -\\nu \\varepsilon_{33}^{(j)}, \n\\quad \\varepsilon_{12}^{(j)} = \\varepsilon_{13}^{(j)} = \\varepsilon_{23}^{(j)} = 0,\n\\end{align}", "eq:horizontal_averages_u3": "\\begin{align}\n \\label{eq:horizontal_averages_u3}\nnt_{x_1^2+x_2^2 < 1} u_3^{(j)} (x,y,z) \\, d\\mathcal{H}^2 (x,y).\n\\end{align}", "eq:EjInExtense": "\\begin{eqnarray}\\label{la1}\nE_{j} [\\boldsymbol{u}, \\alpha] \n&:=&\n\\frac{1}{\\pi}\\int_{\\Omega} a_{\\eta} (\\alpha (\\boldsymbol{x})) \n\\left[\n\\mu \\left( \\frac{\\partial u_1}{\\partial x} \\right)^2\n+ \\mu \\left( \\frac{\\partial u_2}{\\partial y} \\right)^2\n+ \\mu \\left( \\frac{\\partial u_3}{\\partial z} \\right)^2\n+ \\frac{\\lambda}{2} \\left( \\frac{\\partial u_1}{\\partial x} + \\frac{\\partial u_2}{\\partial y} + \\frac{\\partial u_3}{\\partial z} \\right)^2\n\\right.\\nonumber\n\\\\ \n&&\\left.+ \\frac{\\mu}{2}\\left( \\frac{\\partial u_1}{\\partial y}+ \\frac{\\partial u_2}{\\partial x} \\right)^2\n+ \\frac{\\mu}{2} \\left( \\delta_j \\frac{\\partial u_1}{\\partial z} + \\delta_j^{-1} \\frac{\\partial u_3}{\\partial x} \\right)^2 + \\frac{\\mu}{2} \\left( \\delta_j \\frac{\\partial u_2}{\\partial z} + \\delta_j^{-1} \\frac{\\partial u_3}{\\partial y} \\right)^2\\right] \n\\nonumber \n\\\\\n\\label{eq:EjInExtense}\n&&+ w (\\alpha (\\boldsymbol{x})) + \\frac{1}{2} w_1 \\left( \\frac{l}{L} \\right)^2 \\left[ \\delta_j^{-2} \\left( \\frac{\\partial \\alpha}{\\partial x} \\right)^2 + \\delta_j^{-2} \\left( \\frac{\\partial \\alpha}{\\partial y} \\right)^2 + \\left( \\frac{\\partial \\alpha}{\\partial z} \\right)^2 \\right] d\\boldsymbol{x}.\n\\end{eqnarray}", "Main": "\\begin{theorem}\\label{Main}\nSuppose that for each $j\\in\\N$\n\\begin{eqnarray*}\n(\\vec u^{(j)}, \\alpha^{(j)}) \\ \\text{minimizes}\\ E_j[\\vec u, \\alpha]\n\\ \\text{in}\\ \\mathcal A.\n\\end{eqnarray*}\nThen there exists a pair $(\\widehat{\\vec u},\\widehat{\\alpha})$ in $\\mathcal {S}$\nsuch that (for a subsequence)\n\\begin{eqnarray}\n &&\n \\nonumber\n \\int_\\Omega |u_3^{(j)}(x,y,z) - \\widehat{u}_3(z)|^2 d\\vec x \\longrightarrow 0, \\quad\n \\int_\\Omega \\left |\\frac{\\partial u_3^{(j)}}{\\partial z}(x,y,z) - \\frac{d \\widehat{u}_3}{d z}(z)\\right |^2d\\vec x\n \\longrightarrow{0},\n \\\\\n &&\n \\label{eq:strainL2converges}\n \\int_\\Omega \\left |\\frac{\\partial u_1^{(j)}}{\\partial x}(x,y,z) + \\nu \\frac{d \\widehat{u}_3}{d z}(z)\\right |^2d\\vec x\n \\longrightarrow{0},\n \\quad\n \\int_\\Omega \\left |\\frac{\\partial u_2^{(j)}}{\\partial y}(x,y,z) + \\nu \\frac{d \\widehat{u}_3}{d z}(z)\\right |^2d\\vec x\n \\longrightarrow{0},\n \\\\\n &&\n \\nonumber\n \\int_\\Omega \\left | \\frac{\\partial u_1^{(j)}}{\\partial y} + \\frac{\\partial u_2^{(j)}}{\\partial x}\\right|^2\n +\\left |\\delta_j \\frac{\\partial u_1^{(j)}}{\\partial z} + \\delta_j^{-1} \\frac{\\partial u_3^{(j)}}{\\partial x} \\right|^2\n +\\left |\\delta_j \\frac{\\partial u_2^{(j)}}{\\partial z} + \\delta_j^{-1} \\frac{\\partial u_3^{(j)}}{\\partial y} \\right|^2d\\vec x\n \\longrightarrow 0,\n \\\\\n &&\n \\nonumber\n \\qquad \\alpha^{(j)} \\xrightarrow{H^1 (\\Omega)} \\widehat \\alpha,\n \\quad \\text{and}\\quad (\\widehat{\\vec u}, \\widehat{\\alpha})\\ \\text{minimizes}\\ E_\\infty[\\vec u, \\alpha]\\ \\text{in}\\ \\mathcal {S}.\n\\end{eqnarray}\n\\end{theorem}", "eq:horizontal_adapt": "\\begin{align}\n \\label{eq:horizontal_adapt}\nu_1^{(j)}(x, y, z) := -\\nu x\\, \\varepsilon_{33}^{(j)}(z),\n\\quad \nu_2^{(j)}(x, y, z) := -\\nu y\\, \\varepsilon_{33}^{(j)}(z),\n\\quad \nu_3^{(j)}(x,y,z) := \n\\overline{u}_3 (z),\n\\end{align}", "eq:test_function": "\\begin{align}\n \\label{eq:test_function}\nu_1(x,y,z)=\\nu \\varepsilon_zx, \\quad u_2(x,y,z)=\\nu \\varepsilon_zy, \n\\quad u_3(x,y,z)=-\\varepsilon_zz, \\quad \\alpha_{\\text{test}}(x,y,z)\\equiv 0.\n\\end{align}", "eq:strainL2converges": "\\begin{eqnarray}\n &&\n \\nonumber\n \\int_\\Omega |u_3^{(j)}(x,y,z) - \\widehat{u}_3(z)|^2 d\\vec x \\longrightarrow 0, \\quad\n \\int_\\Omega \\left |\\frac{\\partial u_3^{(j)}}{\\partial z}(x,y,z) - \\frac{d \\widehat{u}_3}{d z}(z)\\right |^2d\\vec x\n \\longrightarrow{0},\n \\\\\n &&\n \\label{eq:strainL2converges}\n \\int_\\Omega \\left |\\frac{\\partial u_1^{(j)}}{\\partial x}(x,y,z) + \\nu \\frac{d \\widehat{u}_3}{d z}(z)\\right |^2d\\vec x\n \\longrightarrow{0},\n \\quad\n \\int_\\Omega \\left |\\frac{\\partial u_2^{(j)}}{\\partial y}(x,y,z) + \\nu \\frac{d \\widehat{u}_3}{d z}(z)\\right |^2d\\vec x\n \\longrightarrow{0},\n \\\\\n &&\n \\nonumber\n \\int_\\Omega \\left | \\frac{\\partial u_1^{(j)}}{\\partial y} + \\frac{\\partial u_2^{(j)}}{\\partial x}\\right|^2\n +\\left |\\delta_j \\frac{\\partial u_1^{(j)}}{\\partial z} + \\delta_j^{-1} \\frac{\\partial u_3^{(j)}}{\\partial x} \\right|^2\n +\\left |\\delta_j \\frac{\\partial u_2^{(j)}}{\\partial z} + \\delta_j^{-1} \\frac{\\partial u_3^{(j)}}{\\partial y} \\right|^2d\\vec x\n \\longrightarrow 0,\n \\\\\n &&\n \\nonumber\n \\qquad \\alpha^{(j)} \\xrightarrow{H^1 (\\Omega)} \\widehat \\alpha,\n \\quad \\text{and}\\quad (\\widehat{\\vec u}, \\widehat{\\alpha})\\ \\text{minimizes}\\ E_\\infty[\\vec u, \\alpha]\\ \\text{in}\\ \\mathcal {S}.\n\\end{eqnarray}", "se:minimizers": "\\begin{eqnarray} \\label{eq_Ej}\n E_j[\\vec u^{(j)}, \\alpha]\n&=&\n\\frac{1}{\\pi} \\int_{\\Omega} \na_{\\eta}(\\overline{\\alpha}(z)) \\left[ \\frac{E}{2} \\left| \\varepsilon_{33}^{(j)} \\right|^2 \n+ \\Big( \\mu + \\frac{\\lambda}{2} \\Big) \\Big( |\\overline{u}_3'|^2 - |\\varepsilon_{33}^{(j)}|^2 \\Big) \n+ 2\\lambda\\nu \\varepsilon_{33}^{(j)} \\Big( \\varepsilon_{33}^{(j)} - \\overline{u}_3' \\Big)\n\\right . \n\\nonumber \\\\ \n&& \n\\left . \n+ (x^2 + y^2) \\frac{\\mu}{2} \\nu^2 \\delta_j^2 \n\\left| \\frac{\\partial}{\\partial z} \\varepsilon_{33}^{(j)} \\right|^2 \\right]\n+ w(\\overline{\\alpha}(z)) \n+ w_1 \\frac{\\ell^2}{2L^2} \\left| \\overline{\\alpha}'(z) \\right|^2 \\,d\\boldsymbol{x} \n\\nonumber \\\\\n&=&\n\\frac{1}{\\pi}\\int_{\\Omega} \na_{\\eta} (\\overline{\\alpha}(z))\\frac{E}{2}(\\overline{u}_3'(z))^2\n+ w(\\overline{\\alpha}(z)) + w_1 \\frac{\\ell^2}{2L^2} \\left| \\overline{\\alpha}'(z) \\right|^2 \n\\,d\\boldsymbol{x} \n\\nonumber \\\\ \n&&\n+ \\frac{1}{\\pi}\\int_{\\Omega} a_{\\eta}(\\overline{\\alpha}(z))\\Bigg[\n\\Bigg( \\mu + \\frac{\\lambda}{2} - \\frac{E}{2} \\Bigg)\n\\left( |\\overline{u}_3'|^2 -|\\varepsilon_{33}^{(j)} |^2\\right)\n+ 2\\lambda\\nu \\varepsilon_{33}^{(j)} \\Big( \\varepsilon_{33}^{(j)} - \\overline{u}_3' \\Big)\n\\nonumber \\\\ \n&& \n+ (x^2 + y^2) \\frac{\\mu}{2} \\nu^2 \\delta_j^2 \n\\left| \\frac{\\partial}{\\partial z} \\varepsilon_{33}^{(j)} \\right|^2 \\Bigg]\n\\,d\\boldsymbol{x}.\n\\end{eqnarray}\n\\medskip\n\nWe now proceed to construct the functions $\\varepsilon_{33}^{(j)}$, so that\n$\\varepsilon_{33}^{(j)} \\to \\overline{u}_3^\\prime$ in $L^2(0,1)$,\nand so that \n\\begin{eqnarray*}\n\\frac{\\mu}{2} (x^2 + y^2)\\, \\nu^2 \\left\\|\\delta_j\\, \\frac{\\partial}{\\partial z} \\varepsilon_{33}^{(j)} \\right\\|^2\n&\\rightarrow& 0.\n\\end{eqnarray*}\nThe latter condition requires that $\\delta_j \\partial_z \\varepsilon_{33}^{(j)}$ tends\nto 0, and hence that $\\partial_z \\varepsilon_{33}^{(j)}$ should not grow too rapidly.\nThe specific construction of $\\varepsilon_{33}^{(j)}=v_{k_j}$ must then balance the approximation \nof $\\overline{u}_3'(z)$ with a strict control of its derivative.\nTo this end, we consider a mollifier $\\rho \\in {\\mathcal C}^\\infty_0(\\mathbb{R})$, \nand set\n\\begin{eqnarray*}\nv_k(z) &:=& \\overline{u}_3'*\\rho_{1/\\sqrt{k}}(z),\n\\end{eqnarray*}\nwhere $\\rho_s(z) = s^{-1} \\rho(z/s)$ for $s > 0$, extending the\ndefinition of $\\overline{u}_3^\\prime$ by $0$ outside of $(0,1)$, so that\nthe convolution is well defined. The functions $v_k$ satisfy\n\\begin{eqnarray*}\nv_k &=&\n\\overline{u}_3'*\\rho_{\\frac{1}{\\sqrt{k}}}\n\\;\\to\\; \\overline{u}_3' \n\\quad \\textrm{strongly in}\\; L^2(0,1)\\;\\textrm{as}\\; k\\to \\infty,\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n\\frac{d}{dz}v_k &=& \n\\frac{d}{dz}\\left (\\overline{u}_3'*\\rho_{\\frac{1}{\\sqrt{k}}}\\right)\n\\;=\\;\n\\overline{u}_3'* \\left( k\\rho'\\left(\\sqrt{k}z\\right) \\right).\n\\end{eqnarray*}\nIt follows that\n\\begin{eqnarray*}\n\\left\\|\\frac{d}{dz}v_k \\right\\|_{L^\\infty ( (0,1) )}\n&\\leq& \n\\|\\overline{u}_3'(z)\\|_{L^1} \\left\\|k\\rho'\\left(\\sqrt{k}z\\right) \\right\\|_{L^{\\infty}}\n\\;=\\;\nk\\|\\overline{u}_3'(z)\\|_{L^1}\\left\\|\\rho'\\left(\\sqrt{k}z\\right) \\right\\|_{L^{\\infty}}\n\\;\\leq\\; k M\\|\\overline{u}_3'\\|_{L^1}.\n\\end{eqnarray*}\nGiven that \n$\\overline{u}_3'(z)\\in L^2\\big ( (0,1)\\big ) \\subset L^{1}\\big ( (0,1)\\big )$, we see that\nfor some constant $C >0$, independent of $k$,\n\\begin{eqnarray*}\n\\left\\|\\frac{d}{dz}v_k \\right\\|_{L^{\\infty}((0,1))} &\\leq& Ck.\n\\end{eqnarray*}\n\nWe define $\\eps_{33}^{(j)}$ by taking the diagonal subsequence\n$\\eps_{33}^{(j)} = v_{k_j}$, with the choice\n${k_j}= \\left\\lfloor \\delta_j^{-1/2} \\right\\rfloor$,\nwhich garantees that\n\\begin{eqnarray*}\n\\eps_{33}^{(j)} = v_{k_j} \\to \\overline{u}_3' \\quad \\text{in } L^2(0,1),\n&\\;\\text{and}\\;&\n\\left\\| \\frac{\\partial}{\\partial z} \\varepsilon_{33}^{(j)} \\right\\|_{L^\\infty} \n= \\left\\| \\frac{\\partial}{\\partial z} v_{k_j} \\right\\|_{L^{\\infty}} \n\\;\\leq\\; Ck_j\\approx C\\delta_{j}^{-1/2},\n\\end{eqnarray*}\nso that\n\\begin{eqnarray*}\n\\delta_j^2\\left\\| \\frac{\\partial}{\\partial z} \\varepsilon_{33}^{(j)} \\right\\|_{L^2}^2 \n&\\leq&\n\\delta_j^2 \\cdot \\delta_j^{-1} \\;=\\; \\delta_j\n\\;\\to\\; 0 \\quad \\textrm{as}\\; j \\to \\infty.\n\\end{eqnarray*}\nWe conclude from~\\eqref{eq_Ej}that\n\\begin{eqnarray*}\n\\lim_{j \\to \\infty} E_j[\\vec u^{(j)}, \\alpha] \n&=&\n\\int_0^1 a_\\eta\\left( \\overline{\\alpha}(z) \\right) \\cdot\n\\frac{1}{2} E \\overline{u}_3'(z)^2 + w\\left( \\overline{\\alpha}(z) \\right) +\n\\frac{w_1 \\ell^2}{2 L^2} \\left| \\overline{\\alpha}'(z) \\right|^2 \\, \\mathrm{d}z,\n\\end{eqnarray*}\nas claimed.\\\\\n\\end{proof}\n\n\\section{Convergence of minimizers} \\label{se:minimizers}\n\n\\begin{proof}[Proof of Theorem \\ref{Main}.]\n\nWe consider the functions \n$\\boldsymbol{u}_{\\text{test}}(\\boldsymbol{x})\n=(u_1(\\boldsymbol{x}),u_2(\\boldsymbol{x}),u_3(\\boldsymbol{x}))$, \nand $\\alpha_{\\text{test}}(\\boldsymbol{x})$, defined by\n\\begin{align}\n \\label{eq:test_function}\nu_1(x,y,z)=\\nu \\varepsilon_zx, \\quad u_2(x,y,z)=\\nu \\varepsilon_zy, \n\\quad u_3(x,y,z)=-\\varepsilon_zz, \\quad \\alpha_{\\text{test}}(x,y,z)\\equiv 0.\n\\end{align}\nNote that $\\boldsymbol{u}_{\\text{test}}$ is the minimizer of the energy when no damage is present, and that Equations \\eqref{eq:aspiration_UB} are satisfied.\nSince $a_\\eta(0)=1$ and $w(0)=0$, see~\\eqref{la8},\nwe evaluate the functional $E_j$ at the test pair \n$\\big( \\vec u_{\\text{test}}, \\alpha_{\\text{test}} \\big)$~:\n\\begin{eqnarray*}\nE_j[\\boldsymbol{{u}}_{\\text{test}},\\alpha_{\\text{test}}]\n&=&\n\\frac{1}{\\pi}\\int_{\\Omega} a_{\\eta}(\\alpha_{\\text{test}}(\\boldsymbol{x}))\\left[ \\mu (\\nu \\varepsilon_z)^2+\\mu (\\nu \\varepsilon_z)^2+\\mu \\varepsilon_z^2 + \\frac{\\lambda}{2}\\left(\\nu \\varepsilon_z+\\nu \\varepsilon_z- \\varepsilon_z\\right)^2 \\right]+w(\\alpha_{\\text{test}}(\\boldsymbol{x})) \\,dx\n\\\\\n&=&\n\\frac{1}{\\pi}\\int_{\\Omega}\\left[ \\mu\\varepsilon_z^2(2\\nu^2+1)\n+ \\frac{\\lambda}{2}\\varepsilon_z^2(2\\nu-1)^2 \\right] \\,dx\n\\;=\\;\n\\frac{E}{2}\\varepsilon_z^2.\n\\end{eqnarray*}\nSince by hypothesis, $(\\boldsymbol{u}^{(j)},\\alpha^{(j)})$ minimizes \n$E_{j}[\\boldsymbol{u},\\alpha]$, we infer that\n\\begin{align}\n \\label{eq:aPrioriBound}\nE_{j}[\\boldsymbol{u}^{(j) },\\alpha^{(j)}]\n\\leq \nE_j[\\boldsymbol{u}_{\\text{test}},\\alpha_{\\text{test}}]\n\\;=\\; \\frac{E}{2}\\varepsilon_z^2,\n\\end{align}\nso that $\\displaystyle{E_{j}[\\boldsymbol{u}^{(j) },\\alpha^{(j)}]\\leq M}$ \nis uniformly bounded.\nIt follows from Theorem~\\ref{Teo1} that there exists\n$\\widehat{u}_3,\\hspace{0.1cm} \\widehat \\alpha$ in $H^1(\\Omega)$, depending only on $z$,\nand\n$\\widehat{\\varepsilon}_{11}, \\widehat{\\varepsilon}_{22}, \\widehat{\\varepsilon}_{33}$ in \n$L^2(\\Omega; \\mathbb{R}^{3})$, which satisfy~\\eqref{la0}, such that\n\\[ \\left\\{ \\begin{array}{c}\n\\widehat{u}_3(\\cdot, 0) = 0, \n\\quad \\widehat{u}_3(\\cdot, 1) = -\\varepsilon_z,\n\\qquad 0\\leq \\widehat \\alpha\\leq 1\\quad \\text{a.e. in}\\; (0,1),\n\\\\[3pt]\n\\frac{\\partial u_1^{(j)}}{\\partial x} \\rightharpoonup \\widehat{\\varepsilon}_{11}, \\qquad\n\\frac{\\partial u_2^{(j)}}{\\partial y} \\rightharpoonup \\widehat{\\varepsilon}_{22}, \\qquad\n\\frac{\\partial u_3^{(j)}}{\\partial z} \\rightharpoonup \\widehat{\\varepsilon}_{33},\n\\\\[3pt]\n\\nabla \\alpha^{(j)} \\rightharpoonup \\nabla \\widehat\\alpha, \\qquad\n\\alpha^{(j)} \\to \\widehat\\alpha \\text{ a.e. in}\\; \\Omega,\n\\end{array} \\right.\n\\]\nand such that the sequence $\\overline{u}^{(j)}$, defined in \\eqref{eq:horizontal_averages_u3},\nconverges to $\\widehat u_3$ weakly in $H^1\\big ( (0,1) \\big )$ and strongly in $L^2\\big ((0,1) \\big )$.\nWe define the vector-valued map $\\widehat{\\vec u}\\in H^1(\\Omega;\\R^3)$ by\n\\begin{eqnarray*}\n\\widehat{\\vec u}(x,y,z) &:=& \n\\big(0, 0, \\widehat u_3(z) \\big), \\quad (x,y,z) \\in \\Omega.\n\\end{eqnarray*}\nApplying Proposition \\ref{Prop1}, we obtain\n\\begin{eqnarray}\\label{la9}\n\\lf E_{j} [\\boldsymbol{u}^{(j)}, \\alpha^{(j)}] \n&\\geq&\nE_\\infty [\\widehat {\\vec u}, \\widehat\\alpha] + J + \\lf K^{(j)},\n\\end{eqnarray}"}, "pre_theorem_intro_text_len": 8487, "pre_theorem_intro_text": "Gradient damage models (see, e.g., \\cite{marigo2016overview,ambrosio1990approximation,ambrosio1992approximation,braides1998approximation,bourdin2000numerical,bourdin2008variational,tanne2018crack,kumar2020revisiting,bourdin2025}, \nand the references therein) have been used to describe the nucleation and propagation of cracks \nin brittle and quasi-brittle materials, in very challenging problems such as\nthe formation of regularly spaced cracks with very complex geometries in thin films subject to thermal shocks \\cite{bourdin2014morphogenesis,sicsic2014initiation},\nor such as determining the causes of the damage observed in the ashlar masonry work \nof the French Panth\\'eon \\cite{lancioni2009variational}.\nIn these models, cracks correspond to localized bands, where an internal damage variable is activated\nthat reduces the stiffness of the material.\nA first stage consists in the onset of damage from an initially elastic material, \nwhen the stress reaches a well-defined intrinsic limit, \nwhich can be identified in terms of the model parameters \n(and is independent of the domain size and shape, and of the loading history). \nAs the loading increases, the level of damage raises until the maximal stress \nthat the material can sustain is attained.\nThe response of the body upon further loading depends on its size relative \nto a regularization parameter $\\ell$ specified in the model, \nwhich can be interpreted as internal characteristic length (another material property, \nas the elastic limit stress).\n\\medskip\n\nIn short rods, of length $L \\sim \\ell$, the homogeneous damage solutions\n(where the damage variable is constant throughout the body) remain stable \neven for extreme loading conditions.\nIn contrast, in rods made of a stress-softening material (so that the elastic region in \nstrain space shrinks as the damage progresses), the homogeneous damage solutions lose their \nstability, allowing the internal variable to continue its growth in narrow bands \nof width comparable to $\\ell$.\n\\medskip\n\nGradient damage models have thus proved to be a consistent numerical approximation of\nthe propagation of a pre-existing crack in the Griffith model for brittle fracture~\\cite{griffith1921vi,francfort1998revisiting}, \nto provide a mechanism for crack nucleation in a faultless material without geometrical \nsingularities~\\cite{chambolle2008nucleation},\nand to capture size effects and softening properties which are significant in the \nbehaviour of concrete, rock, and biomaterials.\nIn addition, gradient damage models overcome the spurious mesh dependency observed \nin local damage models~\\cite{marigo1989constitutive,benallal1993,pham2010approche},\nsince the addition of a gradient term on the damage variable leads to a dissipated energy in\nthe localized solutions which is essentially proportional to the area of the crack,\nas opposed to the failure without dissipated energy observed in the local models.\nThe initiation of cracks in more complicated three-dimensional geometries has been \nstudied, e.g., in~\\cite{tanne2018crack,kumar2020revisiting}, and a unified treatment \nof cohesive fracture and nucleation with an arbitrary strength surface is given \nin~\\cite{bourdin2025}.\nIt was shown in \\cite{bonnetier14faults} that variants of gradient damage models including \nplasticity and visco-elasticity are compatible with descriptions of the formation of \ngeological faults.\n\\medskip\n\nIn such gradient damage models,\nthe capability for crack nucleation is associated to the \nloss of stability of the homogeneous damage state and the emergence of \nlocalized damage solutions. \nThe previously mentioned stability \nanalyses~\\cite{pham2011stability,pham2013stability,marigo2016overview} \nhave been conducted mainly for uniaxial tension tests, using the variational inequalities \nof the formally derived one-dimensional gradient damage model. \nA validation of the important dimension reduction from 3D to 1D is thus desirable. \nThis is the purpose of this current work:\nwe prove the $\\Gamma$-convergence \\cite{braides2002gamma,dalMaso2012,ambrosio2000calculus}\nof the three-dimensional model to the one-dimensional gradient damage model in the slender \nrod limit.\n\\medskip\n\n An application of gradient damage models to bulk degradation of the rock mass \n in underground mining, with a numerical observation of surface subsidence, has been proposed in~\\cite{bonnetier2025gradient}.\n The present work has been motivated by the use that will be made of the simplified one-dimensional gradient damage model in the identification of parameters from uniaxial compression tests. That is a necessary step towards the more systematic study of three-dimensional damage models in the block caving problem, which is being studied by the CMM (Center for Mathematical Modeling, Universidad de Chile) Mining group.\n\n\\medskip\nThe precise formulation of the mathematical problem studied in this article is as follows: consider a sequence of cylindrical domains\n\\[\n\\Omega_j := \\{\\boldsymbol{x}=(x,y,z)\\in \\mathbb{R}^3: x^2+y^2\\leq R_j, \\hspace{0.2cm} 0\\leq z\\leq L\\}, \n\\quad j\\in \\mathbb{N}, \\quad \\text{with} \\quad \\delta_j := \\frac{R_j}{L} \\xrightarrow{j\\to\\infty} 0\n\\]\nwhere \\( L > 0 \\) is fixed and \\( R_j \\to 0 \\). \nAn axial displacement \\(\\tilde{t}\\) is imposed at the top boundary \\(z = L\\), \nproducing a fixed, uniform, axial strain\n\\[\n\\varepsilon_z = \\frac{\\tilde{t}}{L}, \\quad \\text{uniform for all } j\\in\\mathbb{N}.\n\\]\n\nThe total energy associated with each configuration \\((\\tilde{\\boldsymbol{u}}, \\tilde{\\alpha})\\), \nwith \\( \\tilde{\\boldsymbol{u}} : \\Omega_j \\to \\mathbb{R}^3 \\) the displacement \nand \\( \\tilde{\\alpha} : \\Omega_j \\to [0,1] \\) the damage variable, is given by the functional:\n\\begin{align}\n\\label{eq:energy_physical}\n\\widetilde{E}_j[\\tilde{\\boldsymbol{u}}, \\tilde{\\alpha} ] = \\int_{\\Omega_j} \\frac{1}{2} a_\\eta(\\tilde{\\alpha} (\\boldsymbol{X})) A \\varepsilon : \\varepsilon + w(\\tilde{\\alpha} (\\boldsymbol{X})) + \\frac{1}{2} w_1 \\ell^2 |\\nabla \\tilde{\\alpha} (\\boldsymbol{X})|^2 \\, d\\boldsymbol{X},\n\\end{align}\nwhere \\(A\\) denotes the linear elasticity tensor, $\\varepsilon$ the strain tensor, \n\\(a_\\eta\\) a degradation function, \\(w\\) a damage energy density, \nand \\(\\ell\\) the fixed length-scale parameter (see Section \\ref{se:preliminaries} for more details.)\nWe define the set of admissible configurations by\n\\begin{align} \\label{eq:admissible}\n\\mathcal{A} := \\left\\{ (\\boldsymbol{u}, \\alpha) \\in H^1\\left(\\Omega, \\mathbb{R}^3\\right) \\times H^1(\\Omega) \\mid 0 \\leq \\alpha \\leq 1 \\text{ a.e.},\\, u_3(\\cdot,0) = 0,\\, u_3(\\cdot,1) = -\\varepsilon_z \\right\\},\n\\end{align}\nwhere $\\Omega$ is the unit cylinder and $\\displaystyle{\\boldsymbol{u}=\n \\big (u_1(\\boldsymbol x), u_2(\\boldsymbol x), u_3(\\boldsymbol x)\\big )= \n\\left(\\frac{\\tilde{u}_1}{R_j}, \\frac{\\tilde{u}_2}{R_j}, \\frac{\\tilde{u}_3}{L}\\right)}$ \nis the non-dimensionalized displacement, defined on the rescaled coordinates \n$\\displaystyle{\\boldsymbol{x}=\\left( \\frac{X}{R_j}, \\frac{Y}{R_j}, \\frac{Z}{L}\\right)}$.\nAdditionally, we introduce the reduced ansatz space: \n\\begin{align} \\label{eq:uniaxial_space}\n\\mathcal{S} := \\left\\{(\\boldsymbol u, \\alpha)\\in \\mathcal{A} \\mid \\exists\\, \\bar{u}, \\bar{\\alpha} \\in H^1(0,1) \\text{ such that } u_3(x,y,z) = \\bar{u}(z),\\ \\alpha(x,y,z) = \\bar{\\alpha}(z) \\text{ a.e.} \\right\\}.\n\\end{align}\nIf a pair $(\\boldsymbol u,\\alpha)$ belongs to $\\mathcal{S}$ then, with a slight abuse of notation, \nwe write $u_3'(z)$ to denote $\\displaystyle \\frac{d \\bar u_3}{dz}(z)$, in view of the definition of $\\mathcal{S}$.\nMore generally, throughout the text, we consider any function $u \\in L^2(0,1)$ as a function in $L^2(\\Omega)$ and write indifferently $u(z) = u(\\boldsymbol{x}) = u(x,y,z)$.\n\\medskip\n\nLet $E_j$ denote the energy per unit volume $\\displaystyle{\\widetilde{E}_j\\setminus |\\Omega_j|}$\n(the expression of which in the new variables is given in \\eqref{eq:EjInExtense}).\nWe prove two main results~:\nFirstly, we show that the sequence of functionals \\((E_j)\\) $\\Gamma$-converges \nto the effective one-dimensional functional \\(E_\\infty\\) defined by\n\\begin{align}\n \\label{eq:defEinfty}\nE_\\infty [\\boldsymbol u, \\alpha] =\n\\begin{cases}\n\\displaystyle \\int_0^1 a_\\eta(\\alpha(z)) \\frac{1}{2} E |u_3'(z)|^2 + w(\\alpha(z)) + \\frac{1}{2} w_1 \\left( \\frac{\\ell}{L} \\right)^2 |\\alpha'(z)|^2 \\, dz,\n& ( \\boldsymbol u,\\alpha)\\in \\mathcal{S},\n\\\\\n+\\infty, & (\\boldsymbol u,\\alpha) \\in \\mathcal{A}\\setminus \\mathcal{S}.\n\\end{cases}\n\\end{align}\nMore precisely, we prove that", "context": "In short rods, of length $L \\sim \\ell$, the homogeneous damage solutions\n(where the damage variable is constant throughout the body) remain stable \neven for extreme loading conditions.\nIn contrast, in rods made of a stress-softening material (so that the elastic region in \nstrain space shrinks as the damage progresses), the homogeneous damage solutions lose their \nstability, allowing the internal variable to continue its growth in narrow bands \nof width comparable to $\\ell$.\n\\medskip\n\nIn such gradient damage models,\nthe capability for crack nucleation is associated to the \nloss of stability of the homogeneous damage state and the emergence of \nlocalized damage solutions. \nThe previously mentioned stability \nanalyses~\\cite{pham2011stability,pham2013stability,marigo2016overview} \nhave been conducted mainly for uniaxial tension tests, using the variational inequalities \nof the formally derived one-dimensional gradient damage model. \nA validation of the important dimension reduction from 3D to 1D is thus desirable. \nThis is the purpose of this current work:\nwe prove the $\\Gamma$-convergence \\cite{braides2002gamma,dalMaso2012,ambrosio2000calculus}\nof the three-dimensional model to the one-dimensional gradient damage model in the slender \nrod limit.\n\\medskip\n\n\\medskip\nThe precise formulation of the mathematical problem studied in this article is as follows: consider a sequence of cylindrical domains\n\\[\n\\Omega_j := \\{\\boldsymbol{x}=(x,y,z)\\in \\mathbb{R}^3: x^2+y^2\\leq R_j, \\hspace{0.2cm} 0\\leq z\\leq L\\}, \n\\quad j\\in \\mathbb{N}, \\quad \\text{with} \\quad \\delta_j := \\frac{R_j}{L} \\xrightarrow{j\\to\\infty} 0\n\\]\nwhere \\( L > 0 \\) is fixed and \\( R_j \\to 0 \\). \nAn axial displacement \\(\\tilde{t}\\) is imposed at the top boundary \\(z = L\\), \nproducing a fixed, uniform, axial strain\n\\[\n\\varepsilon_z = \\frac{\\tilde{t}}{L}, \\quad \\text{uniform for all } j\\in\\mathbb{N}.\n\\]\n\nThe total energy associated with each configuration \\((\\tilde{\\boldsymbol{u}}, \\tilde{\\alpha})\\), \nwith \\( \\tilde{\\boldsymbol{u}} : \\Omega_j \\to \\mathbb{R}^3 \\) the displacement \nand \\( \\tilde{\\alpha} : \\Omega_j \\to [0,1] \\) the damage variable, is given by the functional:\n\\begin{align}\n\\label{eq:energy_physical}\n\\widetilde{E}_j[\\tilde{\\boldsymbol{u}}, \\tilde{\\alpha} ] = \\int_{\\Omega_j} \\frac{1}{2} a_\\eta(\\tilde{\\alpha} (\\boldsymbol{X})) A \\varepsilon : \\varepsilon + w(\\tilde{\\alpha} (\\boldsymbol{X})) + \\frac{1}{2} w_1 \\ell^2 |\\nabla \\tilde{\\alpha} (\\boldsymbol{X})|^2 \\, d\\boldsymbol{X},\n\\end{align}\nwhere \\(A\\) denotes the linear elasticity tensor, $\\varepsilon$ the strain tensor, \n\\(a_\\eta\\) a degradation function, \\(w\\) a damage energy density, \nand \\(\\ell\\) the fixed length-scale parameter (see Section \\ref{se:preliminaries} for more details.)\nWe define the set of admissible configurations by\n\\begin{align} \\label{eq:admissible}\n\\mathcal{A} := \\left\\{ (\\boldsymbol{u}, \\alpha) \\in H^1\\left(\\Omega, \\mathbb{R}^3\\right) \\times H^1(\\Omega) \\mid 0 \\leq \\alpha \\leq 1 \\text{ a.e.},\\, u_3(\\cdot,0) = 0,\\, u_3(\\cdot,1) = -\\varepsilon_z \\right\\},\n\\end{align}\nwhere $\\Omega$ is the unit cylinder and $\\displaystyle{\\boldsymbol{u}=\n \\big (u_1(\\boldsymbol x), u_2(\\boldsymbol x), u_3(\\boldsymbol x)\\big )= \n\\left(\\frac{\\tilde{u}_1}{R_j}, \\frac{\\tilde{u}_2}{R_j}, \\frac{\\tilde{u}_3}{L}\\right)}$ \nis the non-dimensionalized displacement, defined on the rescaled coordinates \n$\\displaystyle{\\boldsymbol{x}=\\left( \\frac{X}{R_j}, \\frac{Y}{R_j}, \\frac{Z}{L}\\right)}$.\nAdditionally, we introduce the reduced ansatz space: \n\\begin{align} \\label{eq:uniaxial_space}\n\\mathcal{S} := \\left\\{(\\boldsymbol u, \\alpha)\\in \\mathcal{A} \\mid \\exists\\, \\bar{u}, \\bar{\\alpha} \\in H^1(0,1) \\text{ such that } u_3(x,y,z) = \\bar{u}(z),\\ \\alpha(x,y,z) = \\bar{\\alpha}(z) \\text{ a.e.} \\right\\}.\n\\end{align}\nIf a pair $(\\boldsymbol u,\\alpha)$ belongs to $\\mathcal{S}$ then, with a slight abuse of notation, \nwe write $u_3'(z)$ to denote $\\displaystyle \\frac{d \\bar u_3}{dz}(z)$, in view of the definition of $\\mathcal{S}$.\nMore generally, throughout the text, we consider any function $u \\in L^2(0,1)$ as a function in $L^2(\\Omega)$ and write indifferently $u(z) = u(\\boldsymbol{x}) = u(x,y,z)$.\n\\medskip\n\nLet $E_j$ denote the energy per unit volume $\\displaystyle{\\widetilde{E}_j\\setminus |\\Omega_j|}$\n(the expression of which in the new variables is given in \\eqref{eq:EjInExtense}).\nWe prove two main results~:\nFirstly, we show that the sequence of functionals \\((E_j)\\) $\\Gamma$-converges \nto the effective one-dimensional functional \\(E_\\infty\\) defined by\n\\begin{align}\n \\label{eq:defEinfty}\nE_\\infty [\\boldsymbol u, \\alpha] =\n\\begin{cases}\n\\displaystyle \\int_0^1 a_\\eta(\\alpha(z)) \\frac{1}{2} E |u_3'(z)|^2 + w(\\alpha(z)) + \\frac{1}{2} w_1 \\left( \\frac{\\ell}{L} \\right)^2 |\\alpha'(z)|^2 \\, dz,\n& ( \\boldsymbol u,\\alpha)\\in \\mathcal{S},\n\\\\\n+\\infty, & (\\boldsymbol u,\\alpha) \\in \\mathcal{A}\\setminus \\mathcal{S}.\n\\end{cases}\n\\end{align}\nMore precisely, we prove that", "full_context": "In short rods, of length $L \\sim \\ell$, the homogeneous damage solutions\n(where the damage variable is constant throughout the body) remain stable \neven for extreme loading conditions.\nIn contrast, in rods made of a stress-softening material (so that the elastic region in \nstrain space shrinks as the damage progresses), the homogeneous damage solutions lose their \nstability, allowing the internal variable to continue its growth in narrow bands \nof width comparable to $\\ell$.\n\\medskip\n\nIn such gradient damage models,\nthe capability for crack nucleation is associated to the \nloss of stability of the homogeneous damage state and the emergence of \nlocalized damage solutions. \nThe previously mentioned stability \nanalyses~\\cite{pham2011stability,pham2013stability,marigo2016overview} \nhave been conducted mainly for uniaxial tension tests, using the variational inequalities \nof the formally derived one-dimensional gradient damage model. \nA validation of the important dimension reduction from 3D to 1D is thus desirable. \nThis is the purpose of this current work:\nwe prove the $\\Gamma$-convergence \\cite{braides2002gamma,dalMaso2012,ambrosio2000calculus}\nof the three-dimensional model to the one-dimensional gradient damage model in the slender \nrod limit.\n\\medskip\n\n\\medskip\nThe precise formulation of the mathematical problem studied in this article is as follows: consider a sequence of cylindrical domains\n\\[\n\\Omega_j := \\{\\boldsymbol{x}=(x,y,z)\\in \\mathbb{R}^3: x^2+y^2\\leq R_j, \\hspace{0.2cm} 0\\leq z\\leq L\\}, \n\\quad j\\in \\mathbb{N}, \\quad \\text{with} \\quad \\delta_j := \\frac{R_j}{L} \\xrightarrow{j\\to\\infty} 0\n\\]\nwhere \\( L > 0 \\) is fixed and \\( R_j \\to 0 \\). \nAn axial displacement \\(\\tilde{t}\\) is imposed at the top boundary \\(z = L\\), \nproducing a fixed, uniform, axial strain\n\\[\n\\varepsilon_z = \\frac{\\tilde{t}}{L}, \\quad \\text{uniform for all } j\\in\\mathbb{N}.\n\\]\n\nThe total energy associated with each configuration \\((\\tilde{\\boldsymbol{u}}, \\tilde{\\alpha})\\), \nwith \\( \\tilde{\\boldsymbol{u}} : \\Omega_j \\to \\mathbb{R}^3 \\) the displacement \nand \\( \\tilde{\\alpha} : \\Omega_j \\to [0,1] \\) the damage variable, is given by the functional:\n\\begin{align}\n\\label{eq:energy_physical}\n\\widetilde{E}_j[\\tilde{\\boldsymbol{u}}, \\tilde{\\alpha} ] = \\int_{\\Omega_j} \\frac{1}{2} a_\\eta(\\tilde{\\alpha} (\\boldsymbol{X})) A \\varepsilon : \\varepsilon + w(\\tilde{\\alpha} (\\boldsymbol{X})) + \\frac{1}{2} w_1 \\ell^2 |\\nabla \\tilde{\\alpha} (\\boldsymbol{X})|^2 \\, d\\boldsymbol{X},\n\\end{align}\nwhere \\(A\\) denotes the linear elasticity tensor, $\\varepsilon$ the strain tensor, \n\\(a_\\eta\\) a degradation function, \\(w\\) a damage energy density, \nand \\(\\ell\\) the fixed length-scale parameter (see Section \\ref{se:preliminaries} for more details.)\nWe define the set of admissible configurations by\n\\begin{align} \\label{eq:admissible}\n\\mathcal{A} := \\left\\{ (\\boldsymbol{u}, \\alpha) \\in H^1\\left(\\Omega, \\mathbb{R}^3\\right) \\times H^1(\\Omega) \\mid 0 \\leq \\alpha \\leq 1 \\text{ a.e.},\\, u_3(\\cdot,0) = 0,\\, u_3(\\cdot,1) = -\\varepsilon_z \\right\\},\n\\end{align}\nwhere $\\Omega$ is the unit cylinder and $\\displaystyle{\\boldsymbol{u}=\n \\big (u_1(\\boldsymbol x), u_2(\\boldsymbol x), u_3(\\boldsymbol x)\\big )= \n\\left(\\frac{\\tilde{u}_1}{R_j}, \\frac{\\tilde{u}_2}{R_j}, \\frac{\\tilde{u}_3}{L}\\right)}$ \nis the non-dimensionalized displacement, defined on the rescaled coordinates \n$\\displaystyle{\\boldsymbol{x}=\\left( \\frac{X}{R_j}, \\frac{Y}{R_j}, \\frac{Z}{L}\\right)}$.\nAdditionally, we introduce the reduced ansatz space: \n\\begin{align} \\label{eq:uniaxial_space}\n\\mathcal{S} := \\left\\{(\\boldsymbol u, \\alpha)\\in \\mathcal{A} \\mid \\exists\\, \\bar{u}, \\bar{\\alpha} \\in H^1(0,1) \\text{ such that } u_3(x,y,z) = \\bar{u}(z),\\ \\alpha(x,y,z) = \\bar{\\alpha}(z) \\text{ a.e.} \\right\\}.\n\\end{align}\nIf a pair $(\\boldsymbol u,\\alpha)$ belongs to $\\mathcal{S}$ then, with a slight abuse of notation, \nwe write $u_3'(z)$ to denote $\\displaystyle \\frac{d \\bar u_3}{dz}(z)$, in view of the definition of $\\mathcal{S}$.\nMore generally, throughout the text, we consider any function $u \\in L^2(0,1)$ as a function in $L^2(\\Omega)$ and write indifferently $u(z) = u(\\boldsymbol{x}) = u(x,y,z)$.\n\\medskip\n\nLet $E_j$ denote the energy per unit volume $\\displaystyle{\\widetilde{E}_j\\setminus |\\Omega_j|}$\n(the expression of which in the new variables is given in \\eqref{eq:EjInExtense}).\nWe prove two main results~:\nFirstly, we show that the sequence of functionals \\((E_j)\\) $\\Gamma$-converges \nto the effective one-dimensional functional \\(E_\\infty\\) defined by\n\\begin{align}\n \\label{eq:defEinfty}\nE_\\infty [\\boldsymbol u, \\alpha] =\n\\begin{cases}\n\\displaystyle \\int_0^1 a_\\eta(\\alpha(z)) \\frac{1}{2} E |u_3'(z)|^2 + w(\\alpha(z)) + \\frac{1}{2} w_1 \\left( \\frac{\\ell}{L} \\right)^2 |\\alpha'(z)|^2 \\, dz,\n& ( \\boldsymbol u,\\alpha)\\in \\mathcal{S},\n\\\\\n+\\infty, & (\\boldsymbol u,\\alpha) \\in \\mathcal{A}\\setminus \\mathcal{S}.\n\\end{cases}\n\\end{align}\nMore precisely, we prove that\n\nWe consider the functions \n$\\boldsymbol{u}_{\\text{test}}(\\boldsymbol{x})\n=(u_1(\\boldsymbol{x}),u_2(\\boldsymbol{x}),u_3(\\boldsymbol{x}))$, \nand $\\alpha_{\\text{test}}(\\boldsymbol{x})$, defined by\n\\begin{align}\n \\label{eq:test_function}\nu_1(x,y,z)=\\nu \\varepsilon_zx, \\quad u_2(x,y,z)=\\nu \\varepsilon_zy, \n\\quad u_3(x,y,z)=-\\varepsilon_zz, \\quad \\alpha_{\\text{test}}(x,y,z)\\equiv 0.\n\\end{align}\nNote that $\\boldsymbol{u}_{\\text{test}}$ is the minimizer of the energy when no damage is present, and that Equations \\eqref{eq:aspiration_UB} are satisfied.\nSince $a_\\eta(0)=1$ and $w(0)=0$, see~\\eqref{la8},\nwe evaluate the functional $E_j$ at the test pair \n$\\big( \\vec u_{\\text{test}}, \\alpha_{\\text{test}} \\big)$~:\n\\begin{eqnarray*}\nE_j[\\boldsymbol{{u}}_{\\text{test}},\\alpha_{\\text{test}}]\n&=&\n\\frac{1}{\\pi}\\int_{\\Omega} a_{\\eta}(\\alpha_{\\text{test}}(\\boldsymbol{x}))\\left[ \\mu (\\nu \\varepsilon_z)^2+\\mu (\\nu \\varepsilon_z)^2+\\mu \\varepsilon_z^2 + \\frac{\\lambda}{2}\\left(\\nu \\varepsilon_z+\\nu \\varepsilon_z- \\varepsilon_z\\right)^2 \\right]+w(\\alpha_{\\text{test}}(\\boldsymbol{x})) \\,dx\n\\\\\n&=&\n\\frac{1}{\\pi}\\int_{\\Omega}\\left[ \\mu\\varepsilon_z^2(2\\nu^2+1)\n+ \\frac{\\lambda}{2}\\varepsilon_z^2(2\\nu-1)^2 \\right] \\,dx\n\\;=\\;\n\\frac{E}{2}\\varepsilon_z^2.\n\\end{eqnarray*}\nSince by hypothesis, $(\\boldsymbol{u}^{(j)},\\alpha^{(j)})$ minimizes \n$E_{j}[\\boldsymbol{u},\\alpha]$, we infer that\n\\begin{align}\n \\label{eq:aPrioriBound}\nE_{j}[\\boldsymbol{u}^{(j) },\\alpha^{(j)}]\n\\leq \nE_j[\\boldsymbol{u}_{\\text{test}},\\alpha_{\\text{test}}]\n\\;=\\; \\frac{E}{2}\\varepsilon_z^2,\n\\end{align}\nso that $\\displaystyle{E_{j}[\\boldsymbol{u}^{(j) },\\alpha^{(j)}]\\leq M}$ \nis uniformly bounded.\nIt follows from Theorem~\\ref{Teo1} that there exists\n$\\widehat{u}_3,\\hspace{0.1cm} \\widehat \\alpha$ in $H^1(\\Omega)$, depending only on $z$,\nand\n$\\widehat{\\varepsilon}_{11}, \\widehat{\\varepsilon}_{22}, \\widehat{\\varepsilon}_{33}$ in \n$L^2(\\Omega; \\mathbb{R}^{3})$, which satisfy~\\eqref{la0}, such that\n\\[ \\left\\{ \\begin{array}{c}\n\\widehat{u}_3(\\cdot, 0) = 0, \n\\quad \\widehat{u}_3(\\cdot, 1) = -\\varepsilon_z,\n\\qquad 0\\leq \\widehat \\alpha\\leq 1\\quad \\text{a.e. in}\\; (0,1),\n\\\\[3pt]\n\\frac{\\partial u_1^{(j)}}{\\partial x} \\rightharpoonup \\widehat{\\varepsilon}_{11}, \\qquad\n\\frac{\\partial u_2^{(j)}}{\\partial y} \\rightharpoonup \\widehat{\\varepsilon}_{22}, \\qquad\n\\frac{\\partial u_3^{(j)}}{\\partial z} \\rightharpoonup \\widehat{\\varepsilon}_{33},\n\\\\[3pt]\n\\nabla \\alpha^{(j)} \\rightharpoonup \\nabla \\widehat\\alpha, \\qquad\n\\alpha^{(j)} \\to \\widehat\\alpha \\text{ a.e. in}\\; \\Omega,\n\\end{array} \\right.\n\\]\nand such that the sequence $\\overline{u}^{(j)}$, defined in \\eqref{eq:horizontal_averages_u3},\nconverges to $\\widehat u_3$ weakly in $H^1\\big ( (0,1) \\big )$ and strongly in $L^2\\big ((0,1) \\big )$.\nWe define the vector-valued map $\\widehat{\\vec u}\\in H^1(\\Omega;\\R^3)$ by\n\\begin{eqnarray*}\n\\widehat{\\vec u}(x,y,z) &:=& \n\\big(0, 0, \\widehat u_3(z) \\big), \\quad (x,y,z) \\in \\Omega.\n\\end{eqnarray*}\nApplying Proposition \\ref{Prop1}, we obtain\n\\begin{eqnarray}\\label{la9}\n\\lf E_{j} [\\boldsymbol{u}^{(j)}, \\alpha^{(j)}] \n&\\geq&\nE_\\infty [\\widehat {\\vec u}, \\widehat\\alpha] + J + \\lf K^{(j)},\n\\end{eqnarray}\nwhere \n\\begin{eqnarray*}\nJ &=&\n\\frac{1}{\\pi} \\int_{\\Omega}a_{\\eta}(\\widehat{\\alpha}(\\boldsymbol{x}))\n\\left[ \\frac{\\mu}{2} (\\widehat{\\varepsilon}_{11} - \\widehat{\\varepsilon}_{22})^2\n+ 2(\\lambda+\\mu)\\left(\\frac{\\widehat{\\varepsilon}_{11} + \\widehat{\\varepsilon}_{22}}{2}+\\nu \\widehat{\\varepsilon}_{33} \\right)^2\\right] \\,d\\boldsymbol{x}\n\\\\\n&&\n+\\frac{1}{2}\\int_{0}^1 a_{\\eta}(\\widehat{\\alpha}(z))E\n\\left(\\intp\\left( \\vareg_{33}(x,y,z)-\\varepsilon_{33}(z) \\right)^2 \\,d\\mathcal{H}^2(x,y) \n\\,\\right)dz \n\\\\\nK^{(j)} &=&\n\\left\\{ \\frac{\\eta \\mu}{2\\pi} \\int_{\\Omega}\n\\left[ \\left( \\frac{\\partial u_1^{(j)}}{\\partial y} \n+ \\frac{\\partial u_2^{(j)}}{\\partial x} \\right)^2\n+ \\left( \\delta_j \\frac{\\partial u_1^{(j)}}{\\partial z} + \\delta_j^{-1} \\frac{\\partial u_3^{(j)}}{\\partial x} \\right)^2 + \\left( \\delta_j \\frac{\\partial u_2^{(j)}}{\\partial z} + \\delta_j^{-1} \\frac{\\partial u_3^{(j)}}{\\partial y} \\right)^2 \\right] \\, d\\boldsymbol{x} \\right.\n\\\\ \n&&\n+ \\frac{\\mu}{\\pi} \n\\int_{\\Omega}\n\\left[ \n\\left( \\sqrt{a_{\\eta} (\\widehat{\\alpha} (\\boldsymbol{x}))} \\widehat{\\varepsilon}_{11}\n- \\sqrt{a_{\\eta} (\\alpha^{(j)} (\\boldsymbol{x}))} \\frac{\\partial u_1^{(j)}}{\\partial x} \\right)^2 \n+\n\\left( \\sqrt{a_{\\eta} (\\widehat{\\alpha} (\\boldsymbol{x}))} \\widehat{\\varepsilon}_{22}\n- \\sqrt{a_{\\eta} (\\alpha^{(j)} (\\boldsymbol{x}))} \\frac{\\partial u_2^{(j)}}{\\partial y} \\right)^2\n\\right.\n\\\\\n&&\\hspace*{10mm}\n+\n\\left.\n\\left( \\sqrt{a_{\\eta} (\\widehat{\\alpha} (\\boldsymbol{x}))} \\widehat{\\varepsilon}_{33}\n- \\sqrt{a_{\\eta} (\\alpha^{(j)} (\\boldsymbol{x}))} \\frac{\\partial u_3^{(j)}}{\\partial z} \\right)^2\n\\right] \n\\,d \\vec{x} \n\\\\\n&&\n+ \\frac{1}{2} w_1 \\left( \\frac{l}{L} \\right)^2\n\\int_{\\Omega}\n\\left[ \\delta_j^{-2} \\left( \\frac{\\partial \\alpha^{(j)}}{\\partial x} \n- \\frac{\\partial {\\widehat \\alpha}}{\\partial x} \\right)^2 \n+ \\delta_j^{-2} \\left( \\frac{\\partial \\alpha^{(j)}}{\\partial y} \n- \\frac{\\partial {\\widehat \\alpha}}{\\partial y}\\right)^2 \n+ \\left( \\frac{\\partial \\alpha^{(j)}}{\\partial z} \n- \\frac{\\partial {\\widehat \\alpha}}{\\partial z}\\right)^2 \\right]\n\\, d\\boldsymbol{x}\n\\Bigg\\}.\n\\end{eqnarray*}\nBy Theorem \\ref{Teo2}-$(ii)$ applied to $(\\widehat{\\vec u},\\widehat{\\alpha})$, there exist $\\left(\\widehat{\\boldsymbol{u}}^{(j)}_{r},\\widehat{\\alpha}^{(j)}_{r}\\right)_{j\\in \\mathbb{N}}$ in $\\mathcal{A}$ such that\n\\begin{eqnarray}\\label{la10}\n\\limsup_{j\\to \\infty} E_{j}\\left[ \\widehat{\\boldsymbol{u}}^{(j)}_{r},\\widehat{\\alpha}_{r}^{(j)}\\right]= E_\\infty [\\widehat{\\vec u},\\widehat{\\alpha}].\n\\end{eqnarray}\nSince, for each $j$, $\\big (\\boldsymbol{u}^{(j)}, \\alpha^{(j)}\\big )$\nis a minimizer,\n\\[\n \\liminf_{j\\to\\infty} E_{j} [\\boldsymbol{u}^{(j)}, \\alpha^{(j)}]\n \\leq\n \\limsup_{j\\to\\infty} E_{j} [\\boldsymbol{u}^{(j)}, \\alpha^{(j)}]\n\\leq\n\\limsup_{j\\to\\infty} E_{j} [\\widehat{\\boldsymbol{u}}_r^{(j)}, \\widehat{\\alpha}^{(j)}_r].\n\\]\nThus we deduce from~\\eqref{la9} and~\\eqref{la10} that\n\\begin{eqnarray*}\nE_\\infty [\\widehat{\\vec u}, \\widehat\\alpha] \\;+\\; J \\;+\\; \\;\\liminf_{j\\to \\infty} K^{(j)}\n&\\leq&\nE_\\infty [\\widehat{\\vec u}, \\widehat\\alpha].\n\\end{eqnarray*}\nPassing to a subsequence, we may assume, without loss of generality, that the lim-inf \nin the above inequalities is actually a limit.\nSince expressions of $J$ and $K^{(j)}$ are sums of squares, we obtain\n\\begin{eqnarray}\n \\label{eq:limit_Poisson}\n\\frac{1}{\\pi} \\int_{\\Omega}a_{\\eta}(\\widehat{\\alpha}(\\boldsymbol{x}))\n\\left[ \\frac{\\mu}{2} (\\widehat{\\varepsilon}_{11} - \\widehat{\\varepsilon}_{22})^2\n + 2(\\lambda+\\mu)\\left(\\frac{\\widehat{\\varepsilon}_{11} + \\widehat{\\varepsilon}_{22}}{2}+\\nu \\widehat{\\varepsilon}_{33} \\right)^2\\right] \\,d\\boldsymbol{x}\n&=& 0.\n\\end{eqnarray}\n\\begin{eqnarray} \\label{eq:limitJensen}\n \\int_{0}^1 a_{\\eta}(\\widehat{\\alpha}(z))E\\left(\\intp\\left( \\vareg_{33}(x,y,z)-\\varepsilon_{33}(z) \\right)^2d\\mathcal{H}^2(x,y)\\right) \\,dz &=& 0,\n\\end{eqnarray}\n\\begin{eqnarray} \\label{eq:limitShearStrains}\n\\int_{\\Omega}\n\\left( \\frac{\\partial u_1^{(j)}}{\\partial y} + \\frac{\\partial u_2^{(j)}}{\\partial x} \\right)^2\n+ \\left( \\delta_j \\frac{\\partial u_1^{(j)}}{\\partial z} + \\delta_j^{-1} \\frac{\\partial u_3^{(j)}}{\\partial x} \\right)^2 + \\left( \\delta_j \\frac{\\partial u_2^{(j)}}{\\partial z} + \\delta_j^{-1} \\frac{\\partial u_3^{(j)}}{\\partial y} \\right)^2 \\,d\\boldsymbol{x}\n&\\to& 0,\n\\end{eqnarray}\n\\begin{eqnarray} \\label{eq:limitNormalStrains1}\n\\int_{\\Omega}\n\\left( \\sqrt{a_{\\eta} (\\widehat{\\alpha} (\\boldsymbol{x}))} \\widehat{\\varepsilon}_{11}\n- \\sqrt{a_{\\eta} (\\alpha^{(j)} (\\boldsymbol{x}))} \\frac{\\partial u_1^{(j)}}{\\partial x} \\right)^2 \\,d\\boldsymbol{x}\n&\\to& 0,\n\\end{eqnarray}", "post_theorem_intro_text_len": 5876, "post_theorem_intro_text": "\\medskip \n\nSecondly, we prove that if~$( \\boldsymbol{u}^{(j)}, \\alpha^{(j)})_{j \\in \\mathbb{N}}$\nis a sequence of minimizers of~$E_j$, a stronger result holds\n\\begin{theorem}\\label{Main}\nSuppose that for each $j\\in\\mathbb{N}$\n\\begin{eqnarray*}\n(\\boldsymbol u^{(j)}, \\alpha^{(j)}) \\ \\text{minimizes}\\ E_j[\\boldsymbol u, \\alpha]\n\\ \\text{in}\\ \\mathcal A.\n\\end{eqnarray*}\nThen there exists a pair $(\\widehat{\\boldsymbol u},\\widehat{\\alpha})$ in $\\mathcal {S}$\nsuch that (for a subsequence)\n\\begin{eqnarray}\n &&\n \\nonumber\n \\int_\\Omega |u_3^{(j)}(x,y,z) - \\widehat{u}_3(z)|^2 d\\boldsymbol x \\longrightarrow 0, \\quad\n \\int_\\Omega \\left |\\frac{\\partial u_3^{(j)}}{\\partial z}(x,y,z) - \\frac{d \\widehat{u}_3}{d z}(z)\\right |^2d\\boldsymbol x\n \\longrightarrow{0},\n \\\\\n &&\n \\label{eq:strainL2converges}\n \\int_\\Omega \\left |\\frac{\\partial u_1^{(j)}}{\\partial x}(x,y,z) + \\nu \\frac{d \\widehat{u}_3}{d z}(z)\\right |^2d\\boldsymbol x\n \\longrightarrow{0},\n \\quad\n \\int_\\Omega \\left |\\frac{\\partial u_2^{(j)}}{\\partial y}(x,y,z) + \\nu \\frac{d \\widehat{u}_3}{d z}(z)\\right |^2d\\boldsymbol x\n \\longrightarrow{0},\n \\\\\n &&\n \\nonumber\n \\int_\\Omega \\left | \\frac{\\partial u_1^{(j)}}{\\partial y} + \\frac{\\partial u_2^{(j)}}{\\partial x}\\right|^2\n +\\left |\\delta_j \\frac{\\partial u_1^{(j)}}{\\partial z} + \\delta_j^{-1} \\frac{\\partial u_3^{(j)}}{\\partial x} \\right|^2\n +\\left |\\delta_j \\frac{\\partial u_2^{(j)}}{\\partial z} + \\delta_j^{-1} \\frac{\\partial u_3^{(j)}}{\\partial y} \\right|^2d\\boldsymbol x\n \\longrightarrow 0,\n \\\\\n &&\n \\nonumber\n \\qquad \\alpha^{(j)} \\xrightarrow{H^1 (\\Omega)} \\widehat \\alpha,\n \\quad \\text{and}\\quad (\\widehat{\\boldsymbol u}, \\widehat{\\alpha})\\ \\text{minimizes}\\ E_\\infty[\\boldsymbol u, \\alpha]\\ \\text{in}\\ \\mathcal {S}.\n\\end{eqnarray}\n\\end{theorem}\n\n\\begin{remark*}\n\\begin{itemize}\n\\item[1.]\n Theorem~\\ref{Teo2} is a $\\Gamma$-convergence result, but for a non-standard topology. Its proof is based\n on Theorem~3 below, a compactness result, which characterizes this `dimension reduction' topology.\n The latter Theorem indeed describes the structure of weak limits of admissible $(\\boldsymbol{u}^{(j)}, \\alpha^{(j)})$'s\nwhen only some of the components of the displacement fields or their derivatives can be expected to be controlled, \nwhen the energies $E(\\boldsymbol{u}^{(j)}, \n\\alpha^{(j)})$ are uniformly bounded.\n\\item[2.] \nConcerning Theorem~2, its proof combines Theorem~3\nwith a precise lower bound on the energies, when the convergences \nin Theorem~3 hold. This lower bound is derived in Proposition~1.\n\\item[3.] In addition, under the hypotheses of Theorem~2,\nwe show strong convergence of the minimizers \n(or, more precisely, on the components $u_3^{(j)}$, on the strain tensors $\\varepsilon^{(j)}$, and on the damage variables $\\alpha^{(j)}$ and their gradients),\nanother particular feature of the underlying `dimension reduction' topology.\n\\item[4.]\nAs another feature of this topology,\none can choose $u_3^{(j)}(x,y,z) = \\widehat{u}_3(z)$ to construct a recovery sequence\nin the proof of Theorem~1, i.e., a function which is independent of $j$ (and of $x,y$). The horizontal components $u_1^{(j)}$, $u_2^{(j)}$ do depend on $j$ in the recovery sequence, even with gradients that grow unbounded as $j\\to\\infty$, but their contribution to the energy becomes negligible in the slender rod limit because of the prefactor $\\delta_j$ in the rescaled shear strains in \\eqref{eq:strainL2converges}, \\eqref{eq:EjInExtense}. The system is dominated by the behaviour of the axial displacement $u_3$, to which the horizontal components are able to adjust, guided by the energy-minimality criterion of reducing, as much as possible, the shear strains (see Equations \\eqref{eq:aspiration_UB}, \\eqref{eq:horizontal_adapt}, \\eqref{eq:test_function}, and \\eqref{eq:uniaxial_strain}).\nThis is consistent with the way in which the limit functional $E_\\infty$ depends on the vectorial displacement field $\\boldsymbol u$: through the axial component $u_3$ only.\n\\item[5.] In this work, the internal characteristic length~$\\ell$ and\nthe regularization parameter $\\eta$ are fixed positive quantities,\nso that for fixed damage $\\alpha$, the associated elastic displacement\nis the solution to an elliptic PDE. It would be interesting to study\nunder which regimes the dimension reduction could be performed, when\nthe parameters $\\ell_j$, $\\eta_j$ are allowed to tend to 0. \n\\end{itemize}\n\\end{remark*}\n\\medskip\n\nAn extensive literature is available for the rigorous derivation of reduced models \nobtained from three-dimensional elasticity (see, e.g., \\cite[Chapter 16]{antman2005} \nfor linear theories of rods; \\cite[Chapters A.1 and B.5]{ciarlet2021volII} for Kirchhoff-Love and von K\\'arm\\'an equations in the theory of plates,\n\\cite{FJM2006} and~\\cite{neukammRichter2024} for models for plates and rods in the context \nof nonlinear elasticity).\nThe techniques we use here are inspired by the rigorous derivation of the model for fracture \nand delamination of a thin plate on an elastic foundation, proposed in~\\cite{baldelli2014},\nand by the dimension reduction analysis of a brittle Kirchhoff-Love plate in the SBD setting\nderived~in \\cite{babadjianHenao2016}.\n\\medskip\n\nThe structure of the article is as follows: \nSection 2 presents preliminary definitions, notation, and the assumptions under\nwhich our analysis is carried out.\nIn Section 3, we detail the non-dimensionalization and rescaling process.\nIn Section~4, we proceed with a discussion on compactness results, while\nSection 5 is dedicated to deriving the lower and upper bounds for the energy \nfunctionals $E_j$.\nFinally, the proof of Theorem~\\ref{Main} is assembled in Section~\\ref{se:minimizers}.", "sketch": "The post-theorem text gives the following proof outline for Theorem~\\ref{Teo2}.\n\n- It is a $\\Gamma$-convergence result “for a non-standard topology”; its proof is “based on Theorem~3 below, a compactness result, which characterizes this `dimension reduction' topology”, describing “the structure of weak limits of admissible $(\\boldsymbol{u}^{(j)},\\alpha^{(j)})$'s when only some of the components of the displacement fields or their derivatives can be expected to be controlled, when the energies $E(\\boldsymbol{u}^{(j)},\\alpha^{(j)})$ are uniformly bounded.”\n\n- The proof “combines Theorem~3 with a precise lower bound on the energies, when the convergences in Theorem~3 hold.” This lower bound is “derived in Proposition~1.”\n\n- For the recovery sequence (the $\\Gamma$-$\\limsup$ part), “one can choose $u_3^{(j)}(x,y,z)=\\widehat{u}_3(z)$ … independent of $j$ (and of $x,y$).” The components $u_1^{(j)},u_2^{(j)}$ “do depend on $j$ … even with gradients that grow unbounded as $j\\to\\infty$, but their contribution to the energy becomes negligible … because of the prefactor $\\delta_j$ in the rescaled shear strains in \\eqref{eq:strainL2converges}, \\eqref{eq:EjInExtense}.” Thus “the system is dominated by the behaviour of the axial displacement $u_3$,” consistent with “the limit functional $E_\\infty$ … through the axial component $u_3$ only.”", "expanded_sketch": "The post-theorem text gives the following proof outline for the main theorem.\n\n- It is a $\\Gamma$-convergence result “for a non-standard topology”; its proof is “based on Theorem~3 below, a compactness result, which characterizes this `dimension reduction' topology”, describing “the structure of weak limits of admissible $(\\boldsymbol{u}^{(j)},\\alpha^{(j)})$'s when only some of the components of the displacement fields or their derivatives can be expected to be controlled, when the energies $E(\\boldsymbol{u}^{(j)},\\alpha^{(j)})$ are uniformly bounded.”\n\n- The proof “combines Theorem~3 with a precise lower bound on the energies, when the convergences in Theorem~3 hold.” This lower bound is “derived in Proposition~1.”\n\n- For the recovery sequence (the $\\Gamma$-$\\limsup$ part), “one can choose $u_3^{(j)}(x,y,z)=\\widehat{u}_3(z)$ … independent of $j$ (and of $x,y$).” The components $u_1^{(j)},u_2^{(j)}$ “do depend on $j$ … even with gradients that grow unbounded as $j\\to\\infty$, but their contribution to the energy becomes negligible … because of the prefactor $\\delta_j$ in the rescaled shear strains in\n\\begin{eqnarray}\n &&\n \\nonumber\n \\int_\\Omega |u_3^{(j)}(x,y,z) - \\widehat{u}_3(z)|^2 d\\vec x \\longrightarrow 0, \\quad\n \\int_\\Omega \\left |\\frac{\\partial u_3^{(j)}}{\\partial z}(x,y,z) - \\frac{d \\widehat{u}_3}{d z}(z)\\right |^2d\\vec x\n \\longrightarrow{0},\n \\\\\n &&\n \\label{eq:strainL2converges}\n \\int_\\Omega \\left |\\frac{\\partial u_1^{(j)}}{\\partial x}(x,y,z) + \\nu \\frac{d \\widehat{u}_3}{d z}(z)\\right |^2d\\vec x\n \\longrightarrow{0},\n \\quad\n \\int_\\Omega \\left |\\frac{\\partial u_2^{(j)}}{\\partial y}(x,y,z) + \\nu \\frac{d \\widehat{u}_3}{d z}(z)\\right |^2d\\vec x\n \\longrightarrow{0},\n \\\\\n &&\n \\nonumber\n \\int_\\Omega \\left | \\frac{\\partial u_1^{(j)}}{\\partial y} + \\frac{\\partial u_2^{(j)}}{\\partial x}\\right|^2\n +\\left |\\delta_j \\frac{\\partial u_1^{(j)}}{\\partial z} + \\delta_j^{-1} \\frac{\\partial u_3^{(j)}}{\\partial x} \\right|^2\n +\\left |\\delta_j \\frac{\\partial u_2^{(j)}}{\\partial z} + \\delta_j^{-1} \\frac{\\partial u_3^{(j)}}{\\partial y} \\right|^2d\\vec x\n \\longrightarrow 0,\n \\\\\n &&\n \\nonumber\n \\qquad \\alpha^{(j)} \\xrightarrow{H^1 (\\Omega)} \\widehat \\alpha,\n \\quad \\text{and}\\quad (\\widehat{\\vec u}, \\widehat{\\alpha})\\ \\text{minimizes}\\ E_\\infty[\\vec u, \\alpha]\\ \\text{in}\\ \\mathcal {S}.\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{la1}\nE_{j} [\\boldsymbol{u}, \\alpha] \n&:=&\n\\frac{1}{\\pi}\\int_{\\Omega} a_{\\eta} (\\alpha (\\boldsymbol{x})) \n\\left[\n\\mu \\left( \\frac{\\partial u_1}{\\partial x} \\right)^2\n+ \\mu \\left( \\frac{\\partial u_2}{\\partial y} \\right)^2\n+ \\mu \\left( \\frac{\\partial u_3}{\\partial z} \\right)^2\n+ \\frac{\\lambda}{2} \\left( \\frac{\\partial u_1}{\\partial x} + \\frac{\\partial u_2}{\\partial y} + \\frac{\\partial u_3}{\\partial z} \\right)^2\n\\right.\\nonumber\n\\\\ \n&&\\left.+ \\frac{\\mu}{2}\\left( \\frac{\\partial u_1}{\\partial y}+ \\frac{\\partial u_2}{\\partial x} \\right)^2\n+ \\frac{\\mu}{2} \\left( \\delta_j \\frac{\\partial u_1}{\\partial z} + \\delta_j^{-1} \\frac{\\partial u_3}{\\partial x} \\right)^2 + \\frac{\\mu}{2} \\left( \\delta_j \\frac{\\partial u_2}{\\partial z} + \\delta_j^{-1} \\frac{\\partial u_3}{\\partial y} \\right)^2\\right] \n\\nonumber \n\\\\\n\\label{eq:EjInExtense}\n&&+ w (\\alpha (\\boldsymbol{x})) + \\frac{1}{2} w_1 \\left( \\frac{l}{L} \\right)^2 \\left[ \\delta_j^{-2} \\left( \\frac{\\partial \\alpha}{\\partial x} \\right)^2 + \\delta_j^{-2} \\left( \\frac{\\partial \\alpha}{\\partial y} \\right)^2 + \\left( \\frac{\\partial \\alpha}{\\partial z} \\right)^2 \\right] d\\boldsymbol{x}.\n\\end{eqnarray}\n.” Thus “the system is dominated by the behaviour of the axial displacement $u_3$,” consistent with “the limit functional $E_\\infty$ … through the axial component $u_3$ only.”", "expanded_theorem": "\\label{Teo2}\n\\begin{enumerate}\n\\item[i)] {($\\Gamma$-$\\liminf$ inequality)}\n\nLet \\( \\{( \\boldsymbol{u}^{(j)}, \\alpha^{(j)})\\}_{j \\in \\mathbb{N}} \\) be a sequence \nin~\\( \\mathcal{A} \\). \nAssume that for some $\\left( \\widehat{\\boldsymbol u}, \\widehat{\\alpha} \\right) \\in \\mathcal{A}$,\nthe sequence $\\Big ( \\overline{u}_3^{(j)} \\Big )_{j\\in\\mathbb{N}}$ defined by\n\\begin{align}\n \\label{eq:horizontal_averages_u3}\nnt_{x_1^2+x_2^2 < 1} u_3^{(j)} (x,y,z) \\, d\\mathcal{H}^2 (x,y).\n\\end{align}\nconverges weakly in $L^2(\\Omega)$ to $\\widehat{u}_3$, and that $\\alpha^{(j)}$ converges strongly in $L^2(\\Omega)$ to $\\widehat{\\alpha}$.\nThen \n\\begin{align*}\n\t E_\\infty[\\widehat {\\boldsymbol u}, \\widehat \\alpha]\n\t \\leq \\liminf_{j\\to\\infty} E_j[\\boldsymbol u^{(j)}, \\alpha^{(j)}].\n\\end{align*}\n\n\\item[ii)] {(\\, $\\Gamma$-$\\limsup$ inequality)}\n\nGiven any $(\\boldsymbol u,\\alpha)\\in \\mathcal{S}$, it is possible to construct a sequence $(\\boldsymbol{u}^{(j)},\\alpha^{(j)})$ in $\\mathcal{A}$ such that\n $$ \\big (u_3^{(j)}, \\alpha^{(j)}\\big) \\quad \\text{converges strongly in } L^{2}(\\Omega)\\times L^2(\\Omega)\n \\text{ to } \\big (u_3, \\alpha\\big ), $$\n and\n \\[\n \\lim_{j \\to \\infty} E_j[{\\boldsymbol u}^{(j)}, \\alpha^{(j)}] =\\int_0^1 a_\\eta\\big(\\alpha(z)\\big) \\cdot \\frac{1}{2} E u_3'(z)^2 + w\\big(\\alpha(z)\\big) + \\frac{w_1 \\ell^2}{2L^2} |\\alpha'(z)|^2 \\, \\mathrm{d}z.\n \\]\n\\end{enumerate}", "theorem_type": ["Inequality or Bound", "Universal–Existential"], "mcq": {"question": "Let \\\\(\\Omega=\\{(x,y,z)\\in\\mathbb R^3: x^2+y^2\\le 1,\\ 0\\le z\\le 1\\}\\\\). Define the admissible class\n\\\\[\n\\mathcal A=\\{(\\boldsymbol u,\\alpha)\\in H^1(\\Omega;\\mathbb R^3)\\times H^1(\\Omega): 0\\le \\alpha\\le 1\\ \\text{a.e.},\\ u_3(\\cdot,0)=0,\\ u_3(\\cdot,1)=-\\varepsilon_z\\},\n\\\\]\nand the reduced one-dimensional class\n\\\\[\n\\mathcal S=\\{(\\boldsymbol u,\\alpha)\\in\\mathcal A: \\exists\\,\\bar u,\\bar\\alpha\\in H^1(0,1)\\text{ with }u_3(x,y,z)=\\bar u(z),\\ \\alpha(x,y,z)=\\bar\\alpha(z)\\text{ a.e.}\\}.\n\\\\]\nLet \\\\(E_j\\\\) be the rescaled energies on \\\\((\\mathcal A)\\\\), and define the effective functional\n\\\\[\nE_\\infty[\\boldsymbol u,\\alpha]=\n\\begin{cases}\n\\displaystyle \\int_0^1 a_\\eta(\\alpha(z))\\,\\frac12 E\\,|u_3'(z)|^2+w(\\alpha(z))+\\frac12 w_1\\left(\\frac\\ell L\\right)^2|\\alpha'(z)|^2\\,dz,&(\\boldsymbol u,\\alpha)\\in\\mathcal S,\\\\[1ex]\n+\\infty,&(\\boldsymbol u,\\alpha)\\in\\mathcal A\\setminus\\mathcal S.\n\\end{cases}\n\\\\]\nFor a sequence \\\\((\\boldsymbol u^{(j)},\\alpha^{(j)})\\in\\mathcal A\\\\), define the cross-sectional average of the axial displacement by\n\\\\[\n\\overline u_3^{(j)}(z):=\\frac1\\pi\\int_{x^2+y^2<1}u_3^{(j)}(x,y,z)\\,d\\mathcal H^2(x,y),\n\\\\]\nviewed as a function on \\\\(\\Omega\\\\) depending only on \\\\(z\\\\). Which quantitative estimate holds?", "correct_choice": {"label": "A", "text": "If for some \\\\((\\widehat{\\boldsymbol u},\\widehat\\alpha)\\in\\mathcal A\\\\), the averages \\\\(\\overline u_3^{(j)}\\\\) converge weakly in \\\\(L^2(\\Omega)\\\\) to \\\\(\\widehat u_3\\\\) and \\\\(\\alpha^{(j)}\\\\) converges strongly in \\\\(L^2(\\Omega)\\\\) to \\\\(\\widehat\\alpha\\\\), then\n\\\\[\nE_\\infty[\\widehat{\\boldsymbol u},\\widehat\\alpha]\\le \\liminf_{j\\to\\infty}E_j[\\boldsymbol u^{(j)},\\alpha^{(j)}].\n\\\\]\nMoreover, for every \\\\((\\boldsymbol u,\\alpha)\\in\\mathcal S\\\\), there exists a sequence \\\\((\\boldsymbol u^{(j)},\\alpha^{(j)})\\in\\mathcal A\\\\) such that \\\\((u_3^{(j)},\\alpha^{(j)})\\\\) converges strongly in \\\\(L^2(\\Omega)\\times L^2(\\Omega)\\\\) to \\\\((u_3,\\alpha)\\\\) and\n\\\\[\n\\lim_{j\\to\\infty}E_j[\\boldsymbol u^{(j)},\\alpha^{(j)}]\n=\\int_0^1 a_\\eta(\\alpha(z))\\,\\frac12 E\\,u_3'(z)^2+w(\\alpha(z))+\\frac{w_1\\ell^2}{2L^2}|\\alpha'(z)|^2\\,dz.\n\\\\]\"\n }\n}"}, "choices": [{"label": "B", "text": "If for some \\((\\widehat{\\boldsymbol u},\\widehat\\alpha)\\in\\mathcal A\\), the full axial components \\(u_3^{(j)}\\) converge weakly in \\(L^2(\\Omega)\\) to \\(\\widehat u_3\\) and \\(\\alpha^{(j)}\\) converges strongly in \\(L^2(\\Omega)\\) to \\(\\widehat\\alpha\\), then\n\\[\nE_\\infty[\\widehat{\\boldsymbol u},\\widehat\\alpha]\\le \\liminf_{j\\to\\infty}E_j[\\boldsymbol u^{(j)},\\alpha^{(j)}].\n\\]\nMoreover, for every \\((\\boldsymbol u,\\alpha)\\in\\mathcal S\\), there exists a sequence \\((\\boldsymbol u^{(j)},\\alpha^{(j)})\\in\\mathcal A\\) such that \\((\\boldsymbol u^{(j)},\\alpha^{(j)})\\) converges strongly in \\(L^2(\\Omega;\\mathbb R^3)\\times L^2(\\Omega)\\) to \\((\\boldsymbol u,\\alpha)\\) and\n\\[\n\\lim_{j\\to\\infty}E_j[\\boldsymbol u^{(j)},\\alpha^{(j)}]\n=\\int_0^1 a_\\eta(\\alpha(z))\\,\\frac12 E\\,u_3'(z)^2+w(\\alpha(z))+\\frac{w_1\\ell^2}{2L^2}|\\alpha'(z)|^2\\,dz.\n\\]"}, {"label": "C", "text": "If for some \\((\\widehat{\\boldsymbol u},\\widehat\\alpha)\\in\\mathcal A\\), the averages \\(\\overline u_3^{(j)}\\) converge weakly in \\(L^2(\\Omega)\\) to \\(\\widehat u_3\\) and \\(\\alpha^{(j)}\\) converges strongly in \\(L^2(\\Omega)\\) to \\(\\widehat\\alpha\\), then\n\\[\nE_\\infty[\\widehat{\\boldsymbol u},\\widehat\\alpha]\\le \\liminf_{j\\to\\infty}E_j[\\boldsymbol u^{(j)},\\alpha^{(j)}].\n\\]"}, {"label": "D", "text": "If for some \\((\\widehat{\\boldsymbol u},\\widehat\\alpha)\\in\\mathcal A\\), the averages \\(\\overline u_3^{(j)}\\) converge weakly in \\(L^2(\\Omega)\\) to \\(\\widehat u_3\\) and \\(\\alpha^{(j)}\\) converges strongly in \\(L^2(\\Omega)\\) to \\(\\widehat\\alpha\\), then\n\\[\nE_\\infty[\\widehat{\\boldsymbol u},\\widehat\\alpha]= \\lim_{j\\to\\infty}E_j[\\boldsymbol u^{(j)},\\alpha^{(j)}].\n\\]\nMoreover, for every \\((\\boldsymbol u,\\alpha)\\in\\mathcal S\\), there exists a sequence \\((\\boldsymbol u^{(j)},\\alpha^{(j)})\\in\\mathcal A\\) such that \\((u_3^{(j)},\\alpha^{(j)})\\) converges strongly in \\(L^2(\\Omega)\\times L^2(\\Omega)\\) to \\((u_3,\\alpha)\\) and\n\\[\n\\lim_{j\\to\\infty}E_j[\\boldsymbol u^{(j)},\\alpha^{(j)}]\n=\\int_0^1 a_\\eta(\\alpha(z))\\,\\frac12 E\\,u_3'(z)^2+w(\\alpha(z))+\\frac{w_1\\ell^2}{2L^2}|\\alpha'(z)|^2\\,dz.\n\\]"}, {"label": "E", "text": "If for some \\((\\widehat{\\boldsymbol u},\\widehat\\alpha)\\in\\mathcal A\\), the averages \\(\\overline u_3^{(j)}\\) converge weakly in \\(L^2(\\Omega)\\) to \\(\\widehat u_3\\) and \\(\\alpha^{(j)}\\) converges weakly in \\(L^2(\\Omega)\\) to \\(\\widehat\\alpha\\), then\n\\[\nE_\\infty[\\widehat{\\boldsymbol u},\\widehat\\alpha]\\le \\liminf_{j\\to\\infty}E_j[\\boldsymbol u^{(j)},\\alpha^{(j)}].\n\\]\nMoreover, for every \\((\\boldsymbol u,\\alpha)\\in\\mathcal S\\), there exists a sequence \\((\\boldsymbol u^{(j)},\\alpha^{(j)})\\in\\mathcal A\\) such that \\((u_3^{(j)},\\alpha^{(j)})\\) converges strongly in \\(L^2(\\Omega)\\times L^2(\\Omega)\\) to \\((u_3,\\alpha)\\) and\n\\[\n\\lim_{j\\to\\infty}E_j[\\boldsymbol u^{(j)},\\alpha^{(j)}]\n=\\int_0^1 a_\\eta(\\alpha(z))\\,\\frac12 E\\,u_3'(z)^2+w(\\alpha(z))+\\frac{w_1\\ell^2}{2L^2}|\\alpha'(z)|^2\\,dz.\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": "nonstandard topology uses cross-sectional averages and only partial displacement control", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the recovery-sequence conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "liminf inequality weakened to false full convergence/equality for arbitrary sequences", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "strong L^2 convergence of damage replaced by weak L^2 convergence", "template_used": "quantifier_dependence"}]}} +{"id": "2601.01164v1", "paper_link": "http://arxiv.org/abs/2601.01164v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{cycle}\nLet $G$ be a connected outerplanar $C_\\ell$-free graph of order $n$ with $\\ell\\ge 3$. Then $q(G)\\le q(K_1\\vee (\\alpha P_{\\ell-2}\\cup P_{r}))$ with equality if and only if $G\\cong K_1\\vee (\\alpha P_{\\ell-2}\\cup P_{r})$, where $\\alpha =\\lfloor\\frac{n-1}{\\ell-2}\\rfloor$ and $r=n-1-\\alpha(\\ell-2)$.", "start_pos": 4925, "end_pos": 5263, "label": "cycle"}, "ref_dict": {"cycle": "\\begin{theorem}\\label{cycle}\nLet $G$ be a connected outerplanar $C_\\ell$-free graph of order $n$ with $\\ell\\ge 3$. Then $q(G)\\le q(K_1\\vee (\\alpha P_{\\ell-2}\\cup P_{r}))$ with equality if and only if $G\\cong K_1\\vee (\\alpha P_{\\ell-2}\\cup P_{r})$, where $\\alpha =\\lfloor\\frac{n-1}{\\ell-2}\\rfloor$ and $r=n-1-\\alpha(\\ell-2)$.\n\\end{theorem}", "path": "\\begin{theorem}\\label{path}\nLet $G$ be a connected outerplanar $tP_\\ell$-free graph of order $n$ with $t\\ge 1$. \n\n(i) If $t=1$, $\\ell\\ge 4$ and $n\\ge \\max\\Big\\{\\lfloor\\frac{\\ell-3}{2} \\rfloor^2+\\lfloor\\frac{\\ell-3}{2} \\rfloor+\\ell-1,30\\lfloor\\frac{\\ell-3}{2} \\rfloor+30\\sqrt{\\lfloor\\frac{\\ell-3}{2} \\rfloor\\big(\\lfloor\\frac{\\ell-3}{2} \\rfloor+1\\big)}\\Big \\}$, then $q(G)\\le q\\big (K_1\\vee \\big(P_{\\lceil\\frac{\\ell-2}{2} \\rceil}\\cup\\alpha P_{\\lfloor\\frac{\\ell-2}{2} \\rfloor}\\cup P_r\\big)\\big)$ with equality if and only if $G\\cong K_1\\vee \\big(P_{\\lceil\\frac{\\ell-2}{2} \\rceil}\\cup\\alpha P_{\\lfloor\\frac{\\ell-2}{2} \\rfloor}\\cup P_r\\big)$, where $\\alpha=\\left \\lfloor \\frac{n-\\ell+1}{\\lfloor\\frac{\\ell-2}{2} \\rfloor}\\right\\rfloor+1$ and $r=n-\\ell+1-(\\alpha-1)\\lfloor\\frac{\\ell-2}{2} \\rfloor$.\n\n(ii) If $t\\ge 2$, $\\ell\\ge 2$ and $n\\ge \\max\\Big\\{\\ell^2+(t-3)\\ell+1,30(\\ell-2)+30\\sqrt{(\\ell-2)(\\ell-1)}\\Big\\}$, then $q(G)\\le q(K_1\\vee (P_{t\\ell-\\ell-1}\\cup\\alpha P_{\\ell-1}\\cup P_r))$ with equality if and only if $G\\cong K_1\\vee (P_{t\\ell-\\ell-1}\\cup\\alpha P_{\\ell-1}\\cup P_r)$, where $\\alpha=\\lfloor \\frac{n-t+2}{\\ell-1}\\rfloor-t+1$ and $r=n-2\\ell+2-(\\alpha-1)(\\ell-1)$.\n\\end{theorem}", "str": "\\begin{theorem}\\label{str}\nLet $F=C_\\ell$ with $3\\le \\ell\\le n$ or $F=tP_k$ with $k\\ge 2$ and $4\\le tk\\le n-1$. If $G$ maximizes the $Q$-index among all connected $n$-vertex $F$-free outerplanar graphs, then $G$ contains a vertex $u$ of degree $n-1$ and $G[N(u)]$ consists of paths. \\end{theorem}"}, "pre_theorem_intro_text_len": 3720, "pre_theorem_intro_text": "For two vertex disjoint graphs $G$ and $H$, $G\\cup H$ denotes the union of $G$ and $H$, and $G\\vee H$ denotes the join of $G$ and $H$. For a positive integer $a$, $aG$ denotes the union of $a$ (vertex disjoint) copies of a graph $G$. A complete graph on $n$ vertices is denoted by $K_n$. A path on $n$ vertices is denoted by $P_n$. A cycle on $n$ vertices is denoted by $C_n$, where $n\\ge 3$.\n\nGiven a graph $F$, a graph $G$ is said to be $F$-free if $G$ does not contain $F$ (as a subgraph). The well known Tur\\'an-type problem is to determine the maximum number of edges of an $n$-vertex $F$-free graph (as well as the $n$-vertex $F$-free graph(s) possessing exactly this number of edges). \nThe classic Tur\\'an theorem \\cite{Tur} states that the balanced complete $r$-partite graph on $n$ vertices is the unique extremal graph maximizing the number of edges among all $n$-vertex $K_{r+1}$-free graphs. The research on Tur\\'an-type problems is well summarized in the survey \\cite{FS}.\n\nIn 2016, Dowden \\cite{Dow} initiated the study of the Tur\\'an-type problem for planar graphs. Given a planar graph $F$, this problem asks for the maximum number of edges of an $n$-vertex $F$-free planar graph. Among others, there are many results for $F$ being cycles or paths, see the survey \\cite{LSS} (and reference therein) and subsequent papers, e.g., \\cite{CLL,GGMPX,GVZ,SWY,SWY2}.\n\nA graph $G$ is outerplanar if it has a planar embedding in which all vertices lie on the boundary of its outer face. Given an outerplanar graph $F$, one may want to know\nthe maximum number of edges of an $n$-vertex $F$-free outerplanar graph. \nRecently, Matolcsi and Nagy \\cite{MN} determined this number for $F=P_3$, Gy\\\"ori et al \\cite{GPX} determined this number for $F=P_4, P_5$. In \\cite{FZ}, Fang and Zhai determined this number for $F=C_\\ell$ with $\\ell\\ge 3$ or $F=P_\\ell$ with $\\ell\\ge 4$. \n\nLet $A(G)$ denote the adjacency matrix of a graph $G$. The spectral radius of $G$ is the largest eigenvalue of $A(G)$. In 1990, Cvetkovi\\'c and Rowlinson \\cite{CR} proposed a conjecture on the maximum spectral radius of outerplanar graphs. Tait and Tobin \\cite{TT} confirmed the conjecture for graphs of sufficiently large order in 2017, and Lin and Ning \\cite{LN} completely confirmed this conjecture in 2021. Recently, Sun et al. \\cite{SWS} determined the maximum spectral radius of $n$-vertex $C_\\ell$-free outerplanar graphs, and Yin and Li \\cite{YL} considered the maximum spectral radius of $n$-vertex $F$-free outerplanar graphs when $F$ is either the graph consisting of $t$-edge disjoint cycles $C_\\ell$ with a common vertex or $F=(t+1)K_2$. \n\nThe signless Laplacian matrix of a graph $G$ is the matrix $Q(G)=D(G)+A(G)$, where $D(G)$ is the degree diagonal matrix of $G$. The largest eigenvalue of $Q(G)$ is known as the $Q$-index of $G$, denoted by $q(G)$. Yu et al. \\cite{YWG} considered the maximum $Q$-index among all $n$-vertex planar graphs. For outerplanar graphs, Yu et al. \\cite{YGW} showed that $K_1\\vee P_{n-1}$ is the unique graph that maximizes the $Q$-index. \nThe study on the $Q$-index version of Tur\\'an-type problem has attracted much attention, see, e.g., \\cite{CJZ,CLZ,NY,ZW}. \n\nMotivated by the above results, we focus on the $Q$-index version of Tur\\'an-type problem on outerplanar graphs in this paper.\n\nSince $K_1 \\vee P_{n-1}$ maximizes the $Q$-index among all $n$-vertex outerplanar graphs, and $K_1 \\vee P_{n-1}$ does not contain two disjoint cycles, it is natural to restrict our attention to graphs forbidding a single cycle of fixed length. Therefore, in the context of cycles, we study the $Q$-index extremal problem for outerplanar graphs that do not contain a given cycle $C_\\ell$.", "context": "For two vertex disjoint graphs $G$ and $H$, $G\\cup H$ denotes the union of $G$ and $H$, and $G\\vee H$ denotes the join of $G$ and $H$. For a positive integer $a$, $aG$ denotes the union of $a$ (vertex disjoint) copies of a graph $G$. A complete graph on $n$ vertices is denoted by $K_n$. A path on $n$ vertices is denoted by $P_n$. A cycle on $n$ vertices is denoted by $C_n$, where $n\\ge 3$.\n\nA graph $G$ is outerplanar if it has a planar embedding in which all vertices lie on the boundary of its outer face. Given an outerplanar graph $F$, one may want to know\nthe maximum number of edges of an $n$-vertex $F$-free outerplanar graph. \nRecently, Matolcsi and Nagy \\cite{MN} determined this number for $F=P_3$, Gy\\\"ori et al \\cite{GPX} determined this number for $F=P_4, P_5$. In \\cite{FZ}, Fang and Zhai determined this number for $F=C_\\ell$ with $\\ell\\ge 3$ or $F=P_\\ell$ with $\\ell\\ge 4$.\n\nLet $A(G)$ denote the adjacency matrix of a graph $G$. The spectral radius of $G$ is the largest eigenvalue of $A(G)$. In 1990, Cvetkovi\\'c and Rowlinson \\cite{CR} proposed a conjecture on the maximum spectral radius of outerplanar graphs. Tait and Tobin \\cite{TT} confirmed the conjecture for graphs of sufficiently large order in 2017, and Lin and Ning \\cite{LN} completely confirmed this conjecture in 2021. Recently, Sun et al. \\cite{SWS} determined the maximum spectral radius of $n$-vertex $C_\\ell$-free outerplanar graphs, and Yin and Li \\cite{YL} considered the maximum spectral radius of $n$-vertex $F$-free outerplanar graphs when $F$ is either the graph consisting of $t$-edge disjoint cycles $C_\\ell$ with a common vertex or $F=(t+1)K_2$.\n\nThe signless Laplacian matrix of a graph $G$ is the matrix $Q(G)=D(G)+A(G)$, where $D(G)$ is the degree diagonal matrix of $G$. The largest eigenvalue of $Q(G)$ is known as the $Q$-index of $G$, denoted by $q(G)$. Yu et al. \\cite{YWG} considered the maximum $Q$-index among all $n$-vertex planar graphs. For outerplanar graphs, Yu et al. \\cite{YGW} showed that $K_1\\vee P_{n-1}$ is the unique graph that maximizes the $Q$-index. \nThe study on the $Q$-index version of Tur\\'an-type problem has attracted much attention, see, e.g., \\cite{CJZ,CLZ,NY,ZW}.\n\nMotivated by the above results, we focus on the $Q$-index version of Tur\\'an-type problem on outerplanar graphs in this paper.\n\nSince $K_1 \\vee P_{n-1}$ maximizes the $Q$-index among all $n$-vertex outerplanar graphs, and $K_1 \\vee P_{n-1}$ does not contain two disjoint cycles, it is natural to restrict our attention to graphs forbidding a single cycle of fixed length. Therefore, in the context of cycles, we study the $Q$-index extremal problem for outerplanar graphs that do not contain a given cycle $C_\\ell$.", "full_context": "For two vertex disjoint graphs $G$ and $H$, $G\\cup H$ denotes the union of $G$ and $H$, and $G\\vee H$ denotes the join of $G$ and $H$. For a positive integer $a$, $aG$ denotes the union of $a$ (vertex disjoint) copies of a graph $G$. A complete graph on $n$ vertices is denoted by $K_n$. A path on $n$ vertices is denoted by $P_n$. A cycle on $n$ vertices is denoted by $C_n$, where $n\\ge 3$.\n\nA graph $G$ is outerplanar if it has a planar embedding in which all vertices lie on the boundary of its outer face. Given an outerplanar graph $F$, one may want to know\nthe maximum number of edges of an $n$-vertex $F$-free outerplanar graph. \nRecently, Matolcsi and Nagy \\cite{MN} determined this number for $F=P_3$, Gy\\\"ori et al \\cite{GPX} determined this number for $F=P_4, P_5$. In \\cite{FZ}, Fang and Zhai determined this number for $F=C_\\ell$ with $\\ell\\ge 3$ or $F=P_\\ell$ with $\\ell\\ge 4$.\n\nLet $A(G)$ denote the adjacency matrix of a graph $G$. The spectral radius of $G$ is the largest eigenvalue of $A(G)$. In 1990, Cvetkovi\\'c and Rowlinson \\cite{CR} proposed a conjecture on the maximum spectral radius of outerplanar graphs. Tait and Tobin \\cite{TT} confirmed the conjecture for graphs of sufficiently large order in 2017, and Lin and Ning \\cite{LN} completely confirmed this conjecture in 2021. Recently, Sun et al. \\cite{SWS} determined the maximum spectral radius of $n$-vertex $C_\\ell$-free outerplanar graphs, and Yin and Li \\cite{YL} considered the maximum spectral radius of $n$-vertex $F$-free outerplanar graphs when $F$ is either the graph consisting of $t$-edge disjoint cycles $C_\\ell$ with a common vertex or $F=(t+1)K_2$.\n\nThe signless Laplacian matrix of a graph $G$ is the matrix $Q(G)=D(G)+A(G)$, where $D(G)$ is the degree diagonal matrix of $G$. The largest eigenvalue of $Q(G)$ is known as the $Q$-index of $G$, denoted by $q(G)$. Yu et al. \\cite{YWG} considered the maximum $Q$-index among all $n$-vertex planar graphs. For outerplanar graphs, Yu et al. \\cite{YGW} showed that $K_1\\vee P_{n-1}$ is the unique graph that maximizes the $Q$-index. \nThe study on the $Q$-index version of Tur\\'an-type problem has attracted much attention, see, e.g., \\cite{CJZ,CLZ,NY,ZW}.\n\nMotivated by the above results, we focus on the $Q$-index version of Tur\\'an-type problem on outerplanar graphs in this paper.\n\nSince $K_1 \\vee P_{n-1}$ maximizes the $Q$-index among all $n$-vertex outerplanar graphs, and $K_1 \\vee P_{n-1}$ does not contain two disjoint cycles, it is natural to restrict our attention to graphs forbidding a single cycle of fixed length. Therefore, in the context of cycles, we study the $Q$-index extremal problem for outerplanar graphs that do not contain a given cycle $C_\\ell$.\n\nSince $K_1 \\vee P_{n-1}$ maximizes the $Q$-index among all $n$-vertex outerplanar graphs, and $K_1 \\vee P_{n-1}$ does not contain two disjoint cycles, it is natural to restrict our attention to graphs forbidding a single cycle of fixed length. Therefore, in the context of cycles, we study the $Q$-index extremal problem for outerplanar graphs that do not contain a given cycle $C_\\ell$.\n\nIn the context of paths, we consider forbidding a single path and forbidding the disjoint union of $t$ paths of the same length. Interestingly, the extremal graphs for these two types of problems behave somewhat differently. Since any connected graph of order $n\\ge 3$ contains $P_2$ and $P_3$, for the case of forbidding a single path, we restrict our attention to paths of order at least four.\n\n(i) If $t=1$, $\\ell\\ge 4$ and $n\\ge \\max\\Big\\{\\lfloor\\frac{\\ell-3}{2} \\rfloor^2+\\lfloor\\frac{\\ell-3}{2} \\rfloor+\\ell-1,30\\lfloor\\frac{\\ell-3}{2} \\rfloor+30\\sqrt{\\lfloor\\frac{\\ell-3}{2} \\rfloor\\big(\\lfloor\\frac{\\ell-3}{2} \\rfloor+1\\big)}\\Big \\}$, then $q(G)\\le q\\big (K_1\\vee \\big(P_{\\lceil\\frac{\\ell-2}{2} \\rceil}\\cup\\alpha P_{\\lfloor\\frac{\\ell-2}{2} \\rfloor}\\cup P_r\\big)\\big)$ with equality if and only if $G\\cong K_1\\vee \\big(P_{\\lceil\\frac{\\ell-2}{2} \\rceil}\\cup\\alpha P_{\\lfloor\\frac{\\ell-2}{2} \\rfloor}\\cup P_r\\big)$, where $\\alpha=\\left \\lfloor \\frac{n-\\ell+1}{\\lfloor\\frac{\\ell-2}{2} \\rfloor}\\right\\rfloor+1$ and $r=n-\\ell+1-(\\alpha-1)\\lfloor\\frac{\\ell-2}{2} \\rfloor$.\n\n(ii) If $t\\ge 2$, $\\ell\\ge 2$ and $n\\ge \\max\\Big\\{\\ell^2+(t-3)\\ell+1,30(\\ell-2)+30\\sqrt{(\\ell-2)(\\ell-1)}\\Big\\}$, then $q(G)\\le q(K_1\\vee (P_{t\\ell-\\ell-1}\\cup\\alpha P_{\\ell-1}\\cup P_r))$ with equality if and only if $G\\cong K_1\\vee (P_{t\\ell-\\ell-1}\\cup\\alpha P_{\\ell-1}\\cup P_r)$, where $\\alpha=\\lfloor \\frac{n-t+2}{\\ell-1}\\rfloor-t+1$ and $r=n-2\\ell+2-(\\alpha-1)(\\ell-1)$.\n\\end{theorem}\n\n\\begin{theorem}\\label{str}\nLet $F=C_\\ell$ with $3\\le \\ell\\le n$ or $F=tP_k$ with $k\\ge 2$ and $4\\le tk\\le n-1$. If $G$ maximizes the $Q$-index among all connected $n$-vertex $F$-free outerplanar graphs, then $G$ contains a vertex $u$ of degree $n-1$ and $G[N(u)]$ consists of paths. \\end{theorem}\n\nBy Theorem \\ref{x}, $\\Delta(G)=n-1$. Let $u$ be the vertex with degree $n-1$.\nAs $G$ is $C_{\\ell}$-free, $G[N(u)]$ is $P_{\\ell-1}$-free. By Lemma \\ref{obv} (i),\n$G[N(u)]$ consists of paths of order at most $\\ell-2$. Note that $r=n-1-\\alpha (\\ell-2)$. \nBy Lemma \\ref{edgeshift} repeatedly, $G[N(u)]\\cong \\alpha P_{\\ell-2}\\cup P_r$, so \n$G\\cong K_1\\vee(\\alpha P_{\\ell-2}\\cup P_r)$.\n\\end{proof}\n\nWe'll show that $a_2=\\lfloor \\frac{\\ell-1}{2}\\rfloor$ if $t=1$, and $a_2=\\ell-1$ if $t\\ge 2$. Suppose to the contrary that $a_2\\le \\lfloor \\frac{\\ell-3}{2}\\rfloor$ if $t=1$, and $a_2\\le \\ell-2$ if $t\\ge 2$. In the former case, we have \\[\ns=\\left\\lceil\\frac{n-1-a_1}{a_2}\\right\\rceil+1=\\left\\lceil\\frac{n-\\ell+1}{a_2}\\right\\rceil+2\\ge \\left\\lceil\\frac{n-\\ell+1}{\\lfloor\\frac{\\ell-3}{2}\\rfloor}\\right\\rceil+2\\ge \\left\\lfloor\\frac{\\ell-3}{2}\\right\\rfloor+3\\ge a_2+3.\n\\]\nIn the latter case, we have \\[\ns=\\left\\lceil \\frac{n-1-a_1}{a_2} \\right\\rceil+1=\\left\\lceil \\frac{n-t\\ell+1}{a_2} \\right\\rceil+2\\ge\\frac{n-t\\ell+1}{\\ell-2}+2\\ge \\ell+1 \\ge a_2+3.\n\\]\nThis shows that $s\\ge a_2+3$ in either case.\nLet $v_i$ denote an end vertex of the path $P_{a_i}$ for $i=1,\\dots,a_2+2$ and $v_1'$ denote the neighbor of $v_1$ in $P_{a_1}$. Let $P_{a_{s-1}}=w_1\\dots w_{a_2}$ and\n\\[\nG'=G-v_1v_1'-\\sum_{i=1}^{a_2-1}w_iw_{i+1}+v_1v_2+\\sum_{i=3}^{a_2+2}v_iw_{i-2}.\n\\]\nThis implies that $G'$ is obtained from $G$ by deleting $a_2$ edges and adding $a_2+1$ edges.\nBy Claim \\ref{vector}, we have\n\\[\n\\mathbf{x}^\\top (Q(G')-Q(G))\\mathbf{x}\\ge 4(a_2+1)\\frac{1}{q^2}-4a_2\\left(\\frac{1}{q}+\\frac{30}{q^2}\\right)^2=\\frac{4}{q^4}\\left(q^2-60a_2q-900a_2\\right).\n\\]\nAs $n\\ge 30\\lfloor\\frac{\\ell-3}{2} \\rfloor+30\\sqrt{\\lfloor\\frac{\\ell-3}{2} \\rfloor \\big (\\lfloor\\frac{\\ell-3}{2} \\rfloor+1\\big )}$ if $t=1$ and $n\\ge 30(\\ell-2)+30\\sqrt{(\\ell-2)(\\ell-1)}$ if $t\\ge 2$, we have $n\\ge 30a_2+30\\sqrt{a_2(a_2+1)}$ and so $q>n\\ge 30a_2+30\\sqrt{a_2(a_2+1)}$, implying that $\\mathbf{x}^\\top (Q(G')-Q(G))\\mathbf{x}>0$. So by Rayleigh's principle, $q(G')>q$, a contradiction. Thus $a_2=\\lfloor \\frac{\\ell-1}{2}\\rfloor$ if $t=1$, and $a_2=\\ell -1$ if $t\\ge 2$.\n\nTherefore, $a_1=\\lceil\\frac{\\ell-1}{2} \\rceil$ and $G\\cong K_1\\vee \\big(P_{\\lceil\\frac{\\ell-1}{2} \\rceil}\\cup \\alpha P_{\\lfloor\\frac{\\ell-1}{2} \\rfloor}\\cup P_r\\big)$ if $t=1$, and $a_1=t\\ell-\\ell-1$ and $G\\cong K_1\\vee (P_{t\\ell-\\ell-1}\\cup \\alpha P_{\\ell-1}\\cup P_r)$ if $t\\ge 2$.\n\\end{proof}", "post_theorem_intro_text_len": 2213, "post_theorem_intro_text": "In the context of paths, we consider forbidding a single path and forbidding the disjoint union of $t$ paths of the same length. Interestingly, the extremal graphs for these two types of problems behave somewhat differently. Since any connected graph of order $n\\ge 3$ contains $P_2$ and $P_3$, for the case of forbidding a single path, we restrict our attention to paths of order at least four.\n\n\\begin{theorem}\\label{path}\nLet $G$ be a connected outerplanar $tP_\\ell$-free graph of order $n$ with $t\\ge 1$. \n\n(i) If $t=1$, $\\ell\\ge 4$ and $n\\ge \\max\\Big\\{\\lfloor\\frac{\\ell-3}{2} \\rfloor^2+\\lfloor\\frac{\\ell-3}{2} \\rfloor+\\ell-1,30\\lfloor\\frac{\\ell-3}{2} \\rfloor+30\\sqrt{\\lfloor\\frac{\\ell-3}{2} \\rfloor\\big(\\lfloor\\frac{\\ell-3}{2} \\rfloor+1\\big)}\\Big \\}$, then $q(G)\\le q\\big (K_1\\vee \\big(P_{\\lceil\\frac{\\ell-2}{2} \\rceil}\\cup\\alpha P_{\\lfloor\\frac{\\ell-2}{2} \\rfloor}\\cup P_r\\big)\\big)$ with equality if and only if $G\\cong K_1\\vee \\big(P_{\\lceil\\frac{\\ell-2}{2} \\rceil}\\cup\\alpha P_{\\lfloor\\frac{\\ell-2}{2} \\rfloor}\\cup P_r\\big)$, where $\\alpha=\\left \\lfloor \\frac{n-\\ell+1}{\\lfloor\\frac{\\ell-2}{2} \\rfloor}\\right\\rfloor+1$ and $r=n-\\ell+1-(\\alpha-1)\\lfloor\\frac{\\ell-2}{2} \\rfloor$.\n\n(ii) If $t\\ge 2$, $\\ell\\ge 2$ and $n\\ge \\max\\Big\\{\\ell^2+(t-3)\\ell+1,30(\\ell-2)+30\\sqrt{(\\ell-2)(\\ell-1)}\\Big\\}$, then $q(G)\\le q(K_1\\vee (P_{t\\ell-\\ell-1}\\cup\\alpha P_{\\ell-1}\\cup P_r))$ with equality if and only if $G\\cong K_1\\vee (P_{t\\ell-\\ell-1}\\cup\\alpha P_{\\ell-1}\\cup P_r)$, where $\\alpha=\\lfloor \\frac{n-t+2}{\\ell-1}\\rfloor-t+1$ and $r=n-2\\ell+2-(\\alpha-1)(\\ell-1)$.\n\\end{theorem}\n\nTo prove the above results, we establish a structural theorem for the extremal graphs. \n\n\\begin{theorem}\\label{str}\nLet $F=C_\\ell$ with $3\\le \\ell\\le n$ or $F=tP_k$ with $k\\ge 2$ and $4\\le tk\\le n-1$. If $G$ maximizes the $Q$-index among all connected $n$-vertex $F$-free outerplanar graphs, then $G$ contains a vertex $u$ of degree $n-1$ and $G[N(u)]$ consists of paths. \\end{theorem}\n\nThe paper is arranged as follows. In Section 2, we introduce the basic notations and preliminary lemmas that will be used throughout the paper. In Section 3, we prove Theorem \\ref{str}. In Section 4, we prove Theorems \\ref{cycle} and \\ref{path}.", "sketch": "To prove Theorem~\\ref{cycle} (and Theorem~\\ref{path}), the authors first \"establish a structural theorem for the extremal graphs\" (Theorem~\\ref{str}). Specifically, Theorem~\\ref{str} states that if $G$ maximizes the $Q$-index among all connected $n$-vertex $F$-free outerplanar graphs (with $F=C_\\ell$ or $F=tP_k$), then $G$ \"contains a vertex $u$ of degree $n-1$\" and the induced subgraph on its neighborhood satisfies that \"$G[N(u)]$ consists of paths.\" The organization then indicates the proof route: Section 3 proves Theorem~\\ref{str}, and Section 4 uses it to prove Theorems~\\ref{cycle} and~\\ref{path}.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{cycle}\nLet $G$ be a connected outerplanar $C_\\ell$-free graph of order $n$ with $\\ell\\ge 3$. Then $q(G)\\le q(K_1\\vee (\\alpha P_{\\ell-2}\\cup P_{r}))$ with equality if and only if $G\\cong K_1\\vee (\\alpha P_{\\ell-2}\\cup P_{r})$, where $\\alpha =\\lfloor\\frac{n-1}{\\ell-2}\\rfloor$ and $r=n-1-\\alpha(\\ell-2)$.", "theorem_type": ["Biconditional or Equivalence", "Inequality or Bound"], "mcq": {"question": "Let $G$ be a connected outerplanar graph of order $n$ that is $C_\\ell$-free, where $\\ell\\ge 3$ and $C_\\ell$ is the cycle on $\\ell$ vertices. For vertex-disjoint graphs $X$ and $Y$, let $X\\cup Y$ denote their union and $X\\vee Y$ their join; let $P_m$ denote the path on $m$ vertices; and let $q(H)$ be the largest eigenvalue of the signless Laplacian $Q(H)=D(H)+A(H)$. Define\n\\[\n\\alpha=\\left\\lfloor \\frac{n-1}{\\ell-2}\\right\\rfloor,\\qquad r=n-1-\\alpha(\\ell-2).\n\\]\nThe theorem gives the bound\n\\[\nq(G)\\le q\\bigl(K_1\\vee (\\alpha P_{\\ell-2}\\cup P_r)\\bigr).\n\\]\nWhich of the following statements is equivalent to equality in this bound?", "correct_choice": {"label": "A", "text": "$G\\cong K_1\\vee (\\alpha P_{\\ell-2}\\cup P_r)$."}, "choices": [{"label": "B", "text": "$G\\cong K_1\\vee \\bigl((\\alpha-1)P_{\\ell-2}\\cup P_{r+\\ell-2}\\bigr)$."}, {"label": "C", "text": "$G$ contains a vertex of degree $n-1$, and $G[N(u)]$ is a disjoint union of paths, where $u$ is such a vertex."}, {"label": "D", "text": "$G\\cong K_1\\vee (\\alpha P_{\\ell-1}\\cup P_r)$."}, {"label": "E", "text": "$G\\cong K_1\\vee (\\alpha P_{\\ell-2}\\cup P_r)$, up to replacing $P_r$ by any path decomposition whose total order is $r$."}], "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": "canonical quotient-remainder decomposition", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the exact path-length decomposition of $G[N(u)]$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "characteristic", "tampered_component": "forbidden path length in the neighborhood of the universal vertex", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "uniqueness of the extremal path structure", "template_used": "wildcard"}]}} +{"id": "2601.01318v1", "paper_link": "http://arxiv.org/abs/2601.01318v1", "theorems_cnt": 4, "theorem": {"env_name": "mainthm", "content": "\\label{mainA}\nThere exists a $C^1$-generic set $\\mathcal{R}\\subset \\mathcal{X}^1(M)$ such that if $\\Lambda$ is a non-trivial sectional-hyperbolic chain-recurrent class, then it is robustly a homoclinic class.", "start_pos": 15349, "end_pos": 15587, "label": "mainA"}, "ref_dict": {"hyplemma": "\\begin{lemma}[Hyperbolic Lemma]\\label{hyplemma}\nEvery compact invariant set without singularities contained in a sectional-hyperbolic set is hyperbolic of saddle type. \n\\end{lemma}", "lemma: hypsing": "\\begin{lemma}\\label{lemma: hypsing}\nLet $\\Lambda$ be a non trivial sectional-hyperbolic chain-recurrent class for a $C^1$ vector field $X$ containing singularities. Then, every $\\sigma\\in \\Lambda$ is hyperbolic. Moreover, if $T_{\\sigma}M=E^{s}_\\sigma\\oplus E^u_\\sigma$ is the hyperbolic splitting of $\\sigma$ and $T_{\\sigma}M=E_\\sigma\\oplus F_\\sigma$ is the hyperbolic splitting of $\\sigma$ given by sectional-hyperbolicity, we have\n\\begin{displaymath}\ndim\\,E_\\sigma+1=dim\\,E^s_\\sigma. \n\\end{displaymath}\n\\end{lemma}", "mainA": "\\begin{mainthm}\\label{mainA}\nThere exists a $C^1$-generic set $\\mathcal{R}\\subset \\mathcal{X}^1(M)$ such that if $\\Lambda$ is a non-trivial sectional-hyperbolic chain-recurrent class, then it is robustly a homoclinic class. \n\\end{mainthm}", "def: sec-hyp": "\\begin{definition}\\label{def: sec-hyp}\n A compact invariant set $\\Lambda\\subset M$ is sectional-hyperbolic if it is partially hyperbolic and its central bundle $F$ is \\textit{sectional-expanding}, i.e, there are $K,\\lambda>0$ such that for every two-dimensional subspace $L_x$ of $F_x$ one has\n\\begin{displaymath}\n\\vert \\text{det}\\Phi_t(x)\\vert_{L_x}\\vert\\geq Ke^{\\lambda t},\\quad\\forall x\\in\\Lambda,\\forall t> 0.\n\\end{displaymath} \n\\end{definition}"}, "pre_theorem_intro_text_len": 11972, "pre_theorem_intro_text": "Differentiable dynamical systems have become a fruitful and important research field in recent decades. The birth of this discipline is attributed to the seminal work of S. Smale \\cite{S}, where the notion of hyperbolic set was introduced, presenting Smale's Horseshoe as a representative example. Since then, a very strong program towards the understanding of the global dynamics of dynamical systems has been initiated, yielding a rich theory where several important advances were achieved, including a better understanding of stability phenomena and of chaotic dynamics from both topological and statistical point of view. \nLater, motivated by the chaotic dynamics of the strange attractor in Lorenz's polynomial system\n\\begin{displaymath}\n\\left\\lbrace \\begin{array}{lll}\n\\dot{x}&=& \\sigma(y-x) \\\\\n\\dot{y}&=& \\rho x-y-xz \\\\\n\\dot{z}&=& xy-\\beta z,\n\\end{array}\\right.\n\\end{displaymath}\nwhere $\\sigma\\approx 10$, $\\beta\\approx 8/3$ and $\\rho\\approx28$, \\cite{ABS} and \\cite{G} introduced, independently, a geometric model now known as the geometric Lorenz attractor (GLA). One of its main features is the presence of a unique singularity that is accumulated by regular orbits, which causes the system to fail to be hyperbolic. Nevertheless, the GLA exhibits rich dynamical behavior reminiscent of hyperbolic sets, including a dense set of periodic orbits and robust transitivity. Moreover, it was shown in \\cite{B} that the GLA is a homoclinic class. Motivated by this example, the notion of singular-hyperbolicity was introduced in \\cite{Mo3}, where it is proven that singular-hyperbolic sets properly extend the classical notion of hyperbolic sets, incorporating the GLA as a prototypical case.\n\nIn order to gain a better understanding of the dynamics of higher-dimensional sets such as the multidimensional Lorenz attractor \\cite{BPV}, C. Morales and R.J. Metzger introduced the concept of sectional-hyperbolic set in \\cite{Me3}. Both notions, sectional-hyperbolicity and singular-hyperbolicity, agree for three-dimensional vector fields, but in the higher-dimensional setting sectional-hyperbolicity is stronger than singular-hyperbolicity (see \\cite{Me3} and \\cite{Sal} for more details).\n\nNow, in topological dynamics, the chain-recurrent set constitutes an important subject of study because it encompasses all the interesting dynamics of the system. Moreover, by considering an equivalence relation, this set is decomposed into pieces called chain-recurrence classes. The fundamental theorem of dynamical systems, due to Conley \\cite{Co}, asserts the existence of a Lyapunov function which is constant along these classes. In this way, this result provides a general procedure for describing the global dynamics of a system. These sets are the central focus of this work and hence we now recall this concept. \n\n Hereafter, we denote by $M$ a $n$-dimensional compact Riemannian manifold, endowed with a Riemannian metric $\\Vert\\cdot\\Vert$. We will always assume $n\\geq 4$. Denote by $d$ the metric on $M$ induced by its Riemannian metric. Throughout this text, $\\mathcal{X}^1(M)$ denotes the set of $C^1$-vector fields on $M$ endowed with the $C^1$-topology. It is well known that any $X\\in \\mathcal{X}^1(M)$ induces a $C^1$-flow that will be denoted by the one-parameter family of maps $\\lbrace X_t\\rbrace_{t\\in\\mathbb{R}}$. The {\\it{orbit}} of a point $x\\in M$ is the set $$\\mathcal{O}(x)=\\lbrace X_t(x) : t\\in\\mathbb{R}\\rbrace.$$ For $a,b\\in\\mathbb{R}$, the \\textit{orbit segment from $a$ to $b$} of a point $x$ is defined by $X_{[a,b]}(x)=\\lbrace X_t(x) : t\\in[a,b]\\rbrace$. A point $x\\in M$ is said to be a \\textit{singularity} of $X$ if $X(x)=0$. We will denote the set of singularities of $X$ by $Sing(X)$. A point $x\\in M\\setminus Sing(X)$ is a {\\it{periodic point }} of $X$ if there is $t>0$ such that $X_t(x)=x$. Denote by $Per(X)$ the set of periodic points of $X$. The set of \\textit{critical elements} of $X$ is given by $Crit(X)=Sing(X)\\cup Per(X)$. An orbit that does not belong to $Crit(X)$ is called a \\textit{regular orbit}. As usual, we say that a subset $\\Lambda$ of $M$ is {\\it{invariant}} if $X_t(\\Lambda)=\\Lambda$ for any $t\\in\\mathbb{R}$. We say that a compact invariant set $\\Lambda$ is {\\it{Lyapunov stable}} if for every neighborhood $U$ of $\\Lambda$ there exists a neighborhood $V\\subset U$ of $\\Lambda$ such that $X_t(V)\\subset U$, for any $t>0$. We say that a compact and invariant set $\\Lambda$ is transitive if it contains a point whose orbit is dense in $\\Lambda$. \n\nLet us now define the chain-recurrent sets. There are several equivalent ways of defining the chain-recurrent set (see \\cite[Theorem 2.7.18]{AN}), here we choose the form that is most suitable to our purposes. For $\\varepsilon>0$ and $T>0$, we say that a finite sequence $( x_i,t_i)_{i=0}^n$ is an \\textit{$(\\varepsilon, T)$-chain} if $t_0+\\cdots+t_n\\geq T$, $t_i\\geq 1$, and $d(X_{t_i}(x_i), x_{i+1}) <\\varepsilon$ for any $i=0,..., n-1$. Besides, we say that $y$ is \\textit{chain attainable from} $x$, and we denote it by $x\\sim y$, if for any $\\varepsilon,T>0$ there exists an $(\\varepsilon,T)$-chain from $x$ to $y$, i.e. $x_0=x$ and $x_n=y$. When $x\\sim x$, we say that $x$ is a \\textit{chain-recurrent point}. The chain-recurrent set of $X$ is defined as $$CR(X)=\\lbrace x\\in M: x\\sim x\\rbrace.$$ \nIt is well known that $CR(X)$ is a compact and invariant set. Moreover, $\\sim$ is an equivalence relation on $CR(X)$. So, each equivalent class under this relation is called \\textit{chain-recurrent class}. It is easy to see that each chain-recurrent class of $X$ is also a compact and invariant set. When $x\\in CR(X)$, we denote by $C(x)$ its chain-recurrence class. We say that a chain-recurrent class is \\textit{non-trivial} if it is not reduced to either a periodic orbit or a singularity. We say that a chain-recurrent class is an \\textit{aperiodic class} if it does not contain periodic orbits.\n\nSeveral generic properties of these chain-recurrence classes were obtained. For instance, the well-known Kupka-Smale Theorem states that critical orbits of generic vector fields are hyperbolic. Later, in \\cite{BC}, it was shown that for $C^1$-generic systems, chain-recurrence classes with periodic orbits coincide with the homoclinic class of some of such periodic orbits. Moreover, when some form of hyperbolicity is present, we obtain more interesting dynamical properties. Indeed, according to \\cite{PYY}, for $C^1$-generic vector fields, a non-trivial Lyapunov stable chain-recurrence class that is sectional-hyperbolic is necessarily a transitive attractor, and hence, a homoclinic class. In \\cite{GYZ}, the same conclusion was obtained for chain-recurrence classes, not necessarily singular-hyperbolic, associated to $C^1$-generic vector fields away from homoclinic tangencies. In \\cite{CY}, S. Crovisier and D. Yang proved that transitivity is present in a robust way for sectional-hyperbolic Lyapunov stable chain-recurrence classes associated to vector fields $X$ in a certain $C^1$-generic set. In light of this result, the authors posed the following question:\n\n\\vspace{0.1in}\n\\textbf{Question:} Let $M$ be a compact Riemannian manifold with dimension at least $4$. Does there exist an open and dense set $\\mathcal{U}\\subset\\mathcal{X}^1(M)$ such that for any $X\\in \\mathcal{U}$, any non-trivial sectional-hyperbolic chain-recurrent class is robustly a homoclinic class?\n\\vspace{0.1in}\n\nIn this work, we explore this question and some interesting implications. \nOne important aspect of all of the aforementioned results is that they assert the non-existence of aperiodic classes under good generic conditions. Indeed, the existence of aperiodic classes represents a major obstacle in several important conjectures. See for instance the weak Palis conjecture which asserts that any $C^1$-system is approximated by either a Morse-Smale system or a system containing a hyperbolic horseshoe. This conjecture was verified for diffeomorphisms in \\cite{Cro}, for nonsingular flows in \\cite{XZ} and for singular three-dimensional flows in \\cite{GY}.\n\nOn the other hand, in \\cite{CY} and \\cite{GYZ} the existence of periodic orbits for generic chain-recurrence classes was obtained under the assumption that the class is Lyapunov stable. Also, in \\cite{PYY2} a dichotomy for chain-recurrence classes of generic star flows was obtained: either they have periodic orbits and positive entropy or they are sectional-hyperbolic aperiodic classes, and hence non-Lyapunov stable, with zero entropy. This evidences that obtaining periodic orbits for generic chain-recurrence classes is challenging, without Lyapunov stability. In this work, we shall see that in many cases, sectional-hyperbolicity is enough to rule out aperiodic classes. \n\nWe now state our main results. Let us begin by recalling the concept of sectional-hyperbolic set. We say that a compact invariant set $\\Lambda$ has a {\\it{dominated splitting}} if there is a continuous invariant splitting $T_{\\Lambda}M=E\\oplus F$ (with respect to the tangent flow $\\Phi_t= DX_t$, $t\\in\\mathbb{R}$) and constants $K,\\lambda>0$ satisfying the relation\n\\begin{displaymath}\n\\frac{\\Vert \\Phi_t(x)\\vert_{E_x}\\Vert}{m(\\Phi_t(x)\\vert_{F_x})}\\leq Ke^{-\\lambda t}, \\quad \\forall x\\in\\Lambda,\\forall t>0,\n\\end{displaymath} \nwhere $m(A)$ denotes the co-norm of a linear transformation $A$. In this case, we say that $E$ is {\\it{dominated}} by $F$. When the subbundle $E$ is uniformly contracting, i.e., $\\Vert \\Phi_t(x)\\vert_{E_x}\\Vert\\leq Ke^{-\\lambda t}$ for every $t>0$ and $x\\in\\Lambda$, we say that $\\Lambda$ is \\textit{partially hyperbolic}. \nIn \\cite{PYY} it was defined the notion of \\textit{sectional-hyperbolic} set as follows:\n\\begin{definition}\\label{def: sec-hyp}\n A compact invariant set $\\Lambda\\subset M$ is sectional-hyperbolic if it is partially hyperbolic and its central bundle $F$ is \\textit{sectional-expanding}, i.e, there are $K,\\lambda>0$ such that for every two-dimensional subspace $L_x$ of $F_x$ one has\n\\begin{displaymath}\n\\vert \\text{det}\\Phi_t(x)\\vert_{L_x}\\vert\\geq Ke^{\\lambda t},\\quad\\forall x\\in\\Lambda,\\forall t> 0.\n\\end{displaymath} \n\\end{definition}\n\n\\begin{remark}\\label{rmk}\nThe notion of sectional-hyperbolicity was first introduced in \\cite{Me3} to describe the dynamical behavior of higher-dimensional systems, such as the multidimensional Lorenz attractor (see \\cite{BPV}). That definition differs from Definition \\ref{def: sec-hyp} in that it assumes the hyperbolicity of singularities, a condition we do not require here. However, as we shall show in Lemma \\ref{lemma: hypsing}, every singularity contained in a sectional-hyperbolic chain-recurrent class (under Definition \\ref{def: sec-hyp}) is necessarily hyperbolic. Thus, this distinction poses no real restriction when dealing with chain-recurrence classes.\n\\end{remark}\n\nNext, let us clarify the notion of a \\emph{robustly transitive sectional-hyperbolic chain-recurrent class}. It is immediate to verify that \\( \\operatorname{Crit}(X) \\subset CR(X) \\), so the chain-recurrent class of any critical element is well defined. Moreover, by the previous remark and the hyperbolic lemma (see Lemma \\ref{hyplemma}), every critical element of a sectional-hyperbolic chain-recurrent class is hyperbolic. If \\( \\gamma_X \\) is a critical element of \\( X \\) and \\( C(\\gamma_X) \\) is sectional-hyperbolic, then for every \\( Y \\in \\mathcal{X}^1(M) \\) sufficiently \\( C^1 \\)-close to \\( X \\), there exists a continuation \\( \\gamma_Y \\) of \\( \\gamma_X \\). The corresponding chain-recurrent class \\( C(\\gamma_Y) \\) is called the \\emph{continuation} of \\( C(\\gamma_X) \\). We say that the chain-recurrent class \\( C(\\gamma_X) \\) is \\emph{robustly a homoclinic class} if there exists an open \\( C^1 \\)-neighborhood \\( \\mathcal{U} \\) of \\( X \\) such that for every \\( Y \\in \\mathcal{U} \\), the continuation \\( C(\\gamma_Y) \\) is a homoclinic class. The first main result of this paper is the following:", "context": "Several generic properties of these chain-recurrence classes were obtained. For instance, the well-known Kupka-Smale Theorem states that critical orbits of generic vector fields are hyperbolic. Later, in \\cite{BC}, it was shown that for $C^1$-generic systems, chain-recurrence classes with periodic orbits coincide with the homoclinic class of some of such periodic orbits. Moreover, when some form of hyperbolicity is present, we obtain more interesting dynamical properties. Indeed, according to \\cite{PYY}, for $C^1$-generic vector fields, a non-trivial Lyapunov stable chain-recurrence class that is sectional-hyperbolic is necessarily a transitive attractor, and hence, a homoclinic class. In \\cite{GYZ}, the same conclusion was obtained for chain-recurrence classes, not necessarily singular-hyperbolic, associated to $C^1$-generic vector fields away from homoclinic tangencies. In \\cite{CY}, S. Crovisier and D. Yang proved that transitivity is present in a robust way for sectional-hyperbolic Lyapunov stable chain-recurrence classes associated to vector fields $X$ in a certain $C^1$-generic set. In light of this result, the authors posed the following question:\n\n\\vspace{0.1in}\n\\textbf{Question:} Let $M$ be a compact Riemannian manifold with dimension at least $4$. Does there exist an open and dense set $\\mathcal{U}\\subset\\mathcal{X}^1(M)$ such that for any $X\\in \\mathcal{U}$, any non-trivial sectional-hyperbolic chain-recurrent class is robustly a homoclinic class?\n\\vspace{0.1in}\n\nWe now state our main results. Let us begin by recalling the concept of sectional-hyperbolic set. We say that a compact invariant set $\\Lambda$ has a {\\it{dominated splitting}} if there is a continuous invariant splitting $T_{\\Lambda}M=E\\oplus F$ (with respect to the tangent flow $\\Phi_t= DX_t$, $t\\in\\mathbb{R}$) and constants $K,\\lambda>0$ satisfying the relation\n\\begin{displaymath}\n\\frac{\\Vert \\Phi_t(x)\\vert_{E_x}\\Vert}{m(\\Phi_t(x)\\vert_{F_x})}\\leq Ke^{-\\lambda t}, \\quad \\forall x\\in\\Lambda,\\forall t>0,\n\\end{displaymath} \nwhere $m(A)$ denotes the co-norm of a linear transformation $A$. In this case, we say that $E$ is {\\it{dominated}} by $F$. When the subbundle $E$ is uniformly contracting, i.e., $\\Vert \\Phi_t(x)\\vert_{E_x}\\Vert\\leq Ke^{-\\lambda t}$ for every $t>0$ and $x\\in\\Lambda$, we say that $\\Lambda$ is \\textit{partially hyperbolic}. \nIn \\cite{PYY} it was defined the notion of \\textit{sectional-hyperbolic} set as follows:\n\\begin{definition}\\label{def: sec-hyp}\n A compact invariant set $\\Lambda\\subset M$ is sectional-hyperbolic if it is partially hyperbolic and its central bundle $F$ is \\textit{sectional-expanding}, i.e, there are $K,\\lambda>0$ such that for every two-dimensional subspace $L_x$ of $F_x$ one has\n\\begin{displaymath}\n\\vert \\text{det}\\Phi_t(x)\\vert_{L_x}\\vert\\geq Ke^{\\lambda t},\\quad\\forall x\\in\\Lambda,\\forall t> 0.\n\\end{displaymath} \n\\end{definition}\n\n\\begin{remark}\\label{rmk}\nThe notion of sectional-hyperbolicity was first introduced in \\cite{Me3} to describe the dynamical behavior of higher-dimensional systems, such as the multidimensional Lorenz attractor (see \\cite{BPV}). That definition differs from Definition \\ref{def: sec-hyp} in that it assumes the hyperbolicity of singularities, a condition we do not require here. However, as we shall show in Lemma \\ref{lemma: hypsing}, every singularity contained in a sectional-hyperbolic chain-recurrent class (under Definition \\ref{def: sec-hyp}) is necessarily hyperbolic. Thus, this distinction poses no real restriction when dealing with chain-recurrence classes.\n\\end{remark}\n\nNext, let us clarify the notion of a \\emph{robustly transitive sectional-hyperbolic chain-recurrent class}. It is immediate to verify that \\( \\operatorname{Crit}(X) \\subset CR(X) \\), so the chain-recurrent class of any critical element is well defined. Moreover, by the previous remark and the hyperbolic lemma (see Lemma \\ref{hyplemma}), every critical element of a sectional-hyperbolic chain-recurrent class is hyperbolic. If \\( \\gamma_X \\) is a critical element of \\( X \\) and \\( C(\\gamma_X) \\) is sectional-hyperbolic, then for every \\( Y \\in \\mathcal{X}^1(M) \\) sufficiently \\( C^1 \\)-close to \\( X \\), there exists a continuation \\( \\gamma_Y \\) of \\( \\gamma_X \\). The corresponding chain-recurrent class \\( C(\\gamma_Y) \\) is called the \\emph{continuation} of \\( C(\\gamma_X) \\). We say that the chain-recurrent class \\( C(\\gamma_X) \\) is \\emph{robustly a homoclinic class} if there exists an open \\( C^1 \\)-neighborhood \\( \\mathcal{U} \\) of \\( X \\) such that for every \\( Y \\in \\mathcal{U} \\), the continuation \\( C(\\gamma_Y) \\) is a homoclinic class. The first main result of this paper is the following:", "full_context": "Several generic properties of these chain-recurrence classes were obtained. For instance, the well-known Kupka-Smale Theorem states that critical orbits of generic vector fields are hyperbolic. Later, in \\cite{BC}, it was shown that for $C^1$-generic systems, chain-recurrence classes with periodic orbits coincide with the homoclinic class of some of such periodic orbits. Moreover, when some form of hyperbolicity is present, we obtain more interesting dynamical properties. Indeed, according to \\cite{PYY}, for $C^1$-generic vector fields, a non-trivial Lyapunov stable chain-recurrence class that is sectional-hyperbolic is necessarily a transitive attractor, and hence, a homoclinic class. In \\cite{GYZ}, the same conclusion was obtained for chain-recurrence classes, not necessarily singular-hyperbolic, associated to $C^1$-generic vector fields away from homoclinic tangencies. In \\cite{CY}, S. Crovisier and D. Yang proved that transitivity is present in a robust way for sectional-hyperbolic Lyapunov stable chain-recurrence classes associated to vector fields $X$ in a certain $C^1$-generic set. In light of this result, the authors posed the following question:\n\n\\vspace{0.1in}\n\\textbf{Question:} Let $M$ be a compact Riemannian manifold with dimension at least $4$. Does there exist an open and dense set $\\mathcal{U}\\subset\\mathcal{X}^1(M)$ such that for any $X\\in \\mathcal{U}$, any non-trivial sectional-hyperbolic chain-recurrent class is robustly a homoclinic class?\n\\vspace{0.1in}\n\nWe now state our main results. Let us begin by recalling the concept of sectional-hyperbolic set. We say that a compact invariant set $\\Lambda$ has a {\\it{dominated splitting}} if there is a continuous invariant splitting $T_{\\Lambda}M=E\\oplus F$ (with respect to the tangent flow $\\Phi_t= DX_t$, $t\\in\\mathbb{R}$) and constants $K,\\lambda>0$ satisfying the relation\n\\begin{displaymath}\n\\frac{\\Vert \\Phi_t(x)\\vert_{E_x}\\Vert}{m(\\Phi_t(x)\\vert_{F_x})}\\leq Ke^{-\\lambda t}, \\quad \\forall x\\in\\Lambda,\\forall t>0,\n\\end{displaymath} \nwhere $m(A)$ denotes the co-norm of a linear transformation $A$. In this case, we say that $E$ is {\\it{dominated}} by $F$. When the subbundle $E$ is uniformly contracting, i.e., $\\Vert \\Phi_t(x)\\vert_{E_x}\\Vert\\leq Ke^{-\\lambda t}$ for every $t>0$ and $x\\in\\Lambda$, we say that $\\Lambda$ is \\textit{partially hyperbolic}. \nIn \\cite{PYY} it was defined the notion of \\textit{sectional-hyperbolic} set as follows:\n\\begin{definition}\\label{def: sec-hyp}\n A compact invariant set $\\Lambda\\subset M$ is sectional-hyperbolic if it is partially hyperbolic and its central bundle $F$ is \\textit{sectional-expanding}, i.e, there are $K,\\lambda>0$ such that for every two-dimensional subspace $L_x$ of $F_x$ one has\n\\begin{displaymath}\n\\vert \\text{det}\\Phi_t(x)\\vert_{L_x}\\vert\\geq Ke^{\\lambda t},\\quad\\forall x\\in\\Lambda,\\forall t> 0.\n\\end{displaymath} \n\\end{definition}\n\n\\begin{remark}\\label{rmk}\nThe notion of sectional-hyperbolicity was first introduced in \\cite{Me3} to describe the dynamical behavior of higher-dimensional systems, such as the multidimensional Lorenz attractor (see \\cite{BPV}). That definition differs from Definition \\ref{def: sec-hyp} in that it assumes the hyperbolicity of singularities, a condition we do not require here. However, as we shall show in Lemma \\ref{lemma: hypsing}, every singularity contained in a sectional-hyperbolic chain-recurrent class (under Definition \\ref{def: sec-hyp}) is necessarily hyperbolic. Thus, this distinction poses no real restriction when dealing with chain-recurrence classes.\n\\end{remark}\n\nNext, let us clarify the notion of a \\emph{robustly transitive sectional-hyperbolic chain-recurrent class}. It is immediate to verify that \\( \\operatorname{Crit}(X) \\subset CR(X) \\), so the chain-recurrent class of any critical element is well defined. Moreover, by the previous remark and the hyperbolic lemma (see Lemma \\ref{hyplemma}), every critical element of a sectional-hyperbolic chain-recurrent class is hyperbolic. If \\( \\gamma_X \\) is a critical element of \\( X \\) and \\( C(\\gamma_X) \\) is sectional-hyperbolic, then for every \\( Y \\in \\mathcal{X}^1(M) \\) sufficiently \\( C^1 \\)-close to \\( X \\), there exists a continuation \\( \\gamma_Y \\) of \\( \\gamma_X \\). The corresponding chain-recurrent class \\( C(\\gamma_Y) \\) is called the \\emph{continuation} of \\( C(\\gamma_X) \\). We say that the chain-recurrent class \\( C(\\gamma_X) \\) is \\emph{robustly a homoclinic class} if there exists an open \\( C^1 \\)-neighborhood \\( \\mathcal{U} \\) of \\( X \\) such that for every \\( Y \\in \\mathcal{U} \\), the continuation \\( C(\\gamma_Y) \\) is a homoclinic class. The first main result of this paper is the following:\n\n\\vspace{0.1in}\n\\textbf{Question:} Let $M$ be a compact Riemannian manifold with dimension at least $4$. Does there exist an open and dense set $\\mathcal{U}\\subset\\mathcal{X}^1(M)$ such that for any $X\\in \\mathcal{U}$, any non-trivial sectional-hyperbolic chain-recurrent class is robustly a homoclinic class?\n\\vspace{0.1in}\n\nNext, let us clarify the notion of a \\emph{robustly transitive sectional-hyperbolic chain-recurrent class}. It is immediate to verify that \\( \\operatorname{Crit}(X) \\subset CR(X) \\), so the chain-recurrent class of any critical element is well defined. Moreover, by the previous remark and the hyperbolic lemma (see Lemma \\ref{hyplemma}), every critical element of a sectional-hyperbolic chain-recurrent class is hyperbolic. If \\( \\gamma_X \\) is a critical element of \\( X \\) and \\( C(\\gamma_X) \\) is sectional-hyperbolic, then for every \\( Y \\in \\mathcal{X}^1(M) \\) sufficiently \\( C^1 \\)-close to \\( X \\), there exists a continuation \\( \\gamma_Y \\) of \\( \\gamma_X \\). The corresponding chain-recurrent class \\( C(\\gamma_Y) \\) is called the \\emph{continuation} of \\( C(\\gamma_X) \\). We say that the chain-recurrent class \\( C(\\gamma_X) \\) is \\emph{robustly a homoclinic class} if there exists an open \\( C^1 \\)-neighborhood \\( \\mathcal{U} \\) of \\( X \\) such that for every \\( Y \\in \\mathcal{U} \\), the continuation \\( C(\\gamma_Y) \\) is a homoclinic class. The first main result of this paper is the following:\n\nNotice that in the previous result we dropped out any stability assumption on $\\Lambda$. In particular, Theorem \\ref{mainA} is a direct improvement of \\cite[Corollary E]{PYY}. Also, an implicit fact behind the statement of Theorem \\ref{mainA} is that any non-trivial sectional-hyperbolic chain-recurrent set within \\(\\mathcal{R}\\) must contain periodic orbits; in other words, sectional-hyperbolicity is enough to rule out the existence of aperiodic classes.\n\n\\begin{mainthm}\\label{conj1.2}\n Let $M$ be an $n$-dimensional manifold with $n\\geq 4$. There exists a \n $C^1$-generic set $\\mathcal{R}\\subset \\mathcal{X}_*^1(M)$ such that for every \n $X\\in \\mathcal{R}$ and every non-trivial chain-recurrent class $\\Lambda$ of $X$, \n the set $\\Lambda$ has positive topological entropy, contains a periodic orbit, \n and is isolated. Consequently, $C^1$-generic singular star flows have only finitely \n many chain-recurrent classes, all of which are either trivial or homoclinic classes of hyperbolic periodic orbits. \n\\end{mainthm}\n\n\\begin{theorem}\\label{Teohtop}\nThere exists a $C^1$-generic set $\\mathcal{R}\\subset \\mathcal{X}^1(M)$ such that if $\\Lambda$ is a non-trivial sectional-hyperbolic chain-recurrent class then it is robustly periodic, that is, there is an open neighborhood $\\mathcal{U}$ of $X$ such that for every $Y\\in \\mathcal{U}$, one has $Per(Y|_{\\Lambda_Y})\\neq\\emptyset$, where $\\Lambda_Y$ denotes the continuation of the chain recurrent class $\\Lambda$ for $X$. \n\\end{theorem}\n\nWe now present the following lemma, which will be used to prove Theorem \\ref{Teohtop}.\n\\begin{lemma}\\label{prop2.4CY}\nLet $\\mathcal{R}$ be the residual set of Lemma \\ref{lemm: genericsets}. Let $X\\in \\mathcal{R}$, $\\sigma_X\\in Sing(X)$ and $C(\\sigma)$ be a non-trivial sectional-hyperbolic chain-recurrent class. There is a neighborhood $\\mathcal{U}_0\\subset\\mathcal{U}_X$ of $X$ such that for every $Y\\in \\mathcal{U}_0$, the continuation $C(\\sigma_Y)$ is sectional-hyperbolic. \n\\end{lemma}\n\\begin{proof}\n Let $X\\in \\mathcal{R}$ and $\\Lambda$ a non-trivial sectional-hyperbolic class of $X$. Recall that sectional-hyperbolicity is a robust property, i.e., there are neighborhoods $U$ of $\\Lambda$ and $\\mathcal{U}'_X$ of $X$ such that for every $Y\\in \\mathcal{U}'_X$ the set $\\cap_{t\\in\\R}Y_t(U)$ is sectional-hyperbolic. Let $\\mathcal{U}_X$ be given by item 3 of Lemma \\ref{lemm: genericsets}. Denote $\\mathcal{U}_0=\\mathcal{U}_X\\cap\\mathcal{U}'_X$. Since the map $Y\\in \\mathcal{U}_X\\to C(\\sigma)$ is continuous, after possibly shrinking $\\mathcal{U}_X$, we can assume $C(\\sigma_Y)\\subset U$. Therefore, $C(\\sigma_Y)$ is contained in the maximal invariant set of $U$ under $Y$. In particular, $C(\\sigma_Y)$ is sectional-hyperbolic. \n\\end{proof}\n\n\\begin{lemma}\\label{thm: homoclinicclass}\n If $X\\in \\mathcal{R}$ and $\\Lambda\\subset M$ is a non-trivial sectional-hyperbolic chain-recurrent class, then it is a homoclinic class.\n\\end{lemma}\n\nOn the other hand, by \\cite[Theorem A]{PYY2}, there is a residual subset $\\mathcal{S}'\\subset\\mathcal{X}^1_*(M)$ such that for any $Y\\in \\mathcal{S}'$ and $\\Lambda$ is a non-trivial chain-recurrent class, one of the following holds:\n\\begin{enumerate}\n \\item $h_{top}(X|_{\\Lambda})>0$, $\\Lambda$ contains a periodic orbit and is isolated; or \n \\item $h_{top}(\\Lambda)=0$ and $\\Lambda$ is sectional-hyperbolic aperiodic class.\n\\end{enumerate}\n By setting $\\mathcal{S}=\\mathcal{R}'\\cap\\mathcal{S}'$, any non-trivial sectional-hyperbolic chain-recurrent class for $Y\\in \\mathcal{S}$ contains a periodic orbit by Theorem \\ref{mainA}. Hence, option 2 cannot occur, and the proof is complete.\n\\end{proof}\n\n\\begin{lemma}[Hyperbolic Lemma]\\label{hyplemma}\nEvery compact invariant set without singularities contained in a sectional-hyperbolic set is hyperbolic of saddle type. \n\\end{lemma}\n\n\\begin{mainthm}\\label{mainA}\nThere exists a $C^1$-generic set $\\mathcal{R}\\subset \\mathcal{X}^1(M)$ such that if $\\Lambda$ is a non-trivial sectional-hyperbolic chain-recurrent class, then it is robustly a homoclinic class. \n\\end{mainthm}", "post_theorem_intro_text_len": 3387, "post_theorem_intro_text": "Notice that in the previous result we dropped out any stability assumption on $\\Lambda$. In particular, Theorem \\ref{mainA} is a direct improvement of \\cite[Corollary E]{PYY}. Also, an implicit fact behind the statement of Theorem \\ref{mainA} is that any non-trivial sectional-hyperbolic chain-recurrent set within \\(\\mathcal{R}\\) must contain periodic orbits; in other words, sectional-hyperbolicity is enough to rule out the existence of aperiodic classes. \n\nAnother application of Theorem~\\ref{mainA} concerns the setting of star flows. \nRecall that a vector field $X\\in \\mathcal{X}^1(M)$ is \\emph{star} if there exists \na neighborhood $\\mathcal{U}$ such that all critical elements of every $Y\\in\\mathcal{U}$ \nare hyperbolic. Star systems play a central role in stability theory. For diffeomorphisms, \nit was proved in \\cite{Ma78,Li81} that $\\Omega$-stability implies the star condition, \nand later in \\cite{Ha92} that the star condition is equivalent to Axiom~A with the \nno-cycle condition. For regular vector fields, the same equivalence was obtained in \n\\cite{GW06}.\n\nBeyond stability, singular star flows are known to exhibit rich dynamics, which \nhas motivated substantial research. In \\cite{ZGW08}, it was conjectured that \n$C^1$-generic star flows and sectional-hyperbolicity should be intrinsically related; namely, they were expected \nto have finitely many chain-recurrent classes, each sectional-hyperbolic. A partial \nconfirmation was obtained in \\cite{SGW14} under the assumption of Lyapunov stability. \nHowever, \\cite{BdL21,dL17} disproved half of the conjecture by showing that \n$C^1$-generic star flows need not be sectional-hyperbolic, thus opening a new line \nof research with the introduction of multi-singular hyperbolic sets. Nevertheless, \nthe other half of the conjecture remains open:\n\n\\begin{conj}[Conjecture 1.2 in \\cite{PYY2}]\\label{conjPYY}\n $C^1$-generic singular star flows do not admit singular aperiodic classes. \n Consequently, they have only finitely many chain-recurrent classes, all of which \n are homoclinic classes of periodic orbits.\n\\end{conj}\n\nA partial answer was given in \\cite{PYY2}, proving that generic star flows admit \nfinitely many Lyapunov chain-recurrent classes and establishing a dichotomy for the \nnon-Lyapunov stable case, based on entropy. As a consequence of our techniques, we \ncan fully confirm the conjecture:\n\n\\begin{mainthm}\\label{conj1.2}\n Let $M$ be an $n$-dimensional manifold with $n\\geq 4$. There exists a \n $C^1$-generic set $\\mathcal{R}\\subset \\mathcal{X}_*^1(M)$ such that for every \n $X\\in \\mathcal{R}$ and every non-trivial chain-recurrent class $\\Lambda$ of $X$, \n the set $\\Lambda$ has positive topological entropy, contains a periodic orbit, \n and is isolated. Consequently, $C^1$-generic singular star flows have only finitely \n many chain-recurrent classes, all of which are either trivial or homoclinic classes of hyperbolic periodic orbits. \n\\end{mainthm}\n\nThe remainder of this text is organized as follows: In Section 2, we present a collection of preliminary results concerning sectional-hyperbolic chain-recurrence classes. In Section 3, we prove that any chain-recurrent class associated with $X\\in\\mathcal{R}$ is robustly periodic, which constitutes a key step toward proving Theorem \\ref{mainA}. Finally, Section 4 is devoted to proving the main results of this article.", "sketch": "The post-theorem text does not give a detailed proof, but it outlines the structure of the argument toward Theorem~\\ref{mainA}: in Section~2 the authors \"present a collection of preliminary results concerning sectional-hyperbolic chain-recurrence classes\"; in Section~3 they \"prove that any chain-recurrent class associated with $X\\in\\mathcal{R}$ is robustly periodic, which constitutes a key step toward proving Theorem~\\ref{mainA}\"; and Section~4 is \"devoted to proving the main results of this article.\" It is also noted as an \"implicit fact behind the statement of Theorem~\\ref{mainA}\" that any non-trivial sectional-hyperbolic chain-recurrent set within $\\mathcal{R}$ \"must contain periodic orbits\" (i.e. sectional-hyperbolicity \"rule[s] out the existence of aperiodic classes\").", "expanded_sketch": "The post-theorem text does not give a detailed proof, but it outlines the structure of the argument toward the main theorem: next the authors present a collection of preliminary results concerning sectional-hyperbolic chain-recurrence classes; then they prove that any chain-recurrent class associated with $X\\in\\mathcal{R}$ is robustly periodic, which constitutes a key step toward establishing the main theorem; and finally they prove the main results of the article. It is also noted as an implicit fact behind the statement of the main theorem that any non-trivial sectional-hyperbolic chain-recurrent set within $\\mathcal{R}$ must contain periodic orbits (i.e. sectional-hyperbolicity rules out the existence of aperiodic classes).", "expanded_theorem": "\\label{mainA}\nThere exists a $C^1$-generic set $\\mathcal{R}\\subset \\mathcal{X}^1(M)$ such that if $\\Lambda$ is a non-trivial sectional-hyperbolic chain-recurrent class, then it is robustly a homoclinic class.,", "theorem_type": ["Existential–Universal", "Implication"], "mcq": {"question": "Let $M$ be a compact Riemannian manifold, and let $\\mathcal{X}^1(M)$ denote the space of $C^1$ vector fields on $M$ with the $C^1$ topology. Call a subset of $\\mathcal{X}^1(M)$ $C^1$-generic if it is residual. A chain-recurrent class is an equivalence class in the chain recurrent set under the chain-recurrence relation. A compact invariant set $\\Lambda$ of a vector field $X$ is called sectional-hyperbolic if it admits a continuous $DX_t$-invariant splitting $T_{\\Lambda}M=E\\oplus F$ such that $E$ is uniformly contracting and there exist constants $K,\\lambda>0$ for which, for every $x\\in\\Lambda$, every $t>0$, and every two-dimensional subspace $L_x\\subset F_x$, one has\n\\[\n\\big|\\det(DX_t(x)|_{L_x})\\big|\\ge Ke^{\\lambda t}.\n\\]\nSay that a chain-recurrent class $\\Lambda$ of $X$ is robustly a homoclinic class if there exists an open $C^1$-neighborhood $\\mathcal U$ of $X$ such that for every $Y\\in\\mathcal U$, the continuation $\\Lambda_Y$ of $\\Lambda$ is a homoclinic class. Under these hypotheses, which conclusion about $C^1$-generic vector fields holds?", "correct_choice": {"label": "A", "text": "There exists a $C^1$-generic set $\\mathcal R\\subset\\mathcal X^1(M)$ such that, for every $X\\in\\mathcal R$, if $\\Lambda$ is a non-trivial sectional-hyperbolic chain-recurrent class of $X$, then $\\Lambda$ is robustly a homoclinic class; that is, there is an open $C^1$-neighborhood $\\mathcal U$ of $X$ such that for every $Y\\in\\mathcal U$, the continuation $\\Lambda_Y$ is a homoclinic class."}, "choices": [{"label": "B", "text": "There exists a $C^1$-generic set $\\mathcal R\\subset\\mathcal X^1(M)$ such that, for every $X\\in\\mathcal R$, if $\\Lambda$ is a sectional-hyperbolic chain-recurrent class of $X$, then $\\Lambda$ is robustly a homoclinic class; that is, there is an open $C^1$-neighborhood $\\mathcal U$ of $X$ such that for every $Y\\in\\mathcal U$, the continuation $\\Lambda_Y$ is a homoclinic class."}, {"label": "C", "text": "There exists a $C^1$-generic set $\\mathcal R\\subset\\mathcal X^1(M)$ such that, for every $X\\in\\mathcal R$, if $\\Lambda$ is a non-trivial sectional-hyperbolic chain-recurrent class of $X$, then $\\Lambda$ is a homoclinic class."}, {"label": "D", "text": "There exists a $C^1$-generic set $\\mathcal R\\subset\\mathcal X^1(M)$ such that, for every $X\\in\\mathcal R$, if $\\Lambda$ is a non-trivial sectional-hyperbolic chain-recurrent class of $X$, then for every $Y$ sufficiently $C^1$-close to $X$, the continuation $\\Lambda_Y$ is a homoclinic class."}, {"label": "E", "text": "There exists an open and dense set $\\mathcal R\\subset\\mathcal X^1(M)$ such that, for every $X\\in\\mathcal R$, if $\\Lambda$ is a non-trivial sectional-hyperbolic chain-recurrent class of $X$, then $\\Lambda$ is robustly a homoclinic class; that is, there is an open $C^1$-neighborhood $\\mathcal U$ of $X$ such that for every $Y\\in\\mathcal U$, the continuation $\\Lambda_Y$ is a homoclinic class."}], "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": "nontriviality_exclusion", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "robust_neighborhood_conclusion", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "uniform_neighborhood_quantifier", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "generic_residual_vs_open_dense", "template_used": "property_confusion"}]}} +{"id": "2601.01447v1", "paper_link": "http://arxiv.org/abs/2601.01447v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{main}\nSuppose that $F(0)=0$, ${\\bf E}[\\xi] < \\infty$ and $$\\gamma := \\dfrac{(a-r)\\kappa + r}{\\kappa^2 \\sigma^2} > 1.$$\nThen the survival probability $\\Phi \\in C([0,\\infty)) \\cap C^2((0,\\infty))$ satisfies the integro-differential equation (\\ref{int-diff}) with boundary conditions $\\Phi(0)=0$, $\\Phi(\\infty)=1$. Moreover, there exists a finite constant $C > 0$ such that $\\Psi(u) \\sim C u^{-\\gamma+1}$ as $u \\to \\infty$.", "start_pos": 10420, "end_pos": 10872, "label": "main"}, "ref_dict": {"one": "\\begin{equation}\\label{one}\nX_t = u + \\int_0^t \\kappa X_s (a\\,ds +\\sigma\\,dW_s) + \\int_0^t r(1-\\kappa)X_s\\,ds + P_t,\n\\end{equation}", "int-diff": "\\begin{multline}\\label{int-diff}\n\\frac{1}{2} \\kappa^2\\sigma^2 u^2 \\Phi''(u) + ((a-r)\\kappa + r) u \\Phi'(u) =\n\\\\\n= c \\Phi'(u) - \\lambda \\int_0^\\infty (\\Phi(u+y) - \\Phi(u)) \\,dF(y),\n\\end{multline}"}, "pre_theorem_intro_text_len": 7674, "pre_theorem_intro_text": "In the classical collective risk theory, initiated by Filip Lundberg in 1903 and developed further by Harald Cram\\'er in the thirties, it was assumed that an insurance company kept its reserve entirely separate from risky financial activities. This was natural for the era: the legislation of the period was based on a paradigm of social responsibility and was strongly influenced by the terrible consequences of stock market disasters. However, evolving economic realities, together with progress in financial theory, led to a relaxation of legal constraints and the emergence of theoretical studies on ruin problems with risky investments. The origin of this theory can be traced back to the seminal 1993 paper by Jostein Paulsen, who applied the Kesten--Goldie implicit renewal theory (also known as the theory of distributional equations), as exposed in Goldie's \\cite{Go91}, to derive asymptotics for ruin probabilities. A number of authors have since contributed to its development along the same lines, leveraging recent progress in the theory of distributional equations as summarized in the book \\cite{BDM}. We mention here only a few recent papers covering Sparre Andersen type models with risky investments; see \\cite{EKS}, \\cite{KPro}, \\cite{KLP}, and references therein. Although this method is powerful and applicable to a variety of models, it has a weakness: it provides only the rate of convergence to zero of the ruin probability, not the exact asymptotic (i.e., it gives no information on the leading constant).\n\nFor the asymptotic analysis of Lundberg--Cram\\'er type models with investment, there is an alternative method based on representing the survival (or ruin) probability as a function satisfying a second-order integro-differential equation. Such an equation seems to have appeared for the first time in a short note by Anna Frolova, \\cite{Fr}, who observed that in the case of exponential claims, it can be reduced to a linear ordinary differential equation (ODE). From its general solution, she conjectured that the ruin probability decays as a power function. This conjecture was later proved in \\cite{FrKP} for the non-life insurance model and in \\cite{KP2016} for the annuity payments model.\n\nIn the present note, we study the integro-differential equation arising in the asymptotic analysis of ruin probabilities for an annuity model more general than the one described in \\cite{KP2016}. Specifically, we suppose that the insurance company pays pensions to its customers at a constant rate $c>0$ and receives, as income, the remaining part of their pension fund when the contracts expire. The company instantaneously invests a fraction $\\kappa$ of its capital reserve in a risky asset whose price $S=(S_t)$ evolves as a geometric Brownian motion (gBm) with parameters $a$ and $\\sigma>0$. The remaining fraction $1-\\kappa$ of the capital reserve is invested in a non-risky asset (a bank account) with a rate of return $r$. Formally, the process $X=X^u$ describing the evolution of the capital reserve is given by the following stochastic differential equation with jumps:\n\\begin{equation}\\label{one}\nX_t = u + \\int_0^t \\kappa X_s (a\\,ds +\\sigma\\,dW_s) + \\int_0^t r(1-\\kappa)X_s\\,ds + P_t,\n\\end{equation}\nwhere $u \\ge 0$ is the initial capital, $W=(W_t)$ is a standard Wiener process and $P=(P_t)$ is the ``business'' process (or Cram\\'er--Lundberg process), independent of $W$. In the classical literature, $P$ is usually represented as\n\\[\nP_t = -ct + \\sum_{i=1}^{N_t} \\xi_i,\n\\]\nwhere $N=(N_t)$ is a Poisson process with intensity $\\lambda > 0$, and $(\\xi_i)_{i\\ge 1}$ is an i.i.d. sequence of strictly positive random variables, independent of $N$, with common distribution function $F$ such that $F(0)=0$. \nIn the literature on stochastic calculus, it is standard to represent the sum as a stochastic integral:\n\\[\n\\sum_{i=1}^{N_t} \\xi_i = \\int_0^t \\int_0^\\infty x\\,p(ds,dx) ,\n\\]\nwhere $p(ds,dx)$ is a Poisson random measure with mean measure $q(ds,dx) = \\lambda ds\\,F(dx)$.\n\nSuch a model can also describe the capital evolution of a venture company paying current expenditures and receiving incomes $\\xi_i$ from selling innovations.\n\nAfter regrouping terms, the equation (\\ref{one}), written in the traditional form of a stochastic differential equation, becomes:\n\\begin{equation}\\label{two}\ndX_t = \\big( (a-r)\\kappa + r\\big) X_t\\,dt + \\kappa\\sigma X_t\\,dW_t + dP_t, \\qquad X_0 = u.\n\\end{equation}\n\nLet $\\tau^u := \\inf \\{t: X^u_t \\le 0\\}$ (the ruin time), $\\Psi(u) := {\\bf P}(\\tau^u < \\infty)$ (the ruin probability), and $\\Phi(u) := 1 - \\Psi(u)$ (the survival probability).\n\nThe aforementioned paper \\cite{KP2016}, together with its complements \\cite{PZ} and \\cite{PerErr} (treating the case $\\beta=0$), deals with the asymptotic analysis of the ruin probability under the assumption of a riskless rate $r=0$. For the case where $\\kappa=1$ and $F$ is an exponential distribution, it was shown that if $\\beta := 2a/\\sigma^2 - 1 > 0$, then $\\Psi(u) \\sim C u^{-\\beta}$ as $u \\to \\infty$; if $\\beta \\le 0$, ruin is imminent. If $\\kappa \\in (0,1)$, the same conclusion holds with $\\beta_\\kappa := 2a/(\\kappa\\sigma^2) - 1$. The equation $\\beta_\\kappa = 0$ defines the threshold for the fraction of risky investment above which ruin is certain. The first step of the argument, which is technically difficult and delicate, is to prove that the survival probability is smooth under appropriate assumptions on $F$. The second, easier step is to prove that $\\Phi$ satisfies an integro-differential equation. In the specific case of an exponential jump distribution, this equation can be reduced--by differentiating and eliminating the integral term--to a linear ODE of third order. This ODE turns out to be of second order for the derivative of the survival probability. Asymptotic results for solutions of such equations are available in the literature. The property that the ruin probability cannot decrease faster than some power function, established by Kalashnikov and Norberg in \\cite{Kalash-Nor}, then yields the aforementioned result.\n\nIt is worth noting that the method of eliminating integral terms works not only for exponential jump distributions but also for other classes (e.g., Pareto distributions) and can lead to asymptotic results via Laplace transforms and Tauberian theorems; see \\cite{ABL}, \\cite{ACT}, and \\cite{A2025}.\n\nImportantly, if the survival probability is smooth, i.e., if $\\Phi \\in C^2$, then straightforward arguments based on It\\^o's formula lead to the conclusion that $\\Phi$ satisfies, in the classical sense, the integro-differential equation (IDE)\n\\begin{multline}\\label{int-diff}\n\\frac{1}{2} \\kappa^2\\sigma^2 u^2 \\Phi''(u) + ((a-r)\\kappa + r) u \\Phi'(u) =\n\\\\\n= c \\Phi'(u) - \\lambda \\int_0^\\infty (\\Phi(u+y) - \\Phi(u)) \\,dF(y),\n\\end{multline}\nsee, e.g., \\cite{KP2016}. The same equation holds for the ruin probability.\n\nAvailable sufficient conditions on $F$ ensuring the smoothness of $\\Phi$ are rather restrictive: the property holds if $F$ has a density $f \\in C^2$ such that $f' \\in L^1(\\mathbb{R}_+)$; see \\cite{KPukh}. For this reason, some authors prefer to assume smoothness a priori. On the other hand, it is not difficult to show that $\\Phi$ is a viscosity solution of the above equation under much more general conditions, but proving uniqueness for boundary value problems in the viscosity sense is rather involved; see \\cite{Bel-Kab} and, for an equation arising in a model based on a generalized Ornstein--Uhlenbeck process, the recent study \\cite{AK2025}.\n\n\\smallskip\n\\textbf{Main result.} \nIn this note, we employ a completely different proof strategy. Our main result is the following theorem.", "context": "In the present note, we study the integro-differential equation arising in the asymptotic analysis of ruin probabilities for an annuity model more general than the one described in \\cite{KP2016}. Specifically, we suppose that the insurance company pays pensions to its customers at a constant rate $c>0$ and receives, as income, the remaining part of their pension fund when the contracts expire. The company instantaneously invests a fraction $\\kappa$ of its capital reserve in a risky asset whose price $S=(S_t)$ evolves as a geometric Brownian motion (gBm) with parameters $a$ and $\\sigma>0$. The remaining fraction $1-\\kappa$ of the capital reserve is invested in a non-risky asset (a bank account) with a rate of return $r$. Formally, the process $X=X^u$ describing the evolution of the capital reserve is given by the following stochastic differential equation with jumps:\n\\begin{equation}\\label{one}\nX_t = u + \\int_0^t \\kappa X_s (a\\,ds +\\sigma\\,dW_s) + \\int_0^t r(1-\\kappa)X_s\\,ds + P_t,\n\\end{equation}\nwhere $u \\ge 0$ is the initial capital, $W=(W_t)$ is a standard Wiener process and $P=(P_t)$ is the ``business'' process (or Cram\\'er--Lundberg process), independent of $W$. In the classical literature, $P$ is usually represented as\n\\[\nP_t = -ct + \\sum_{i=1}^{N_t} \\xi_i,\n\\]\nwhere $N=(N_t)$ is a Poisson process with intensity $\\lambda > 0$, and $(\\xi_i)_{i\\ge 1}$ is an i.i.d. sequence of strictly positive random variables, independent of $N$, with common distribution function $F$ such that $F(0)=0$. \nIn the literature on stochastic calculus, it is standard to represent the sum as a stochastic integral:\n\\[\n\\sum_{i=1}^{N_t} \\xi_i = \\int_0^t \\int_0^\\infty x\\,p(ds,dx) ,\n\\]\nwhere $p(ds,dx)$ is a Poisson random measure with mean measure $q(ds,dx) = \\lambda ds\\,F(dx)$.\n\nLet $\\tau^u := \\inf \\{t: X^u_t \\le 0\\}$ (the ruin time), $\\Psi(u) := {\\bf P}(\\tau^u < \\infty)$ (the ruin probability), and $\\Phi(u) := 1 - \\Psi(u)$ (the survival probability).\n\nThe aforementioned paper \\cite{KP2016}, together with its complements \\cite{PZ} and \\cite{PerErr} (treating the case $\\beta=0$), deals with the asymptotic analysis of the ruin probability under the assumption of a riskless rate $r=0$. For the case where $\\kappa=1$ and $F$ is an exponential distribution, it was shown that if $\\beta := 2a/\\sigma^2 - 1 > 0$, then $\\Psi(u) \\sim C u^{-\\beta}$ as $u \\to \\infty$; if $\\beta \\le 0$, ruin is imminent. If $\\kappa \\in (0,1)$, the same conclusion holds with $\\beta_\\kappa := 2a/(\\kappa\\sigma^2) - 1$. The equation $\\beta_\\kappa = 0$ defines the threshold for the fraction of risky investment above which ruin is certain. The first step of the argument, which is technically difficult and delicate, is to prove that the survival probability is smooth under appropriate assumptions on $F$. The second, easier step is to prove that $\\Phi$ satisfies an integro-differential equation. In the specific case of an exponential jump distribution, this equation can be reduced--by differentiating and eliminating the integral term--to a linear ODE of third order. This ODE turns out to be of second order for the derivative of the survival probability. Asymptotic results for solutions of such equations are available in the literature. The property that the ruin probability cannot decrease faster than some power function, established by Kalashnikov and Norberg in \\cite{Kalash-Nor}, then yields the aforementioned result.\n\nImportantly, if the survival probability is smooth, i.e., if $\\Phi \\in C^2$, then straightforward arguments based on It\\^o's formula lead to the conclusion that $\\Phi$ satisfies, in the classical sense, the integro-differential equation (IDE)\n\\begin{multline}\\label{int-diff}\n\\frac{1}{2} \\kappa^2\\sigma^2 u^2 \\Phi''(u) + ((a-r)\\kappa + r) u \\Phi'(u) =\n\\\\\n= c \\Phi'(u) - \\lambda \\int_0^\\infty (\\Phi(u+y) - \\Phi(u)) \\,dF(y),\n\\end{multline}\nsee, e.g., \\cite{KP2016}. The same equation holds for the ruin probability.\n\nAvailable sufficient conditions on $F$ ensuring the smoothness of $\\Phi$ are rather restrictive: the property holds if $F$ has a density $f \\in C^2$ such that $f' \\in L^1(\\mathbb{R}_+)$; see \\cite{KPukh}. For this reason, some authors prefer to assume smoothness a priori. On the other hand, it is not difficult to show that $\\Phi$ is a viscosity solution of the above equation under much more general conditions, but proving uniqueness for boundary value problems in the viscosity sense is rather involved; see \\cite{Bel-Kab} and, for an equation arising in a model based on a generalized Ornstein--Uhlenbeck process, the recent study \\cite{AK2025}.\n\n\\smallskip\n\\textbf{Main result.} \nIn this note, we employ a completely different proof strategy. Our main result is the following theorem.", "full_context": "In the present note, we study the integro-differential equation arising in the asymptotic analysis of ruin probabilities for an annuity model more general than the one described in \\cite{KP2016}. Specifically, we suppose that the insurance company pays pensions to its customers at a constant rate $c>0$ and receives, as income, the remaining part of their pension fund when the contracts expire. The company instantaneously invests a fraction $\\kappa$ of its capital reserve in a risky asset whose price $S=(S_t)$ evolves as a geometric Brownian motion (gBm) with parameters $a$ and $\\sigma>0$. The remaining fraction $1-\\kappa$ of the capital reserve is invested in a non-risky asset (a bank account) with a rate of return $r$. Formally, the process $X=X^u$ describing the evolution of the capital reserve is given by the following stochastic differential equation with jumps:\n\\begin{equation}\\label{one}\nX_t = u + \\int_0^t \\kappa X_s (a\\,ds +\\sigma\\,dW_s) + \\int_0^t r(1-\\kappa)X_s\\,ds + P_t,\n\\end{equation}\nwhere $u \\ge 0$ is the initial capital, $W=(W_t)$ is a standard Wiener process and $P=(P_t)$ is the ``business'' process (or Cram\\'er--Lundberg process), independent of $W$. In the classical literature, $P$ is usually represented as\n\\[\nP_t = -ct + \\sum_{i=1}^{N_t} \\xi_i,\n\\]\nwhere $N=(N_t)$ is a Poisson process with intensity $\\lambda > 0$, and $(\\xi_i)_{i\\ge 1}$ is an i.i.d. sequence of strictly positive random variables, independent of $N$, with common distribution function $F$ such that $F(0)=0$. \nIn the literature on stochastic calculus, it is standard to represent the sum as a stochastic integral:\n\\[\n\\sum_{i=1}^{N_t} \\xi_i = \\int_0^t \\int_0^\\infty x\\,p(ds,dx) ,\n\\]\nwhere $p(ds,dx)$ is a Poisson random measure with mean measure $q(ds,dx) = \\lambda ds\\,F(dx)$.\n\nLet $\\tau^u := \\inf \\{t: X^u_t \\le 0\\}$ (the ruin time), $\\Psi(u) := {\\bf P}(\\tau^u < \\infty)$ (the ruin probability), and $\\Phi(u) := 1 - \\Psi(u)$ (the survival probability).\n\nThe aforementioned paper \\cite{KP2016}, together with its complements \\cite{PZ} and \\cite{PerErr} (treating the case $\\beta=0$), deals with the asymptotic analysis of the ruin probability under the assumption of a riskless rate $r=0$. For the case where $\\kappa=1$ and $F$ is an exponential distribution, it was shown that if $\\beta := 2a/\\sigma^2 - 1 > 0$, then $\\Psi(u) \\sim C u^{-\\beta}$ as $u \\to \\infty$; if $\\beta \\le 0$, ruin is imminent. If $\\kappa \\in (0,1)$, the same conclusion holds with $\\beta_\\kappa := 2a/(\\kappa\\sigma^2) - 1$. The equation $\\beta_\\kappa = 0$ defines the threshold for the fraction of risky investment above which ruin is certain. The first step of the argument, which is technically difficult and delicate, is to prove that the survival probability is smooth under appropriate assumptions on $F$. The second, easier step is to prove that $\\Phi$ satisfies an integro-differential equation. In the specific case of an exponential jump distribution, this equation can be reduced--by differentiating and eliminating the integral term--to a linear ODE of third order. This ODE turns out to be of second order for the derivative of the survival probability. Asymptotic results for solutions of such equations are available in the literature. The property that the ruin probability cannot decrease faster than some power function, established by Kalashnikov and Norberg in \\cite{Kalash-Nor}, then yields the aforementioned result.\n\nImportantly, if the survival probability is smooth, i.e., if $\\Phi \\in C^2$, then straightforward arguments based on It\\^o's formula lead to the conclusion that $\\Phi$ satisfies, in the classical sense, the integro-differential equation (IDE)\n\\begin{multline}\\label{int-diff}\n\\frac{1}{2} \\kappa^2\\sigma^2 u^2 \\Phi''(u) + ((a-r)\\kappa + r) u \\Phi'(u) =\n\\\\\n= c \\Phi'(u) - \\lambda \\int_0^\\infty (\\Phi(u+y) - \\Phi(u)) \\,dF(y),\n\\end{multline}\nsee, e.g., \\cite{KP2016}. The same equation holds for the ruin probability.\n\nAvailable sufficient conditions on $F$ ensuring the smoothness of $\\Phi$ are rather restrictive: the property holds if $F$ has a density $f \\in C^2$ such that $f' \\in L^1(\\mathbb{R}_+)$; see \\cite{KPukh}. For this reason, some authors prefer to assume smoothness a priori. On the other hand, it is not difficult to show that $\\Phi$ is a viscosity solution of the above equation under much more general conditions, but proving uniqueness for boundary value problems in the viscosity sense is rather involved; see \\cite{Bel-Kab} and, for an equation arising in a model based on a generalized Ornstein--Uhlenbeck process, the recent study \\cite{AK2025}.\n\n\\smallskip\n\\textbf{Main result.} \nIn this note, we employ a completely different proof strategy. Our main result is the following theorem.\n\nIn the present note, we study the integro-differential equation arising in the asymptotic analysis of ruin probabilities for an annuity model more general than the one described in \\cite{KP2016}. Specifically, we suppose that the insurance company pays pensions to its customers at a constant rate $c>0$ and receives, as income, the remaining part of their pension fund when the contracts expire. The company instantaneously invests a fraction $\\kappa$ of its capital reserve in a risky asset whose price $S=(S_t)$ evolves as a geometric Brownian motion (gBm) with parameters $a$ and $\\sigma>0$. The remaining fraction $1-\\kappa$ of the capital reserve is invested in a non-risky asset (a bank account) with a rate of return $r$. Formally, the process $X=X^u$ describing the evolution of the capital reserve is given by the following stochastic differential equation with jumps:\n\\begin{equation}\\label{one}\nX_t = u + \\int_0^t \\kappa X_s (a\\,ds +\\sigma\\,dW_s) + \\int_0^t r(1-\\kappa)X_s\\,ds + P_t,\n\\end{equation}\nwhere $u \\ge 0$ is the initial capital, $W=(W_t)$ is a standard Wiener process and $P=(P_t)$ is the ``business'' process (or Cram\\'er--Lundberg process), independent of $W$. In the classical literature, $P$ is usually represented as\n\\[\nP_t = -ct + \\sum_{i=1}^{N_t} \\xi_i,\n\\]\nwhere $N=(N_t)$ is a Poisson process with intensity $\\lambda > 0$, and $(\\xi_i)_{i\\ge 1}$ is an i.i.d. sequence of strictly positive random variables, independent of $N$, with common distribution function $F$ such that $F(0)=0$. \nIn the literature on stochastic calculus, it is standard to represent the sum as a stochastic integral:\n\\[\n\\sum_{i=1}^{N_t} \\xi_i = \\int_0^t \\int_0^\\infty x\\,p(ds,dx) ,\n\\]\nwhere $p(ds,dx)$ is a Poisson random measure with mean measure $q(ds,dx) = \\lambda ds\\,F(dx)$.\n\nThe aforementioned paper \\cite{KP2016}, together with its complements \\cite{PZ} and \\cite{PerErr} (treating the case $\\beta=0$), deals with the asymptotic analysis of the ruin probability under the assumption of a riskless rate $r=0$. For the case where $\\kappa=1$ and $F$ is an exponential distribution, it was shown that if $\\beta := 2a/\\sigma^2 - 1 > 0$, then $\\Psi(u) \\sim C u^{-\\beta}$ as $u \\to \\infty$; if $\\beta \\le 0$, ruin is imminent. If $\\kappa \\in (0,1)$, the same conclusion holds with $\\beta_\\kappa := 2a/(\\kappa\\sigma^2) - 1$. The equation $\\beta_\\kappa = 0$ defines the threshold for the fraction of risky investment above which ruin is certain. The first step of the argument, which is technically difficult and delicate, is to prove that the survival probability is smooth under appropriate assumptions on $F$. The second, easier step is to prove that $\\Phi$ satisfies an integro-differential equation. In the specific case of an exponential jump distribution, this equation can be reduced--by differentiating and eliminating the integral term--to a linear ODE of third order. This ODE turns out to be of second order for the derivative of the survival probability. Asymptotic results for solutions of such equations are available in the literature. The property that the ruin probability cannot decrease faster than some power function, established by Kalashnikov and Norberg in \\cite{Kalash-Nor}, then yields the aforementioned result.\n\nImportantly, if the survival probability is smooth, i.e., if $\\Phi \\in C^2$, then straightforward arguments based on It\\^o's formula lead to the conclusion that $\\Phi$ satisfies, in the classical sense, the integro-differential equation (IDE)\n\\begin{multline}\\label{int-diff}\n\\frac{1}{2} \\kappa^2\\sigma^2 u^2 \\Phi''(u) + ((a-r)\\kappa + r) u \\Phi'(u) =\n\\\\\n= c \\Phi'(u) - \\lambda \\int_0^\\infty (\\Phi(u+y) - \\Phi(u)) \\,dF(y),\n\\end{multline}\nsee, e.g., \\cite{KP2016}. The same equation holds for the ruin probability.\n\n\\smallskip\n\\textbf{Main result.} \nIn this note, we employ a completely different proof strategy. Our main result is the following theorem.\n\nOur result substantially relaxes the existing conditions on the jump size distribution. Our approach is based on reducing the integro-differential equation to a pair of integral equations for the derivative of its solution. These integral equations are coupled at a suitably chosen point $u_0 \\in (0,\\infty)$. For the first integral equation on $[u_0,\\infty)$, we establish uniqueness for a somewhat exotic terminal value problem. Specifically, we seek a solution with asymptotic behavior $u^{-\\gamma}$ as $u \\to \\infty$, which is consistent with earlier results such as those in \\cite{KP2016} (note that $\\gamma=\\beta+1$ in the case where $\\kappa=1$ and $r=0$). It is important to note that we do not rely on previously established asymptotic formulas in our proof; they serve only as motivation. For a sufficiently large $u_0$, we find the solution using the Banach fixed point theorem. The second equation for the derivative is a Volterra integral equation of the second kind on $(0, u_0]$ with a terminal condition at $u_0$ defined via the solution of the first equation. Gluing the solutions together yields a solution for the derivative of the desired function, which includes a multiplicative constant $C$ to be determined. To satisfy the zero boundary condition at zero, the additive constant $C_0$ arising from integration to obtain the solution $\\Phi$ of the original IDE must be taken as zero. The constant $C$ is then chosen to ensure that the limit at infinity equals one. Note also that $C$ is ``calculable'' in the sense that it is expressed as an integral involving the solutions of the aforementioned equations, which are themselves at least numerically computable. Finally, using It\\^o's formula, we easily conclude that $\\Phi$ coincides with the survival probability.\n\n\\section{Analysis of the integro-differential equation} \nRecall that we assume that $F$ is continuous at zero ($F(0)=0$) and $\\E \\xi_1 <\\infty$. \nUsing integration by parts for functions of bounded variation (or Fubini's theorem), we get:\n$$\n\\int_0^\\infty \\Phi(u+y) F(dy) = -\\int_0^\\infty \\Phi(u+y) d\\bar{F}(y) = \\Phi(u) + \\int_0^\\infty \\Phi'(u+y)\\bar F(y)dy,\n$$\nwhere $\\bar F:=1-F$. With this relation, the equation (\\ref{int-diff}) can be written in the equivalent form \n\\beq\n\\label{IDE-F}\n\\frac 12 \\kappa^2\\sigma^2u^2 \\Phi''(u)+ ((a-r)\\kappa +r) u\\Phi'(u) -c\\Phi' (u) \n=-\\lambda \\int_0^\\infty\\Phi'(u+y)\\bar F(y)dy,\n\\eeq\nsuitable for our purposes. We are interested in the classical solution of this equation, that is, in $C([0,\\infty])\\cap C^2((0,\\infty))$ with the boundary conditions at infinity and zero $\\Phi(\\infty)=1$ and $\\Phi (0)=0$.\n\nThe equation (\\ref{IDE-F}) is reduced to a homogeneous integro-differential equation of the first order with respect to the derivative $g(u):=\\Phi'(u)$. Introducing the notation \n$$\n\\gamma:=\\frac{2((a-r)\\kappa+r)}{\\kappa^2 \\sigma^2},\\quad \\alpha:=\\frac{2c}{\\kappa^2 \\sigma^2}, \\quad \\mu:=\\frac{2\\lambda}{\\kappa^2 \\sigma^2}, \n $$\nwe get the following IDE for $g$: \n\\begin{equation}\n\\label{IDE}\n u^2 g'(u) + (\\gamma u-\\alpha) g(u)= - Ag(u),\n\\end{equation}\nwhere\n\\beq\n\\label{Ag(u)}\n Ag(u):=\\mu \\int_0^\\infty g(u+z)\\,\\overline{F}(z)\\,dz.\n\\eeq\n\n\\section{The probability of non-ruin and the solution of IDE} \nDue to linearity of the equation (\\ref{IDE}) all its solutions with the asymptotic behavior at infinity $C u^{-\\gamma}$, $C>0$, are of the form $C\\hat g(u)$. The corresponding general solution of the equation (\\ref{int-diff}) has the form \n\\beq\n\\label{GC0C}\nG(u)=C_0+C\\int_0^u \\hat g(t)\\,dt. \n\\eeq\nOur aim is to choose the constants to ensure that $G=\\Phi$. The following lemma shows that the constant $C_0$ must be equal to zero.\n\\begin{lemma}\n$\\Phi(0+)=\\Phi(0):=0$.\n\\end{lemma}\n\\begin{proof} \nConsider the stochastic exponential $Z=(Z_t)_{t\\ge 0}$ given by the linear SDE \n$$\ndZ_t=Z_t \\left( ((a-r)\\kappa + r) \\,d t + \\sigma \\,dW_t \\right), \\qquad Z_0=1, \n$$\nand define the process $\\hat X^u$ \n$$\n\\hat X^u_t=Z_t\\Big( u-c\\int_0^t Z_s^{-1}ds\\Big).\n$$", "post_theorem_intro_text_len": 2148, "post_theorem_intro_text": "Our result substantially relaxes the existing conditions on the jump size distribution. Our approach is based on reducing the integro-differential equation to a pair of integral equations for the derivative of its solution. These integral equations are coupled at a suitably chosen point $u_0 \\in (0,\\infty)$. For the first integral equation on $[u_0,\\infty)$, we establish uniqueness for a somewhat exotic terminal value problem. Specifically, we seek a solution with asymptotic behavior $u^{-\\gamma}$ as $u \\to \\infty$, which is consistent with earlier results such as those in \\cite{KP2016} (note that $\\gamma=\\beta+1$ in the case where $\\kappa=1$ and $r=0$). It is important to note that we do not rely on previously established asymptotic formulas in our proof; they serve only as motivation. For a sufficiently large $u_0$, we find the solution using the Banach fixed point theorem. The second equation for the derivative is a Volterra integral equation of the second kind on $(0, u_0]$ with a terminal condition at $u_0$ defined via the solution of the first equation. Gluing the solutions together yields a solution for the derivative of the desired function, which includes a multiplicative constant $C$ to be determined. To satisfy the zero boundary condition at zero, the additive constant $C_0$ arising from integration to obtain the solution $\\Phi$ of the original IDE must be taken as zero. The constant $C$ is then chosen to ensure that the limit at infinity equals one. Note also that $C$ is ``calculable'' in the sense that it is expressed as an integral involving the solutions of the aforementioned equations, which are themselves at least numerically computable. Finally, using It\\^o's formula, we easily conclude that $\\Phi$ coincides with the survival probability.\n\nOur argument is remarkably simple. We avoid reducing the IDE to a higher-order ODE, which would introduce an additional constant, and instead rely only on standard results concerning the uniqueness of solutions to IDEs. Although conceptually inspired by the paper \\cite{Grandits2004} on the non-life insurance model, our proofs are technically quite different.", "sketch": "To prove Theorem~\\ref{main}, the approach is to reduce the integro-differential equation to “a pair of integral equations for the derivative of its solution,” coupled at a point $u_0\\in(0,\\infty)$. On $[u_0,\\infty)$, one treats “a somewhat exotic terminal value problem,” seeking a solution with asymptotic behavior $u^{-\\gamma}$ as $u\\to\\infty$; for sufficiently large $u_0$ this solution is obtained “using the Banach fixed point theorem,” and uniqueness is established. On $(0,u_0]$, the derivative satisfies “a Volterra integral equation of the second kind” with a terminal condition at $u_0$ coming from the first part. “Gluing the solutions together” gives the derivative (with a multiplicative constant $C$ to determine), then integrating yields $\\Phi$ with an additive constant $C_0$, which “must be taken as zero” to enforce $\\Phi(0)=0$. The constant $C$ is chosen so that “the limit at infinity equals one,” and $C$ is “calculable” via an integral involving the constructed solutions. Finally, “using It\\^o's formula,” one concludes that $\\Phi$ coincides with the survival probability. The method “avoid[s] reducing the IDE to a higher-order ODE” and instead uses “standard results concerning the uniqueness of solutions to IDEs.”", "expanded_sketch": "To prove the main theorem, the approach is to reduce the integro-differential equation to “a pair of integral equations for the derivative of its solution,” coupled at a point $u_0\\in(0,\\infty)$. On $[u_0,\\infty)$, one treats “a somewhat exotic terminal value problem,” seeking a solution with asymptotic behavior $u^{-\\gamma}$ as $u\\to\\infty$; for sufficiently large $u_0$ this solution is obtained “using the Banach fixed point theorem,” and uniqueness is established. On $(0,u_0]$, the derivative satisfies “a Volterra integral equation of the second kind” with a terminal condition at $u_0$ coming from the first part. “Gluing the solutions together” gives the derivative (with a multiplicative constant $C$ to determine), then integrating yields $\\Phi$ with an additive constant $C_0$, which “must be taken as zero” to enforce $\\Phi(0)=0$. The constant $C$ is chosen so that “the limit at infinity equals one,” and $C$ is “calculable” via an integral involving the constructed solutions. Finally, “using It\\^o's formula,” one concludes that $\\Phi$ coincides with the survival probability. The method “avoid[s] reducing the IDE to a higher-order ODE” and instead uses “standard results concerning the uniqueness of solutions to IDEs.”", "expanded_theorem": "\\label{main}\nSuppose that $F(0)=0$, ${\\bf E}[\\xi] < \\infty$ and $$\\gamma := \\dfrac{(a-r)\\kappa + r}{\\kappa^2 \\sigma^2} > 1.$$\nThen the survival probability $\\Phi \\in C([0,\\infty)) \\cap C^2((0,\\infty))$ satisfies the integro-differential equation\n\\begin{multline}\\label{int-diff}\n\\frac{1}{2} \\kappa^2\\sigma^2 u^2 \\Phi''(u) + ((a-r)\\kappa + r) u \\Phi'(u) =\n\\\\\n= c \\Phi'(u) - \\lambda \\int_0^\\infty (\\Phi(u+y) - \\Phi(u)) \\,dF(y),\n\\end{multline}\nwith boundary conditions $\\Phi(0)=0$, $\\Phi(\\infty)=1$. Moreover, there exists a finite constant $C > 0$ such that $\\Psi(u) \\sim C u^{-\\gamma+1}$ as $u \\to \\infty$.", "theorem_type": ["Existential–Universal", "Asymptotic or Limit"], "mcq": {"question": "Consider the annuity-model reserve process\n\\[\nX_t^u=u+\\int_0^t \\kappa X_s(a\\,ds+\\sigma\\,dW_s)+\\int_0^t r(1-\\kappa)X_s\\,ds-ct+\\sum_{i=1}^{N_t}\\xi_i,\n\\]\nwhere $u\\ge 0$, $\\sigma>0$, $c>0$, $N=(N_t)$ is a Poisson process with intensity $\\lambda>0$, and $(\\xi_i)$ are i.i.d. strictly positive random variables with distribution function $F$, independent of $W$ and $N$. Let\n\\[\n\\tau^u:=\\inf\\{t:X_t^u\\le 0\\},\\qquad \\Psi(u):={\\bf P}(\\tau^u<\\infty),\\qquad \\Phi(u):=1-\\Psi(u).\n\\]\nAssume that $F(0)=0$, ${\\bf E}[\\xi]<\\infty$, and\n\\[\n\\gamma:=\\frac{(a-r)\\kappa+r}{\\kappa^2\\sigma^2}>1.\n\\]\nWhich statement holds for the survival probability $\\Phi$ and the ruin probability $\\Psi$?", "correct_choice": {"label": "A", "text": "The survival probability satisfies $\\Phi\\in C([0,\\infty))\\cap C^2((0,\\infty))$ and, for every $u>0$, solves\n\\[\n\\frac{1}{2}\\kappa^2\\sigma^2 u^2 \\Phi''(u)+((a-r)\\kappa+r)u\\Phi'(u)=c\\Phi'(u)-\\lambda\\int_0^\\infty(\\Phi(u+y)-\\Phi(u))\\,dF(y),\n\\]\nwith boundary conditions $\\Phi(0)=0$ and $\\lim_{u\\to\\infty}\\Phi(u)=1$. Moreover, there exists a finite constant $C>0$ such that\n\\[\n\\Psi(u)\\sim C u^{-\\gamma+1}\\qquad\\text{as }u\\to\\infty.\n\\]"}, "choices": [{"label": "B", "text": "The survival probability satisfies $\\Phi\\in C([0,\\infty))\\cap C^2((0,\\infty))$ and, for every $u>0$, solves\n\\[\n\\frac{1}{2}\\kappa^2\\sigma^2 u^2 \\Phi''(u)+((a-r)\\kappa+r)u\\Phi'(u)=c\\Phi'(u)-\\lambda\\int_0^\\infty(\\Phi(u+y)-\\Phi(u))\\,dF(y),\n\\]\nwith boundary conditions $\\Phi(0)=0$ and $\\lim_{u\\to\\infty}\\Phi(u)=1$. Moreover, there exists a finite constant $C>0$ such that\n\\[\n\\Psi(u)\\sim C u^{-\\gamma}\\qquad\\text{as }u\\to\\infty.\n\\]"}, {"label": "C", "text": "The survival probability satisfies $\\Phi\\in C([0,\\infty))\\cap C^2((0,\\infty))$ and, for every $u>0$, solves\n\\[\n\\frac{1}{2}\\kappa^2\\sigma^2 u^2 \\Phi''(u)+((a-r)\\kappa+r)u\\Phi'(u)=c\\Phi'(u)-\\lambda\\int_0^\\infty(\\Phi(u+y)-\\Phi(u))\\,dF(y),\n\\]\nwith boundary conditions $\\Phi(0)=0$ and $\\lim_{u\\to\\infty}\\Phi(u)=1$."}, {"label": "D", "text": "The survival probability satisfies $\\Phi\\in C([0,\\infty))\\cap C^2((0,\\infty))$ and, for every $u>0$, solves\n\\[\n\\frac{1}{2}\\kappa^2\\sigma^2 u^2 \\Phi''(u)+((a-r)\\kappa+r)u\\Phi'(u)=c\\Phi'(u)-\\lambda\\int_0^\\infty(\\Phi(u+y)-\\Phi(u))\\,dF(y),\n\\]\nwith boundary conditions $\\Phi(0)=0$ and $\\lim_{u\\to\\infty}\\Phi(u)=1$. Moreover, there exists a finite constant $C>0$, depending only on $(a,r,\\kappa,\\sigma,c,\\lambda)$ and not on $F$, such that\n\\[\n\\Psi(u)\\sim C u^{-\\gamma+1}\\qquad\\text{as }u\\to\\infty.\n\\]"}, {"label": "E", "text": "There exists a unique function $\\Phi\\in C([0,\\infty))\\cap C^2((0,\\infty))$ satisfying\n\\[\n\\frac{1}{2}\\kappa^2\\sigma^2 u^2 \\Phi''(u)+((a-r)\\kappa+r)u\\Phi'(u)=c\\Phi'(u)-\\lambda\\int_0^\\infty(\\Phi(u+y)-\\Phi(u))\\,dF(y),\n\\]\nfor every $u>0$, together with the boundary conditions $\\Phi(0)=0$ and $\\lim_{u\\to\\infty}\\Phi(u)=1$, and this unique solution automatically coincides with the survival probability. Moreover, there exists a finite constant $C>0$ such that\n\\[\n\\Psi(u)=C u^{-\\gamma+1}+o\\big(u^{-\\gamma+1}\\big)\n\\]\nand the multiplicative constant $C$ is uniquely determined by the differential equation and the asymptotic condition at infinity alone."}], "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": "asymptotic exponent from derivative-to-solution integration", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the tail asymptotic conclusion for \\Psi", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence of asymptotic constant on glued integral-equation solution and F", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "need to fix additive constant at zero and multiplicative constant via normalization at infinity, not from asymptotic condition alone", "template_used": "wildcard"}]}} +{"id": "2601.01447v1", "paper_link": "http://arxiv.org/abs/2601.01447v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{main}\nSuppose that $F(0)=0$, ${\\bf E}[\\xi] < \\infty$ and $$\\gamma := \\dfrac{(a-r)\\kappa + r}{\\kappa^2 \\sigma^2} > 1.$$\nThen the survival probability $\\Phi \\in C([0,\\infty)) \\cap C^2((0,\\infty))$ satisfies the integro-differential equation (\\ref{int-diff}) with boundary conditions $\\Phi(0)=0$, $\\Phi(\\infty)=1$. Moreover, there exists a finite constant $C > 0$ such that $\\Psi(u) \\sim C u^{-\\gamma+1}$ as $u \\to \\infty$.", "start_pos": 10420, "end_pos": 10872, "label": "main"}, "ref_dict": {"one": "\\begin{equation}\\label{one}\nX_t = u + \\int_0^t \\kappa X_s (a\\,ds +\\sigma\\,dW_s) + \\int_0^t r(1-\\kappa)X_s\\,ds + P_t,\n\\end{equation}", "int-diff": "\\begin{multline}\\label{int-diff}\n\\frac{1}{2} \\kappa^2\\sigma^2 u^2 \\Phi''(u) + ((a-r)\\kappa + r) u \\Phi'(u) =\n\\\\\n= c \\Phi'(u) - \\lambda \\int_0^\\infty (\\Phi(u+y) - \\Phi(u)) \\,dF(y),\n\\end{multline}"}, "pre_theorem_intro_text_len": 7674, "pre_theorem_intro_text": "In the classical collective risk theory, initiated by Filip Lundberg in 1903 and developed further by Harald Cram\\'er in the thirties, it was assumed that an insurance company kept its reserve entirely separate from risky financial activities. This was natural for the era: the legislation of the period was based on a paradigm of social responsibility and was strongly influenced by the terrible consequences of stock market disasters. However, evolving economic realities, together with progress in financial theory, led to a relaxation of legal constraints and the emergence of theoretical studies on ruin problems with risky investments. The origin of this theory can be traced back to the seminal 1993 paper by Jostein Paulsen, who applied the Kesten--Goldie implicit renewal theory (also known as the theory of distributional equations), as exposed in Goldie's \\cite{Go91}, to derive asymptotics for ruin probabilities. A number of authors have since contributed to its development along the same lines, leveraging recent progress in the theory of distributional equations as summarized in the book \\cite{BDM}. We mention here only a few recent papers covering Sparre Andersen type models with risky investments; see \\cite{EKS}, \\cite{KPro}, \\cite{KLP}, and references therein. Although this method is powerful and applicable to a variety of models, it has a weakness: it provides only the rate of convergence to zero of the ruin probability, not the exact asymptotic (i.e., it gives no information on the leading constant).\n\nFor the asymptotic analysis of Lundberg--Cram\\'er type models with investment, there is an alternative method based on representing the survival (or ruin) probability as a function satisfying a second-order integro-differential equation. Such an equation seems to have appeared for the first time in a short note by Anna Frolova, \\cite{Fr}, who observed that in the case of exponential claims, it can be reduced to a linear ordinary differential equation (ODE). From its general solution, she conjectured that the ruin probability decays as a power function. This conjecture was later proved in \\cite{FrKP} for the non-life insurance model and in \\cite{KP2016} for the annuity payments model.\n\nIn the present note, we study the integro-differential equation arising in the asymptotic analysis of ruin probabilities for an annuity model more general than the one described in \\cite{KP2016}. Specifically, we suppose that the insurance company pays pensions to its customers at a constant rate $c>0$ and receives, as income, the remaining part of their pension fund when the contracts expire. The company instantaneously invests a fraction $\\kappa$ of its capital reserve in a risky asset whose price $S=(S_t)$ evolves as a geometric Brownian motion (gBm) with parameters $a$ and $\\sigma>0$. The remaining fraction $1-\\kappa$ of the capital reserve is invested in a non-risky asset (a bank account) with a rate of return $r$. Formally, the process $X=X^u$ describing the evolution of the capital reserve is given by the following stochastic differential equation with jumps:\n\\begin{equation}\\label{one}\nX_t = u + \\int_0^t \\kappa X_s (a\\,ds +\\sigma\\,dW_s) + \\int_0^t r(1-\\kappa)X_s\\,ds + P_t,\n\\end{equation}\nwhere $u \\ge 0$ is the initial capital, $W=(W_t)$ is a standard Wiener process and $P=(P_t)$ is the ``business'' process (or Cram\\'er--Lundberg process), independent of $W$. In the classical literature, $P$ is usually represented as\n\\[\nP_t = -ct + \\sum_{i=1}^{N_t} \\xi_i,\n\\]\nwhere $N=(N_t)$ is a Poisson process with intensity $\\lambda > 0$, and $(\\xi_i)_{i\\ge 1}$ is an i.i.d. sequence of strictly positive random variables, independent of $N$, with common distribution function $F$ such that $F(0)=0$. \nIn the literature on stochastic calculus, it is standard to represent the sum as a stochastic integral:\n\\[\n\\sum_{i=1}^{N_t} \\xi_i = \\int_0^t \\int_0^\\infty x\\,p(ds,dx) ,\n\\]\nwhere $p(ds,dx)$ is a Poisson random measure with mean measure $q(ds,dx) = \\lambda ds\\,F(dx)$.\n\nSuch a model can also describe the capital evolution of a venture company paying current expenditures and receiving incomes $\\xi_i$ from selling innovations.\n\nAfter regrouping terms, the equation (\\ref{one}), written in the traditional form of a stochastic differential equation, becomes:\n\\begin{equation}\\label{two}\ndX_t = \\big( (a-r)\\kappa + r\\big) X_t\\,dt + \\kappa\\sigma X_t\\,dW_t + dP_t, \\qquad X_0 = u.\n\\end{equation}\n\nLet $\\tau^u := \\inf \\{t: X^u_t \\le 0\\}$ (the ruin time), $\\Psi(u) := {\\bf P}(\\tau^u < \\infty)$ (the ruin probability), and $\\Phi(u) := 1 - \\Psi(u)$ (the survival probability).\n\nThe aforementioned paper \\cite{KP2016}, together with its complements \\cite{PZ} and \\cite{PerErr} (treating the case $\\beta=0$), deals with the asymptotic analysis of the ruin probability under the assumption of a riskless rate $r=0$. For the case where $\\kappa=1$ and $F$ is an exponential distribution, it was shown that if $\\beta := 2a/\\sigma^2 - 1 > 0$, then $\\Psi(u) \\sim C u^{-\\beta}$ as $u \\to \\infty$; if $\\beta \\le 0$, ruin is imminent. If $\\kappa \\in (0,1)$, the same conclusion holds with $\\beta_\\kappa := 2a/(\\kappa\\sigma^2) - 1$. The equation $\\beta_\\kappa = 0$ defines the threshold for the fraction of risky investment above which ruin is certain. The first step of the argument, which is technically difficult and delicate, is to prove that the survival probability is smooth under appropriate assumptions on $F$. The second, easier step is to prove that $\\Phi$ satisfies an integro-differential equation. In the specific case of an exponential jump distribution, this equation can be reduced--by differentiating and eliminating the integral term--to a linear ODE of third order. This ODE turns out to be of second order for the derivative of the survival probability. Asymptotic results for solutions of such equations are available in the literature. The property that the ruin probability cannot decrease faster than some power function, established by Kalashnikov and Norberg in \\cite{Kalash-Nor}, then yields the aforementioned result.\n\nIt is worth noting that the method of eliminating integral terms works not only for exponential jump distributions but also for other classes (e.g., Pareto distributions) and can lead to asymptotic results via Laplace transforms and Tauberian theorems; see \\cite{ABL}, \\cite{ACT}, and \\cite{A2025}.\n\nImportantly, if the survival probability is smooth, i.e., if $\\Phi \\in C^2$, then straightforward arguments based on It\\^o's formula lead to the conclusion that $\\Phi$ satisfies, in the classical sense, the integro-differential equation (IDE)\n\\begin{multline}\\label{int-diff}\n\\frac{1}{2} \\kappa^2\\sigma^2 u^2 \\Phi''(u) + ((a-r)\\kappa + r) u \\Phi'(u) =\n\\\\\n= c \\Phi'(u) - \\lambda \\int_0^\\infty (\\Phi(u+y) - \\Phi(u)) \\,dF(y),\n\\end{multline}\nsee, e.g., \\cite{KP2016}. The same equation holds for the ruin probability.\n\nAvailable sufficient conditions on $F$ ensuring the smoothness of $\\Phi$ are rather restrictive: the property holds if $F$ has a density $f \\in C^2$ such that $f' \\in L^1(\\mathbb{R}_+)$; see \\cite{KPukh}. For this reason, some authors prefer to assume smoothness a priori. On the other hand, it is not difficult to show that $\\Phi$ is a viscosity solution of the above equation under much more general conditions, but proving uniqueness for boundary value problems in the viscosity sense is rather involved; see \\cite{Bel-Kab} and, for an equation arising in a model based on a generalized Ornstein--Uhlenbeck process, the recent study \\cite{AK2025}.\n\n\\smallskip\n\\textbf{Main result.} \nIn this note, we employ a completely different proof strategy. Our main result is the following theorem.", "context": "In the present note, we study the integro-differential equation arising in the asymptotic analysis of ruin probabilities for an annuity model more general than the one described in \\cite{KP2016}. Specifically, we suppose that the insurance company pays pensions to its customers at a constant rate $c>0$ and receives, as income, the remaining part of their pension fund when the contracts expire. The company instantaneously invests a fraction $\\kappa$ of its capital reserve in a risky asset whose price $S=(S_t)$ evolves as a geometric Brownian motion (gBm) with parameters $a$ and $\\sigma>0$. The remaining fraction $1-\\kappa$ of the capital reserve is invested in a non-risky asset (a bank account) with a rate of return $r$. Formally, the process $X=X^u$ describing the evolution of the capital reserve is given by the following stochastic differential equation with jumps:\n\\begin{equation}\\label{one}\nX_t = u + \\int_0^t \\kappa X_s (a\\,ds +\\sigma\\,dW_s) + \\int_0^t r(1-\\kappa)X_s\\,ds + P_t,\n\\end{equation}\nwhere $u \\ge 0$ is the initial capital, $W=(W_t)$ is a standard Wiener process and $P=(P_t)$ is the ``business'' process (or Cram\\'er--Lundberg process), independent of $W$. In the classical literature, $P$ is usually represented as\n\\[\nP_t = -ct + \\sum_{i=1}^{N_t} \\xi_i,\n\\]\nwhere $N=(N_t)$ is a Poisson process with intensity $\\lambda > 0$, and $(\\xi_i)_{i\\ge 1}$ is an i.i.d. sequence of strictly positive random variables, independent of $N$, with common distribution function $F$ such that $F(0)=0$. \nIn the literature on stochastic calculus, it is standard to represent the sum as a stochastic integral:\n\\[\n\\sum_{i=1}^{N_t} \\xi_i = \\int_0^t \\int_0^\\infty x\\,p(ds,dx) ,\n\\]\nwhere $p(ds,dx)$ is a Poisson random measure with mean measure $q(ds,dx) = \\lambda ds\\,F(dx)$.\n\nLet $\\tau^u := \\inf \\{t: X^u_t \\le 0\\}$ (the ruin time), $\\Psi(u) := {\\bf P}(\\tau^u < \\infty)$ (the ruin probability), and $\\Phi(u) := 1 - \\Psi(u)$ (the survival probability).\n\nThe aforementioned paper \\cite{KP2016}, together with its complements \\cite{PZ} and \\cite{PerErr} (treating the case $\\beta=0$), deals with the asymptotic analysis of the ruin probability under the assumption of a riskless rate $r=0$. For the case where $\\kappa=1$ and $F$ is an exponential distribution, it was shown that if $\\beta := 2a/\\sigma^2 - 1 > 0$, then $\\Psi(u) \\sim C u^{-\\beta}$ as $u \\to \\infty$; if $\\beta \\le 0$, ruin is imminent. If $\\kappa \\in (0,1)$, the same conclusion holds with $\\beta_\\kappa := 2a/(\\kappa\\sigma^2) - 1$. The equation $\\beta_\\kappa = 0$ defines the threshold for the fraction of risky investment above which ruin is certain. The first step of the argument, which is technically difficult and delicate, is to prove that the survival probability is smooth under appropriate assumptions on $F$. The second, easier step is to prove that $\\Phi$ satisfies an integro-differential equation. In the specific case of an exponential jump distribution, this equation can be reduced--by differentiating and eliminating the integral term--to a linear ODE of third order. This ODE turns out to be of second order for the derivative of the survival probability. Asymptotic results for solutions of such equations are available in the literature. The property that the ruin probability cannot decrease faster than some power function, established by Kalashnikov and Norberg in \\cite{Kalash-Nor}, then yields the aforementioned result.\n\nImportantly, if the survival probability is smooth, i.e., if $\\Phi \\in C^2$, then straightforward arguments based on It\\^o's formula lead to the conclusion that $\\Phi$ satisfies, in the classical sense, the integro-differential equation (IDE)\n\\begin{multline}\\label{int-diff}\n\\frac{1}{2} \\kappa^2\\sigma^2 u^2 \\Phi''(u) + ((a-r)\\kappa + r) u \\Phi'(u) =\n\\\\\n= c \\Phi'(u) - \\lambda \\int_0^\\infty (\\Phi(u+y) - \\Phi(u)) \\,dF(y),\n\\end{multline}\nsee, e.g., \\cite{KP2016}. The same equation holds for the ruin probability.\n\nAvailable sufficient conditions on $F$ ensuring the smoothness of $\\Phi$ are rather restrictive: the property holds if $F$ has a density $f \\in C^2$ such that $f' \\in L^1(\\mathbb{R}_+)$; see \\cite{KPukh}. For this reason, some authors prefer to assume smoothness a priori. On the other hand, it is not difficult to show that $\\Phi$ is a viscosity solution of the above equation under much more general conditions, but proving uniqueness for boundary value problems in the viscosity sense is rather involved; see \\cite{Bel-Kab} and, for an equation arising in a model based on a generalized Ornstein--Uhlenbeck process, the recent study \\cite{AK2025}.\n\n\\smallskip\n\\textbf{Main result.} \nIn this note, we employ a completely different proof strategy. Our main result is the following theorem.", "full_context": "In the present note, we study the integro-differential equation arising in the asymptotic analysis of ruin probabilities for an annuity model more general than the one described in \\cite{KP2016}. Specifically, we suppose that the insurance company pays pensions to its customers at a constant rate $c>0$ and receives, as income, the remaining part of their pension fund when the contracts expire. The company instantaneously invests a fraction $\\kappa$ of its capital reserve in a risky asset whose price $S=(S_t)$ evolves as a geometric Brownian motion (gBm) with parameters $a$ and $\\sigma>0$. The remaining fraction $1-\\kappa$ of the capital reserve is invested in a non-risky asset (a bank account) with a rate of return $r$. Formally, the process $X=X^u$ describing the evolution of the capital reserve is given by the following stochastic differential equation with jumps:\n\\begin{equation}\\label{one}\nX_t = u + \\int_0^t \\kappa X_s (a\\,ds +\\sigma\\,dW_s) + \\int_0^t r(1-\\kappa)X_s\\,ds + P_t,\n\\end{equation}\nwhere $u \\ge 0$ is the initial capital, $W=(W_t)$ is a standard Wiener process and $P=(P_t)$ is the ``business'' process (or Cram\\'er--Lundberg process), independent of $W$. In the classical literature, $P$ is usually represented as\n\\[\nP_t = -ct + \\sum_{i=1}^{N_t} \\xi_i,\n\\]\nwhere $N=(N_t)$ is a Poisson process with intensity $\\lambda > 0$, and $(\\xi_i)_{i\\ge 1}$ is an i.i.d. sequence of strictly positive random variables, independent of $N$, with common distribution function $F$ such that $F(0)=0$. \nIn the literature on stochastic calculus, it is standard to represent the sum as a stochastic integral:\n\\[\n\\sum_{i=1}^{N_t} \\xi_i = \\int_0^t \\int_0^\\infty x\\,p(ds,dx) ,\n\\]\nwhere $p(ds,dx)$ is a Poisson random measure with mean measure $q(ds,dx) = \\lambda ds\\,F(dx)$.\n\nLet $\\tau^u := \\inf \\{t: X^u_t \\le 0\\}$ (the ruin time), $\\Psi(u) := {\\bf P}(\\tau^u < \\infty)$ (the ruin probability), and $\\Phi(u) := 1 - \\Psi(u)$ (the survival probability).\n\nThe aforementioned paper \\cite{KP2016}, together with its complements \\cite{PZ} and \\cite{PerErr} (treating the case $\\beta=0$), deals with the asymptotic analysis of the ruin probability under the assumption of a riskless rate $r=0$. For the case where $\\kappa=1$ and $F$ is an exponential distribution, it was shown that if $\\beta := 2a/\\sigma^2 - 1 > 0$, then $\\Psi(u) \\sim C u^{-\\beta}$ as $u \\to \\infty$; if $\\beta \\le 0$, ruin is imminent. If $\\kappa \\in (0,1)$, the same conclusion holds with $\\beta_\\kappa := 2a/(\\kappa\\sigma^2) - 1$. The equation $\\beta_\\kappa = 0$ defines the threshold for the fraction of risky investment above which ruin is certain. The first step of the argument, which is technically difficult and delicate, is to prove that the survival probability is smooth under appropriate assumptions on $F$. The second, easier step is to prove that $\\Phi$ satisfies an integro-differential equation. In the specific case of an exponential jump distribution, this equation can be reduced--by differentiating and eliminating the integral term--to a linear ODE of third order. This ODE turns out to be of second order for the derivative of the survival probability. Asymptotic results for solutions of such equations are available in the literature. The property that the ruin probability cannot decrease faster than some power function, established by Kalashnikov and Norberg in \\cite{Kalash-Nor}, then yields the aforementioned result.\n\nImportantly, if the survival probability is smooth, i.e., if $\\Phi \\in C^2$, then straightforward arguments based on It\\^o's formula lead to the conclusion that $\\Phi$ satisfies, in the classical sense, the integro-differential equation (IDE)\n\\begin{multline}\\label{int-diff}\n\\frac{1}{2} \\kappa^2\\sigma^2 u^2 \\Phi''(u) + ((a-r)\\kappa + r) u \\Phi'(u) =\n\\\\\n= c \\Phi'(u) - \\lambda \\int_0^\\infty (\\Phi(u+y) - \\Phi(u)) \\,dF(y),\n\\end{multline}\nsee, e.g., \\cite{KP2016}. The same equation holds for the ruin probability.\n\nAvailable sufficient conditions on $F$ ensuring the smoothness of $\\Phi$ are rather restrictive: the property holds if $F$ has a density $f \\in C^2$ such that $f' \\in L^1(\\mathbb{R}_+)$; see \\cite{KPukh}. For this reason, some authors prefer to assume smoothness a priori. On the other hand, it is not difficult to show that $\\Phi$ is a viscosity solution of the above equation under much more general conditions, but proving uniqueness for boundary value problems in the viscosity sense is rather involved; see \\cite{Bel-Kab} and, for an equation arising in a model based on a generalized Ornstein--Uhlenbeck process, the recent study \\cite{AK2025}.\n\n\\smallskip\n\\textbf{Main result.} \nIn this note, we employ a completely different proof strategy. Our main result is the following theorem.\n\nIn the present note, we study the integro-differential equation arising in the asymptotic analysis of ruin probabilities for an annuity model more general than the one described in \\cite{KP2016}. Specifically, we suppose that the insurance company pays pensions to its customers at a constant rate $c>0$ and receives, as income, the remaining part of their pension fund when the contracts expire. The company instantaneously invests a fraction $\\kappa$ of its capital reserve in a risky asset whose price $S=(S_t)$ evolves as a geometric Brownian motion (gBm) with parameters $a$ and $\\sigma>0$. The remaining fraction $1-\\kappa$ of the capital reserve is invested in a non-risky asset (a bank account) with a rate of return $r$. Formally, the process $X=X^u$ describing the evolution of the capital reserve is given by the following stochastic differential equation with jumps:\n\\begin{equation}\\label{one}\nX_t = u + \\int_0^t \\kappa X_s (a\\,ds +\\sigma\\,dW_s) + \\int_0^t r(1-\\kappa)X_s\\,ds + P_t,\n\\end{equation}\nwhere $u \\ge 0$ is the initial capital, $W=(W_t)$ is a standard Wiener process and $P=(P_t)$ is the ``business'' process (or Cram\\'er--Lundberg process), independent of $W$. In the classical literature, $P$ is usually represented as\n\\[\nP_t = -ct + \\sum_{i=1}^{N_t} \\xi_i,\n\\]\nwhere $N=(N_t)$ is a Poisson process with intensity $\\lambda > 0$, and $(\\xi_i)_{i\\ge 1}$ is an i.i.d. sequence of strictly positive random variables, independent of $N$, with common distribution function $F$ such that $F(0)=0$. \nIn the literature on stochastic calculus, it is standard to represent the sum as a stochastic integral:\n\\[\n\\sum_{i=1}^{N_t} \\xi_i = \\int_0^t \\int_0^\\infty x\\,p(ds,dx) ,\n\\]\nwhere $p(ds,dx)$ is a Poisson random measure with mean measure $q(ds,dx) = \\lambda ds\\,F(dx)$.\n\nThe aforementioned paper \\cite{KP2016}, together with its complements \\cite{PZ} and \\cite{PerErr} (treating the case $\\beta=0$), deals with the asymptotic analysis of the ruin probability under the assumption of a riskless rate $r=0$. For the case where $\\kappa=1$ and $F$ is an exponential distribution, it was shown that if $\\beta := 2a/\\sigma^2 - 1 > 0$, then $\\Psi(u) \\sim C u^{-\\beta}$ as $u \\to \\infty$; if $\\beta \\le 0$, ruin is imminent. If $\\kappa \\in (0,1)$, the same conclusion holds with $\\beta_\\kappa := 2a/(\\kappa\\sigma^2) - 1$. The equation $\\beta_\\kappa = 0$ defines the threshold for the fraction of risky investment above which ruin is certain. The first step of the argument, which is technically difficult and delicate, is to prove that the survival probability is smooth under appropriate assumptions on $F$. The second, easier step is to prove that $\\Phi$ satisfies an integro-differential equation. In the specific case of an exponential jump distribution, this equation can be reduced--by differentiating and eliminating the integral term--to a linear ODE of third order. This ODE turns out to be of second order for the derivative of the survival probability. Asymptotic results for solutions of such equations are available in the literature. The property that the ruin probability cannot decrease faster than some power function, established by Kalashnikov and Norberg in \\cite{Kalash-Nor}, then yields the aforementioned result.\n\nImportantly, if the survival probability is smooth, i.e., if $\\Phi \\in C^2$, then straightforward arguments based on It\\^o's formula lead to the conclusion that $\\Phi$ satisfies, in the classical sense, the integro-differential equation (IDE)\n\\begin{multline}\\label{int-diff}\n\\frac{1}{2} \\kappa^2\\sigma^2 u^2 \\Phi''(u) + ((a-r)\\kappa + r) u \\Phi'(u) =\n\\\\\n= c \\Phi'(u) - \\lambda \\int_0^\\infty (\\Phi(u+y) - \\Phi(u)) \\,dF(y),\n\\end{multline}\nsee, e.g., \\cite{KP2016}. The same equation holds for the ruin probability.\n\n\\smallskip\n\\textbf{Main result.} \nIn this note, we employ a completely different proof strategy. Our main result is the following theorem.\n\nOur result substantially relaxes the existing conditions on the jump size distribution. Our approach is based on reducing the integro-differential equation to a pair of integral equations for the derivative of its solution. These integral equations are coupled at a suitably chosen point $u_0 \\in (0,\\infty)$. For the first integral equation on $[u_0,\\infty)$, we establish uniqueness for a somewhat exotic terminal value problem. Specifically, we seek a solution with asymptotic behavior $u^{-\\gamma}$ as $u \\to \\infty$, which is consistent with earlier results such as those in \\cite{KP2016} (note that $\\gamma=\\beta+1$ in the case where $\\kappa=1$ and $r=0$). It is important to note that we do not rely on previously established asymptotic formulas in our proof; they serve only as motivation. For a sufficiently large $u_0$, we find the solution using the Banach fixed point theorem. The second equation for the derivative is a Volterra integral equation of the second kind on $(0, u_0]$ with a terminal condition at $u_0$ defined via the solution of the first equation. Gluing the solutions together yields a solution for the derivative of the desired function, which includes a multiplicative constant $C$ to be determined. To satisfy the zero boundary condition at zero, the additive constant $C_0$ arising from integration to obtain the solution $\\Phi$ of the original IDE must be taken as zero. The constant $C$ is then chosen to ensure that the limit at infinity equals one. Note also that $C$ is ``calculable'' in the sense that it is expressed as an integral involving the solutions of the aforementioned equations, which are themselves at least numerically computable. Finally, using It\\^o's formula, we easily conclude that $\\Phi$ coincides with the survival probability.\n\n\\section{Analysis of the integro-differential equation} \nRecall that we assume that $F$ is continuous at zero ($F(0)=0$) and $\\E \\xi_1 <\\infty$. \nUsing integration by parts for functions of bounded variation (or Fubini's theorem), we get:\n$$\n\\int_0^\\infty \\Phi(u+y) F(dy) = -\\int_0^\\infty \\Phi(u+y) d\\bar{F}(y) = \\Phi(u) + \\int_0^\\infty \\Phi'(u+y)\\bar F(y)dy,\n$$\nwhere $\\bar F:=1-F$. With this relation, the equation (\\ref{int-diff}) can be written in the equivalent form \n\\beq\n\\label{IDE-F}\n\\frac 12 \\kappa^2\\sigma^2u^2 \\Phi''(u)+ ((a-r)\\kappa +r) u\\Phi'(u) -c\\Phi' (u) \n=-\\lambda \\int_0^\\infty\\Phi'(u+y)\\bar F(y)dy,\n\\eeq\nsuitable for our purposes. We are interested in the classical solution of this equation, that is, in $C([0,\\infty])\\cap C^2((0,\\infty))$ with the boundary conditions at infinity and zero $\\Phi(\\infty)=1$ and $\\Phi (0)=0$.\n\nThe equation (\\ref{IDE-F}) is reduced to a homogeneous integro-differential equation of the first order with respect to the derivative $g(u):=\\Phi'(u)$. Introducing the notation \n$$\n\\gamma:=\\frac{2((a-r)\\kappa+r)}{\\kappa^2 \\sigma^2},\\quad \\alpha:=\\frac{2c}{\\kappa^2 \\sigma^2}, \\quad \\mu:=\\frac{2\\lambda}{\\kappa^2 \\sigma^2}, \n $$\nwe get the following IDE for $g$: \n\\begin{equation}\n\\label{IDE}\n u^2 g'(u) + (\\gamma u-\\alpha) g(u)= - Ag(u),\n\\end{equation}\nwhere\n\\beq\n\\label{Ag(u)}\n Ag(u):=\\mu \\int_0^\\infty g(u+z)\\,\\overline{F}(z)\\,dz.\n\\eeq\n\n\\section{The probability of non-ruin and the solution of IDE} \nDue to linearity of the equation (\\ref{IDE}) all its solutions with the asymptotic behavior at infinity $C u^{-\\gamma}$, $C>0$, are of the form $C\\hat g(u)$. The corresponding general solution of the equation (\\ref{int-diff}) has the form \n\\beq\n\\label{GC0C}\nG(u)=C_0+C\\int_0^u \\hat g(t)\\,dt. \n\\eeq\nOur aim is to choose the constants to ensure that $G=\\Phi$. The following lemma shows that the constant $C_0$ must be equal to zero.\n\\begin{lemma}\n$\\Phi(0+)=\\Phi(0):=0$.\n\\end{lemma}\n\\begin{proof} \nConsider the stochastic exponential $Z=(Z_t)_{t\\ge 0}$ given by the linear SDE \n$$\ndZ_t=Z_t \\left( ((a-r)\\kappa + r) \\,d t + \\sigma \\,dW_t \\right), \\qquad Z_0=1, \n$$\nand define the process $\\hat X^u$ \n$$\n\\hat X^u_t=Z_t\\Big( u-c\\int_0^t Z_s^{-1}ds\\Big).\n$$", "post_theorem_intro_text_len": 2148, "post_theorem_intro_text": "Our result substantially relaxes the existing conditions on the jump size distribution. Our approach is based on reducing the integro-differential equation to a pair of integral equations for the derivative of its solution. These integral equations are coupled at a suitably chosen point $u_0 \\in (0,\\infty)$. For the first integral equation on $[u_0,\\infty)$, we establish uniqueness for a somewhat exotic terminal value problem. Specifically, we seek a solution with asymptotic behavior $u^{-\\gamma}$ as $u \\to \\infty$, which is consistent with earlier results such as those in \\cite{KP2016} (note that $\\gamma=\\beta+1$ in the case where $\\kappa=1$ and $r=0$). It is important to note that we do not rely on previously established asymptotic formulas in our proof; they serve only as motivation. For a sufficiently large $u_0$, we find the solution using the Banach fixed point theorem. The second equation for the derivative is a Volterra integral equation of the second kind on $(0, u_0]$ with a terminal condition at $u_0$ defined via the solution of the first equation. Gluing the solutions together yields a solution for the derivative of the desired function, which includes a multiplicative constant $C$ to be determined. To satisfy the zero boundary condition at zero, the additive constant $C_0$ arising from integration to obtain the solution $\\Phi$ of the original IDE must be taken as zero. The constant $C$ is then chosen to ensure that the limit at infinity equals one. Note also that $C$ is ``calculable'' in the sense that it is expressed as an integral involving the solutions of the aforementioned equations, which are themselves at least numerically computable. Finally, using It\\^o's formula, we easily conclude that $\\Phi$ coincides with the survival probability.\n\nOur argument is remarkably simple. We avoid reducing the IDE to a higher-order ODE, which would introduce an additional constant, and instead rely only on standard results concerning the uniqueness of solutions to IDEs. Although conceptually inspired by the paper \\cite{Grandits2004} on the non-life insurance model, our proofs are technically quite different.", "sketch": "To prove Theorem~\\ref{main}, the approach is to reduce the integro-differential equation to “a pair of integral equations for the derivative of its solution,” coupled at a point $u_0\\in(0,\\infty)$. On $[u_0,\\infty)$, one treats “a somewhat exotic terminal value problem,” seeking a solution with asymptotic behavior $u^{-\\gamma}$ as $u\\to\\infty$; for sufficiently large $u_0$ this solution is obtained “using the Banach fixed point theorem,” and uniqueness is established. On $(0,u_0]$, the derivative satisfies “a Volterra integral equation of the second kind” with a terminal condition at $u_0$ coming from the first part. “Gluing the solutions together” gives the derivative (with a multiplicative constant $C$ to determine), then integrating yields $\\Phi$ with an additive constant $C_0$, which “must be taken as zero” to enforce $\\Phi(0)=0$. The constant $C$ is chosen so that “the limit at infinity equals one,” and $C$ is “calculable” via an integral involving the constructed solutions. Finally, “using It\\^o's formula,” one concludes that $\\Phi$ coincides with the survival probability. The method “avoid[s] reducing the IDE to a higher-order ODE” and instead uses “standard results concerning the uniqueness of solutions to IDEs.”", "expanded_sketch": "To prove the main theorem, the approach is to reduce the integro-differential equation to “a pair of integral equations for the derivative of its solution,” coupled at a point $u_0\\in(0,\\infty)$. On $[u_0,\\infty)$, one treats “a somewhat exotic terminal value problem,” seeking a solution with asymptotic behavior $u^{-\\gamma}$ as $u\\to\\infty$; for sufficiently large $u_0$ this solution is obtained “using the Banach fixed point theorem,” and uniqueness is established. On $(0,u_0]$, the derivative satisfies “a Volterra integral equation of the second kind” with a terminal condition at $u_0$ coming from the first part. “Gluing the solutions together” gives the derivative (with a multiplicative constant $C$ to determine), then integrating yields $\\Phi$ with an additive constant $C_0$, which “must be taken as zero” to enforce $\\Phi(0)=0$. The constant $C$ is chosen so that “the limit at infinity equals one,” and $C$ is “calculable” via an integral involving the constructed solutions. Finally, “using It\\^o's formula,” one concludes that $\\Phi$ coincides with the survival probability. The method “avoid[s] reducing the IDE to a higher-order ODE” and instead uses “standard results concerning the uniqueness of solutions to IDEs.”", "expanded_theorem": "\\label{main}\nSuppose that $F(0)=0$, ${\\bf E}[\\xi] < \\infty$ and $$\\gamma := \\dfrac{(a-r)\\kappa + r}{\\kappa^2 \\sigma^2} > 1.$$\nThen the survival probability $\\Phi \\in C([0,\\infty)) \\cap C^2((0,\\infty))$ satisfies the integro-differential equation\n\\begin{multline}\\label{int-diff}\n\\frac{1}{2} \\kappa^2\\sigma^2 u^2 \\Phi''(u) + ((a-r)\\kappa + r) u \\Phi'(u) =\n\\\\\n= c \\Phi'(u) - \\lambda \\int_0^\\infty (\\Phi(u+y) - \\Phi(u)) \\,dF(y),\n\\end{multline}\nwith boundary conditions $\\Phi(0)=0$, $\\Phi(\\infty)=1$. Moreover, there exists a finite constant $C > 0$ such that $\\Psi(u) \\sim C u^{-\\gamma+1}$ as $u \\to \\infty$.", "theorem_type": ["Existential–Universal", "Asymptotic or Limit"], "mcq": {"question": "Let \\(X^u=(X_t^u)_{t\\ge 0}\\) be the capital reserve process defined by\n\\[\nX_t^u = u + \\int_0^t \\kappa X_s^u(a\\,ds + \\sigma\\,dW_s) + \\int_0^t r(1-\\kappa)X_s^u\\,ds + P_t,\n\\]\nwhere \\(\\sigma>0\\), \\(W\\) is a standard Wiener process, and\n\\[\nP_t=-ct+\\sum_{i=1}^{N_t}\\xi_i,\n\\]\nwith \\(N\\) a Poisson process of intensity \\(\\lambda>0\\) and \\((\\xi_i)\\) i.i.d. positive random variables with distribution function \\(F\\). Define the ruin time\n\\[\n\\tau^u:=\\inf\\{t:X_t^u\\le 0\\},\n\\]\nthe ruin probability \\(\\Psi(u):={\\bf P}(\\tau^u<\\infty)\\), and the survival probability \\(\\Phi(u):=1-\\Psi(u)\\). Assume that \\(F(0)=0\\), \\({\\bf E}[\\xi]<\\infty\\), and\n\\[\n\\gamma:=\\frac{(a-r)\\kappa+r}{\\kappa^2\\sigma^2}>1.\n\\]\nUnder these assumptions, which asymptotic and regularity statement holds for \\(\\Phi\\) and \\(\\Psi\\)?", "correct_choice": {"label": "A", "text": "The survival probability satisfies \\(\\Phi\\in C([0,\\infty))\\cap C^2((0,\\infty))\\) and solves\n\\[\n\\frac12\\kappa^2\\sigma^2 u^2\\Phi''(u)+((a-r)\\kappa+r)u\\Phi'(u)\n= c\\Phi'(u)-\\lambda\\int_0^\\infty (\\Phi(u+y)-\\Phi(u))\\,dF(y),\n\\]\nwith boundary conditions \\(\\Phi(0)=0\\) and \\(\\Phi(\\infty)=1\\) (equivalently, \\(\\lim_{u\\to\\infty}\\Phi(u)=1\\)). Moreover, there exists a finite constant \\(C>0\\) such that\n\\[\n\\Psi(u)\\sim C u^{-\\gamma+1}\\qquad\\text{as }u\\to\\infty.\n\\]"}, "choices": [{"label": "B", "text": "The survival probability satisfies \\(\\Phi\\in C([0,\\infty))\\cap C^2((0,\\infty))\\) and solves\n\\[\n\\frac12\\kappa^2\\sigma^2 u^2\\Phi''(u)+((a-r)\\kappa+r)u\\Phi'(u)\n= c\\Phi'(u)-\\lambda\\int_0^\\infty (\\Phi(u+y)-\\Phi(u))\\,dF(y),\n\\]\nwith boundary conditions \\(\\Phi(0)=0\\) and \\(\\Phi(\\infty)=1\\). Moreover, there exists a finite constant \\(C>0\\) such that\n\\[\n\\Psi(u)\\sim C u^{-\\gamma}\\qquad\\text{as }u\\to\\infty.\n\\]"}, {"label": "C", "text": "There exists a finite constant \\(C>0\\) such that\n\\[\n\\Psi(u)=O\\big(u^{-\\gamma+1}\\big)\\qquad\\text{as }u\\to\\infty.\n\\]"}, {"label": "D", "text": "The survival probability satisfies \\(\\Phi\\in C([0,\\infty))\\cap C^2((0,\\infty))\\) and solves\n\\[\n\\frac12\\kappa^2\\sigma^2 u^2\\Phi''(u)+((a-r)\\kappa+r)u\\Phi'(u)\n= c\\Phi'(u)-\\lambda\\int_0^\\infty (\\Phi(u+y)-\\Phi(u))\\,dF(y),\n\\]\nwith boundary conditions \\(\\Phi(0)=0\\) and \\(\\Phi(\\infty)=1\\). Moreover, for every sufficiently large \\(u_0>0\\) there exists a finite constant \\(C(u_0)>0\\) such that\n\\[\n\\Psi(u)\\sim C(u_0)\\,u^{-\\gamma+1}\\qquad\\text{as }u\\to\\infty.\n\\]"}, {"label": "E", "text": "The survival probability satisfies \\(\\Phi\\in C([0,\\infty))\\cap C^1((0,\\infty))\\) and solves the above integro-differential equation in the viscosity sense with boundary conditions \\(\\Phi(0)=0\\) and \\(\\Phi(\\infty)=1\\). Moreover, there exists a finite constant \\(C>0\\) such that\n\\[\n\\Psi(u)\\sim C u^{-\\gamma+1}\\qquad\\text{as }u\\to\\infty.\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": "asymptotic exponent from integrating derivative behaving like u^{-\\gamma}", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped precise asymptotic equivalence and regularity/IDE conclusion, keeping only polynomial upper asymptotic order", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "global constant determined by normalization at infinity replaced by dependence on gluing point u_0", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "classical C^2 solution identification via gluing and Ito replaced by weaker viscosity/C^1 statement", "template_used": "wildcard"}]}} +{"id": "2601.02655v2", "paper_link": "http://arxiv.org/abs/2601.02655v2", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\n There exists a sequence of compact hyperbolic $3$-manifolds $M_n$ with totally geodesic boundary such that the boundary cone-offs $\\coneoff M_n$ satisfy the following:\n\\begin{enumerate}\n \\item \\label{item:negative} $\\coneoff M_n$ is a closed negatively curved $3$-pseudomanifold.\n \\item \\label{item:T} $\\pi_1(\\coneoff M_n)$ contains an infinite quasiconvex subgroup with property (T).\n \\item \\label{item:char} The genus of the components of $\\partial M_n$ and the Euler characteristic of $\\coneoff M_n$ both go to infinity with $n$.\n\\end{enumerate}", "start_pos": 6973, "end_pos": 7580, "label": "thm:main"}, "ref_dict": {"cor:main": "\\begin{corollary}\\label{cor:main}\n In the same setting as Theorem~\\ref{thm:main}, we have the following.\n\\begin{enumerate}\n \\item \\label{item:notcubulable} $\\pi_1(\\coneoff M_n)$ does not act properly by cubical isometries on a $\\CAT 0$ cube complex. In particular, it is not cubulable.\n\n \\item \\label{item:notHaagerup} $\\pi_1(\\coneoff M_n)$ does not have the Haagerup property. In particular, it does not act properly by isometries on a real or complex hyperbolic space.\n\n \\item \\label{item:notRFRS} $\\pi_1(\\coneoff M_n)$ is not virtually RFRS.\n\\end{enumerate}\n\\end{corollary}", "prop:agol": "\\begin{proposition}\\label{prop:agol}\n Let $(\\Gamma,\\mathcal P)<(\\piorb B,\\{\\piorb{\\partial B}\\})$ be finite index.\n Let $X$ be a $\\CAT 0$ cube complex, and let $(\\Gamma,\\mathcal P) \\acts X$ be a relatively geometric action.\n Then there exist $P\\in \\mathcal P$ and a hyperplane stabilizer $H$ such that $H\\cap P$ is infinite.\n \\end{proposition}", "item:notcubulable": "\\begin{enumerate}\n \\item \\label{item:notcubulable} $\\pi_1(\\coneoff M_n)$ does not act properly by cubical isometries on a $\\CAT 0$ cube complex. In particular, it is not cubulable.\n\n \\item \\label{item:notHaagerup} $\\pi_1(\\coneoff M_n)$ does not have the Haagerup property. In particular, it does not act properly by isometries on a real or complex hyperbolic space.\n\n \\item \\label{item:notRFRS} $\\pi_1(\\coneoff M_n)$ is not virtually RFRS.\n\\end{enumerate}", "prop:split": "\\begin{proposition}\\label{prop:split}\nThe $3$-pseudomanifolds $\\coneoff M_n$ constructed in Theorem~\\ref{thm:main} contain embedded locally convex and separating surfaces.\nIn particular, for large enough $n$, the fundamental group $\\pi_1(\\coneoff M_n)$\nsplits.\n\\end{proposition}", "q:cannon": "\\begin{question}\\label{q:cannon}\n Is every word hyperbolic group with $2$-sphere boundary cubulable?\n\\end{question}", "thm:main": "\\begin{theorem}\\label{thm:main}\n There exists a sequence of compact hyperbolic $3$-manifolds $M_n$ with totally geodesic boundary such that the boundary cone-offs $\\coneoff M_n$ satisfy the following:\n\\begin{enumerate}\n \\item \\label{item:negative} $\\coneoff M_n$ is a closed negatively curved $3$-pseudomanifold.\n \\item \\label{item:T} $\\pi_1(\\coneoff M_n)$ contains an infinite quasiconvex subgroup with property (T).\n \\item \\label{item:char} The genus of the components of $\\partial M_n$ and the Euler characteristic of $\\coneoff M_n$ both go to infinity with $n$.\n\\end{enumerate}\n\\end{theorem}", "q:pcannon": "\\begin{question}\\label{q:pcannon}\n Is every word hyperbolic group with Pontryagin sphere boundary cubulable?\n\\end{question}"}, "pre_theorem_intro_text_len": 3174, "pre_theorem_intro_text": "One of the most important open problems in geometric group theory is the Cannon Conjecture, which asserts that any word hyperbolic group with $2$-sphere boundary is virtually cocompact Kleinian. By work of Bergeron--Wise in one direction, and Markovi\\'c and Ha\\\"issinsky in the other, this conjecture is equivalent to a positive answer to the following.\n\\begin{question}\\label{q:cannon}\n Is every word hyperbolic group with $2$-sphere boundary cubulable?\n\\end{question}\nTo say a group is \\emph{cubulable} is to say it admits a proper and cocompact action by isometries on some $\\textrm{CAT}(0)$ cube complex. \nEvery cocompact Kleinian group is cubulable by \\cite[Theorem 1.5]{BW12}. And a cubulable word hyperbolic group with $2$-sphere boundary is virtually cocompact Kleinian by~\\cite[Theorem 1.10]{Ha15} (generalizing~\\cite{Mar13}).\n\nWe do not directly speak to Question~\\ref{q:cannon}, but we study the corresponding question for groups with \\emph{Pontryagin sphere} boundary. The Pontryagin sphere is a certain inverse limit of surfaces which appears as the boundary at infinity of many $\\textrm{CAT}(0)$ $3$-dimensional complexes (universal covers of $3$-\\emph{pseudomanifolds}, see below). \nIt is like a sphere in that it is a connected $2$-dimensional compactum whose $2$-dimensional \\v{C}ech cohomology is cyclic. However its $1$-dimensional cohomology is infinitely generated. We refer the reader to \\cite{JA91,DR99,FI03,SW20}.\n\\begin{question}\\label{q:pcannon}\n Is every word hyperbolic group with Pontryagin sphere boundary cubulable?\n\\end{question}\nClassical examples of word hyperbolic groups with Pontryagin sphere boundary occur for instance as finite index subgroups of a right-angled Coxeter group (RACG) whose nerve is a flag-no-square triangulation of a connected closed orientable surface, and via Charney--Davis strict hyperbolization \\cite{DA08,CD95}. \nBoth sources of examples yield groups that are cubulable (see \\cite{DA08,LR24}). \nNonetheless, we answer Question~\\ref{q:pcannon} in the negative; see \\eqref{item:notcubulable} in Corollary~\\ref{cor:main}.\n\nOur examples, as well as the aforementioned classical examples, are fundamental groups of negatively curved pseudomanifolds.\nA \\textit{$3$-pseudomanifold} is a $3$-dimensional polyhedral complex $P$ in which the links of vertices are connected closed orientable surfaces of arbitrary genus.\nThese arise by taking a compact $3$-manifold $M$ with orientable boundary and coning off the boundary components (one cone point for each component) to obtain a compact space $\\coneoff M$; see \\S\\ref{sec:coneoff} for details.\n\nA \\emph{negatively curved} $3$-pseudomanifold is one which admits a complete locally $\\textrm{CAT}(-1)$ metric. \nSuch a pseudomanifold is aspherical. If it is also compact, then the fundamental group is word hyperbolic. Our definition does not permit pseudomanifolds with boundary, so for us a pseudomanifold will be \\emph{closed} if and only if it is compact. A closed pseudomanifold has a fundamental class over $\\mathbb Z/2\\mathbb Z$, and so the fundamental group of a closed aspherical $3$-pseudomanifold is $3$-dimensional.\n\nOur main result is the following.", "context": "One of the most important open problems in geometric group theory is the Cannon Conjecture, which asserts that any word hyperbolic group with $2$-sphere boundary is virtually cocompact Kleinian. By work of Bergeron--Wise in one direction, and Markovi\\'c and Ha\\\"issinsky in the other, this conjecture is equivalent to a positive answer to the following.\n\\begin{question}\\label{q:cannon}\n Is every word hyperbolic group with $2$-sphere boundary cubulable?\n\\end{question}\nTo say a group is \\emph{cubulable} is to say it admits a proper and cocompact action by isometries on some $\\textrm{CAT}(0)$ cube complex. \nEvery cocompact Kleinian group is cubulable by \\cite[Theorem 1.5]{BW12}. And a cubulable word hyperbolic group with $2$-sphere boundary is virtually cocompact Kleinian by~\\cite[Theorem 1.10]{Ha15} (generalizing~\\cite{Mar13}).\n\nWe do not directly speak to Question~\\ref{q:cannon}, but we study the corresponding question for groups with \\emph{Pontryagin sphere} boundary. The Pontryagin sphere is a certain inverse limit of surfaces which appears as the boundary at infinity of many $\\textrm{CAT}(0)$ $3$-dimensional complexes (universal covers of $3$-\\emph{pseudomanifolds}, see below). \nIt is like a sphere in that it is a connected $2$-dimensional compactum whose $2$-dimensional \\v{C}ech cohomology is cyclic. However its $1$-dimensional cohomology is infinitely generated. We refer the reader to \\cite{JA91,DR99,FI03,SW20}.\n\\begin{question}\\label{q:pcannon}\n Is every word hyperbolic group with Pontryagin sphere boundary cubulable?\n\\end{question}\nClassical examples of word hyperbolic groups with Pontryagin sphere boundary occur for instance as finite index subgroups of a right-angled Coxeter group (RACG) whose nerve is a flag-no-square triangulation of a connected closed orientable surface, and via Charney--Davis strict hyperbolization \\cite{DA08,CD95}. \nBoth sources of examples yield groups that are cubulable (see \\cite{DA08,LR24}). \nNonetheless, we answer Question~\\ref{q:pcannon} in the negative; see \\eqref{item:notcubulable} in Corollary~\\ref{cor:main}.\n\nOur examples, as well as the aforementioned classical examples, are fundamental groups of negatively curved pseudomanifolds.\nA \\textit{$3$-pseudomanifold} is a $3$-dimensional polyhedral complex $P$ in which the links of vertices are connected closed orientable surfaces of arbitrary genus.\nThese arise by taking a compact $3$-manifold $M$ with orientable boundary and coning off the boundary components (one cone point for each component) to obtain a compact space $\\coneoff M$; see \\S\\ref{sec:coneoff} for details.\n\nA \\emph{negatively curved} $3$-pseudomanifold is one which admits a complete locally $\\textrm{CAT}(-1)$ metric. \nSuch a pseudomanifold is aspherical. If it is also compact, then the fundamental group is word hyperbolic. Our definition does not permit pseudomanifolds with boundary, so for us a pseudomanifold will be \\emph{closed} if and only if it is compact. A closed pseudomanifold has a fundamental class over $\\mathbb Z/2\\mathbb Z$, and so the fundamental group of a closed aspherical $3$-pseudomanifold is $3$-dimensional.\n\nOur main result is the following.\n\n\\begin{corollary}\\label{cor:main}\n In the same setting as Theorem~\\ref{thm:main}, we have the following.\n\\begin{enumerate}\n \\item \\label{item:notcubulable} $\\pi_1(\\coneoff M_n)$ does not act properly by cubical isometries on a $\\CAT 0$ cube complex. In particular, it is not cubulable.\n\n \\item \\label{item:notHaagerup} $\\pi_1(\\coneoff M_n)$ does not have the Haagerup property. In particular, it does not act properly by isometries on a real or complex hyperbolic space.\n\n \\item \\label{item:notRFRS} $\\pi_1(\\coneoff M_n)$ is not virtually RFRS.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{enumerate}\n \\item \\label{item:notcubulable} $\\pi_1(\\coneoff M_n)$ does not act properly by cubical isometries on a $\\CAT 0$ cube complex. In particular, it is not cubulable.\n\n \\item \\label{item:notHaagerup} $\\pi_1(\\coneoff M_n)$ does not have the Haagerup property. In particular, it does not act properly by isometries on a real or complex hyperbolic space.\n\n \\item \\label{item:notRFRS} $\\pi_1(\\coneoff M_n)$ is not virtually RFRS.\n\\end{enumerate}\n\n\\begin{question}\\label{q:cannon}\n Is every word hyperbolic group with $2$-sphere boundary cubulable?\n\\end{question}\n\n\\begin{question}\\label{q:pcannon}\n Is every word hyperbolic group with Pontryagin sphere boundary cubulable?\n\\end{question}", "full_context": "One of the most important open problems in geometric group theory is the Cannon Conjecture, which asserts that any word hyperbolic group with $2$-sphere boundary is virtually cocompact Kleinian. By work of Bergeron--Wise in one direction, and Markovi\\'c and Ha\\\"issinsky in the other, this conjecture is equivalent to a positive answer to the following.\n\\begin{question}\\label{q:cannon}\n Is every word hyperbolic group with $2$-sphere boundary cubulable?\n\\end{question}\nTo say a group is \\emph{cubulable} is to say it admits a proper and cocompact action by isometries on some $\\textrm{CAT}(0)$ cube complex. \nEvery cocompact Kleinian group is cubulable by \\cite[Theorem 1.5]{BW12}. And a cubulable word hyperbolic group with $2$-sphere boundary is virtually cocompact Kleinian by~\\cite[Theorem 1.10]{Ha15} (generalizing~\\cite{Mar13}).\n\nWe do not directly speak to Question~\\ref{q:cannon}, but we study the corresponding question for groups with \\emph{Pontryagin sphere} boundary. The Pontryagin sphere is a certain inverse limit of surfaces which appears as the boundary at infinity of many $\\textrm{CAT}(0)$ $3$-dimensional complexes (universal covers of $3$-\\emph{pseudomanifolds}, see below). \nIt is like a sphere in that it is a connected $2$-dimensional compactum whose $2$-dimensional \\v{C}ech cohomology is cyclic. However its $1$-dimensional cohomology is infinitely generated. We refer the reader to \\cite{JA91,DR99,FI03,SW20}.\n\\begin{question}\\label{q:pcannon}\n Is every word hyperbolic group with Pontryagin sphere boundary cubulable?\n\\end{question}\nClassical examples of word hyperbolic groups with Pontryagin sphere boundary occur for instance as finite index subgroups of a right-angled Coxeter group (RACG) whose nerve is a flag-no-square triangulation of a connected closed orientable surface, and via Charney--Davis strict hyperbolization \\cite{DA08,CD95}. \nBoth sources of examples yield groups that are cubulable (see \\cite{DA08,LR24}). \nNonetheless, we answer Question~\\ref{q:pcannon} in the negative; see \\eqref{item:notcubulable} in Corollary~\\ref{cor:main}.\n\nOur examples, as well as the aforementioned classical examples, are fundamental groups of negatively curved pseudomanifolds.\nA \\textit{$3$-pseudomanifold} is a $3$-dimensional polyhedral complex $P$ in which the links of vertices are connected closed orientable surfaces of arbitrary genus.\nThese arise by taking a compact $3$-manifold $M$ with orientable boundary and coning off the boundary components (one cone point for each component) to obtain a compact space $\\coneoff M$; see \\S\\ref{sec:coneoff} for details.\n\nA \\emph{negatively curved} $3$-pseudomanifold is one which admits a complete locally $\\textrm{CAT}(-1)$ metric. \nSuch a pseudomanifold is aspherical. If it is also compact, then the fundamental group is word hyperbolic. Our definition does not permit pseudomanifolds with boundary, so for us a pseudomanifold will be \\emph{closed} if and only if it is compact. A closed pseudomanifold has a fundamental class over $\\mathbb Z/2\\mathbb Z$, and so the fundamental group of a closed aspherical $3$-pseudomanifold is $3$-dimensional.\n\nOur main result is the following.\n\n\\begin{corollary}\\label{cor:main}\n In the same setting as Theorem~\\ref{thm:main}, we have the following.\n\\begin{enumerate}\n \\item \\label{item:notcubulable} $\\pi_1(\\coneoff M_n)$ does not act properly by cubical isometries on a $\\CAT 0$ cube complex. In particular, it is not cubulable.\n\n \\item \\label{item:notHaagerup} $\\pi_1(\\coneoff M_n)$ does not have the Haagerup property. In particular, it does not act properly by isometries on a real or complex hyperbolic space.\n\n \\item \\label{item:notRFRS} $\\pi_1(\\coneoff M_n)$ is not virtually RFRS.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{enumerate}\n \\item \\label{item:notcubulable} $\\pi_1(\\coneoff M_n)$ does not act properly by cubical isometries on a $\\CAT 0$ cube complex. In particular, it is not cubulable.\n\n \\item \\label{item:notHaagerup} $\\pi_1(\\coneoff M_n)$ does not have the Haagerup property. In particular, it does not act properly by isometries on a real or complex hyperbolic space.\n\n \\item \\label{item:notRFRS} $\\pi_1(\\coneoff M_n)$ is not virtually RFRS.\n\\end{enumerate}\n\n\\begin{question}\\label{q:cannon}\n Is every word hyperbolic group with $2$-sphere boundary cubulable?\n\\end{question}\n\n\\begin{question}\\label{q:pcannon}\n Is every word hyperbolic group with Pontryagin sphere boundary cubulable?\n\\end{question}\n\n\\begin{abstract}\nWe construct compact hyperbolic $3$-manifolds with totally geodesic boundary, such that the closed $3$-pseudomanifolds obtained by coning off the boundary components are negatively curved and contain locally convex subspaces whose fundamental groups have property (T).\nIn particular, the fundamental groups of these $3$-pseudomanifolds are word hyperbolic but not cubulable. We deduce that in any relative cubulation of one of these hyperbolic $3$-manifold groups some hyperplane stabilizer has infinite intersection with the fundamental group of some boundary component.\n\\end{abstract}\n\nWe do not directly speak to Question~\\ref{q:cannon}, but we study the corresponding question for groups with \\emph{Pontryagin sphere} boundary. The Pontryagin sphere is a certain inverse limit of surfaces which appears as the boundary at infinity of many $\\CAT{0}$ $3$-dimensional complexes (universal covers of $3$-\\emph{pseudomanifolds}, see below). \nIt is like a sphere in that it is a connected $2$-dimensional compactum whose $2$-dimensional \\v{C}ech cohomology is cyclic. However its $1$-dimensional cohomology is infinitely generated. We refer the reader to \\cite{JA91,DR99,FI03,SW20}.\n\\begin{question}\\label{q:pcannon}\n Is every word hyperbolic group with Pontryagin sphere boundary cubulable?\n\\end{question}\nClassical examples of word hyperbolic groups with Pontryagin sphere boundary occur for instance as finite index subgroups of a right-angled Coxeter group (RACG) whose nerve is a flag-no-square triangulation of a connected closed orientable surface, and via Charney--Davis strict hyperbolization \\cite{DA08,CD95}. \nBoth sources of examples yield groups that are cubulable (see \\cite{DA08,LR24}). \nNonetheless, we answer Question~\\ref{q:pcannon} in the negative; see \\eqref{item:notcubulable} in Corollary~\\ref{cor:main}.\n\nOur main result is the following.\n\nThe existence of an infinite subgroup with property (T) is a well-known obstruction to the existence of nice geometric actions, and makes these groups quite different from $3$-manifold groups.\nMore precisely, we have the following.\n(See \\cite{BHV08} for the definitions of the Haagerup and (T) properties, and \\cite{AG08} for the definition of RFRS.)\n\n\\begin{remark}[Gromov boundary]\\label{rmk:boundary}\n The Gromov boundary of the groups $\\pi_1(\\coneoff M_n)$ in Theorem~\\ref{thm:main} is the tree of manifolds defined by closed orientable surfaces of positive genus, i.e., \n a Pontryagin sphere, see \\cite[Theorem A.1]{KM} and \\cite{SW20}.\n Moreover, the Gromov boundary of the infinite quasiconvex subgroup with property (T) has limit set a Menger curve, see \\cite{KK00}.\n\\end{remark}\n\n\\subsection{Coning off}\\label{sec:coneoff}\nHere we state the results from~\\cite{KM} which allow us to put negatively curved (i.e., locally $\\CAT{k}$ for some $k<0$) metrics on our $3$-pseu\\-do\\-man\\-i\\-folds. \nLet $M$ be a compact $3$-manifold with boundary. \nThe \\textit{cone-off} is the space\n\\[ \\coneoff{M} = M \\sqcup (\\partial M\\times [0,1])/\\sim \\]\nwhere $\\sim$ is the equivalence relation generated by\n\\begin{itemize}\n\\item $x\\sim (x, 1)$ if $x\\in \\partial M$; and\n\\item $(x, 0)\\sim (y,0)$ if $x$ and $y$ lie in the same component of $\\partial M$.\n\\end{itemize}\nThe resulting space $\\coneoff M$ is a closed $3$-pseudomanifold (since $3$-manifolds can be triangulated).\nIf $Z\\subset M$, then we define the \\textit{induced cone-off} $\\coneoff{Z}$ to be the subset which is the image of\n\\[ Z\\sqcup (\\partial M\\cap Z) \\times [0,1] \\]\nin the quotient space $\\coneoff{M}$.\n\nThe first result is about when the boundary cone-off $\\coneoff{M}$ of a compact hyperbolic $3$-manifold $M$ can be given a negatively curved metric.\nFor $A\\subset M$, we denote by $N_b(A)$ the open neighborhood of radius $b$ around $A$.\n\\begin{theorem} \\label{thm:KM1} \\cite[Theorem A]{KM}\n Let $M$ be a compact hyperbolic manifold with totally geodesic boundary, let $b$ be a positive number less than $\\bw{M}{\\partial M}$ and let $c>\\pi/\\sinh(b)$.\n Suppose \\[\\injrad(\\partial M) > c.\\]\n Then there is a negatively curved metric $\\hat{d}$ on $\\coneoff{M}$ and an embedding of $M\\smallsetminus N_b(\\partial M)$ into $(\\widehat{M},\\hat{d})$ which is a local isometry with image equal $M \\subset \\coneoff M$.\n\\end{theorem}\n\n\\begin{proposition}\\label{prop:manifold}\n There exists a finite-index torsion-free normal subgroup $K\\lhd W$ such that the orbifold cover $p_M: M\\to H$ corresponding to $\\Gamma = f^{-1}(K)$ satisfies the following.\n \\begin{enumerate}\n \\item \\label{item:manifold} $M$ has empty orbifold locus (in particular $M$ is a compact hyperbolic $3$-manifold, whose totally geodesic boundary is tiled by copies of the components of $\\partial H$).\n\n\\begin{corollary}\\label{cor:main}\n In the same setting as Theorem~\\ref{thm:main}, we have the following.\n\\begin{enumerate}\n \\item \\label{item:notcubulable} $\\pi_1(\\coneoff M_n)$ does not act properly by cubical isometries on a $\\CAT 0$ cube complex. In particular, it is not cubulable.\n\n \\item \\label{item:notHaagerup} $\\pi_1(\\coneoff M_n)$ does not have the Haagerup property. In particular, it does not act properly by isometries on a real or complex hyperbolic space.\n\n \\item \\label{item:notRFRS} $\\pi_1(\\coneoff M_n)$ is not virtually RFRS.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{enumerate}\n \\item \\label{item:notcubulable} $\\pi_1(\\coneoff M_n)$ does not act properly by cubical isometries on a $\\CAT 0$ cube complex. In particular, it is not cubulable.\n\n \\item \\label{item:notHaagerup} $\\pi_1(\\coneoff M_n)$ does not have the Haagerup property. In particular, it does not act properly by isometries on a real or complex hyperbolic space.\n\n \\item \\label{item:notRFRS} $\\pi_1(\\coneoff M_n)$ is not virtually RFRS.\n\\end{enumerate}\n\n\\begin{question}\\label{q:cannon}\n Is every word hyperbolic group with $2$-sphere boundary cubulable?\n\\end{question}\n\n\\begin{question}\\label{q:pcannon}\n Is every word hyperbolic group with Pontryagin sphere boundary cubulable?\n\\end{question}\n\n\\begin{theorem}\\label{thm:main}\n There exists a sequence of compact hyperbolic $3$-manifolds $M_n$ with totally geodesic boundary such that the boundary cone-offs $\\coneoff M_n$ satisfy the following:\n\\begin{enumerate}\n \\item \\label{item:negative} $\\coneoff M_n$ is a closed negatively curved $3$-pseudomanifold.\n \\item \\label{item:T} $\\pi_1(\\coneoff M_n)$ contains an infinite quasiconvex subgroup with property (T).\n \\item \\label{item:char} The genus of the components of $\\partial M_n$ and the Euler characteristic of $\\coneoff M_n$ both go to infinity with $n$.\n\\end{enumerate}\n\\end{theorem}", "post_theorem_intro_text_len": 6039, "post_theorem_intro_text": "The existence of an infinite subgroup with property (T) is a well-known obstruction to the existence of nice geometric actions, and makes these groups quite different from $3$-manifold groups.\nMore precisely, we have the following.\n(See \\cite{BHV08} for the definitions of the Haagerup and (T) properties, and \\cite{AG08} for the definition of RFRS.)\n\n\\begin{corollary}\\label{cor:main}\n In the same setting as Theorem~\\ref{thm:main}, we have the following.\n\\begin{enumerate}\n \\item \\label{item:notcubulable} $\\pi_1(\\coneoff M_n)$ does not act properly by cubical isometries on a $\\CAT 0$ cube complex. In particular, it is not cubulable.\n\n \\item \\label{item:notHaagerup} $\\pi_1(\\coneoff M_n)$ does not have the Haagerup property. In particular, it does not act properly by isometries on a real or complex hyperbolic space.\n\n \\item \\label{item:notRFRS} $\\pi_1(\\coneoff M_n)$ is not virtually RFRS.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{remark}[Virtual cubulability]\nThe non-cubulability of $\\pi_1(\\coneoff M_n)$ does not automatically imply the non-cubulability of fundamental groups of $3$-pseudomanifolds obtained by coning off the boundary components of an arbitrary finite-sheeted cover of $M_n$. Indeed, one may use Wise's Malnormal Special Quotient Theorem \\cite[Theorem 12.2]{Wise21} together with the cubulability of $\\pi_1(M_n)$ \\cite[Theorem 17.14]{Wise21} to see that each $M_n$ has some finite sheeted cover $M_n'$ so that $\\pi_1(\\coneoff M_n')$ \\emph{is} cubulable. \n\\end{remark}\n\n\\begin{remark}[Gromov boundary]\\label{rmk:boundary}\n The Gromov boundary of the groups $\\pi_1(\\coneoff M_n)$ in Theorem~\\ref{thm:main} is the tree of manifolds defined by closed orientable surfaces of positive genus, i.e., \n a Pontryagin sphere, see \\cite[Theorem A.1]{KM} and \\cite{SW20}.\n Moreover, the Gromov boundary of the infinite quasiconvex subgroup with property (T) has limit set a Menger curve, see \\cite{KK00}.\n\\end{remark}\n\n\\begin{remark}[Property (T) vs Haagerup Property]\\label{rmk:T}\nThe groups $\\pi_1(\\coneoff M_n)$ in Theorem~\\ref{thm:main} \nsplit over quasiconvex surface subgroups (see Proposition~\\ref{prop:split}), so they do not have property (T).\nIn particular, they are $3$-dimensional word hyperbolic groups without (T) and without Haagerup, whose boundary is connected and has cyclic top-dimensional \\v Cech cohomology (since the boundary is a Pontryagin sphere, see \\cite[Theorem 6.2]{SW20}).\nSee \\cite[Remark 5.6]{LR25} for other examples in dimension $\\geq 9$, whose boundary is a topological sphere.\nFor examples of non-Kleinian hyperbolic 3-pseudomanifold groups that have the Haagerup property, and even act convex cocompactly on $\\mathbb H^n$ for some $n>3$, see the RACGs considered in \\cite{DLMR25}. \n\\end{remark}\n\n\\begin{remark}[Relatively geometric cubulation]\\label{rem:agol}\n Ian Agol pointed out the following consequence of our construction (see Proposition~\\ref{prop:agol}):\n For any relatively geometric cubulation (in the sense of \\cite{EG20}) of any of the Kleinian groups $\\pi_1(M_n)$ that we construct, there must be a hyperplane stabilizer with infinite intersection with some boundary subgroup of $\\pi_1(M_n)$.\n This is in contrast to the situation for finite volume cusped $3$--manifolds. The fundamental groups of such manifolds admit relatively geometric cubulations by quasi-Fuchsian closed surface subgroups by the ubiquity results in~\\cite{CF19,KW21}.\n\\end{remark}\n\n\\subsection*{Outline of the paper}\nOur proof strategy can be summarized as follows. We first describe a sequence $T_n$ of simplicial $2$-complexes whose fundamental groups have property (T), and so that as $n$ tends to infinity, the girth of links of vertices in $T_n$ also tends to infinity. For large $n$ we will embed these complexes $\\pi_1$-injectively into negatively curved $3$-pseudomanifolds. To embed such a complex, we first delete a regular neighborhood of the vertices, and thicken the resulting hexagon complex to a $3$-dimensional handlebody. Mirroring the boundary of this handlebody appropriately, we obtain a $3$-orbifold with boundary $H_n$ (usually referred to in the sequel as $H$). The boundary of $H_n$ contains a $\\pi_1$-injective graph corresponding to the union of links of vertices of $T_n$. The $3$-orbifold $H_n$ is (orbifold) covered by a hyperbolic $3$-manifold $M$ with totally geodesic boundary, which we cone off to obtain our $3$-pseudomanifold. The handlebody $H_n$ lifts to $M$, giving rise to a subset of the cone-off homotopy equivalent to $T_n$.\n\nIn \\S\\ref{sec:prelim} we fix our notation and terminology about the construction of pseudomanifolds as cone-offs of manifolds and about orbifolds. Moreover, we collect some preliminary material from \\cite{LMW19} about the $2$-dimensional simplicial complexes $T_n$ whose fundamental groups have property (T).\nIn \\S\\ref{sec:main orbifold} we present the construction of a particular hyperbolic $3$-orbifold with boundary, and in \\S\\ref{sec:covers} we show how to construct suitable finite covers for which we can obtain quantitative control on various geometric quantities. \nThe orbifold structure constructed in \\S\\ref{sec:main orbifold} is given by orthogonal mirrors. This is reminiscent of the Davis reflection trick \\cite{DA83}. However, we use the mirror structure to construct manifolds with boundary instead of closed manifolds.\nThe geometric control enables us to use the results from \\cite{KM} to construct negatively curved metrics on the cone-offs of such covers. The proofs of the main theorem and corollary stated in the Introduction are presented in \\S\\ref{sec:proofs}. We end the paper with some open questions in \\S\\ref{sec:questions}.\n\n\\vspace{.25cm}\n\n\\noindent \\textbf{Acknowledgments.}\nJ.M. was partially supported by the Simons Foundation, grant \\#942496.\nL.R. was partially supported by INDAM-GNSAGA.\nWe thank Chris Hruska for pointing us to useful references. We thank Ian Agol for pointing out Proposition~\\ref{prop:agol}, and Daniel Groves for useful conversations.", "sketch": "Our proof strategy can be summarized as follows. We first describe a sequence $T_n$ of simplicial $2$-complexes whose fundamental groups have property (T), and so that as $n$ tends to infinity, the girth of links of vertices in $T_n$ also tends to infinity. For large $n$ we will embed these complexes $\\pi_1$-injectively into negatively curved $3$-pseudomanifolds. To embed such a complex, we first delete a regular neighborhood of the vertices, and thicken the resulting hexagon complex to a $3$-dimensional handlebody. Mirroring the boundary of this handlebody appropriately, we obtain a $3$-orbifold with boundary $H_n$.\n\nThe boundary of $H_n$ contains a $\\pi_1$-injective graph corresponding to the union of links of vertices of $T_n$. The $3$-orbifold $H_n$ is (orbifold) covered by a hyperbolic $3$-manifold $M$ with totally geodesic boundary, which we cone off to obtain our $3$-pseudomanifold. The handlebody $H_n$ lifts to $M$, giving rise to a subset of the cone-off homotopy equivalent to $T_n$.\n\nThey then use “quantitative control on various geometric quantities” in suitable finite covers together with “the results from \\cite{KM} to construct negatively curved metrics on the cone-offs of such covers.”", "expanded_sketch": "Our proof strategy can be summarized as follows. We first describe a sequence $T_n$ of simplicial $2$-complexes whose fundamental groups have property (T), and so that as $n$ tends to infinity, the girth of links of vertices in $T_n$ also tends to infinity. For large $n$ we will embed these complexes $\\pi_1$-injectively into negatively curved $3$-pseudomanifolds. To embed such a complex, we first delete a regular neighborhood of the vertices, and thicken the resulting hexagon complex to a $3$-dimensional handlebody. Mirroring the boundary of this handlebody appropriately, we obtain a $3$-orbifold with boundary $H_n$.\n\nThe boundary of $H_n$ contains a $\\pi_1$-injective graph corresponding to the union of links of vertices of $T_n$. The $3$-orbifold $H_n$ is (orbifold) covered by a hyperbolic $3$-manifold $M$ with totally geodesic boundary, which we cone off to obtain our $3$-pseudomanifold. The handlebody $H_n$ lifts to $M$, giving rise to a subset of the cone-off homotopy equivalent to $T_n$.\n\nThey then use “quantitative control on various geometric quantities” in suitable finite covers together with “the results from \\cite{KM} to construct negatively curved metrics on the cone-offs of such covers.”", "expanded_theorem": "\\label{thm:main}\n There exists a sequence of compact hyperbolic $3$-manifolds $M_n$ with totally geodesic boundary such that the boundary cone-offs $\\coneoff M_n$ satisfy the following:\n\\begin{enumerate}\n \\item \\label{item:negative} $\\coneoff M_n$ is a closed negatively curved $3$-pseudomanifold.\n \\item \\label{item:T} $\\pi_1(\\coneoff M_n)$ contains an infinite quasiconvex subgroup with property (T).\n \\item \\label{item:char} The genus of the components of $\\partial M_n$ and the Euler characteristic of $\\coneoff M_n$ both go to infinity with $n$.\n\\end{enumerate}", "theorem_type": ["Existence", "Universal"], "mcq": {"question": "Which statement is true about the existence of compact hyperbolic 3-manifolds with totally geodesic boundary? Here, for a compact 3-manifold $M$ with boundary, $\\coneoff M$ denotes the closed space obtained by coning off each boundary component of $M$ to a point; a $3$-pseudomanifold is a 3-dimensional polyhedral complex whose vertex links are connected closed orientable surfaces; and “negatively curved” means admitting a complete locally $\\mathrm{CAT}(-1)$ metric.", "correct_choice": {"label": "A", "text": "There exists a sequence of compact hyperbolic $3$-manifolds $M_n$ with totally geodesic boundary such that, for every $n$, the boundary cone-off $\\coneoff M_n$ is a closed negatively curved $3$-pseudomanifold, and $\\pi_1(\\coneoff M_n)$ contains an infinite quasiconvex subgroup with property (T); moreover, the genus of the components of $\\partial M_n$ and the Euler characteristic of $\\coneoff M_n$ both go to infinity with $n$."}, "choices": [{"label": "B", "text": "There exists a sequence of compact hyperbolic $3$-manifolds $M_n$ with totally geodesic boundary such that, for every $n$, the boundary cone-off $\\coneoff M_n$ is a closed negatively curved $3$-pseudomanifold, and $\\pi_1(\\coneoff M_n)$ contains an infinite subgroup with property (T); moreover, the genus of the components of $\\partial M_n$ and the Euler characteristic of $\\coneoff M_n$ are both bounded independently of $n$."}, {"label": "C", "text": "There exists a sequence of compact hyperbolic $3$-manifolds $M_n$ with totally geodesic boundary such that, for every $n$, the boundary cone-off $\\coneoff M_n$ is a closed negatively curved $3$-pseudomanifold, and $\\pi_1(\\coneoff M_n)$ contains an infinite quasiconvex subgroup with property (T)."}, {"label": "D", "text": "There exists a sequence of compact hyperbolic $3$-manifolds $M_n$ with totally geodesic boundary such that, for every $n$, the boundary cone-off $\\coneoff M_n$ is a closed negatively curved $3$-pseudomanifold, and $\\pi_1(\\coneoff M_n)$ has property (T); moreover, the genus of the components of $\\partial M_n$ and the Euler characteristic of $\\coneoff M_n$ both go to infinity with $n$."}, {"label": "E", "text": "For every sequence of compact hyperbolic $3$-manifolds $M_n$ with totally geodesic boundary for which the genus of the components of $\\partial M_n$ goes to infinity with $n$, the boundary cone-offs $\\coneoff M_n$ are closed negatively curved $3$-pseudomanifolds and $\\pi_1(\\coneoff M_n)$ contains an infinite quasiconvex subgroup with property (T); moreover, the Euler characteristic of $\\coneoff M_n$ also goes to infinity with $n$."}], "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": "growth_of_boundary_genus_and_euler_characteristic", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "asymptotic divergence of genus and Euler characteristic", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "subgroup_property_T_vs_whole_group_property_T", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "existentially constructed_special_sequence_vs_all_sequences", "template_used": "quantifier_dependence"}]}} +{"id": "2601.02859v1", "paper_link": "http://arxiv.org/abs/2601.02859v1", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "\\label{MainTheorem}\n Let ${\\mathbf K}$ be ${\\mathbf N}$-$2$-AD regular. Suppose that the coefficients of the operator ${\\mathcal L}$ belong to $C^3$. Then function \n $f$ defined on ${\\mathbf K}$ \n belongs to the class $\\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$ if \nand only if\nfor any $\\delta< \\frac14\\operatorname{diam\\,}({\\mathbf K})$, there exists an approximating \nfunction\n $v_\\delta(x), \\, x\\in\\Kb_\\delta,$ such that, with some constant\n$\\mathbbm{c}>0$,\n\\begin{gather}\\label{appr.function}\n \\Lc_y v_\\delta(y)=0, \\, y\\in \\Kb_{\\delta};\\\\\\nonumber\n |v_\\delta(x)-f(x)|\\le \\mathbbm{c} \\delta^{\\mathbf r}\\omega(\\delta), \\, x\\in \n {\\mathbf K};\\\\\\nonumber\n |v_\\delta(y)-v_{\\delta/2}(y)|\\le \\mathbbm{c}\\delta^{\\mathbf r}\\omega(\\delta), \\, \n y\\in\\Kb_{\\frac{\\delta}{2}}.\n\\end{gather}", "start_pos": 25144, "end_pos": 25808, "label": "MainTheorem"}, "ref_dict": {"Thm.quality": "\\begin{thm}\\label{Thm.quality} Suppose that $\\rb\\ge 1$ and the \ncoefficients $\\ab(x)=(a_{\\jt\\jt'}(x))_{\\jt,\\jt'\\le \\Nb}$ belong to \n$C^{k_0+3}(\\Om)$ for a certain $k_0\\le\\rb$. Let the \nfunction $f$, defined on \nthe compact set $\\Kb$, belong to the class $H_{\\Lc}^{\\rb+\\om}(\\Kb)$ and \n$v_\\de$ be its \napproximation, as in \\eqref{appr.function}. Then derivatives of \n$v_\\de$ satisfy\n\\begin{equation}\\label{appr.deriv}\n \\|\\nabla^{k_0+1}v_\\de\\|_{\\Kb_{\\de/2}}\\le c\\frac{\\om(\\de)}{\\de}.\n\\end{equation}\nmoreover, surrogate derivatives $f_{(\\a)}(x)$ can be defined, so that\n\\begin{equation}\\label{appr.deriv.2}\n |f_{(\\a)}(x)-\\partial^\\a v_{\\de}(x)|\\le C \n \\de^{\\rb-|\\a|}\\om(\\de),\\, x\\in\\Kb, \\, \n 1\\le |\\a|\\le k_0.\n\\end{equation}\n\\end{thm}", "intro.repres": "\\begin{equation}\\label{intro.repres}\n f_0(x)=\\int_{\\Om}\\Lc f_0(y)G^{\\circ}(x,y)dy, \\, x\\in \\Om.\n\\end{equation}", "MainTheorem": "\\begin{thm}\\label{MainTheorem}\n Let $\\Kb$ be $\\Nb$-$2$-AD regular. Suppose that the coefficients of the operator $\\Lc$ belong to $C^3$. Then function \n $f$ defined on $\\Kb$ \n belongs to the class $\\Hc_{\\Lc}^{\\rb+\\om}(\\Kb)$ if \nand only if\nfor any $\\de< \\frac14\\diam(\\Kb)$, there exists an approximating \nfunction\n $v_\\de(x), \\, x\\in\\Kb_\\de,$ such that, with some constant\n$\\mathbbm{c}>0$,\n\\begin{gather}\\label{appr.function}\n \\Lc_y v_\\de(y)=0, \\, y\\in \\Kb_{\\de};\\\\\\nonumber\n |v_\\de(x)-f(x)|\\le \\mathbbm{c} \\de^\\rb\\om(\\de), \\, x\\in \n \\Kb;\\\\\\nonumber\n |v_\\de(y)-v_{\\de/2}(y)|\\le \\mathbbm{c}\\de^\\rb\\om(\\de), \\, \n y\\in\\Kb_{\\frac{\\de}{2}}.\n\\end{gather}\n\\end{thm}", "35": "\\begin{equation}\\label{35}\n v_{\\de}(x)=c_\\Nb\\int\\limits_{\\Om\\setminus \\Kb'_\\de}G(x,y)\\Lc \n f_0(y)dy+\\sum_{\\n=1}^{N} F_\\n(x).\n\\end{equation}", "appr.function": "\\begin{gather}\\label{appr.function}\n \\Lc_y v_\\de(y)=0, \\, y\\in \\Kb_{\\de};\\\\\\nonumber\n |v_\\de(x)-f(x)|\\le \\mathbbm{c} \\de^\\rb\\om(\\de), \\, x\\in \n \\Kb;\\\\\\nonumber\n |v_\\de(y)-v_{\\de/2}(y)|\\le \\mathbbm{c}\\de^\\rb\\om(\\de), \\, \n y\\in\\Kb_{\\frac{\\de}{2}}.\n\\end{gather}"}, "pre_theorem_intro_text_len": 11594, "pre_theorem_intro_text": "\\label{Intro}\n \\subsection{The approximation problem}\\label{1.1} Approximating\n 'bad' functions by 'good' ones is one of classical topics in \n Analysis. The \n qualitative direction\n has started with the Weierstrass Theorem on the possibility of \n polynomial \n approximation\n of continuous functions. An important further development here concerns \n approximating \n continuous functions by solutions of differential equations. A \n fundamental result \n for rather general differential equations (possessing a kind of \n unique \n continuation property) was obtained by F.E. Browder, \\cite{BR1}, \n \\cite{BR2}.\n\n The studies in the quantitative direction began later.\n Generally speaking, quantitative approximation results can be \n expected to have \n the following common structure:\n \\begin{enumerate}\n \\item A class $\\sc\\mbox{F}\\hspace{1.0pt}$ of functions to be approximated is \n described; \n \\item A class $\\sc\\mbox{G}\\hspace{1.0pt}$ of functions used for approximation is \n proposed;\n \\item The result: a quantitative relation between the rate of \n approximation \n and the properties of the \n approximating function.\n \\end{enumerate}\n For example, the order of the error in the approximation of a \n continuous function \n by polynomials\n of a given degree is determined by the smoothness of this \n function, understood in \n a proper sense.\n\n When considering approximation by solutions of elliptic \n equations, it is \n reasonable to consider as $\\sc\\mbox{F}\\hspace{1.0pt}$, a class of functions defined \n on a nowhere dense \n set ${\\mathbf K}$. In fact, if, on the opposite, ${\\mathbf K}$ possesses \n interior points, it is \n only solutions of the equation that can be approximated by \n solutions. \n So, we are interested in approximating a given continuous \n function $f$ defined \n on a nowhere dense compact\nset ${\\mathbf K}\\subset {\\mathbb R}^{\\mathbf N}$ by solutions of a second order elliptic \nequation. When the \napproximating functions are harmonic,\nand the set ${\\mathbf K}$ is nice, say, a Lipschitz surface, there are many \nresults in this \ndirection, see, e.g. \\cite{Bliedtner}, \\cite{AlSh1}, \n\\cite{Andrievskii}, \\cite{Gardiner Book}, \\cite{Gardiner}, \n\\cite{Gardiner Goldstein}, \\cite{Hausmann}, \\cite{Khav} and many \nmore.\n\n When the conditions on ${\\mathbf K}$ are less restrictive, one can cite \n \\cite{AlSh2}, \\cite{Pavlov}, \\cite{RZ25}. Here, \n one needs to decide, which terms should be used \nto describe properties of the function $f$ in order to determine \nthe rate of \n approximation.\n If we only know that the given function $f$ is continuous on a compact set \n ${\\mathbf K}$, then the \n quality of this continuity, and consequently,\n the quality of approximation, can be described by the modulus \n of continuity of \n $f$. In this direction, in the paper \\cite{RZ25}, the authors \n considered the \n problem on approximating a continuous function $f$ on \n ${\\mathbf K}\\subset{\\mathbb R}^{{\\mathbf N}}$, possessing the continuity\n modulus $\\omega(\\delta)$, by solutions of a second order \n elliptic equation ${\\mathcal L} u=0$ (\\emph{${\\mathcal L}$-harmonic functions}).\n It was established there that if the set \n ${\\mathbf K}$ is \n Ahlfors-David ${\\mathbf N}$-$2$-regular (which means, almost exactly \n speaking, that\n it has one and the same Hausdorff dimension ${\\mathbf N}$-$2$ in any \n neighborhood of any \n of its points), then the function $f$ can be,\n for any $\\delta>0$, approximated in $C({\\mathbf K})$ by a function $v_\\delta$, \n so that \n $|f(x)-v_{\\delta}(x)|\\le c \\omega(\\delta)$ for all $x\\in{\\mathbf K}$,\n the function $v_{\\delta}$ is ${\\mathcal L}$-harmonic in a \n $\\delta$-neighbourhood $\\Kb_{\\delta}$ \n of the set ${\\mathbf K}$, moreover,\n the quality of this function $v_\\delta$ is controlled by $\\delta$, \n namely, $|\\nabla \n v_\\delta(x)|\\le C \\frac{\\omega(\\delta)}{\\delta}$ in $\\Kb_{\\delta}$. This matches the \n general \n principle: the smaller $\\delta$, the better is the aproximation, i.e., the smaller, \n is the \n approximation error, but the worse is the approximating\n function $v_\\delta$: it is ${\\mathcal L}$-harmonic on a smaller set, and \n its gradient may \n grow with $\\delta$ decreasing. Moreover, a converse result was \n established: if a \n continuous function $f$ on ${\\mathbf K}$ can be approximated in the \n above sense, with \n some function $\\omega(\\delta)$, by solutions of a second order \n elliptic differential \n equation, then $f$ possesses the continuity modulus majorated \n by $\\omega(\\delta)$.\n\n It is natural to expect that if we wish to have a better \n approximation (the one better than with \n $O(\\omega(\\delta))$ error), with the same quality of\n the approximating function, we should suppose some better \n properties of the given\n function $f$. If the set ${\\mathbf K}$ were a smooth\n surface (of codimension 2), such 'better' properties would \n naturally involve a \n higher classical smoothness of $f$. However, if we only know \n that the set \n ${\\mathbf K}$ is Ahlfors-David ${\\mathbf N}$-$2$-regular, some other terms \n should be used.\n\n In the literature, there exist methods of defining spaces of \n 'nice' functions on \n arbitrary compacts. One of them is based upon\n describing\n classes of functions via their local approximations by \n polynomials or other\n sufficiently regular functions, see \\cite{Brudny}, \n \\cite{Brudny2}, \n \\cite{Shvartsman}, and many sources afterwards.\n\n So, the expected approximation results should sound like 'if a \n function admits \n local approximation of a certain kind, it admits\n the corresponding quality of global approximation' by \n ${\\mathcal L}$-harmonic \n functions. \n\n This is, in fact, the contents of the present paper. Namely, \n in our main \n result,\n if $f$ is a continuous function on ${\\mathbf K}$, which can, for any \n $\\delta>0$, be \n locally, in a $\\delta$-neighborhood of any point $x\\in{\\mathbf K}$, \n approximated by a function $\\F_{x,\\delta}(y)$ which is a solution \n of the \n second-order elliptic equation \n${\\mathcal L}(y,\\partial_y)u(y)=0$ in a $2\\delta$-neighbourhood of $x$, with \nerror \n$O(\\delta^{\\mathbf r}\\omega(\\delta)), $ ${\\mathbf r}\\ge1$ (with some natural compatibility\n conditions concerning the functions $\\F_{x,\\delta}$ for different values of\n $\\delta$ and different close-lying points\n $x$),\n then $f$ can be approximated on the whole ${\\mathbf K}$, with error of \n the same order, by \n a solution $v_\\delta$ of the same equation in the \n $\\delta$-neighborhood of ${\\mathbf K}$.\nNote that the above compatibility conditions, mentioned in \nparentheses, are \nunavoidable: they are proved to be necessary \nfor the existence of the global approximation. \n\nWhen comparing these results with our previous paper \\cite{RZ25}, \nwhere we \nestablished this kind of properties for ${\\mathbf r}=0$,\n one can notice that an additional restricting condition appears: \n the locally approximating\nfunctions $\\F_{x,\\delta}(y)$ are required here to be solutions of the \nelliptic \nequation, while in \n\\cite{RZ25} no such restriction has been imposed.\nThis restriction is, unfortunately, unavoidable. An example we \npresent in the \npaper demonstrates a function which admits a nice polynomial \nlocal approximation but \ndoes not admit a global approximation by harmonic functions.\n This effect is caused by a visible wildness of the set ${\\mathbf K}$ in \n our example: it is easy to show \n that for a nicer ${\\mathbf K}$, e.g., for a Lipschitz\n surface of codimension 2, such counter-examples are impossible \n and a local \n approximation by smooth functions is sufficient (and, of course, \n necessary)\n for existence of a global approximation by ${\\mathcal L}$-harmonic \n functions. \n\n The elliptic differential operation ${\\mathcal L}(x,\\partial_x)$ is \n supposed to have \n coefficients of certain finite smoothness,\n $C^m(\\Omega)$. The main approximation result, Theorem 1.2, is proved for $m=3$. Under \n additional \n smoothness conditions, the main result can be somewhat\n strengthened: not only the approximating functions $v_\\delta$ \n converge on ${\\mathbf K}$ to \n the initial function $f$, but their derivatives $\\partial^\\alpha \n v_{\\delta}$ (up to \n some order, depending on the smoothness of coefficients of \n ${\\mathcal L}$) converge on \n ${\\mathbf K}$ to some functions $f_{(\\alpha)}$ which can be understood as \n generalized \n derivatives of the given function $f$. The greater $m$, to the higher order these surrogate \n derivatives of $f$ can be defined, see Theorem 1.3. \n\n \\subsection{The main results.1}\\label{Sect.Main. Result1}\n We present here the exact formulation of our main approximation \n result. It is the \n following.\n Let $\\omega(t),$ $t>0$, be a continuity modulus satisfying the \n condition\n \\begin{equation*}\n \\int_0^{\\tau}\\frac{\\omega(t)}{t}dt+\\tau\\int_{\\tau}^\\infty\\frac{\\omega(t)}{t^2}dt \\le c\\omega(\\tau), \\, 0<\\tau<\\infty. \\end{equation*}\nLet, further, ${\\mathbf K}$ be a compact set in ${\\mathbb R}^{\\mathbf N}$, \n${\\mathbf N}$-$2$-Ahlfors-David regular (see, \ne.g., \\cite{David}).\nLet $\\Omega\\supset{\\mathbf K}$ be a bounded open connected set, where a \nformally self-adjoint \nsecond-order elliptic operator \n\\begin{equation*}\n{\\mathcal L} u (x)=-\\sum_{\\mathrm{j},\\mathrm{j}'} \\partial_\\mathrm{j}(a_{\\mathrm{j}\\mathrm{j}'}(x)\\partial_{\\mathrm{j}'}u(x))\\equiv \n-\\nabla\\cdot({\\mathbf a}(x)\\nabla u(x)),\n\\end{equation*}\nwith $C^m$-coefficients $a_{\\mathrm{j}\\mathrm{j}'}$, $m\\ge 3$,\nis defined.\n\nWith the continuity modulus $\\omega$ fixed, for an integer ${\\mathbf r}\\ge0$, \nthe local \n${\\mathcal L}$-H{\\\"o}lder class $\\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$ is defined in \nthe \nfollowing way.\n\\begin{defin}\\label{defin.class}\n The continuous function $f(x)$, $x\\in{\\mathbf K},$ is said to belong to \n$\\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$, if there exist constants \n$\\cb_1=\\cb_1(f)$, \n$\\cb_2=\\cb_2(f)$, such that for any $x\\in{\\mathbf K}$ and any $\\delta,\\, \n0<\\delta\\le \n2\\operatorname{diam\\,}({\\mathbf K})$, there exists \na function $\\F_{x,\\delta}(y)$ defined in the ball $B_\\delta(x)$ such \nthat\n\\begin{equation*} \\Lc_y\\F_{x,\\delta}(y)=0, \\, y\\in B_\\delta(x),\n\\end{equation*}\n\\begin{equation*} |f(y)-\\F_{x,\\delta}(y)|\\le \\cb_1\\delta^{\\mathbf r}\\omega(\\delta), \\, y\\in \n B_\\delta(x)\\cap{\\mathbf K}.\n\\end{equation*}\nFor close-lying points $x_1$, $x_2$, the approximating functions \n should be consistent in the following sense: for some constants \n $\\g_1,\\g_2,$ \n $\\frac18\\le \\g_1\\le 1\\le \\g_2\\le 8,$\n if $\\g_1\\de_1\\le \\de_2\\le \\g_2\\de_1$, given any points \n $x_1,x_2\\in {\\mathbf K}$, such that the \n balls $B_{\\de_1}(x_1), B_{\\de_2}(x_2)$ are not disjoint, the \n inequality\n\\begin{equation}\\label{F2}\n |\\F_{x_1,\\de_1}(y)-\\F_{x_2,\\de_2}(y)|\\le \n \\cb_2\\de_1^{\\mathbf r}\\omega(\\de_1). \n\\end{equation}\nmust hold for all $y\\in B_{\\de_1}(x_1)\\cap B_{\\de_2}(x_2)$,\n\\end{defin}\n\nWe recall the definition of Ahlfors-David regularity. The compact \nset ${\\mathbf K}$ is called 'AD regular' of dimension $\\varkappa$ if for some \nconstants\n$c', c'',$ $00$, be \n locally, in a $\\delta$-neighborhood of any point $x\\in{\\mathbf K}$, \n approximated by a function $\\F_{x,\\delta}(y)$ which is a solution \n of the \n second-order elliptic equation \n${\\mathcal L}(y,\\partial_y)u(y)=0$ in a $2\\delta$-neighbourhood of $x$, with \nerror \n$O(\\delta^{\\mathbf r}\\omega(\\delta)), $ ${\\mathbf r}\\ge1$ (with some natural compatibility\n conditions concerning the functions $\\F_{x,\\delta}$ for different values of\n $\\delta$ and different close-lying points\n $x$),\n then $f$ can be approximated on the whole ${\\mathbf K}$, with error of \n the same order, by \n a solution $v_\\delta$ of the same equation in the \n $\\delta$-neighborhood of ${\\mathbf K}$.\nNote that the above compatibility conditions, mentioned in \nparentheses, are \nunavoidable: they are proved to be necessary \nfor the existence of the global approximation.\n\n\\subsection{The main results.1}\\label{Sect.Main. Result1}\n We present here the exact formulation of our main approximation \n result. It is the \n following.\n Let $\\omega(t),$ $t>0$, be a continuity modulus satisfying the \n condition\n \\begin{equation*}\n \\int_0^{\\tau}\\frac{\\omega(t)}{t}dt+\\tau\\int_{\\tau}^\\infty\\frac{\\omega(t)}{t^2}dt \\le c\\omega(\\tau), \\, 0<\\tau<\\infty. \\end{equation*}\nLet, further, ${\\mathbf K}$ be a compact set in ${\\mathbb R}^{\\mathbf N}$, \n${\\mathbf N}$-$2$-Ahlfors-David regular (see, \ne.g., \\cite{David}).\nLet $\\Omega\\supset{\\mathbf K}$ be a bounded open connected set, where a \nformally self-adjoint \nsecond-order elliptic operator \n\\begin{equation*}\n{\\mathcal L} u (x)=-\\sum_{\\mathrm{j},\\mathrm{j}'} \\partial_\\mathrm{j}(a_{\\mathrm{j}\\mathrm{j}'}(x)\\partial_{\\mathrm{j}'}u(x))\\equiv \n-\\nabla\\cdot({\\mathbf a}(x)\\nabla u(x)),\n\\end{equation*}\nwith $C^m$-coefficients $a_{\\mathrm{j}\\mathrm{j}'}$, $m\\ge 3$,\nis defined.\n\nWith the continuity modulus $\\omega$ fixed, for an integer ${\\mathbf r}\\ge0$, \nthe local \n${\\mathcal L}$-H{\\\"o}lder class $\\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$ is defined in \nthe \nfollowing way.\n\\begin{defin}\\label{defin.class}\n The continuous function $f(x)$, $x\\in{\\mathbf K},$ is said to belong to \n$\\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$, if there exist constants \n$\\cb_1=\\cb_1(f)$, \n$\\cb_2=\\cb_2(f)$, such that for any $x\\in{\\mathbf K}$ and any $\\delta,\\, \n0<\\delta\\le \n2\\operatorname{diam\\,}({\\mathbf K})$, there exists \na function $\\F_{x,\\delta}(y)$ defined in the ball $B_\\delta(x)$ such \nthat\n\\begin{equation*} \\Lc_y\\F_{x,\\delta}(y)=0, \\, y\\in B_\\delta(x),\n\\end{equation*}\n\\begin{equation*} |f(y)-\\F_{x,\\delta}(y)|\\le \\cb_1\\delta^{\\mathbf r}\\omega(\\delta), \\, y\\in \n B_\\delta(x)\\cap{\\mathbf K}.\n\\end{equation*}\nFor close-lying points $x_1$, $x_2$, the approximating functions \n should be consistent in the following sense: for some constants \n $\\g_1,\\g_2,$ \n $\\frac18\\le \\g_1\\le 1\\le \\g_2\\le 8,$\n if $\\g_1\\de_1\\le \\de_2\\le \\g_2\\de_1$, given any points \n $x_1,x_2\\in {\\mathbf K}$, such that the \n balls $B_{\\de_1}(x_1), B_{\\de_2}(x_2)$ are not disjoint, the \n inequality\n\\begin{equation}\\label{F2}\n |\\F_{x_1,\\de_1}(y)-\\F_{x_2,\\de_2}(y)|\\le \n \\cb_2\\de_1^{\\mathbf r}\\omega(\\de_1). \n\\end{equation}\nmust hold for all $y\\in B_{\\de_1}(x_1)\\cap B_{\\de_2}(x_2)$,\n\\end{defin}\n\nWe recall the definition of Ahlfors-David regularity. The compact \nset ${\\mathbf K}$ is called 'AD regular' of dimension $\\varkappa$ if for some \nconstants\n$c', c'',$ $00$, be \n locally, in a $\\delta$-neighborhood of any point $x\\in{\\mathbf K}$, \n approximated by a function $\\F_{x,\\delta}(y)$ which is a solution \n of the \n second-order elliptic equation \n${\\mathcal L}(y,\\partial_y)u(y)=0$ in a $2\\delta$-neighbourhood of $x$, with \nerror \n$O(\\delta^{\\mathbf r}\\omega(\\delta)), $ ${\\mathbf r}\\ge1$ (with some natural compatibility\n conditions concerning the functions $\\F_{x,\\delta}$ for different values of\n $\\delta$ and different close-lying points\n $x$),\n then $f$ can be approximated on the whole ${\\mathbf K}$, with error of \n the same order, by \n a solution $v_\\delta$ of the same equation in the \n $\\delta$-neighborhood of ${\\mathbf K}$.\nNote that the above compatibility conditions, mentioned in \nparentheses, are \nunavoidable: they are proved to be necessary \nfor the existence of the global approximation.\n\n\\subsection{The main results.1}\\label{Sect.Main. Result1}\n We present here the exact formulation of our main approximation \n result. It is the \n following.\n Let $\\omega(t),$ $t>0$, be a continuity modulus satisfying the \n condition\n \\begin{equation*}\n \\int_0^{\\tau}\\frac{\\omega(t)}{t}dt+\\tau\\int_{\\tau}^\\infty\\frac{\\omega(t)}{t^2}dt \\le c\\omega(\\tau), \\, 0<\\tau<\\infty. \\end{equation*}\nLet, further, ${\\mathbf K}$ be a compact set in ${\\mathbb R}^{\\mathbf N}$, \n${\\mathbf N}$-$2$-Ahlfors-David regular (see, \ne.g., \\cite{David}).\nLet $\\Omega\\supset{\\mathbf K}$ be a bounded open connected set, where a \nformally self-adjoint \nsecond-order elliptic operator \n\\begin{equation*}\n{\\mathcal L} u (x)=-\\sum_{\\mathrm{j},\\mathrm{j}'} \\partial_\\mathrm{j}(a_{\\mathrm{j}\\mathrm{j}'}(x)\\partial_{\\mathrm{j}'}u(x))\\equiv \n-\\nabla\\cdot({\\mathbf a}(x)\\nabla u(x)),\n\\end{equation*}\nwith $C^m$-coefficients $a_{\\mathrm{j}\\mathrm{j}'}$, $m\\ge 3$,\nis defined.\n\nWith the continuity modulus $\\omega$ fixed, for an integer ${\\mathbf r}\\ge0$, \nthe local \n${\\mathcal L}$-H{\\\"o}lder class $\\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$ is defined in \nthe \nfollowing way.\n\\begin{defin}\\label{defin.class}\n The continuous function $f(x)$, $x\\in{\\mathbf K},$ is said to belong to \n$\\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$, if there exist constants \n$\\cb_1=\\cb_1(f)$, \n$\\cb_2=\\cb_2(f)$, such that for any $x\\in{\\mathbf K}$ and any $\\delta,\\, \n0<\\delta\\le \n2\\operatorname{diam\\,}({\\mathbf K})$, there exists \na function $\\F_{x,\\delta}(y)$ defined in the ball $B_\\delta(x)$ such \nthat\n\\begin{equation*} \\Lc_y\\F_{x,\\delta}(y)=0, \\, y\\in B_\\delta(x),\n\\end{equation*}\n\\begin{equation*} |f(y)-\\F_{x,\\delta}(y)|\\le \\cb_1\\delta^{\\mathbf r}\\omega(\\delta), \\, y\\in \n B_\\delta(x)\\cap{\\mathbf K}.\n\\end{equation*}\nFor close-lying points $x_1$, $x_2$, the approximating functions \n should be consistent in the following sense: for some constants \n $\\g_1,\\g_2,$ \n $\\frac18\\le \\g_1\\le 1\\le \\g_2\\le 8,$\n if $\\g_1\\de_1\\le \\de_2\\le \\g_2\\de_1$, given any points \n $x_1,x_2\\in {\\mathbf K}$, such that the \n balls $B_{\\de_1}(x_1), B_{\\de_2}(x_2)$ are not disjoint, the \n inequality\n\\begin{equation}\\label{F2}\n |\\F_{x_1,\\de_1}(y)-\\F_{x_2,\\de_2}(y)|\\le \n \\cb_2\\de_1^{\\mathbf r}\\omega(\\de_1). \n\\end{equation}\nmust hold for all $y\\in B_{\\de_1}(x_1)\\cap B_{\\de_2}(x_2)$,\n\\end{defin}\n\nWe recall the definition of Ahlfors-David regularity. The compact \nset ${\\mathbf K}$ is called 'AD regular' of dimension $\\varkappa$ if for some \nconstants\n$c', c'',$ $0 2 \\diam\\Kb,$ we set $ \n\\fT_0(y)=0.$\n The function $\\fT_0,$ thus defined, is piecewise $\\Lc$-harmonic, \n however it may \n be discontinuous along the boundaries\n of cubes in $\\Qc$\n in an uncontrollable manner. We use now the averaging kernel \n $K(x,y)$, to be \n constructed later in this section,\n and set\n \\begin{equation}\\label{function f}\n f_0(x)=\\int_{\\R^{\\Nb}} \\fT_0(y) K(x,y) dy.\n \\end{equation}\nWe will show that this function is a continuous extension of $f$ \nto a neighborhood \nof $K,$\n with controlled behavior of derivatives when approaching $\\Kb$; the function $f_0(x)$ will \n later serve for \n constructing the required \n approximation.\n\\subsection{Construction of the kernel \n$K$}\\label{subs.construction K} Our \nreasoning will be constructive. The first \nstep will be describing a proper averaging kernel. We denote by \n$\\dn(x)$ \nthe \\emph{regularized} distance from the point $x\\in \\Ct\\Kb$ \nto $\\Kb$, namely $\\dn(x)\\in C^3(\\Ct\\Kb)$, $c\\,\\dist(x,\\Kb)\\le \n\\dn(x)\\le \nc'\\,\\dist(x,\\Kb),\\, c'<\\frac14,$ $|\\grad^k\\dn(x)|\\le c \n\\dn(x)^{1-k}, \\, k=1,2,3.$ \nLet $\\hb(t)$ be a function in \n$ C^{\\infty}(\\overline{\\R_+}),$ $\\hb(t)\\ge0,$ $\\supp \n\\hb\\subset[\\frac12,1]$, normalized by $\\int_0^1 \\hb(t)dt=1$. The scaled \nfunction \n$h_r(t)=r^{-1}\\hb(t/r)$ is normalized in such a way that \n$\\int_0^\\infty h_r(t)dt=1.$ \nFurther on, for $x\\in \\Ct\\Kb,\\,t< r=\\dn(x) $, we denote by \n$S_t(x)$ the sphere \n$\\{y:|y-x|=t\\}$, \n and by $B_t(x)$ the corresponding open ball;\n they do not touch $\\Kb$, moreover, they are on a controlled \n distance from $\\Kb$.\n\n\\begin{gather*}\n \\|\\grad\\left(\\F_{x,2^{k-1}\\de}-\\F_{x,2^{k}\\de}\\right)\\|_{C\\left(\\overline{B_{\\frac{\\de}{2}}(x)}\\right)}\\le\\\\\\nonumber\n C (2^{k}\\de)^{-1}\\times 2^{k\\rb}\\de^{\\rb}\\om(2^k)=\n c 2^{k(\\rb-1)}\\de^{\\rb-1}\\om(2^k\\de),\\, \\rb\\ge 1.\n \\end{gather*}\n It follows now that\n \\begin{gather*}\n \\|\\nabla \\F_{x,\\de}\n \\|_{C(\\overline{B_{\\frac{\\de}{2}}(x)})}\\le \n \\\\\\nonumber\n \\sum_{k=1}^{N}\\|\\grad(\\F_{x,2^{k-1}\\de}-\\F_{x,2^{k}\\de})\\|_{C(\\overline{B_{\\frac{\\de}{2}}(x)})}+\n \\|\\grad\\F_{x,2^N\\de}\\|_{C(\\overline{B_{\\frac{\\de}{2}}(x)})}\\le \n \\\\\\nonumber \nc\\sum_{k=1}^{N}2^{k(\\rb-1)}\\de^{\\rb-1} \\om(2^k\\de)+c\\le \n\\de^{\\rb-1}\\int_0^N \n2^{s(\\rb-1)}\\om(2^s\\de)ds+c=\n\\\\\\nonumber c\\de^{\\rb-1}\\int_1^{2^{N}}t^{\\rb-1}\\om(\\de \nt)\\frac{dt}{t}+c=\nc\\de^{\\rb-1}\\left(\\de^{-1}\\right)^{\\rb-1}\\int_0^{2^N\\de}\\tau^{\\rb-2}\\om(\\z)d\\tau+c\\\\\\nonumber\n\\le C(\\diam \\Kb)^{\\rb-1}\\int_0^{2\\diam\\Kb}\\om(\\tau)\\tau^{-1}d\\z+c\\le \nC.\n \\end{gather*} \n\\end{proof}\n \\subsection{Estimating $\\Lc f_0(x)$}\\label{Sect.estimat.f0(x)}\n Now, using our estimates for derivatives of the kernel \n $K(x,y)$, \n obtained in the previous section, we establish estimates for \n the function $f_0(x)$ constructed \n in Section 3.1, see \\eqref{function f},\n and for the result of the action of the operator $\\Lc$ on \n this function.\n For a fixed point $x_0\\in\\Ct \\Kb$, we consider\n the open cube $Q$ in the Whitney cover $\\Qc$, whose closure contains $x_0$. Let $x_{Q}$ be \n the point in $\\Kb$, closest to the\n center of this cube (or one of such points).\n Recall that the function $\\fT_0(x)$ (defined in \\eqref{f-tilde}) \n equals $0$ on the boundary of $Q$.\n By construction, $\\Lc_x\\F_{x_Q,2\\de(Q)}(x)=0$, for $x$ \n in the ball $B_{2\\de(Q)}(x_Q)$, in particular,\n this holds in a small neighborhood of the ball \n $\\overline{{B_{\\dn(x_0)}(x_0)}}$. Therefore, for $x$ in a small \n neighborhood of \n the point $x_0$, \n we obtain, recalling the definition of the kernel $K(x,y)$ \n (which acts as an \n $\\Lc$-replacement for the mean value kernel):\n\\begin{gather}\\label{18.p11}\n |\\Lc_x f_0(x)|= |\\Lc_x(f_0(x)-\\F_{x_Q,2\\de(Q)}(x))|=\\\\\\nonumber\n \\left|\\Lc_x\\left(\\int_{\\R^\\Nb} \n \\fT_0(y)K(x,y)dy-\\int_{\\R^{\\Nb}}\\F_{x_Q,2\\de(Q)}(y)K(x,y)dy\\right)\\right|=\\\\\\nonumber\n\\left| \\Lc_x\\left( \n\\int_{\\R^\\Nb}(\\fT_0(y)-\\F_{x_Q,2\\de(Q)}(y))K(x,y) dy \n\\right)\\right|\\le \\\\\\nonumber\nC\\int\\limits_{{B_{\\dn(x_0)+\\ve}(x_0)}}|\\fT_0(y)-\\F_{x_Q,2\\de(Q)}(y)||\\nabla^2_{xx}K(x,y)|dy\\le\\\\\\nonumber\nC \n\\dn(x)^{-\\Nb-2}\\int\\limits_{{B_{\\dn(x_0)+\\ve}(x_0)}}|\\fT_0(y)-\\F_{x_Q,2\\de(Q)}(y)|dy,\n\\end{gather}\n using, on the last step, our estimates for derivatives of the \n kernel $K(x,y)$. We apply estimate \\eqref{17} now. For\n $x\\in \\overline{B_{\\dn(x_0)+\\ve}(x_0)},$ the \ndifference $\\fT_0(y)-\\F_{x_Q,2\\de(Q)}(y)$ has the form \n $\\F_{x_{Q_1},2\\de(Q_1)}(y)-\\F_{x_Q,\\de(Q)}(y)$, for a certain \n cube $Q_1$, and satisfies \n the conditions of the main theorem with parameters $\\de(Q), \n \\de(Q_1)$. Therefore,\n\\begin{equation}\\label{19}\n|\\fT_0(y)-\\F_{x_Q,2\\de(Q)}(y)|=|\\F_{x_{Q_1},2\\de(Q_1)}(y)-\\F_{x_Q,\\de(Q)}(y)|\\le \nc \n\\dn(x_0)^{\\rb}\\om(\\dn(x_0)).\n\\end{equation}\nWe set $x=x_0$ in \\eqref{18.p11}, \\eqref{19} and obtain the \nrequired estimate for \n$\\Lc f_0$ outside $\\Kb$:\n\\begin{equation}\\label{20}\n|(\\Lc f_0)(x_0)|\\le c \\dn^{\\rb-2}(x_0)\\om(\\dn(x_0)).\n\\end{equation}\n We stress here that by \\eqref{20}, the larger $\\rb$ in the conditions of the Theorem, the faster the function $\\Lc f_0(x)$ \n decays as the point $x$ approaches $\\Kb$. This fact will be essentially used further on.\n\n\\begin{thm}\\label{counterex} It is impossible to approximate \n$f(x)$ in the sense \nof Theorem \\ref{MainTheorem} with $\\rb=2,$ and \n$\\om(\\de)=\\de^{\\s},$ $0<\\s<1$, by \nharmonic functions, this means,\n by solutions of the equation $\\Lc v_\\de\\equiv-\\D v_\\de=0$,\n\\end{thm}\nIn other words, for such a wild set $\\Kb,$ one cannot approximate on $\\Kb$ the non-harmonic function $f(x)$ by harmonic functions, even locally. \n\\begin{proof}\nSuppose that the approximation in question is possible, thus, for \nany \n$\\de\\in(0,1),$ there exists a function\n$v_\\de$ such that, \n\\begin{equation}\\label{49}\n |v_\\de(x)-f(x)|\\le \\mathbbm{c}\\de^{2+\\s}, \\, x\\in\\Kb,\n\\end{equation}\n\\begin{equation}\\label{50}\n |v_\\de(x)-v_{\\frac{\\de}{2}}(x)|\\le \\mathbbm{c}\\de^{2+\\s}, \\, \n x\\in\\Kb_{\\frac{\\de}{2}},\n\\end{equation}\nwith some $\\mathbbm{c}$ not depending on $\\de,$ and\n\\begin{equation}\\label{51}\n \\D v_{\\de}(x)=0, \\, x\\in\\Kb_{\\de}.\n\\end{equation}\n We establish the following property.\n \\begin{lem}\\label{lem.counter} Under the assumptions \n \\eqref{49}-\\eqref{51}, the \n function $v_{\\de}$ must\n satisfy the estimate: \n \\begin{gather}\\label{52}\n |\\nabla^3 v_{2\\de_{\\pk}}(x)|\\le c \n \\mathbbm{c}\\de_{\\pk}^{\\s-1},\\,\\\\\\nonumber \n x\\in U_{\\pk}:=(\\tb_{\\pk}+B_{\\de_{\\pk}}(\\mathbb{0}), \\, \n \\tb_{\\pk}=(\\frac{1}{2}(2^{-\\pk-1}+2^{-\\pk}),0,\\dots,0), \\, \n \\de_{\\pk}=2^{-\\pk-2}.\n \\end{gather}\n \\end{lem}\n \\begin{proof}\n To prove \\eqref{52}, we denote by $\\f_k(x), \\, x\\in U_{\\pk}$, \n the function \n $\\f_k(x)=v_{2^k\\de_m}(x)-v_{2^{k+1}\\de_m}(x)$,\\,\n $k=1,\\dots,N$, where $N$ is chosen so that $1<2^N\\de\\le 2$. \n Using this function, \n we can represent $v_{2\\de_{\\pk}}$ and its order 3 gradient as", "post_theorem_intro_text_len": 5943, "post_theorem_intro_text": "\\subsection{The ideas of the proof} The proof of the main \ntheorem is fairly \ntechnical, therefore we consider it reasonable to explain here its \nstructure.\n\n Given \na function $f\\in \\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$ on the compact set \n${\\mathbf K}$, we construct its \nspecial extension $f_0$ to a fixed neighborhood $\\Omega$ of ${\\mathbf K}$ \n(the particular form of this\n neighbourhood is not essential, and we suppose further on that it is the \n unit ball containing the set ${\\mathbf K}$ which is contained in the concentric ball with radius $\\frac13$). For this function \n$f_0$, using the Green function $ G^{\\circ}(x,y)$ of the operator ${\\mathcal L}$ \nin $\\Omega,$ the \nintegral representation is established:\n\\begin{equation}\\label{intro.repres}\n f_0(x)=\\int_{\\Omega}{\\mathcal L} f_0(y)G^{\\circ}(x,y)dy, \\, x\\in \\Omega.\n\\end{equation}\nAlthough this representation looks quite usual if $f_0$ is \nsufficiently smooth, \nthis is not the case for \\emph{our} function $f_0$ for which the \nderivatives may \nbehave badly when approaching ${\\mathbf K}$. Therefore, to justify \n\\eqref{intro.repres}, we need a detailed control of the behavior \nof ${\\mathcal L} f_0(x)$ \nand of derivatives of $f_0(x)$ near ${\\mathbf K}$. Obtaining this \ncontrol requires \ncomplicated estimates of the Green function $G_{x,r}(x,y)$ for ${\\mathcal L}$ \nin balls \n$B_r(x)$ centered at $x$, together with their derivatives, up to \nthe third \norder, in the variables $x,y$, as well as in the additional variable $\\varsigma$ on which the operator ${\\mathcal L}$ depends as a parameter. \nUnder the condition \nof a sufficient smoothness of coefficients of the operation ${\\mathcal L},$ \nwe derive some of these estimates directly, using Schauder-type \n approach, and borrow the other ones \n from the results by Ju.~Krasovskii \n \\cite{Kras.Sing}, and M.~Gr{\\\"u}ter--K.-O.~Widman \\cite{Widman}. \nFinally, having established the representation \n\\eqref{intro.repres}, we define \nthe approximation function $v_\\delta(x)$, looked for, by the integral\n\\begin{equation*}\n v_{\\delta}(x)=\\int_{\\Omega\\setminus \\Kb_{\\delta}}{\\mathcal L} f_0(y)G^{\\circ}(x,y)dy,\n\\end{equation*}\nwith addition of a collection of several compensatory ${\\mathcal L}$-harmonic terms of a simpler nature, see \\eqref{35}.\n The fact that $v_{\\delta}$ is \n${\\mathcal L}$-harmonic in $\\Kb_{\\delta}$ is obvious, it follows from the \ndefinition of the \nGreen function $G^{\\circ}(x,y)$, while the estimates producing the \nquality of the \napproximation follow from our estimates for the function \n$f_0(x)$ and its \nderivatives.\n\n\\subsection{The main result. 2}\\label{Sect.Main. Result2}\nThe second theorem describes the properties of the approximating \nfunctions $v_\\delta:$ \ntheir derivatives, up to a prescribed order \n$k\\le{\\mathbf r}+1$ can be controlled. Moreover, we \ncan define in a consistent way the generalized derivatives $f_{(\\alpha)}$ of the \ninitial function \n$f$ on ${\\mathbf K}$,\nso that the derivatives of $v_{\\delta}$ approximate these derivatives \nof $f$. This \nproperty requires a certain additional smoothness of coefficients \nof the operator \n${\\mathcal L}$. \n\\begin{thm}\\label{Thm.quality} Suppose that ${\\mathbf r}\\ge 1$ and the \ncoefficients ${\\mathbf a}(x)=(a_{\\mathrm{j}\\mathrm{j}'}(x))_{\\mathrm{j},\\mathrm{j}'\\le {\\mathbf N}}$ belong to \n$C^{k_0+3}(\\Omega)$ for a certain $k_0\\le{\\mathbf r}$. Let the \nfunction $f$, defined on \nthe compact set ${\\mathbf K}$, belong to the class $H_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$ and \n$v_\\delta$ be its \napproximation, as in \\eqref{appr.function}. Then derivatives of \n$v_\\delta$ satisfy\n\\begin{equation}\\label{appr.deriv}\n \\|\\nabla^{k_0+1}v_\\delta\\|_{\\Kb_{\\delta/2}}\\le c\\frac{\\omega(\\delta)}{\\delta}.\n\\end{equation}\nmoreover, surrogate derivatives $f_{(\\alpha)}(x)$ can be defined, so that\n\\begin{equation}\\label{appr.deriv.2}\n |f_{(\\alpha)}(x)-\\partial^\\alpha v_{\\delta}(x)|\\le C \n \\delta^{{\\mathbf r}-|\\alpha|}\\omega(\\delta),\\, x\\in{\\mathbf K}, \\, \n 1\\le |\\alpha|\\le k_0.\n\\end{equation}\n\\end{thm}\n\\subsection{Structure of the paper} We start in Sect. \\ref{Sect.Prep} by \npresenting general\nmaterial concerning certain geometry considerations, and formulate estimates \nof important integrals used in further analysis and of derivatives of the Green function, \nincluding the results of \\cite{Kras.Sing} and \\cite{Widman},\n In Sect. 3, we introduce the \n averaging kernel $K(x,y)$ and prove \n estimates of its derivatives. This is the most technical part of \n the paper. Next, in Sect. 4, we construct the\n extension function $f_0$, derive its important properties and \n prove its integral representation, which\n results in presenting the required approximation of the given function \n $f(x)\\in H_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$, thus proving Theorem \n \\ref{MainTheorem}. In Sect 5, we \n discuss generalized derivatives of the function $f$, and prove \n Theorem \\ref{Thm.quality}. Then, in Sect.6, we present the \n example\n showing that for a wild set ${\\mathbf K}$, the condition on local \n approximation cannot, generally, be relaxed.\n\n Proofs of our estimates for derivatives of the Green function and of important integral inequalities\n are placed \n in the Appendix.\n\n\\subsection{Conventions}In the course of the paper, we denote by \nthe same symbol \n$c$ or $C$ various constants whose particular value is of no \nimportance, as long as \nthis does not cause confusion; sometimes, subscripts or superscripts \nare used in \norder to distinguish between such constants in the same formula. \nMore important \nconstants may be highlighted by a different font. By \n$f'_x=\\partial_x f=\\nabla_x f$ we denote the $x$-gradient of a \nfunction $f$;\nfor a vector function $F$, $\\nabla_x F$ stands for the Jacobi \nmatrix of $F$. The symbol $| \\cdot |$ denotes the Euclidean norm of \nthe vector involved, $\\mathbf{E}$ denotes the unit matrix.", "sketch": "The introduction explains the structure of the proof of Theorem~\\ref{MainTheorem} as follows.\n\n1. Start with $f\\in \\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$ and construct a “special extension” $f_0$ to a fixed neighborhood $\\Omega$ of ${\\mathbf K}$.\n\n2. For $f_0$, use the Green function $G^{\\circ}(x,y)$ of ${\\mathcal L}$ in $\\Omega$ to establish the integral representation\n\\[\n f_0(x)=\\int_{\\Omega}{\\mathcal L} f_0(y)G^{\\circ}(x,y)\\,dy, \\qquad x\\in \\Omega.\\tag{\\ref{intro.repres}}\n\\]\nBecause $f_0$ may have derivatives that “behave badly when approaching ${\\mathbf K}$,” the representation requires “detailed control” of ${\\mathcal L}f_0(x)$ and derivatives of $f_0(x)$ near ${\\mathbf K}$.\n\n3. Obtain this control via “complicated estimates” for the Green function $G_{x,r}(x,y)$ in balls $B_r(x)$ and its derivatives “up to the third order” in $x,y$, and also in the parameter variable $\\varsigma$. Some estimates are derived “using Schauder-type approach,” and others are taken from Ju.~Krasovskii \\cite{Kras.Sing} and M.~Gr\\\"uter--K.-O.~Widman \\cite{Widman}.\n\n4. Define the approximants by\n\\[\n v_{\\delta}(x)=\\int_{\\Omega\\setminus \\Kb_{\\delta}}{\\mathcal L} f_0(y)G^{\\circ}(x,y)\\,dy,\n\\]\n“with addition of a collection of several compensatory ${\\mathcal L}$-harmonic terms of a simpler nature” (see \\eqref{35}). Then “the fact that $v_{\\delta}$ is ${\\mathcal L}$-harmonic in $\\Kb_{\\delta}$ is obvious” from the definition of $G^{\\circ}(x,y)$.\n\n5. The estimates giving the approximation quality (the bounds in \\eqref{appr.function}) “follow from our estimates for the function $f_0(x)$ and its derivatives.”\n\nThe paper outline reiterates that Sect.~4 constructs $f_0$, proves the integral representation, and thereby “present[s] the required approximation… thus proving Theorem~\\ref{MainTheorem}.”", "expanded_sketch": "The introduction explains the structure of the proof of the main theorem as follows.\n\n1. Start with $f\\in \\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$ and construct a “special extension” $f_0$ to a fixed neighborhood $\\Omega$ of ${\\mathbf K}$.\n\n2. For $f_0$, use the Green function $G^{\\circ}(x,y)$ of ${\\mathcal L}$ in $\\Omega$ to establish the integral representation\n\\begin{equation}\\label{intro.repres}\n f_0(x)=\\int_{\\Om}\\Lc f_0(y)G^{\\circ}(x,y)dy, \\, x\\in \\Om.\n\\end{equation}\nBecause $f_0$ may have derivatives that “behave badly when approaching ${\\mathbf K}$,” the representation requires “detailed control” of ${\\mathcal L}f_0(x)$ and derivatives of $f_0(x)$ near ${\\mathbf K}$.\n\n3. Obtain this control via “complicated estimates” for the Green function $G_{x,r}(x,y)$ in balls $B_r(x)$ and its derivatives “up to the third order” in $x,y$, and also in the parameter variable $\\varsigma$. Some estimates are derived “using Schauder-type approach,” and others are taken from Ju.~Krasovskii \\cite{Kras.Sing} and M.~Gr\\\"uter--K.-O.~Widman \\cite{Widman}.\n\n4. Define the approximants by\n\\[\n v_{\\delta}(x)=\\int_{\\Omega\\setminus \\Kb_{\\delta}}{\\mathcal L} f_0(y)G^{\\circ}(x,y)\\,dy,\n\\]\n“with addition of a collection of several compensatory ${\\mathcal L}$-harmonic terms of a simpler nature”, namely (as indicated by the reference)\n\\begin{equation}\\label{35}\n v_{\\de}(x)=c_\\Nb\\int\\limits_{\\Om\\setminus \\Kb'_\\de}G(x,y)\\Lc \n f_0(y)dy+\\sum_{\\n=1}^{N} F_\\n(x).\n\\end{equation}\nThen “the fact that $v_{\\delta}$ is ${\\mathcal L}$-harmonic in $\\Kb_{\\delta}$ is obvious” from the definition of $G^{\\circ}(x,y)$.\n\n5. The estimates giving the approximation quality (the bounds in\n\\begin{gather}\\label{appr.function}\n \\Lc_y v_\\de(y)=0, \\, y\\in \\Kb_{\\de};\\\\\\nonumber\n |v_\\de(x)-f(x)|\\le \\mathbbm{c} \\de^\\rb\\om(\\de), \\, x\\in \n \\Kb;\\\\\\nonumber\n |v_\\de(y)-v_{\\de/2}(y)|\\le \\mathbbm{c}\\de^\\rb\\om(\\de), \\, \n y\\in\\Kb_{\\frac{\\de}{2}}.\n\\end{gather}\n) “follow from our estimates for the function $f_0(x)$ and its derivatives.”\n\nThe paper outline reiterates that next it constructs $f_0$, proves the integral representation, and thereby “present[s] the required approximation… thus proving the main theorem.”", "expanded_theorem": "\\label{MainTheorem}\n Let ${\\mathbf K}$ be ${\\mathbf N}$-$2$-AD regular. Suppose that the coefficients of the operator ${\\mathcal L}$ belong to $C^3$. Then function \n $f$ defined on ${\\mathbf K}$ \n belongs to the class $\\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$ if \nand only if\nfor any $\\delta< \\frac14\\operatorname{diam\\,}({\\mathbf K})$, there exists an approximating \nfunction\n $v_\\delta(x), \\, x\\in\\Kb_\\delta,$ such that, with some constant\n$\\mathbbm{c}>0$,\n\\begin{gather}\\label{appr.function}\n \\Lc_y v_\\delta(y)=0, \\, y\\in \\Kb_{\\delta};\\\\\\nonumber\n |v_\\delta(x)-f(x)|\\le \\mathbbm{c} \\delta^{\\mathbf r}\\omega(\\delta), \\, x\\in \n {\\mathbf K};\\\\\\nonumber\n |v_\\delta(y)-v_{\\delta/2}(y)|\\le \\mathbbm{c}\\delta^{\\mathbf r}\\omega(\\delta), \\, \n y\\in\\Kb_{\\frac{\\delta}{2}}.\n\\end{gather}", "theorem_type": ["Biconditional or Equivalence", "Universal–Existential"], "mcq": {"question": "Let \\(\\omega:(0,\\infty)\\to[0,\\infty)\\) be a continuity modulus satisfying \\[\\int_0^{\\tau}\\frac{\\omega(t)}{t}\\,dt+\\tau\\int_{\\tau}^{\\infty}\\frac{\\omega(t)}{t^2}\\,dt\\le c\\,\\omega(\\tau)\\qquad(0<\\tau<\\infty).\\] Let \\(\\mathbf r\\ge 0\\) be an integer, let \\({\\mathbf K}\\subset\\mathbb R^{\\mathbf N}\\) be compact and \\((\\mathbf N-2)\\)-Ahlfors-David regular, and let \\({\\mathcal L}u=-\\nabla\\!\\cdot({\\mathbf a}(x)\\nabla u)\\) be a formally self-adjoint second-order elliptic operator with \\(C^3\\)-coefficients on a neighborhood of \\({\\mathbf K}\\). Write \\({\\mathbf K}_\\delta=\\{x:\\operatorname{dist}(x,{\\mathbf K})<\\delta\\}\\). A continuous function \\(f\\) on \\({\\mathbf K}\\) belongs to \\(\\mathcal H_{\\mathcal L}^{{\\mathbf r}+\\omega}({\\mathbf K})\\) if there exist constants \\(c_1,c_2\\) such that for every \\(x\\in{\\mathbf K}\\) and every \\(0<\\delta\\le 2\\operatorname{diam}({\\mathbf K})\\) there is a function \\(\\mathcal F_{x,\\delta}\\) on \\(B_\\delta(x)\\) with \\({\\mathcal L}_y\\mathcal F_{x,\\delta}(y)=0\\) in \\(B_\\delta(x)\\), \\(|f(y)-\\mathcal F_{x,\\delta}(y)|\\le c_1\\delta^{\\mathbf r}\\omega(\\delta)\\) on \\(B_\\delta(x)\\cap{\\mathbf K}\\), and the local approximants satisfy the consistency condition \\(|\\mathcal F_{x_1,\\delta_1}(y)-\\mathcal F_{x_2,\\delta_2}(y)|\\le c_2\\delta_1^{\\mathbf r}\\omega(\\delta_1)\\) on overlaps whenever \\(B_{\\delta_1}(x_1)\\cap B_{\\delta_2}(x_2)\\neq\\varnothing\\) and \\(\\gamma_1\\delta_1\\le\\delta_2\\le\\gamma_2\\delta_1\\) for some fixed constants \\(\\tfrac18\\le\\gamma_1\\le1\\le\\gamma_2\\le8\\). Under these assumptions, which statement about \\(f\\) is valid?", "correct_choice": {"label": "A", "text": "The function \\(f\\) belongs to \\(\\mathcal H_{\\mathcal L}^{{\\mathbf r}+\\omega}({\\mathbf K})\\) if and only if for every \\(\\delta<\\tfrac14\\operatorname{diam}({\\mathbf K})\\) there exists a function \\(v_\\delta\\) on \\({\\mathbf K}_\\delta\\) and a constant \\(\\mathbbm c>0\\), independent of \\(\\delta\\), such that \\({\\mathcal L}_y v_\\delta(y)=0\\) for all \\(y\\in{\\mathbf K}_\\delta\\), \\(|v_\\delta(x)-f(x)|\\le \\mathbbm c\\,\\delta^{\\mathbf r}\\omega(\\delta)\\) for all \\(x\\in{\\mathbf K}\\), and \\(|v_\\delta(y)-v_{\\delta/2}(y)|\\le \\mathbbm c\\,\\delta^{\\mathbf r}\\omega(\\delta)\\) for all \\(y\\in{\\mathbf K}_{\\delta/2}\\)."}, "choices": [{"label": "B", "text": "The function \\(f\\) belongs to \\(\\mathcal H_{\\mathcal L}^{{\\mathbf r}+\\omega}({\\mathbf K})\\) if and only if for every \\(\\delta<\\tfrac14\\operatorname{diam}({\\mathbf K})\\) there exists a function \\(v_\\delta\\) on \\({\\mathbf K}_\\delta\\) and a constant \\(\\mathbbm c>0\\), independent of \\(\\delta\\), such that \\({\\mathcal L}_y v_\\delta(y)=0\\) for all \\(y\\in{\\mathbf K}_\\delta\\), \\(|v_\\delta(x)-f(x)|\\le \\mathbbm c\\,\\delta^{{\\mathbf r}+1}\\omega(\\delta)\\) for all \\(x\\in{\\mathbf K}\\), and \\(|v_\\delta(y)-v_{\\delta/2}(y)|\\le \\mathbbm c\\,\\delta^{{\\mathbf r}+1}\\omega(\\delta)\\) for all \\(y\\in{\\mathbf K}_{\\delta/2}\\)."}, {"label": "C", "text": "If for every \\(\\delta<\\tfrac14\\operatorname{diam}({\\mathbf K})\\) there exists a function \\(v_\\delta\\) on \\({\\mathbf K}_\\delta\\) and a constant \\(\\mathbbm c>0\\), independent of \\(\\delta\\), such that \\({\\mathcal L}_y v_\\delta(y)=0\\) for all \\(y\\in{\\mathbf K}_\\delta\\), \\(|v_\\delta(x)-f(x)|\\le \\mathbbm c\\,\\delta^{\\mathbf r}\\omega(\\delta)\\) for all \\(x\\in{\\mathbf K}\\), and \\(|v_\\delta(y)-v_{\\delta/2}(y)|\\le \\mathbbm c\\,\\delta^{\\mathbf r}\\omega(\\delta)\\) for all \\(y\\in{\\mathbf K}_{\\delta/2}\\), then \\(f\\in\\mathcal H_{\\mathcal L}^{{\\mathbf r}+\\omega}({\\mathbf K})\\)."}, {"label": "D", "text": "The function \\(f\\) belongs to \\(\\mathcal H_{\\mathcal L}^{{\\mathbf r}+\\omega}({\\mathbf K})\\) if and only if for every \\(\\delta<\\tfrac14\\operatorname{diam}({\\mathbf K})\\) there exists a function \\(v_\\delta\\) on \\({\\mathbf K}_\\delta\\) such that \\({\\mathcal L}_y v_\\delta(y)=0\\) for all \\(y\\in{\\mathbf K}_\\delta\\), \\(|v_\\delta(x)-f(x)|\\le \\mathbbm c_\\delta\\,\\delta^{\\mathbf r}\\omega(\\delta)\\) for all \\(x\\in{\\mathbf K}\\), and \\(|v_\\delta(y)-v_{\\delta/2}(y)|\\le \\mathbbm c_\\delta\\,\\delta^{\\mathbf r}\\omega(\\delta)\\) for all \\(y\\in{\\mathbf K}_{\\delta/2}\\), where the constant \\(\\mathbbm c_\\delta\\) may depend on \\(\\delta\\)."}, {"label": "E", "text": "The function \\(f\\) belongs to \\(\\mathcal H_{\\mathcal L}^{{\\mathbf r}+\\omega}({\\mathbf K})\\) if and only if for every \\(\\delta<\\tfrac14\\operatorname{diam}({\\mathbf K})\\) there exists a function \\(v_\\delta\\) on \\({\\mathbf K}_{\\delta/2}\\) and a constant \\(\\mathbbm c>0\\), independent of \\(\\delta\\), such that \\({\\mathcal L}_y v_\\delta(y)=0\\) for all \\(y\\in{\\mathbf K}_{\\delta/2}\\), \\(|v_\\delta(x)-f(x)|\\le \\mathbbm c\\,\\delta^{\\mathbf r}\\omega(\\delta)\\) for all \\(x\\in{\\mathbf K}\\), and \\(|v_\\delta(y)-v_{\\delta/2}(y)|\\le \\mathbbm c\\,\\delta^{\\mathbf r}\\omega(\\delta)\\) for all \\(y\\in{\\mathbf K}_{\\delta/2}\\)."}], "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": "approximation_order_delta_power", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped_only_if_direction", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "uniform_constant_independent_of_delta", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "harmonicity_domain_K_delta_vs_K_delta_over_2", "template_used": "wildcard"}]}} +{"id": "2601.02859v1", "paper_link": "http://arxiv.org/abs/2601.02859v1", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "\\label{MainTheorem}\n Let ${\\mathbf K}$ be ${\\mathbf N}$-$2$-AD regular. Suppose that the coefficients of the operator ${\\mathcal L}$ belong to $C^3$. Then function \n $f$ defined on ${\\mathbf K}$ \n belongs to the class $\\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$ if \nand only if\nfor any $\\delta< \\frac14\\operatorname{diam\\,}({\\mathbf K})$, there exists an approximating \nfunction\n $v_\\delta(x), \\, x\\in\\Kb_\\delta,$ such that, with some constant\n$\\mathbbm{c}>0$,\n\\begin{gather}\\label{appr.function}\n \\Lc_y v_\\delta(y)=0, \\, y\\in \\Kb_{\\delta};\\\\\\nonumber\n |v_\\delta(x)-f(x)|\\le \\mathbbm{c} \\delta^{\\mathbf r}\\omega(\\delta), \\, x\\in \n {\\mathbf K};\\\\\\nonumber\n |v_\\delta(y)-v_{\\delta/2}(y)|\\le \\mathbbm{c}\\delta^{\\mathbf r}\\omega(\\delta), \\, \n y\\in\\Kb_{\\frac{\\delta}{2}}.\n\\end{gather}", "start_pos": 25144, "end_pos": 25808, "label": "MainTheorem"}, "ref_dict": {"Thm.quality": "\\begin{thm}\\label{Thm.quality} Suppose that $\\rb\\ge 1$ and the \ncoefficients $\\ab(x)=(a_{\\jt\\jt'}(x))_{\\jt,\\jt'\\le \\Nb}$ belong to \n$C^{k_0+3}(\\Om)$ for a certain $k_0\\le\\rb$. Let the \nfunction $f$, defined on \nthe compact set $\\Kb$, belong to the class $H_{\\Lc}^{\\rb+\\om}(\\Kb)$ and \n$v_\\de$ be its \napproximation, as in \\eqref{appr.function}. Then derivatives of \n$v_\\de$ satisfy\n\\begin{equation}\\label{appr.deriv}\n \\|\\nabla^{k_0+1}v_\\de\\|_{\\Kb_{\\de/2}}\\le c\\frac{\\om(\\de)}{\\de}.\n\\end{equation}\nmoreover, surrogate derivatives $f_{(\\a)}(x)$ can be defined, so that\n\\begin{equation}\\label{appr.deriv.2}\n |f_{(\\a)}(x)-\\partial^\\a v_{\\de}(x)|\\le C \n \\de^{\\rb-|\\a|}\\om(\\de),\\, x\\in\\Kb, \\, \n 1\\le |\\a|\\le k_0.\n\\end{equation}\n\\end{thm}", "intro.repres": "\\begin{equation}\\label{intro.repres}\n f_0(x)=\\int_{\\Om}\\Lc f_0(y)G^{\\circ}(x,y)dy, \\, x\\in \\Om.\n\\end{equation}", "MainTheorem": "\\begin{thm}\\label{MainTheorem}\n Let $\\Kb$ be $\\Nb$-$2$-AD regular. Suppose that the coefficients of the operator $\\Lc$ belong to $C^3$. Then function \n $f$ defined on $\\Kb$ \n belongs to the class $\\Hc_{\\Lc}^{\\rb+\\om}(\\Kb)$ if \nand only if\nfor any $\\de< \\frac14\\diam(\\Kb)$, there exists an approximating \nfunction\n $v_\\de(x), \\, x\\in\\Kb_\\de,$ such that, with some constant\n$\\mathbbm{c}>0$,\n\\begin{gather}\\label{appr.function}\n \\Lc_y v_\\de(y)=0, \\, y\\in \\Kb_{\\de};\\\\\\nonumber\n |v_\\de(x)-f(x)|\\le \\mathbbm{c} \\de^\\rb\\om(\\de), \\, x\\in \n \\Kb;\\\\\\nonumber\n |v_\\de(y)-v_{\\de/2}(y)|\\le \\mathbbm{c}\\de^\\rb\\om(\\de), \\, \n y\\in\\Kb_{\\frac{\\de}{2}}.\n\\end{gather}\n\\end{thm}", "35": "\\begin{equation}\\label{35}\n v_{\\de}(x)=c_\\Nb\\int\\limits_{\\Om\\setminus \\Kb'_\\de}G(x,y)\\Lc \n f_0(y)dy+\\sum_{\\n=1}^{N} F_\\n(x).\n\\end{equation}", "appr.function": "\\begin{gather}\\label{appr.function}\n \\Lc_y v_\\de(y)=0, \\, y\\in \\Kb_{\\de};\\\\\\nonumber\n |v_\\de(x)-f(x)|\\le \\mathbbm{c} \\de^\\rb\\om(\\de), \\, x\\in \n \\Kb;\\\\\\nonumber\n |v_\\de(y)-v_{\\de/2}(y)|\\le \\mathbbm{c}\\de^\\rb\\om(\\de), \\, \n y\\in\\Kb_{\\frac{\\de}{2}}.\n\\end{gather}"}, "pre_theorem_intro_text_len": 11594, "pre_theorem_intro_text": "\\label{Intro}\n \\subsection{The approximation problem}\\label{1.1} Approximating\n 'bad' functions by 'good' ones is one of classical topics in \n Analysis. The \n qualitative direction\n has started with the Weierstrass Theorem on the possibility of \n polynomial \n approximation\n of continuous functions. An important further development here concerns \n approximating \n continuous functions by solutions of differential equations. A \n fundamental result \n for rather general differential equations (possessing a kind of \n unique \n continuation property) was obtained by F.E. Browder, \\cite{BR1}, \n \\cite{BR2}.\n\n The studies in the quantitative direction began later.\n Generally speaking, quantitative approximation results can be \n expected to have \n the following common structure:\n \\begin{enumerate}\n \\item A class $\\sc\\mbox{F}\\hspace{1.0pt}$ of functions to be approximated is \n described; \n \\item A class $\\sc\\mbox{G}\\hspace{1.0pt}$ of functions used for approximation is \n proposed;\n \\item The result: a quantitative relation between the rate of \n approximation \n and the properties of the \n approximating function.\n \\end{enumerate}\n For example, the order of the error in the approximation of a \n continuous function \n by polynomials\n of a given degree is determined by the smoothness of this \n function, understood in \n a proper sense.\n\n When considering approximation by solutions of elliptic \n equations, it is \n reasonable to consider as $\\sc\\mbox{F}\\hspace{1.0pt}$, a class of functions defined \n on a nowhere dense \n set ${\\mathbf K}$. In fact, if, on the opposite, ${\\mathbf K}$ possesses \n interior points, it is \n only solutions of the equation that can be approximated by \n solutions. \n So, we are interested in approximating a given continuous \n function $f$ defined \n on a nowhere dense compact\nset ${\\mathbf K}\\subset {\\mathbb R}^{\\mathbf N}$ by solutions of a second order elliptic \nequation. When the \napproximating functions are harmonic,\nand the set ${\\mathbf K}$ is nice, say, a Lipschitz surface, there are many \nresults in this \ndirection, see, e.g. \\cite{Bliedtner}, \\cite{AlSh1}, \n\\cite{Andrievskii}, \\cite{Gardiner Book}, \\cite{Gardiner}, \n\\cite{Gardiner Goldstein}, \\cite{Hausmann}, \\cite{Khav} and many \nmore.\n\n When the conditions on ${\\mathbf K}$ are less restrictive, one can cite \n \\cite{AlSh2}, \\cite{Pavlov}, \\cite{RZ25}. Here, \n one needs to decide, which terms should be used \nto describe properties of the function $f$ in order to determine \nthe rate of \n approximation.\n If we only know that the given function $f$ is continuous on a compact set \n ${\\mathbf K}$, then the \n quality of this continuity, and consequently,\n the quality of approximation, can be described by the modulus \n of continuity of \n $f$. In this direction, in the paper \\cite{RZ25}, the authors \n considered the \n problem on approximating a continuous function $f$ on \n ${\\mathbf K}\\subset{\\mathbb R}^{{\\mathbf N}}$, possessing the continuity\n modulus $\\omega(\\delta)$, by solutions of a second order \n elliptic equation ${\\mathcal L} u=0$ (\\emph{${\\mathcal L}$-harmonic functions}).\n It was established there that if the set \n ${\\mathbf K}$ is \n Ahlfors-David ${\\mathbf N}$-$2$-regular (which means, almost exactly \n speaking, that\n it has one and the same Hausdorff dimension ${\\mathbf N}$-$2$ in any \n neighborhood of any \n of its points), then the function $f$ can be,\n for any $\\delta>0$, approximated in $C({\\mathbf K})$ by a function $v_\\delta$, \n so that \n $|f(x)-v_{\\delta}(x)|\\le c \\omega(\\delta)$ for all $x\\in{\\mathbf K}$,\n the function $v_{\\delta}$ is ${\\mathcal L}$-harmonic in a \n $\\delta$-neighbourhood $\\Kb_{\\delta}$ \n of the set ${\\mathbf K}$, moreover,\n the quality of this function $v_\\delta$ is controlled by $\\delta$, \n namely, $|\\nabla \n v_\\delta(x)|\\le C \\frac{\\omega(\\delta)}{\\delta}$ in $\\Kb_{\\delta}$. This matches the \n general \n principle: the smaller $\\delta$, the better is the aproximation, i.e., the smaller, \n is the \n approximation error, but the worse is the approximating\n function $v_\\delta$: it is ${\\mathcal L}$-harmonic on a smaller set, and \n its gradient may \n grow with $\\delta$ decreasing. Moreover, a converse result was \n established: if a \n continuous function $f$ on ${\\mathbf K}$ can be approximated in the \n above sense, with \n some function $\\omega(\\delta)$, by solutions of a second order \n elliptic differential \n equation, then $f$ possesses the continuity modulus majorated \n by $\\omega(\\delta)$.\n\n It is natural to expect that if we wish to have a better \n approximation (the one better than with \n $O(\\omega(\\delta))$ error), with the same quality of\n the approximating function, we should suppose some better \n properties of the given\n function $f$. If the set ${\\mathbf K}$ were a smooth\n surface (of codimension 2), such 'better' properties would \n naturally involve a \n higher classical smoothness of $f$. However, if we only know \n that the set \n ${\\mathbf K}$ is Ahlfors-David ${\\mathbf N}$-$2$-regular, some other terms \n should be used.\n\n In the literature, there exist methods of defining spaces of \n 'nice' functions on \n arbitrary compacts. One of them is based upon\n describing\n classes of functions via their local approximations by \n polynomials or other\n sufficiently regular functions, see \\cite{Brudny}, \n \\cite{Brudny2}, \n \\cite{Shvartsman}, and many sources afterwards.\n\n So, the expected approximation results should sound like 'if a \n function admits \n local approximation of a certain kind, it admits\n the corresponding quality of global approximation' by \n ${\\mathcal L}$-harmonic \n functions. \n\n This is, in fact, the contents of the present paper. Namely, \n in our main \n result,\n if $f$ is a continuous function on ${\\mathbf K}$, which can, for any \n $\\delta>0$, be \n locally, in a $\\delta$-neighborhood of any point $x\\in{\\mathbf K}$, \n approximated by a function $\\F_{x,\\delta}(y)$ which is a solution \n of the \n second-order elliptic equation \n${\\mathcal L}(y,\\partial_y)u(y)=0$ in a $2\\delta$-neighbourhood of $x$, with \nerror \n$O(\\delta^{\\mathbf r}\\omega(\\delta)), $ ${\\mathbf r}\\ge1$ (with some natural compatibility\n conditions concerning the functions $\\F_{x,\\delta}$ for different values of\n $\\delta$ and different close-lying points\n $x$),\n then $f$ can be approximated on the whole ${\\mathbf K}$, with error of \n the same order, by \n a solution $v_\\delta$ of the same equation in the \n $\\delta$-neighborhood of ${\\mathbf K}$.\nNote that the above compatibility conditions, mentioned in \nparentheses, are \nunavoidable: they are proved to be necessary \nfor the existence of the global approximation. \n\nWhen comparing these results with our previous paper \\cite{RZ25}, \nwhere we \nestablished this kind of properties for ${\\mathbf r}=0$,\n one can notice that an additional restricting condition appears: \n the locally approximating\nfunctions $\\F_{x,\\delta}(y)$ are required here to be solutions of the \nelliptic \nequation, while in \n\\cite{RZ25} no such restriction has been imposed.\nThis restriction is, unfortunately, unavoidable. An example we \npresent in the \npaper demonstrates a function which admits a nice polynomial \nlocal approximation but \ndoes not admit a global approximation by harmonic functions.\n This effect is caused by a visible wildness of the set ${\\mathbf K}$ in \n our example: it is easy to show \n that for a nicer ${\\mathbf K}$, e.g., for a Lipschitz\n surface of codimension 2, such counter-examples are impossible \n and a local \n approximation by smooth functions is sufficient (and, of course, \n necessary)\n for existence of a global approximation by ${\\mathcal L}$-harmonic \n functions. \n\n The elliptic differential operation ${\\mathcal L}(x,\\partial_x)$ is \n supposed to have \n coefficients of certain finite smoothness,\n $C^m(\\Omega)$. The main approximation result, Theorem 1.2, is proved for $m=3$. Under \n additional \n smoothness conditions, the main result can be somewhat\n strengthened: not only the approximating functions $v_\\delta$ \n converge on ${\\mathbf K}$ to \n the initial function $f$, but their derivatives $\\partial^\\alpha \n v_{\\delta}$ (up to \n some order, depending on the smoothness of coefficients of \n ${\\mathcal L}$) converge on \n ${\\mathbf K}$ to some functions $f_{(\\alpha)}$ which can be understood as \n generalized \n derivatives of the given function $f$. The greater $m$, to the higher order these surrogate \n derivatives of $f$ can be defined, see Theorem 1.3. \n\n \\subsection{The main results.1}\\label{Sect.Main. Result1}\n We present here the exact formulation of our main approximation \n result. It is the \n following.\n Let $\\omega(t),$ $t>0$, be a continuity modulus satisfying the \n condition\n \\begin{equation*}\n \\int_0^{\\tau}\\frac{\\omega(t)}{t}dt+\\tau\\int_{\\tau}^\\infty\\frac{\\omega(t)}{t^2}dt \\le c\\omega(\\tau), \\, 0<\\tau<\\infty. \\end{equation*}\nLet, further, ${\\mathbf K}$ be a compact set in ${\\mathbb R}^{\\mathbf N}$, \n${\\mathbf N}$-$2$-Ahlfors-David regular (see, \ne.g., \\cite{David}).\nLet $\\Omega\\supset{\\mathbf K}$ be a bounded open connected set, where a \nformally self-adjoint \nsecond-order elliptic operator \n\\begin{equation*}\n{\\mathcal L} u (x)=-\\sum_{\\mathrm{j},\\mathrm{j}'} \\partial_\\mathrm{j}(a_{\\mathrm{j}\\mathrm{j}'}(x)\\partial_{\\mathrm{j}'}u(x))\\equiv \n-\\nabla\\cdot({\\mathbf a}(x)\\nabla u(x)),\n\\end{equation*}\nwith $C^m$-coefficients $a_{\\mathrm{j}\\mathrm{j}'}$, $m\\ge 3$,\nis defined.\n\nWith the continuity modulus $\\omega$ fixed, for an integer ${\\mathbf r}\\ge0$, \nthe local \n${\\mathcal L}$-H{\\\"o}lder class $\\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$ is defined in \nthe \nfollowing way.\n\\begin{defin}\\label{defin.class}\n The continuous function $f(x)$, $x\\in{\\mathbf K},$ is said to belong to \n$\\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$, if there exist constants \n$\\cb_1=\\cb_1(f)$, \n$\\cb_2=\\cb_2(f)$, such that for any $x\\in{\\mathbf K}$ and any $\\delta,\\, \n0<\\delta\\le \n2\\operatorname{diam\\,}({\\mathbf K})$, there exists \na function $\\F_{x,\\delta}(y)$ defined in the ball $B_\\delta(x)$ such \nthat\n\\begin{equation*} \\Lc_y\\F_{x,\\delta}(y)=0, \\, y\\in B_\\delta(x),\n\\end{equation*}\n\\begin{equation*} |f(y)-\\F_{x,\\delta}(y)|\\le \\cb_1\\delta^{\\mathbf r}\\omega(\\delta), \\, y\\in \n B_\\delta(x)\\cap{\\mathbf K}.\n\\end{equation*}\nFor close-lying points $x_1$, $x_2$, the approximating functions \n should be consistent in the following sense: for some constants \n $\\g_1,\\g_2,$ \n $\\frac18\\le \\g_1\\le 1\\le \\g_2\\le 8,$\n if $\\g_1\\de_1\\le \\de_2\\le \\g_2\\de_1$, given any points \n $x_1,x_2\\in {\\mathbf K}$, such that the \n balls $B_{\\de_1}(x_1), B_{\\de_2}(x_2)$ are not disjoint, the \n inequality\n\\begin{equation}\\label{F2}\n |\\F_{x_1,\\de_1}(y)-\\F_{x_2,\\de_2}(y)|\\le \n \\cb_2\\de_1^{\\mathbf r}\\omega(\\de_1). \n\\end{equation}\nmust hold for all $y\\in B_{\\de_1}(x_1)\\cap B_{\\de_2}(x_2)$,\n\\end{defin}\n\nWe recall the definition of Ahlfors-David regularity. The compact \nset ${\\mathbf K}$ is called 'AD regular' of dimension $\\varkappa$ if for some \nconstants\n$c', c'',$ $00$, be \n locally, in a $\\delta$-neighborhood of any point $x\\in{\\mathbf K}$, \n approximated by a function $\\F_{x,\\delta}(y)$ which is a solution \n of the \n second-order elliptic equation \n${\\mathcal L}(y,\\partial_y)u(y)=0$ in a $2\\delta$-neighbourhood of $x$, with \nerror \n$O(\\delta^{\\mathbf r}\\omega(\\delta)), $ ${\\mathbf r}\\ge1$ (with some natural compatibility\n conditions concerning the functions $\\F_{x,\\delta}$ for different values of\n $\\delta$ and different close-lying points\n $x$),\n then $f$ can be approximated on the whole ${\\mathbf K}$, with error of \n the same order, by \n a solution $v_\\delta$ of the same equation in the \n $\\delta$-neighborhood of ${\\mathbf K}$.\nNote that the above compatibility conditions, mentioned in \nparentheses, are \nunavoidable: they are proved to be necessary \nfor the existence of the global approximation.\n\n\\subsection{The main results.1}\\label{Sect.Main. Result1}\n We present here the exact formulation of our main approximation \n result. It is the \n following.\n Let $\\omega(t),$ $t>0$, be a continuity modulus satisfying the \n condition\n \\begin{equation*}\n \\int_0^{\\tau}\\frac{\\omega(t)}{t}dt+\\tau\\int_{\\tau}^\\infty\\frac{\\omega(t)}{t^2}dt \\le c\\omega(\\tau), \\, 0<\\tau<\\infty. \\end{equation*}\nLet, further, ${\\mathbf K}$ be a compact set in ${\\mathbb R}^{\\mathbf N}$, \n${\\mathbf N}$-$2$-Ahlfors-David regular (see, \ne.g., \\cite{David}).\nLet $\\Omega\\supset{\\mathbf K}$ be a bounded open connected set, where a \nformally self-adjoint \nsecond-order elliptic operator \n\\begin{equation*}\n{\\mathcal L} u (x)=-\\sum_{\\mathrm{j},\\mathrm{j}'} \\partial_\\mathrm{j}(a_{\\mathrm{j}\\mathrm{j}'}(x)\\partial_{\\mathrm{j}'}u(x))\\equiv \n-\\nabla\\cdot({\\mathbf a}(x)\\nabla u(x)),\n\\end{equation*}\nwith $C^m$-coefficients $a_{\\mathrm{j}\\mathrm{j}'}$, $m\\ge 3$,\nis defined.\n\nWith the continuity modulus $\\omega$ fixed, for an integer ${\\mathbf r}\\ge0$, \nthe local \n${\\mathcal L}$-H{\\\"o}lder class $\\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$ is defined in \nthe \nfollowing way.\n\\begin{defin}\\label{defin.class}\n The continuous function $f(x)$, $x\\in{\\mathbf K},$ is said to belong to \n$\\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$, if there exist constants \n$\\cb_1=\\cb_1(f)$, \n$\\cb_2=\\cb_2(f)$, such that for any $x\\in{\\mathbf K}$ and any $\\delta,\\, \n0<\\delta\\le \n2\\operatorname{diam\\,}({\\mathbf K})$, there exists \na function $\\F_{x,\\delta}(y)$ defined in the ball $B_\\delta(x)$ such \nthat\n\\begin{equation*} \\Lc_y\\F_{x,\\delta}(y)=0, \\, y\\in B_\\delta(x),\n\\end{equation*}\n\\begin{equation*} |f(y)-\\F_{x,\\delta}(y)|\\le \\cb_1\\delta^{\\mathbf r}\\omega(\\delta), \\, y\\in \n B_\\delta(x)\\cap{\\mathbf K}.\n\\end{equation*}\nFor close-lying points $x_1$, $x_2$, the approximating functions \n should be consistent in the following sense: for some constants \n $\\g_1,\\g_2,$ \n $\\frac18\\le \\g_1\\le 1\\le \\g_2\\le 8,$\n if $\\g_1\\de_1\\le \\de_2\\le \\g_2\\de_1$, given any points \n $x_1,x_2\\in {\\mathbf K}$, such that the \n balls $B_{\\de_1}(x_1), B_{\\de_2}(x_2)$ are not disjoint, the \n inequality\n\\begin{equation}\\label{F2}\n |\\F_{x_1,\\de_1}(y)-\\F_{x_2,\\de_2}(y)|\\le \n \\cb_2\\de_1^{\\mathbf r}\\omega(\\de_1). \n\\end{equation}\nmust hold for all $y\\in B_{\\de_1}(x_1)\\cap B_{\\de_2}(x_2)$,\n\\end{defin}\n\nWe recall the definition of Ahlfors-David regularity. The compact \nset ${\\mathbf K}$ is called 'AD regular' of dimension $\\varkappa$ if for some \nconstants\n$c', c'',$ $00$, be \n locally, in a $\\delta$-neighborhood of any point $x\\in{\\mathbf K}$, \n approximated by a function $\\F_{x,\\delta}(y)$ which is a solution \n of the \n second-order elliptic equation \n${\\mathcal L}(y,\\partial_y)u(y)=0$ in a $2\\delta$-neighbourhood of $x$, with \nerror \n$O(\\delta^{\\mathbf r}\\omega(\\delta)), $ ${\\mathbf r}\\ge1$ (with some natural compatibility\n conditions concerning the functions $\\F_{x,\\delta}$ for different values of\n $\\delta$ and different close-lying points\n $x$),\n then $f$ can be approximated on the whole ${\\mathbf K}$, with error of \n the same order, by \n a solution $v_\\delta$ of the same equation in the \n $\\delta$-neighborhood of ${\\mathbf K}$.\nNote that the above compatibility conditions, mentioned in \nparentheses, are \nunavoidable: they are proved to be necessary \nfor the existence of the global approximation.\n\n\\subsection{The main results.1}\\label{Sect.Main. Result1}\n We present here the exact formulation of our main approximation \n result. It is the \n following.\n Let $\\omega(t),$ $t>0$, be a continuity modulus satisfying the \n condition\n \\begin{equation*}\n \\int_0^{\\tau}\\frac{\\omega(t)}{t}dt+\\tau\\int_{\\tau}^\\infty\\frac{\\omega(t)}{t^2}dt \\le c\\omega(\\tau), \\, 0<\\tau<\\infty. \\end{equation*}\nLet, further, ${\\mathbf K}$ be a compact set in ${\\mathbb R}^{\\mathbf N}$, \n${\\mathbf N}$-$2$-Ahlfors-David regular (see, \ne.g., \\cite{David}).\nLet $\\Omega\\supset{\\mathbf K}$ be a bounded open connected set, where a \nformally self-adjoint \nsecond-order elliptic operator \n\\begin{equation*}\n{\\mathcal L} u (x)=-\\sum_{\\mathrm{j},\\mathrm{j}'} \\partial_\\mathrm{j}(a_{\\mathrm{j}\\mathrm{j}'}(x)\\partial_{\\mathrm{j}'}u(x))\\equiv \n-\\nabla\\cdot({\\mathbf a}(x)\\nabla u(x)),\n\\end{equation*}\nwith $C^m$-coefficients $a_{\\mathrm{j}\\mathrm{j}'}$, $m\\ge 3$,\nis defined.\n\nWith the continuity modulus $\\omega$ fixed, for an integer ${\\mathbf r}\\ge0$, \nthe local \n${\\mathcal L}$-H{\\\"o}lder class $\\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$ is defined in \nthe \nfollowing way.\n\\begin{defin}\\label{defin.class}\n The continuous function $f(x)$, $x\\in{\\mathbf K},$ is said to belong to \n$\\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$, if there exist constants \n$\\cb_1=\\cb_1(f)$, \n$\\cb_2=\\cb_2(f)$, such that for any $x\\in{\\mathbf K}$ and any $\\delta,\\, \n0<\\delta\\le \n2\\operatorname{diam\\,}({\\mathbf K})$, there exists \na function $\\F_{x,\\delta}(y)$ defined in the ball $B_\\delta(x)$ such \nthat\n\\begin{equation*} \\Lc_y\\F_{x,\\delta}(y)=0, \\, y\\in B_\\delta(x),\n\\end{equation*}\n\\begin{equation*} |f(y)-\\F_{x,\\delta}(y)|\\le \\cb_1\\delta^{\\mathbf r}\\omega(\\delta), \\, y\\in \n B_\\delta(x)\\cap{\\mathbf K}.\n\\end{equation*}\nFor close-lying points $x_1$, $x_2$, the approximating functions \n should be consistent in the following sense: for some constants \n $\\g_1,\\g_2,$ \n $\\frac18\\le \\g_1\\le 1\\le \\g_2\\le 8,$\n if $\\g_1\\de_1\\le \\de_2\\le \\g_2\\de_1$, given any points \n $x_1,x_2\\in {\\mathbf K}$, such that the \n balls $B_{\\de_1}(x_1), B_{\\de_2}(x_2)$ are not disjoint, the \n inequality\n\\begin{equation}\\label{F2}\n |\\F_{x_1,\\de_1}(y)-\\F_{x_2,\\de_2}(y)|\\le \n \\cb_2\\de_1^{\\mathbf r}\\omega(\\de_1). \n\\end{equation}\nmust hold for all $y\\in B_{\\de_1}(x_1)\\cap B_{\\de_2}(x_2)$,\n\\end{defin}\n\nWe recall the definition of Ahlfors-David regularity. The compact \nset ${\\mathbf K}$ is called 'AD regular' of dimension $\\varkappa$ if for some \nconstants\n$c', c'',$ $0 2 \\diam\\Kb,$ we set $ \n\\fT_0(y)=0.$\n The function $\\fT_0,$ thus defined, is piecewise $\\Lc$-harmonic, \n however it may \n be discontinuous along the boundaries\n of cubes in $\\Qc$\n in an uncontrollable manner. We use now the averaging kernel \n $K(x,y)$, to be \n constructed later in this section,\n and set\n \\begin{equation}\\label{function f}\n f_0(x)=\\int_{\\R^{\\Nb}} \\fT_0(y) K(x,y) dy.\n \\end{equation}\nWe will show that this function is a continuous extension of $f$ \nto a neighborhood \nof $K,$\n with controlled behavior of derivatives when approaching $\\Kb$; the function $f_0(x)$ will \n later serve for \n constructing the required \n approximation.\n\\subsection{Construction of the kernel \n$K$}\\label{subs.construction K} Our \nreasoning will be constructive. The first \nstep will be describing a proper averaging kernel. We denote by \n$\\dn(x)$ \nthe \\emph{regularized} distance from the point $x\\in \\Ct\\Kb$ \nto $\\Kb$, namely $\\dn(x)\\in C^3(\\Ct\\Kb)$, $c\\,\\dist(x,\\Kb)\\le \n\\dn(x)\\le \nc'\\,\\dist(x,\\Kb),\\, c'<\\frac14,$ $|\\grad^k\\dn(x)|\\le c \n\\dn(x)^{1-k}, \\, k=1,2,3.$ \nLet $\\hb(t)$ be a function in \n$ C^{\\infty}(\\overline{\\R_+}),$ $\\hb(t)\\ge0,$ $\\supp \n\\hb\\subset[\\frac12,1]$, normalized by $\\int_0^1 \\hb(t)dt=1$. The scaled \nfunction \n$h_r(t)=r^{-1}\\hb(t/r)$ is normalized in such a way that \n$\\int_0^\\infty h_r(t)dt=1.$ \nFurther on, for $x\\in \\Ct\\Kb,\\,t< r=\\dn(x) $, we denote by \n$S_t(x)$ the sphere \n$\\{y:|y-x|=t\\}$, \n and by $B_t(x)$ the corresponding open ball;\n they do not touch $\\Kb$, moreover, they are on a controlled \n distance from $\\Kb$.\n\n\\begin{gather*}\n \\|\\grad\\left(\\F_{x,2^{k-1}\\de}-\\F_{x,2^{k}\\de}\\right)\\|_{C\\left(\\overline{B_{\\frac{\\de}{2}}(x)}\\right)}\\le\\\\\\nonumber\n C (2^{k}\\de)^{-1}\\times 2^{k\\rb}\\de^{\\rb}\\om(2^k)=\n c 2^{k(\\rb-1)}\\de^{\\rb-1}\\om(2^k\\de),\\, \\rb\\ge 1.\n \\end{gather*}\n It follows now that\n \\begin{gather*}\n \\|\\nabla \\F_{x,\\de}\n \\|_{C(\\overline{B_{\\frac{\\de}{2}}(x)})}\\le \n \\\\\\nonumber\n \\sum_{k=1}^{N}\\|\\grad(\\F_{x,2^{k-1}\\de}-\\F_{x,2^{k}\\de})\\|_{C(\\overline{B_{\\frac{\\de}{2}}(x)})}+\n \\|\\grad\\F_{x,2^N\\de}\\|_{C(\\overline{B_{\\frac{\\de}{2}}(x)})}\\le \n \\\\\\nonumber \nc\\sum_{k=1}^{N}2^{k(\\rb-1)}\\de^{\\rb-1} \\om(2^k\\de)+c\\le \n\\de^{\\rb-1}\\int_0^N \n2^{s(\\rb-1)}\\om(2^s\\de)ds+c=\n\\\\\\nonumber c\\de^{\\rb-1}\\int_1^{2^{N}}t^{\\rb-1}\\om(\\de \nt)\\frac{dt}{t}+c=\nc\\de^{\\rb-1}\\left(\\de^{-1}\\right)^{\\rb-1}\\int_0^{2^N\\de}\\tau^{\\rb-2}\\om(\\z)d\\tau+c\\\\\\nonumber\n\\le C(\\diam \\Kb)^{\\rb-1}\\int_0^{2\\diam\\Kb}\\om(\\tau)\\tau^{-1}d\\z+c\\le \nC.\n \\end{gather*} \n\\end{proof}\n \\subsection{Estimating $\\Lc f_0(x)$}\\label{Sect.estimat.f0(x)}\n Now, using our estimates for derivatives of the kernel \n $K(x,y)$, \n obtained in the previous section, we establish estimates for \n the function $f_0(x)$ constructed \n in Section 3.1, see \\eqref{function f},\n and for the result of the action of the operator $\\Lc$ on \n this function.\n For a fixed point $x_0\\in\\Ct \\Kb$, we consider\n the open cube $Q$ in the Whitney cover $\\Qc$, whose closure contains $x_0$. Let $x_{Q}$ be \n the point in $\\Kb$, closest to the\n center of this cube (or one of such points).\n Recall that the function $\\fT_0(x)$ (defined in \\eqref{f-tilde}) \n equals $0$ on the boundary of $Q$.\n By construction, $\\Lc_x\\F_{x_Q,2\\de(Q)}(x)=0$, for $x$ \n in the ball $B_{2\\de(Q)}(x_Q)$, in particular,\n this holds in a small neighborhood of the ball \n $\\overline{{B_{\\dn(x_0)}(x_0)}}$. Therefore, for $x$ in a small \n neighborhood of \n the point $x_0$, \n we obtain, recalling the definition of the kernel $K(x,y)$ \n (which acts as an \n $\\Lc$-replacement for the mean value kernel):\n\\begin{gather}\\label{18.p11}\n |\\Lc_x f_0(x)|= |\\Lc_x(f_0(x)-\\F_{x_Q,2\\de(Q)}(x))|=\\\\\\nonumber\n \\left|\\Lc_x\\left(\\int_{\\R^\\Nb} \n \\fT_0(y)K(x,y)dy-\\int_{\\R^{\\Nb}}\\F_{x_Q,2\\de(Q)}(y)K(x,y)dy\\right)\\right|=\\\\\\nonumber\n\\left| \\Lc_x\\left( \n\\int_{\\R^\\Nb}(\\fT_0(y)-\\F_{x_Q,2\\de(Q)}(y))K(x,y) dy \n\\right)\\right|\\le \\\\\\nonumber\nC\\int\\limits_{{B_{\\dn(x_0)+\\ve}(x_0)}}|\\fT_0(y)-\\F_{x_Q,2\\de(Q)}(y)||\\nabla^2_{xx}K(x,y)|dy\\le\\\\\\nonumber\nC \n\\dn(x)^{-\\Nb-2}\\int\\limits_{{B_{\\dn(x_0)+\\ve}(x_0)}}|\\fT_0(y)-\\F_{x_Q,2\\de(Q)}(y)|dy,\n\\end{gather}\n using, on the last step, our estimates for derivatives of the \n kernel $K(x,y)$. We apply estimate \\eqref{17} now. For\n $x\\in \\overline{B_{\\dn(x_0)+\\ve}(x_0)},$ the \ndifference $\\fT_0(y)-\\F_{x_Q,2\\de(Q)}(y)$ has the form \n $\\F_{x_{Q_1},2\\de(Q_1)}(y)-\\F_{x_Q,\\de(Q)}(y)$, for a certain \n cube $Q_1$, and satisfies \n the conditions of the main theorem with parameters $\\de(Q), \n \\de(Q_1)$. Therefore,\n\\begin{equation}\\label{19}\n|\\fT_0(y)-\\F_{x_Q,2\\de(Q)}(y)|=|\\F_{x_{Q_1},2\\de(Q_1)}(y)-\\F_{x_Q,\\de(Q)}(y)|\\le \nc \n\\dn(x_0)^{\\rb}\\om(\\dn(x_0)).\n\\end{equation}\nWe set $x=x_0$ in \\eqref{18.p11}, \\eqref{19} and obtain the \nrequired estimate for \n$\\Lc f_0$ outside $\\Kb$:\n\\begin{equation}\\label{20}\n|(\\Lc f_0)(x_0)|\\le c \\dn^{\\rb-2}(x_0)\\om(\\dn(x_0)).\n\\end{equation}\n We stress here that by \\eqref{20}, the larger $\\rb$ in the conditions of the Theorem, the faster the function $\\Lc f_0(x)$ \n decays as the point $x$ approaches $\\Kb$. This fact will be essentially used further on.\n\n\\begin{thm}\\label{counterex} It is impossible to approximate \n$f(x)$ in the sense \nof Theorem \\ref{MainTheorem} with $\\rb=2,$ and \n$\\om(\\de)=\\de^{\\s},$ $0<\\s<1$, by \nharmonic functions, this means,\n by solutions of the equation $\\Lc v_\\de\\equiv-\\D v_\\de=0$,\n\\end{thm}\nIn other words, for such a wild set $\\Kb,$ one cannot approximate on $\\Kb$ the non-harmonic function $f(x)$ by harmonic functions, even locally. \n\\begin{proof}\nSuppose that the approximation in question is possible, thus, for \nany \n$\\de\\in(0,1),$ there exists a function\n$v_\\de$ such that, \n\\begin{equation}\\label{49}\n |v_\\de(x)-f(x)|\\le \\mathbbm{c}\\de^{2+\\s}, \\, x\\in\\Kb,\n\\end{equation}\n\\begin{equation}\\label{50}\n |v_\\de(x)-v_{\\frac{\\de}{2}}(x)|\\le \\mathbbm{c}\\de^{2+\\s}, \\, \n x\\in\\Kb_{\\frac{\\de}{2}},\n\\end{equation}\nwith some $\\mathbbm{c}$ not depending on $\\de,$ and\n\\begin{equation}\\label{51}\n \\D v_{\\de}(x)=0, \\, x\\in\\Kb_{\\de}.\n\\end{equation}\n We establish the following property.\n \\begin{lem}\\label{lem.counter} Under the assumptions \n \\eqref{49}-\\eqref{51}, the \n function $v_{\\de}$ must\n satisfy the estimate: \n \\begin{gather}\\label{52}\n |\\nabla^3 v_{2\\de_{\\pk}}(x)|\\le c \n \\mathbbm{c}\\de_{\\pk}^{\\s-1},\\,\\\\\\nonumber \n x\\in U_{\\pk}:=(\\tb_{\\pk}+B_{\\de_{\\pk}}(\\mathbb{0}), \\, \n \\tb_{\\pk}=(\\frac{1}{2}(2^{-\\pk-1}+2^{-\\pk}),0,\\dots,0), \\, \n \\de_{\\pk}=2^{-\\pk-2}.\n \\end{gather}\n \\end{lem}\n \\begin{proof}\n To prove \\eqref{52}, we denote by $\\f_k(x), \\, x\\in U_{\\pk}$, \n the function \n $\\f_k(x)=v_{2^k\\de_m}(x)-v_{2^{k+1}\\de_m}(x)$,\\,\n $k=1,\\dots,N$, where $N$ is chosen so that $1<2^N\\de\\le 2$. \n Using this function, \n we can represent $v_{2\\de_{\\pk}}$ and its order 3 gradient as", "post_theorem_intro_text_len": 5943, "post_theorem_intro_text": "\\subsection{The ideas of the proof} The proof of the main \ntheorem is fairly \ntechnical, therefore we consider it reasonable to explain here its \nstructure.\n\n Given \na function $f\\in \\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$ on the compact set \n${\\mathbf K}$, we construct its \nspecial extension $f_0$ to a fixed neighborhood $\\Omega$ of ${\\mathbf K}$ \n(the particular form of this\n neighbourhood is not essential, and we suppose further on that it is the \n unit ball containing the set ${\\mathbf K}$ which is contained in the concentric ball with radius $\\frac13$). For this function \n$f_0$, using the Green function $ G^{\\circ}(x,y)$ of the operator ${\\mathcal L}$ \nin $\\Omega,$ the \nintegral representation is established:\n\\begin{equation}\\label{intro.repres}\n f_0(x)=\\int_{\\Omega}{\\mathcal L} f_0(y)G^{\\circ}(x,y)dy, \\, x\\in \\Omega.\n\\end{equation}\nAlthough this representation looks quite usual if $f_0$ is \nsufficiently smooth, \nthis is not the case for \\emph{our} function $f_0$ for which the \nderivatives may \nbehave badly when approaching ${\\mathbf K}$. Therefore, to justify \n\\eqref{intro.repres}, we need a detailed control of the behavior \nof ${\\mathcal L} f_0(x)$ \nand of derivatives of $f_0(x)$ near ${\\mathbf K}$. Obtaining this \ncontrol requires \ncomplicated estimates of the Green function $G_{x,r}(x,y)$ for ${\\mathcal L}$ \nin balls \n$B_r(x)$ centered at $x$, together with their derivatives, up to \nthe third \norder, in the variables $x,y$, as well as in the additional variable $\\varsigma$ on which the operator ${\\mathcal L}$ depends as a parameter. \nUnder the condition \nof a sufficient smoothness of coefficients of the operation ${\\mathcal L},$ \nwe derive some of these estimates directly, using Schauder-type \n approach, and borrow the other ones \n from the results by Ju.~Krasovskii \n \\cite{Kras.Sing}, and M.~Gr{\\\"u}ter--K.-O.~Widman \\cite{Widman}. \nFinally, having established the representation \n\\eqref{intro.repres}, we define \nthe approximation function $v_\\delta(x)$, looked for, by the integral\n\\begin{equation*}\n v_{\\delta}(x)=\\int_{\\Omega\\setminus \\Kb_{\\delta}}{\\mathcal L} f_0(y)G^{\\circ}(x,y)dy,\n\\end{equation*}\nwith addition of a collection of several compensatory ${\\mathcal L}$-harmonic terms of a simpler nature, see \\eqref{35}.\n The fact that $v_{\\delta}$ is \n${\\mathcal L}$-harmonic in $\\Kb_{\\delta}$ is obvious, it follows from the \ndefinition of the \nGreen function $G^{\\circ}(x,y)$, while the estimates producing the \nquality of the \napproximation follow from our estimates for the function \n$f_0(x)$ and its \nderivatives.\n\n\\subsection{The main result. 2}\\label{Sect.Main. Result2}\nThe second theorem describes the properties of the approximating \nfunctions $v_\\delta:$ \ntheir derivatives, up to a prescribed order \n$k\\le{\\mathbf r}+1$ can be controlled. Moreover, we \ncan define in a consistent way the generalized derivatives $f_{(\\alpha)}$ of the \ninitial function \n$f$ on ${\\mathbf K}$,\nso that the derivatives of $v_{\\delta}$ approximate these derivatives \nof $f$. This \nproperty requires a certain additional smoothness of coefficients \nof the operator \n${\\mathcal L}$. \n\\begin{thm}\\label{Thm.quality} Suppose that ${\\mathbf r}\\ge 1$ and the \ncoefficients ${\\mathbf a}(x)=(a_{\\mathrm{j}\\mathrm{j}'}(x))_{\\mathrm{j},\\mathrm{j}'\\le {\\mathbf N}}$ belong to \n$C^{k_0+3}(\\Omega)$ for a certain $k_0\\le{\\mathbf r}$. Let the \nfunction $f$, defined on \nthe compact set ${\\mathbf K}$, belong to the class $H_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$ and \n$v_\\delta$ be its \napproximation, as in \\eqref{appr.function}. Then derivatives of \n$v_\\delta$ satisfy\n\\begin{equation}\\label{appr.deriv}\n \\|\\nabla^{k_0+1}v_\\delta\\|_{\\Kb_{\\delta/2}}\\le c\\frac{\\omega(\\delta)}{\\delta}.\n\\end{equation}\nmoreover, surrogate derivatives $f_{(\\alpha)}(x)$ can be defined, so that\n\\begin{equation}\\label{appr.deriv.2}\n |f_{(\\alpha)}(x)-\\partial^\\alpha v_{\\delta}(x)|\\le C \n \\delta^{{\\mathbf r}-|\\alpha|}\\omega(\\delta),\\, x\\in{\\mathbf K}, \\, \n 1\\le |\\alpha|\\le k_0.\n\\end{equation}\n\\end{thm}\n\\subsection{Structure of the paper} We start in Sect. \\ref{Sect.Prep} by \npresenting general\nmaterial concerning certain geometry considerations, and formulate estimates \nof important integrals used in further analysis and of derivatives of the Green function, \nincluding the results of \\cite{Kras.Sing} and \\cite{Widman},\n In Sect. 3, we introduce the \n averaging kernel $K(x,y)$ and prove \n estimates of its derivatives. This is the most technical part of \n the paper. Next, in Sect. 4, we construct the\n extension function $f_0$, derive its important properties and \n prove its integral representation, which\n results in presenting the required approximation of the given function \n $f(x)\\in H_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$, thus proving Theorem \n \\ref{MainTheorem}. In Sect 5, we \n discuss generalized derivatives of the function $f$, and prove \n Theorem \\ref{Thm.quality}. Then, in Sect.6, we present the \n example\n showing that for a wild set ${\\mathbf K}$, the condition on local \n approximation cannot, generally, be relaxed.\n\n Proofs of our estimates for derivatives of the Green function and of important integral inequalities\n are placed \n in the Appendix.\n\n\\subsection{Conventions}In the course of the paper, we denote by \nthe same symbol \n$c$ or $C$ various constants whose particular value is of no \nimportance, as long as \nthis does not cause confusion; sometimes, subscripts or superscripts \nare used in \norder to distinguish between such constants in the same formula. \nMore important \nconstants may be highlighted by a different font. By \n$f'_x=\\partial_x f=\\nabla_x f$ we denote the $x$-gradient of a \nfunction $f$;\nfor a vector function $F$, $\\nabla_x F$ stands for the Jacobi \nmatrix of $F$. The symbol $| \\cdot |$ denotes the Euclidean norm of \nthe vector involved, $\\mathbf{E}$ denotes the unit matrix.", "sketch": "The introduction explains the structure of the proof of Theorem~\\ref{MainTheorem} as follows.\n\n1. Start with $f\\in \\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$ and construct a “special extension” $f_0$ to a fixed neighborhood $\\Omega$ of ${\\mathbf K}$.\n\n2. For $f_0$, use the Green function $G^{\\circ}(x,y)$ of ${\\mathcal L}$ in $\\Omega$ to establish the integral representation\n\\[\n f_0(x)=\\int_{\\Omega}{\\mathcal L} f_0(y)G^{\\circ}(x,y)\\,dy, \\qquad x\\in \\Omega.\\tag{\\ref{intro.repres}}\n\\]\nBecause $f_0$ may have derivatives that “behave badly when approaching ${\\mathbf K}$,” the representation requires “detailed control” of ${\\mathcal L}f_0(x)$ and derivatives of $f_0(x)$ near ${\\mathbf K}$.\n\n3. Obtain this control via “complicated estimates” for the Green function $G_{x,r}(x,y)$ in balls $B_r(x)$ and its derivatives “up to the third order” in $x,y$, and also in the parameter variable $\\varsigma$. Some estimates are derived “using Schauder-type approach,” and others are taken from Ju.~Krasovskii \\cite{Kras.Sing} and M.~Gr\\\"uter--K.-O.~Widman \\cite{Widman}.\n\n4. Define the approximants by\n\\[\n v_{\\delta}(x)=\\int_{\\Omega\\setminus \\Kb_{\\delta}}{\\mathcal L} f_0(y)G^{\\circ}(x,y)\\,dy,\n\\]\n“with addition of a collection of several compensatory ${\\mathcal L}$-harmonic terms of a simpler nature” (see \\eqref{35}). Then “the fact that $v_{\\delta}$ is ${\\mathcal L}$-harmonic in $\\Kb_{\\delta}$ is obvious” from the definition of $G^{\\circ}(x,y)$.\n\n5. The estimates giving the approximation quality (the bounds in \\eqref{appr.function}) “follow from our estimates for the function $f_0(x)$ and its derivatives.”\n\nThe paper outline reiterates that Sect.~4 constructs $f_0$, proves the integral representation, and thereby “present[s] the required approximation… thus proving Theorem~\\ref{MainTheorem}.”", "expanded_sketch": "The introduction explains the structure of the proof of the main theorem as follows.\n\n1. Start with $f\\in \\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$ and construct a “special extension” $f_0$ to a fixed neighborhood $\\Omega$ of ${\\mathbf K}$.\n\n2. For $f_0$, use the Green function $G^{\\circ}(x,y)$ of ${\\mathcal L}$ in $\\Omega$ to establish the integral representation\n\\begin{equation}\\label{intro.repres}\n f_0(x)=\\int_{\\Om}\\Lc f_0(y)G^{\\circ}(x,y)dy, \\, x\\in \\Om.\n\\end{equation}\nBecause $f_0$ may have derivatives that “behave badly when approaching ${\\mathbf K}$,” the representation requires “detailed control” of ${\\mathcal L}f_0(x)$ and derivatives of $f_0(x)$ near ${\\mathbf K}$.\n\n3. Obtain this control via “complicated estimates” for the Green function $G_{x,r}(x,y)$ in balls $B_r(x)$ and its derivatives “up to the third order” in $x,y$, and also in the parameter variable $\\varsigma$. Some estimates are derived “using Schauder-type approach,” and others are taken from Ju.~Krasovskii \\cite{Kras.Sing} and M.~Gr\\\"uter--K.-O.~Widman \\cite{Widman}.\n\n4. Define the approximants by\n\\[\n v_{\\delta}(x)=\\int_{\\Omega\\setminus \\Kb_{\\delta}}{\\mathcal L} f_0(y)G^{\\circ}(x,y)\\,dy,\n\\]\n“with addition of a collection of several compensatory ${\\mathcal L}$-harmonic terms of a simpler nature”, namely (as indicated by the reference)\n\\begin{equation}\\label{35}\n v_{\\de}(x)=c_\\Nb\\int\\limits_{\\Om\\setminus \\Kb'_\\de}G(x,y)\\Lc \n f_0(y)dy+\\sum_{\\n=1}^{N} F_\\n(x).\n\\end{equation}\nThen “the fact that $v_{\\delta}$ is ${\\mathcal L}$-harmonic in $\\Kb_{\\delta}$ is obvious” from the definition of $G^{\\circ}(x,y)$.\n\n5. The estimates giving the approximation quality (the bounds in\n\\begin{gather}\\label{appr.function}\n \\Lc_y v_\\de(y)=0, \\, y\\in \\Kb_{\\de};\\\\\\nonumber\n |v_\\de(x)-f(x)|\\le \\mathbbm{c} \\de^\\rb\\om(\\de), \\, x\\in \n \\Kb;\\\\\\nonumber\n |v_\\de(y)-v_{\\de/2}(y)|\\le \\mathbbm{c}\\de^\\rb\\om(\\de), \\, \n y\\in\\Kb_{\\frac{\\de}{2}}.\n\\end{gather}\n) “follow from our estimates for the function $f_0(x)$ and its derivatives.”\n\nThe paper outline reiterates that next it constructs $f_0$, proves the integral representation, and thereby “present[s] the required approximation… thus proving the main theorem.”", "expanded_theorem": "\\label{MainTheorem}\n Let ${\\mathbf K}$ be ${\\mathbf N}$-$2$-AD regular. Suppose that the coefficients of the operator ${\\mathcal L}$ belong to $C^3$. Then function \n $f$ defined on ${\\mathbf K}$ \n belongs to the class $\\Hc_{{\\mathcal L}}^{{\\mathbf r}+\\omega}({\\mathbf K})$ if \nand only if\nfor any $\\delta< \\frac14\\operatorname{diam\\,}({\\mathbf K})$, there exists an approximating \nfunction\n $v_\\delta(x), \\, x\\in\\Kb_\\delta,$ such that, with some constant\n$\\mathbbm{c}>0$,\n\\begin{gather}\\label{appr.function}\n \\Lc_y v_\\delta(y)=0, \\, y\\in \\Kb_{\\delta};\\\\\\nonumber\n |v_\\delta(x)-f(x)|\\le \\mathbbm{c} \\delta^{\\mathbf r}\\omega(\\delta), \\, x\\in \n {\\mathbf K};\\\\\\nonumber\n |v_\\delta(y)-v_{\\delta/2}(y)|\\le \\mathbbm{c}\\delta^{\\mathbf r}\\omega(\\delta), \\, \n y\\in\\Kb_{\\frac{\\delta}{2}}.\n\\end{gather}", "theorem_type": ["Biconditional or Equivalence", "Universal–Existential"], "mcq": {"question": "Let K ⊂ R^N be a compact (N−2)-Ahlfors–David regular set, let K_δ := {x : dist(x,K) < δ}, and let L be a formally self-adjoint second-order elliptic operator L u(x) = −∇·(a(x)∇u(x)) with C^3 coefficients. Assume ω is a continuity modulus satisfying ∫_0^τ ω(t)/t dt + τ∫_τ^∞ ω(t)/t^2 dt ≤ c ω(τ) for all τ > 0. For an integer r ≥ 0, the local L-Hölder class H_L^{r+ω}(K) consists of continuous functions f on K for which there exist constants c1,c2 such that for every x ∈ K and 0 < δ ≤ 2 diam(K), there is a function F_{x,δ} on B_δ(x) with L_y F_{x,δ}(y)=0 in B_δ(x), |f(y)−F_{x,δ}(y)| ≤ c1 δ^r ω(δ) on B_δ(x)∩K, and whenever x1,x2 ∈ K and comparable radii δ1,δ2 have intersecting balls, one has |F_{x1,δ1}(y)−F_{x2,δ2}(y)| ≤ c2 δ1^r ω(δ1) on the overlap. Which statement is equivalent to f ∈ H_L^{r+ω}(K)?", "correct_choice": {"label": "A", "text": "For every δ < (1/4) diam(K), there exists a function v_δ defined on K_δ such that, for some constant c > 0 independent of δ, L_y v_δ(y)=0 for all y ∈ K_δ, |v_δ(x)−f(x)| ≤ c δ^r ω(δ) for all x ∈ K, and |v_δ(y)−v_{δ/2}(y)| ≤ c δ^r ω(δ) for all y ∈ K_{δ/2}."}, "choices": [{"label": "B", "text": "For every < (1/4)\\operatorname{diam}(K), there exists a function v_\\delta defined on K_{2\\delta} such that, for some constant \\ud835\\udd20 > 0 independent of \\delta, L_y v_\\delta(y)=0 for all y \\in K_{2\\delta}, |v_\\delta(x)-f(x)| \\le \\ud835\\udd20 ^r \\omega(\\delta) for all x \\in K, and |v_\\delta(y)-v_{\\delta/2}(y)| \\le \\ud835\\udd20 ^r \\omega(\\delta) for all y \\in K_\\delta."}, {"label": "C", "text": "For every < (1/4)\\operatorname{diam}(K), there exists a function v_\\delta defined on K_\\delta such that, for some constant \\ud835\\udd20 > 0 independent of \\delta, L_y v_\\delta(y)=0 for all y \\in K_\\delta and |v_\\delta(x)-f(x)| \\le \\ud835\\udd20 ^r \\omega(\\delta) for all x \\in K."}, {"label": "D", "text": "For every < (1/4)\\operatorname{diam}(K), there exists a function v_\\delta defined on K_\\delta such that L_y v_\\delta(y)=0 for all y \\in K_\\delta, |v_\\delta(x)-f(x)| \\le \\ud835\\udd20_\\delta ^r \\omega(\\delta) for all x \\in K, and |v_\\delta(y)-v_{\\delta/2}(y)| \\le \\ud835\\udd20_\\delta ^r \\omega(\\delta) for all y \\in K_{\\delta/2}, where \\ud835\\udd20_\\delta>0 may depend on \\delta."}, {"label": "E", "text": "For every < (1/4)\\operatorname{diam}(K), there exists a function v_\\delta defined on K_\\delta such that, for some constant \\ud835\\udd20 > 0 independent of \\delta, L_y v_\\delta(y)=0 for all y \\in K_\\delta, |v_\\delta(x)-f(x)| \\le \\ud835\\udd20 ^r \\omega(\\delta) for all x \\in K, and |v_\\delta(y)-v_{\\eta}(y)| \\le \\ud835\\udd20 ^r \\omega(\\delta) for all y \\in K_{\\min\\{\\delta,\\eta\\}} and all 0<\\eta\\le \\delta."}], "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": "harmonicity/overlap neighborhood scale K_\\delta vs K_{2\\delta} and K_{\\delta/2} vs K_\\delta", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the inter-scale consistency estimate |v_\\delta-v_{\\delta/2}|", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "uniform constant independent of \\delta", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "specific dyadic comparison with v_{\\delta/2}", "template_used": "wildcard"}]}} +{"id": "2601.04747v1", "paper_link": "http://arxiv.org/abs/2601.04747v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm:main-intro}\n\tLet $\\mathbf{V}$ be a pseudovariety of semigroups.\n\tThe following are equivalent.\n\t\\begin{bracketenumerate}\n\t\t\\item The pseudovariety $\\mathbf{V}$ affords efficient compression.\n\t\t\\item The pseudovariety $\\mathbf{V}$ contains neither $\\mathbf{LRB}$, $\\mathbf{RRB}$, nor $\\mathbf{T}$.\n\t\t\\item The pseudovariety $\\mathbf{V}$ admits straight-line programs of length $\\mathcal{O}(\\smash{\\log}^2 N)$.\n\t\\end{bracketenumerate}\n Furthermore, if all groups in $\\mathbf{V}$ are solvable, then the above are equivalent to $\\mathbf{V}$ admitting straight-line programs of length $\\mathcal{O}(\\log N)$ as well as of width $\\mathcal{O}(1)$ and length $\\mathcal{O}(\\polylog N)$.", "start_pos": 14862, "end_pos": 15543, "label": "thm:main-intro"}, "ref_dict": {"thm:main-intro": "\\begin{theorem}\\label{thm:main-intro}\n\tLet $\\pv{V}$ be a pseudovariety of semigroups.\n\tThe following are equivalent.\n\t\\begin{bracketenumerate}\n\t\t\\item The pseudovariety $\\pv{V}$ affords efficient compression.\n\t\t\\item The pseudovariety $\\pv{V}$ contains neither $\\pv{LRB}$, $\\pv{RRB}$, nor $\\pv{T}$.\n\t\t\\item The pseudovariety $\\pv{V}$ admits straight-line programs of length $\\mathcal{O}(\\smash{\\log}^2 N)$.\n\t\\end{bracketenumerate}\n Furthermore, if all groups in $\\pv{V}$ are solvable, then the above are equivalent to $\\pv{V}$ admitting straight-line programs of length $\\mathcal{O}(\\log N)$ as well as of width $\\mathcal{O}(1)$ and length $\\mathcal{O}(\\polylog N)$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 4704, "pre_theorem_intro_text": "The \\emph{membership problem} asks, given a (finite) algebraic structure $S$, a set $\\Sigma \\subseteq S$, and a target element $t \\in S$, whether $t$ belongs to the substructure of $S$ generated by $\\Sigma$. \nThinking of groups, semigroups, or vector spaces, this is a very fundamental problem in computational algebra with many applications.\nFor permutation groups, Sims gave an efficient solution already in 1967 \\cite{Sims67}, later refined to an \\NC algorithm by Babai, Luks, and Seress \\cite{BabaiLS87}.\nFor transformation semigroups, Kozen \\cite{koz77} showed in 1977 that the problem is \\PSPACE-complete, as hard as intersection non-emptiness for deterministic finite automata (DFAs). \n\nAs a different variant, the membership problem \\dMemb[CT]{} in the \\emph{Cayley table model} was introduced by Jones, Lien, and Laaser \\cite{JonesLL76}, where the semigroup is given by its multiplication table; here the problem is \\NL-complete.\nFor groups, Barrington and McKenzie \\cite{BarringtonM91} showed that $\\dMemb[CT]{\\mathbf{G}}$ can be solved in \\LOGSPACE with an oracle to undirected graph reachability \\cite{Reingold08}, and conjectured it might be $\\LOGSPACE$-hard.\nFleischer \\cite{Fleischer19diss,Fleischer22} refuted the latter (under \\ACz-reductions) by placing the problem in \\NPOLYLOGTIME. \nHis proof is based on \\emph{straight-line programs} (algebraic circuits or context-free grammars producing precisely one word), a tool central in data compression~\\cite{KiefferY00,Bannai16}, algebraic complexity~\\cite{BuergisserCS97}, and in solving algebraic equations~\\cite{Jez15,CiobanuDE16}. \nA key feature is their support for efficient manipulation of compressed data \\cite{Lohrey2012survey,Lohrey14compressed,GanardiJL19,VanderhoevenL25}.\n\nBabai and Szemer\\'edi \\cite{BabaiS84} showed that finite groups afford \\emph{efficient compression}: every element can be expressed by a straight-line program of polylogarithmic length in the size of the group.\nFleischer used this to place $\\dMemb[CT]{\\mathbf{G}}$ in \\NPOLYLOGTIME and extended the result to pseudovarieties of monoids: efficient compression occurs precisely for Clifford monoids (which comprise both groups and semilattices) and commutative monoids.\nIn contrast, there is no maximal pseudovariety of semigroups that affords efficient compression.\\footnote{At the first glance, the difference between monoids and semigroups might seem negligible; however, the landscape of pseudovarieties of semigroups is much richer than the one of monoids.}\n\nThe membership problem has also been studied restricted to other pseudovarieties: Beaudry, McKenzie, and Thérien \\cite{BeaudryMT92} investigated aperiodic monoids, while Fleischer, Stober, and the authors~\\cite{FleischerSTW25} considered inverse semigroups.\n\nAn important variant are straight-line programs of polylogarithmic length and \\emph{bounded width}. \nFleischer showed that they yield membership algorithms in \\FOLL (polynomial-size Boolean circuits of depth $\\log \\log n$), which applies to all commutative semigroups.\nEarlier, Barrington, Kadau, Lange, and McKenzie \\cite{BarringtonKLM01} placed membership in solvable groups of bounded derived length in \\FOLL, and conjectured that the membership problem for \\emph{all} solvable groups may also be in \\FOLL.\nThis was partially confirmed by Collins, Grochow, Levet, and the second author~\\cite{CollinsGLW25} showing this to be true for the class of all nilpotent groups.\n\nIn this work, we complete Fleischer's program to characterize pseudovarieties of finite semigroups that afford efficient compression.\nMoreover, we also improve upon some of the best previously-known length and width bounds for semigroups, monoids, and groups.\nFinally, we apply our findings to the membership problem, in particular, resolving Barrington, Kadau, Lange, and McKenzie's conjecture.\nIn more detail, our results are as follows.\n\n\\vspace{-2mm}\n\n\\subparagraph*{Our Contribution.}\n\nOur main theorem completely characterizes those pseudovarieties $\\mathbf{V}$ of semigroups that afford efficient compression~--~meaning that, for some $k \\in \\mathbb{N}$, all $S \\in \\mathbf{V}$ of size~$\\ensuremath\\left|S\\right| \\leqslant N$ admit straight-line programs of length $\\mathcal{O}(\\smash{\\log}^k N)$; see \\cref{sec:compression}.\nHere, the following three pseudovarieties~--~each requiring straight-line programs of length $\\Omega(\\sqrt{N})$~--~play a crucial role, since they serve as primary obstructions:\n\\begin{equation*}\n \\mathbf{LRB} = \\llbracketx^2 \\approx x, xyx \\approx xy\\rrbracket,\n \\quad \n \\mathbf{RRB} = \\llbracketx^2 \\approx x, xyx \\approx yx\\rrbracket, \n \\quad\n \\mathbf{T} = \\llbracketx^2 \\approx xyx \\approx 0\\rrbracket.\n\\end{equation*}", "context": "As a different variant, the membership problem \\dMemb[CT]{} in the \\emph{Cayley table model} was introduced by Jones, Lien, and Laaser \\cite{JonesLL76}, where the semigroup is given by its multiplication table; here the problem is \\NL-complete.\nFor groups, Barrington and McKenzie \\cite{BarringtonM91} showed that $\\dMemb[CT]{\\mathbf{G}}$ can be solved in \\LOGSPACE with an oracle to undirected graph reachability \\cite{Reingold08}, and conjectured it might be $\\LOGSPACE$-hard.\nFleischer \\cite{Fleischer19diss,Fleischer22} refuted the latter (under \\ACz-reductions) by placing the problem in \\NPOLYLOGTIME. \nHis proof is based on \\emph{straight-line programs} (algebraic circuits or context-free grammars producing precisely one word), a tool central in data compression~\\cite{KiefferY00,Bannai16}, algebraic complexity~\\cite{BuergisserCS97}, and in solving algebraic equations~\\cite{Jez15,CiobanuDE16}. \nA key feature is their support for efficient manipulation of compressed data \\cite{Lohrey2012survey,Lohrey14compressed,GanardiJL19,VanderhoevenL25}.\n\nBabai and Szemer\\'edi \\cite{BabaiS84} showed that finite groups afford \\emph{efficient compression}: every element can be expressed by a straight-line program of polylogarithmic length in the size of the group.\nFleischer used this to place $\\dMemb[CT]{\\mathbf{G}}$ in \\NPOLYLOGTIME and extended the result to pseudovarieties of monoids: efficient compression occurs precisely for Clifford monoids (which comprise both groups and semilattices) and commutative monoids.\nIn contrast, there is no maximal pseudovariety of semigroups that affords efficient compression.\\footnote{At the first glance, the difference between monoids and semigroups might seem negligible; however, the landscape of pseudovarieties of semigroups is much richer than the one of monoids.}\n\nAn important variant are straight-line programs of polylogarithmic length and \\emph{bounded width}. \nFleischer showed that they yield membership algorithms in \\FOLL (polynomial-size Boolean circuits of depth $\\log \\log n$), which applies to all commutative semigroups.\nEarlier, Barrington, Kadau, Lange, and McKenzie \\cite{BarringtonKLM01} placed membership in solvable groups of bounded derived length in \\FOLL, and conjectured that the membership problem for \\emph{all} solvable groups may also be in \\FOLL.\nThis was partially confirmed by Collins, Grochow, Levet, and the second author~\\cite{CollinsGLW25} showing this to be true for the class of all nilpotent groups.\n\nIn this work, we complete Fleischer's program to characterize pseudovarieties of finite semigroups that afford efficient compression.\nMoreover, we also improve upon some of the best previously-known length and width bounds for semigroups, monoids, and groups.\nFinally, we apply our findings to the membership problem, in particular, resolving Barrington, Kadau, Lange, and McKenzie's conjecture.\nIn more detail, our results are as follows.\n\n\\subparagraph*{Our Contribution.}\n\nOur main theorem completely characterizes those pseudovarieties $\\mathbf{V}$ of semigroups that afford efficient compression~--~meaning that, for some $k \\in \\mathbb{N}$, all $S \\in \\mathbf{V}$ of size~$\\ensuremath\\left|S\\right| \\leqslant N$ admit straight-line programs of length $\\mathcal{O}(\\smash{\\log}^k N)$; see \\cref{sec:compression}.\nHere, the following three pseudovarieties~--~each requiring straight-line programs of length $\\Omega(\\sqrt{N})$~--~play a crucial role, since they serve as primary obstructions:\n\\begin{equation*}\n \\mathbf{LRB} = \\llbracketx^2 \\approx x, xyx \\approx xy\\rrbracket,\n \\quad \n \\mathbf{RRB} = \\llbracketx^2 \\approx x, xyx \\approx yx\\rrbracket, \n \\quad\n \\mathbf{T} = \\llbracketx^2 \\approx xyx \\approx 0\\rrbracket.\n\\end{equation*}", "full_context": "As a different variant, the membership problem \\dMemb[CT]{} in the \\emph{Cayley table model} was introduced by Jones, Lien, and Laaser \\cite{JonesLL76}, where the semigroup is given by its multiplication table; here the problem is \\NL-complete.\nFor groups, Barrington and McKenzie \\cite{BarringtonM91} showed that $\\dMemb[CT]{\\mathbf{G}}$ can be solved in \\LOGSPACE with an oracle to undirected graph reachability \\cite{Reingold08}, and conjectured it might be $\\LOGSPACE$-hard.\nFleischer \\cite{Fleischer19diss,Fleischer22} refuted the latter (under \\ACz-reductions) by placing the problem in \\NPOLYLOGTIME. \nHis proof is based on \\emph{straight-line programs} (algebraic circuits or context-free grammars producing precisely one word), a tool central in data compression~\\cite{KiefferY00,Bannai16}, algebraic complexity~\\cite{BuergisserCS97}, and in solving algebraic equations~\\cite{Jez15,CiobanuDE16}. \nA key feature is their support for efficient manipulation of compressed data \\cite{Lohrey2012survey,Lohrey14compressed,GanardiJL19,VanderhoevenL25}.\n\nBabai and Szemer\\'edi \\cite{BabaiS84} showed that finite groups afford \\emph{efficient compression}: every element can be expressed by a straight-line program of polylogarithmic length in the size of the group.\nFleischer used this to place $\\dMemb[CT]{\\mathbf{G}}$ in \\NPOLYLOGTIME and extended the result to pseudovarieties of monoids: efficient compression occurs precisely for Clifford monoids (which comprise both groups and semilattices) and commutative monoids.\nIn contrast, there is no maximal pseudovariety of semigroups that affords efficient compression.\\footnote{At the first glance, the difference between monoids and semigroups might seem negligible; however, the landscape of pseudovarieties of semigroups is much richer than the one of monoids.}\n\nAn important variant are straight-line programs of polylogarithmic length and \\emph{bounded width}. \nFleischer showed that they yield membership algorithms in \\FOLL (polynomial-size Boolean circuits of depth $\\log \\log n$), which applies to all commutative semigroups.\nEarlier, Barrington, Kadau, Lange, and McKenzie \\cite{BarringtonKLM01} placed membership in solvable groups of bounded derived length in \\FOLL, and conjectured that the membership problem for \\emph{all} solvable groups may also be in \\FOLL.\nThis was partially confirmed by Collins, Grochow, Levet, and the second author~\\cite{CollinsGLW25} showing this to be true for the class of all nilpotent groups.\n\nIn this work, we complete Fleischer's program to characterize pseudovarieties of finite semigroups that afford efficient compression.\nMoreover, we also improve upon some of the best previously-known length and width bounds for semigroups, monoids, and groups.\nFinally, we apply our findings to the membership problem, in particular, resolving Barrington, Kadau, Lange, and McKenzie's conjecture.\nIn more detail, our results are as follows.\n\n\\subparagraph*{Our Contribution.}\n\nOur main theorem completely characterizes those pseudovarieties $\\mathbf{V}$ of semigroups that afford efficient compression~--~meaning that, for some $k \\in \\mathbb{N}$, all $S \\in \\mathbf{V}$ of size~$\\ensuremath\\left|S\\right| \\leqslant N$ admit straight-line programs of length $\\mathcal{O}(\\smash{\\log}^k N)$; see \\cref{sec:compression}.\nHere, the following three pseudovarieties~--~each requiring straight-line programs of length $\\Omega(\\sqrt{N})$~--~play a crucial role, since they serve as primary obstructions:\n\\begin{equation*}\n \\mathbf{LRB} = \\llbracketx^2 \\approx x, xyx \\approx xy\\rrbracket,\n \\quad \n \\mathbf{RRB} = \\llbracketx^2 \\approx x, xyx \\approx yx\\rrbracket, \n \\quad\n \\mathbf{T} = \\llbracketx^2 \\approx xyx \\approx 0\\rrbracket.\n\\end{equation*}\n\nOur main theorem completely characterizes those pseudovarieties $\\pv{V}$ of semigroups that afford efficient compression~--~meaning that, for some $k \\in \\N$, all $S \\in \\pv{V}$ of size~$\\abs{S} \\leq N$ admit straight-line programs of length $\\mathcal{O}(\\smash{\\log}^k N)$; see \\cref{sec:compression}.\nHere, the following three pseudovarieties~--~each requiring straight-line programs of length $\\Omega(\\sqrt{N})$~--~play a crucial role, since they serve as primary obstructions:\n\\begin{equation*}\n \\pv{LRB} = \\pvI{x^2 \\approx x, xyx \\approx xy},\n \\quad \n \\pv{RRB} = \\pvI{x^2 \\approx x, xyx \\approx yx}, \n \\quad\n \\pv{T} = \\pvI{x^2 \\approx xyx \\approx 0}.\n\\end{equation*}\n\nMoreover, if $\\pv{V} \\subseteq \\pv{RB} \\pvM \\pv{N}_k$ for some $k \\geq 1$, then the pseudovariety admits straight-line programs of bounded width and length.\nExcept in the case that $\\pv{V}$ contains a nonsolvable group, we show that these bounds are essentially asymptotically optimal.\nOur proofs are fully constructive and can be found in Sections~\\ref{sec:obstructions}--\\ref{sec:general}.\n\n\\begin{itemize}\n \\item The pseudovariety $\\pv{V}$ is \\emph{almost completely regular}, meaning that all its members satisfy an identity of the form $x_1 \\cdots x_n \\approx x_1 \\cdots x_{i-1} (x_i \\cdots x_j)^{\\omega+1} x_{j+1} \\cdots x_n$.\n This condition properly generalizes complete regularity~--~equivalently, being a union of groups~--~which is characterized by the identity $x \\approx x^{\\omega + 1}$.\n In \\cref{sec:general} we show that the general problem for almost completely regular pseudovarieties reduces to this special case.\n If, in addition, the pseudovariety satisfies $\\pv{LRB}, \\pv{RRB} \\not\\sse \\pv{V}$, then the completely regular members of $\\pv{V}$ are necessarily normal bands of groups \\cite[Proposition~4]{Rasin81}.\n Exploiting this structural restriction, we further reduce the problem to the group case in \\cref{sec:completely_regular}.\n Combined with Babai and Szemerédi's result for groups~\\cite{BabaiS84}, this completes the proof.\n \\item The pseudovariety $\\pv{V}$ is \\emph{permutative}, meaning that an identity $x_1 \\cdots x_n \\approx x_{\\sigma(1)} \\cdots x_{\\sigma(n)}$ holds for all members of $\\pv{V}$, where $\\sigma \\in \\mathrm{Sym}(n)$ is some nontrivial permutation of the symbols $1, \\dots, n$.\n This notion properly generalizes commutativity, which is characterized by the identity $xy \\approx yx$.\n In \\cref{sec:permutative} we present a direct proof that such pseudovarieties afford efficient compression, refining an earlier argument for commutative semigroups due to Fleischer~\\cite{Fleischer19diss}.\n Our construction yields straight-line programs of essentially optimal length $\\bigO(\\log N)$ and width two. \n (Matching lower bounds are established in \\cref{sec:constant_length}.)\n\n\\begin{theorem}\\label{thm:bounded-diameter}\n Let $\\pv{V}$ be a pseudovariety.\n Then exactly one of the following holds.\n \\begin{itemize}\n \\item The pseudovariety $\\pv{V}$ has bounded diameter.\n In particular, $C^2_{\\pv{V}}(N) \\in \\mathcal{O}(1)$.\n \\item The pseudovariety $\\pv{V}$ requires straight-line programs of length $\\Omega(\\log N)$.\n \\end{itemize}\n Moreover, the former is the case if and only if $\\pv{V} \\subseteq \\pv{RB} \\pvM \\pv{N}_k$ for some $k \\geq 1$.\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:main-aperiodic}\n Let $\\pv{V} \\subseteq \\pv{A}$ be a pseudovariety.\n The following are equivalent.\n \\begin{bracketenumerate}\n \\item The pseudovariety $\\pv{V}$ affords efficient compression.\n \\item The pseudovariety $\\pv{V}$ contains neither $\\pv{LRB}$, $\\pv{RRB}$, nor $\\pv{T}$.\n \\item The pseudovariety $\\pv{V}$ is permutative.\n \\item The pseudovariety $\\pv{V}$ admits straight-line programs of width two and length $\\mathcal{O}(\\log N)$.\n \\end{bracketenumerate}\n\\end{theorem}\n\\begin{proof}\n See \\cref{lem:obstructions} for $(1) \\Rightarrow (2)$, \\Cref{pro:permutative} entails $(3) \\Rightarrow (4)$, and $(4) \\Rightarrow (1)$ holds by definition.\n For the implication $(2) \\Rightarrow (3)$ we refer to \\cite[Corollary~16]{Thumm2025}.\n\\end{proof}\n\n\\begin{theorem}\\label{thm:main-cregular}\n Let $\\pv{V} \\sse \\pv{CR}$ be a pseudovariety.\n The following are equivalent.\n \\begin{bracketenumerate}\n \\item The pseudovariety $\\pv{V}$ affords efficient compression.\n \\item The pseudovariety $\\pv{V}$ contains neither $\\pv{LRB}$ nor $\\pv{RRB}$.\n \\item The pseudovariety $\\pv{V}$ comprises only normal bands of groups; that is, $\\pv{V} \\sse \\pv{G}\\pvM\\pv{NB}$.\n \\item The pseudovariety $\\pv{V}$ admits straight-line programs of length $\\mathcal{O}(\\smash{\\log}^2 N)$.\n \\end{bracketenumerate}\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:main}\n Let $\\pv{V}$ be a pseudovariety, and let $\\pv{H} = \\pv{V} \\cap \\pv{G}$.\n The following are equivalent.\n \\begin{bracketenumerate}\n \\item The pseudovariety $\\pv{V}$ affords efficient compression.\n \\item The pseudovariety $\\pv{V}$ contains neither $\\pv{LRB}$, $\\pv{RRB}$, nor $\\pv{T}$.\n \\item The pseudovariety $\\pv{V}$ satisfies the following for every $2 \\leq w \\leq \\infty$:\n \\[\n C^{w+3}_{\\pv{V}}(N) \\in \\mathcal{O}(C^w_{\\pv{H}}(N) + \\log N)\n \\quad\\text{and}\\quad\n C^{w+2}_{\\pv{V}}(N) \\in \\mathcal{O}(C^w_{\\pv{H}}(N) \\cdot \\log N).\n \\]\n In particular, $\\pv{V}$ admits straight-line programs of length $\\mathcal{O}(\\smash{\\log}^2 N)$. \n \\end{bracketenumerate}\n\\end{theorem}\n\n\\begin{corollary}\\label{cor:main_membership_restated}\n Let $\\pv{V}$ be a pseudovariety. The following are equivalent.\n \\begin{bracketenumerate}\n \\item\\label{cor_efficient_compression} The pseudovariety $\\pv{V}$ affords efficient compression.\n \\item The pseudovariety $\\pv{V}$ contains neither $\\pv{LRB}$, $\\pv{RRB}$, nor $\\pv{T}$.\n \\item\\label{cor_mem_npolylog} The membership problem \\dMemb[CT]{\\pv{V}} is in \\NPOLYLOGTIME.\n \\end{bracketenumerate}\n Furthermore, if $\\pv{V}$ contains neither $\\pv{LRB}$, $\\pv{RRB}$, $\\pv{T}$, nor any nonsolvable group, then the membership problem \\dMemb[CT]{\\pv{V}} is in $\\NTISP(\\polylog n, \\log n)$ and, hence, in $\\FOLL$. \n\\end{corollary}", "post_theorem_intro_text_len": 4481, "post_theorem_intro_text": "Moreover, if $\\mathbf{V} \\subseteq \\mathbf{RB} \\pvM \\mathbf{N}_k$ for some $k \\geqslant 1$, then the pseudovariety admits straight-line programs of bounded width and length.\nExcept in the case that $\\mathbf{V}$ contains a nonsolvable group, we show that these bounds are essentially asymptotically optimal.\nOur proofs are fully constructive and can be found in Sections~\\ref{sec:obstructions}--\\ref{sec:general}.\n\nIn \\cref{sec:membership}, we apply our findings to the membership problem proving the following result, where \\dMemb[CT]{\\mathbf{V}} denotes the membership problem for $\\mathbf{V}$ in the Cayley table model.\n\n\\begin{corollary}\\label{cor:main_membership}\nLet $\\mathbf{V}$ be a pseudovariety of semigroups with $\\mathbf{LRB}$, $\\mathbf{RRB}$, $\\mathbf{T} \\not \\subseteq \\mathbf{V}$.\n\\begin{bracketenumerate}\n\\item The membership problem \\dMemb[CT]{\\mathbf{V}} is in $\\NPOLYLOGTIME \\subseteq \\qACz$.\n\t\\item If, moreover, $\\mathbf{V}$ contains no nonsolvable group, then \\dMemb[CT]{\\mathbf{V}} is in \\FOLL. \n\\end{bracketenumerate}\n\\end{corollary}\n\nOur results almost completely answer an open problem due to Fleischer \\cite{Fleischer22}, who deemed it ``interesting to see whether the Cayley semigroup membership problem\ncan be shown to be in \\FOLL for all classes of semigroups with the polylogarithmic circuits\nproperty.''\nMoreover, we positively resolve Barrington, Kadau, Lange, and McKenzie's conjecture \\cite{BarringtonKLM01} concerning the membership problem for the pseudovarierty $\\Gsol$ of all finite solvable groups.\n\n\\begin{corollary}\n The problem \\dMemb[CT]{\\Gsol} is in \\FOLL. \n\\end{corollary}\n\n\\subparagraph*{Outline of the Proof.}\n\nOur proof of \\cref{thm:main-intro} combines structural results on semigroups with explicit constructions.\nThe negative results for $\\mathbf{LRB}$, $\\mathbf{RRB}$, and $\\mathbf{T}$ were already established by Fleischer~\\cite{Fleischer19diss}.\nFor the positive direction, we rely on a recent characterization of pseudovarieties $\\mathbf{V}$ satisfying $\\mathbf{T} \\not\\subseteq \\mathbf{V}$ due to the first author~\\cite{Thumm2025}, which shows that any such pseudovariety necessarily falls into one (or both) of the following two classes.\n\n\\begin{itemize}\n \\item The pseudovariety $\\mathbf{V}$ is \\emph{almost completely regular}, meaning that all its members satisfy an identity of the form $x_1 \\cdots x_n \\approx x_1 \\cdots x_{i-1} (x_i \\cdots x_j)^{\\omega+1} x_{j+1} \\cdots x_n$.\n This condition properly generalizes complete regularity~--~equivalently, being a union of groups~--~which is characterized by the identity $x \\approx x^{\\omega + 1}$.\n In \\cref{sec:general} we show that the general problem for almost completely regular pseudovarieties reduces to this special case.\n If, in addition, the pseudovariety satisfies $\\mathbf{LRB}, \\mathbf{RRB} \\not\\subseteq \\mathbf{V}$, then the completely regular members of $\\mathbf{V}$ are necessarily normal bands of groups \\cite[Proposition~4]{Rasin81}.\n Exploiting this structural restriction, we further reduce the problem to the group case in \\cref{sec:completely_regular}.\n Combined with Babai and Szemerédi's result for groups~\\cite{BabaiS84}, this completes the proof.\n \\item The pseudovariety $\\mathbf{V}$ is \\emph{permutative}, meaning that an identity $x_1 \\cdots x_n \\approx x_{\\sigma(1)} \\cdots x_{\\sigma(n)}$ holds for all members of $\\mathbf{V}$, where $\\sigma \\in \\mathrm{Sym}(n)$ is some nontrivial permutation of the symbols $1, \\dots, n$.\n This notion properly generalizes commutativity, which is characterized by the identity $xy \\approx yx$.\n In \\cref{sec:permutative} we present a direct proof that such pseudovarieties afford efficient compression, refining an earlier argument for commutative semigroups due to Fleischer~\\cite{Fleischer19diss}.\n Our construction yields straight-line programs of essentially optimal length $\\mathcal{O}(\\log N)$ and width two. \n (Matching lower bounds are established in \\cref{sec:constant_length}.)\n\n\\end{itemize}\n\nFor pseudovarieties in the first class, our reductions are efficient in that questions about asymptotically optimal straight-line program length (and width) reduce to the group case.\nThe latter is discussed in \\cref{sec:groups}, where we additionally present two new constructions for solvable groups, yielding straight-line programs of asymptotically optimal length $\\mathcal{O}(\\log N)$ but unbounded width, and of polylogarithmic length and bounded width, respectively.", "sketch": "Our proof of \\cref{thm:main-intro} \\emph{“combines structural results on semigroups with explicit constructions.”} The \\emph{negative} direction is taken from prior work: \\emph{“The negative results for $\\mathbf{LRB}$, $\\mathbf{RRB}$, and $\\mathbf{T}$ were already established by Fleischer.”}\n\nFor the \\emph{positive} direction, we use a characterization (for $\\mathbf{T} \\not\\subseteq \\mathbf{V}$) showing that any such pseudovariety \\emph{“necessarily falls into one (or both) of the following two classes”}:\n\\begin{itemize}\n\\item \\textbf{Almost completely regular case.} If $\\mathbf{V}$ is \\emph{“almost completely regular”} (members satisfy an identity $x_1 \\cdots x_n \\approx x_1 \\cdots x_{i-1} (x_i \\cdots x_j)^{\\omega+1} x_{j+1} \\cdots x_n$), then \\emph{“in \\cref{sec:general} we show that the general problem for almost completely regular pseudovarieties reduces to”} the completely regular/union-of-groups special case (characterized by $x \\approx x^{\\omega+1}$). If also $\\mathbf{LRB}, \\mathbf{RRB} \\not\\subseteq \\mathbf{V}$, then \\emph{“the completely regular members of $\\mathbf{V}$ are necessarily normal bands of groups”}; \\emph{“exploiting this structural restriction,”} the problem is \\emph{“further reduce[d] … to the group case in \\cref{sec:completely_regular}.”} \\emph{“Combined with Babai and Szemer\\'edi's result for groups, this completes the proof.”}\n\\item \\textbf{Permutative case.} If $\\mathbf{V}$ is \\emph{“permutative”} (an identity $x_1\\cdots x_n \\approx x_{\\sigma(1)}\\cdots x_{\\sigma(n)}$ holds for some nontrivial $\\sigma$), then \\emph{“in \\cref{sec:permutative} we present a direct proof that such pseudovarieties afford efficient compression,”} refining an earlier commutative argument. The construction \\emph{“yields straight-line programs of essentially optimal length $\\mathcal{O}(\\log N)$ and width two,”} with \\emph{“matching lower bounds … in \\cref{sec:constant_length}.”}\n\\end{itemize}\n\nFinally, for pseudovarieties in the first class, the reductions are \\emph{“efficient … [so that] questions about asymptotically optimal straight-line program length (and width) reduce to the group case,”} and in \\cref{sec:groups} they \\emph{“present two new constructions for solvable groups,”} giving (i) \\emph{“asymptotically optimal length $\\mathcal{O}(\\log N)$ but unbounded width,”} and (ii) \\emph{“polylogarithmic length and bounded width.”}", "expanded_sketch": "Our proof of the main theorem \\emph{“combines structural results on semigroups with explicit constructions.”} The \\emph{negative} direction is taken from prior work: \\emph{“The negative results for $\\mathbf{LRB}$, $\\mathbf{RRB}$, and $\\mathbf{T}$ were already established by Fleischer.”}\n\nFor the \\emph{positive} direction, we use a characterization (for $\\mathbf{T} \\not\\subseteq \\mathbf{V}$) showing that any such pseudovariety \\emph{“necessarily falls into one (or both) of the following two classes”}:\n\\begin{itemize}\n\\item \\textbf{Almost completely regular case.} If $\\mathbf{V}$ is \\emph{“almost completely regular”} (members satisfy an identity $x_1 \\cdots x_n \\approx x_1 \\cdots x_{i-1} (x_i \\cdots x_j)^{\\omega+1} x_{j+1} \\cdots x_n$), then \\emph{“next we show that the general problem for almost completely regular pseudovarieties reduces to”} the completely regular/union-of-groups special case (characterized by $x \\approx x^{\\omega+1}$). If also $\\mathbf{LRB}, \\mathbf{RRB} \\not\\subseteq \\mathbf{V}$, then \\emph{“the completely regular members of $\\mathbf{V}$ are necessarily normal bands of groups”}; \\emph{“exploiting this structural restriction,”} the problem is \\emph{“further reduce[d] … to the group case later.”} \\emph{“Combined with Babai and Szemer\\'edi's result for groups, this completes the proof.”}\n\\item \\textbf{Permutative case.} If $\\mathbf{V}$ is \\emph{“permutative”} (an identity $x_1\\cdots x_n \\approx x_{\\sigma(1)}\\cdots x_{\\sigma(n)}$ holds for some nontrivial $\\sigma$), then \\emph{“later we present a direct proof that such pseudovarieties afford efficient compression,”} refining an earlier commutative argument. The construction \\emph{“yields straight-line programs of essentially optimal length $\\mathcal{O}(\\log N)$ and width two,”} with \\emph{“matching lower bounds … proved later.”}\n\\end{itemize}\n\nFinally, for pseudovarieties in the first class, the reductions are \\emph{“efficient … [so that] questions about asymptotically optimal straight-line program length (and width) reduce to the group case,”} and later they \\emph{“present two new constructions for solvable groups,”} giving (i) \\emph{“asymptotically optimal length $\\mathcal{O}(\\log N)$ but unbounded width,”} and (ii) \\emph{“polylogarithmic length and bounded width.”}", "expanded_theorem": "\\label{thm:main-intro}\n\tLet $\\mathbf{V}$ be a pseudovariety of semigroups.\n\tThe following are equivalent.\n\t\\begin{bracketenumerate}\n\t\t\\item The pseudovariety $\\mathbf{V}$ affords efficient compression.\n\t\t\\item The pseudovariety $\\mathbf{V}$ contains neither $\\mathbf{LRB}$, $\\mathbf{RRB}$, nor $\\mathbf{T}$.\n\t\t\\item The pseudovariety $\\mathbf{V}$ admits straight-line programs of length $\\mathcal{O}(\\smash{\\log}^2 N)$.\n\t\\end{bracketenumerate}\n Furthermore, if all groups in $\\mathbf{V}$ are solvable, then the above are equivalent to $\\mathbf{V}$ admitting straight-line programs of length $\\mathcal{O}(\\log N)$ as well as of width $\\mathcal{O}(1)$ and length $\\mathcal{O}(\\polylog N)$ as well as of width $\\mathcal{O}(1)$ and length $\\mathcal{O}(\\polylog N)$.,", "theorem_type": ["Biconditional or Equivalence", "Implication"], "mcq": {"question": "Let V be a pseudovariety of semigroups, i.e. a class of finite semigroups closed under finite direct products, subsemigroups, and homomorphic images. Say that V affords efficient compression if there exists k in N such that every semigroup S in V with |S| <= N admits straight-line programs of length O((log N)^k). Let\nLRB = [[x^2 ≈ x, xyx ≈ xy]],\nRRB = [[x^2 ≈ x, xyx ≈ yx]], and\nT = [[x^2 ≈ xyx ≈ 0]].\nAlso, saying that V admits straight-line programs of width O(1) and length O(polylog N) means that there is a constant width bound and a length bound O((log N)^k) for some fixed k.\nUnder these assumptions, which statement about V is valid?", "correct_choice": {"label": "A", "text": "The following are equivalent: (1) V affords efficient compression; (2) V contains neither LRB, RRB, nor T; and (3) V admits straight-line programs of length O((log N)^2). Furthermore, if every group contained in V is solvable, then these statements are also equivalent to (4) V admitting straight-line programs of length O(log N), and to (5) V admitting straight-line programs of bounded width O(1) and length O(polylog N)."}, "choices": [{"label": "B", "text": "The following are equivalent: (1) $V$ affords efficient compression; (2) $V$ contains neither $\\mathrm{LRB}$ nor $\\mathrm{RRB}$; and (3) $V$ admits straight-line programs of length $O((\\log N)^2)$. Furthermore, if every group contained in $V$ is solvable, then these statements are also equivalent to (4) $V$ admitting straight-line programs of length $O(\\log N)$, and to (5) $V$ admitting straight-line programs of bounded width $O(1)$ and length $O(\\mathrm{polylog}\\, N)$."}, {"label": "C", "text": "If $V$ affords efficient compression, then $V$ contains neither $\\mathrm{LRB}$, $\\mathrm{RRB}$, nor $\\mathrm{T}$, and $V$ admits straight-line programs of length $O((\\log N)^2)$. Furthermore, if every group contained in $V$ is solvable and $V$ affords efficient compression, then $V$ also admits straight-line programs of length $O(\\log N)$, and straight-line programs of bounded width $O(1)$ and length $O(\\mathrm{polylog}\\, N)$."}, {"label": "D", "text": "The following are equivalent: (1) $V$ affords efficient compression; (2) $V$ contains neither $\\mathrm{LRB}$, $\\mathrm{RRB}$, nor $\\mathrm{T}$; and (3) $V$ admits straight-line programs of length $O(\\log N)$. Furthermore, if every group contained in $V$ is solvable, then these statements are also equivalent to (4) $V$ admitting straight-line programs of bounded width $O(1)$ and length $O(\\mathrm{polylog}\\, N)$."}, {"label": "E", "text": "The following are equivalent: (1) $V$ affords efficient compression; (2) $V$ contains neither $\\mathrm{LRB}$, $\\mathrm{RRB}$, nor $\\mathrm{T}$; and (3) $V$ admits straight-line programs of bounded width $O(1)$ and length $O((\\log N)^2)$. Furthermore, if every group contained in $V$ is solvable, then these statements are also equivalent to (4) $V$ admitting straight-line programs of length $O(\\log N)$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "need to exclude T as a primary obstruction", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "replaced equivalence by one-way implications from efficient compression", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "log-squared bound upgraded to uniform logarithmic bound without solvable-group hypothesis", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "bounded-width conclusion extended beyond the solvable-group case", "template_used": "wildcard"}]}} +{"id": "2601.04756v2", "paper_link": "http://arxiv.org/abs/2601.04756v2", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "[label=thm:main-simplified]\n Let $n>1$ be an integer and $f:2^V\\to \\mathbb Z$ a connectivity function on an $n$-element set~$V$.\n Let $\\gamma$ be the time to compute $f(X)$ for any subset $X$ of~$V$.\nIn time\n $2^{O(k)}\\gamma n^6 \\log n + 2^{O(k^2)} \\gamma n$,\nwe can either find a branch-decomposition of~$f$ of width at most~$k$, \n or confirm that the branch-width of~$f$ is larger than~$k$.", "start_pos": 5777, "end_pos": 6211, "label": null}, "ref_dict": {}, "pre_theorem_intro_text_len": 2808, "pre_theorem_intro_text": "\\label{sec:intro}\nIn 2005, Hlin\\v{e}n\\'y~\\cite{Hlineny2002} asked the following questions for matroids.\n\n\\begin{quote}\n What is the parameterized complexity of the problem to determine the branch-width of a matroid $M$?\n \\begin{enumerate}[label=\\rm(\\arabic*)]\n \\item If $M=M(A)$ is given by a matrix representation over an infinite field?\n \\item If $M$ is given by a rank oracle?\n \\end{enumerate}\n\\end{quote}\nWe resolve this long-standing open problem completely, in a more general setting, by showing that the branch-width of connectivity functions is fixed-parameter tractable (FPT).\nA \\emph{connectivity function} is an integer-valued function $f$ defined on all subsets of a finite set $V$ such that \n\\begin{enumerate}[label=\\rm (\\roman*)]\n \\item (symmetric) $f(X)=f(V-X)$ for all sets $X$,\n \\item (submodular) $f(X)+f(Y)\\ge f(X\\cap Y)+f(X\\cup Y)$ for all sets~$X$ and $Y$, and \n \\item $f(\\emptyset)=0$.\n\\end{enumerate}\n\nBranch-width of graphs and matroids was introduced by Robertson and Seymour in their Graph Minors series~\\cite{RS1991}.\nIt was later generalized for connectivity functions by Oum and Seymour~\\cite{OS2004,OS2005}.\nLet us provide the general definition of branch-width of connectivity functions.\n\nA tree is \\emph{subcubic} if every vertex has degree $3$ or $1$.\nLet $f:2^V\\to\\mathbb Z$ be a connectivity function on a set $V$.\nA \\emph{branch-decomposition} of~$f$ is a pair $(T,L)$ of a subcubic tree $T$ and a bijection $L$ from the set of leaves of $T$ to $V$.\nFor a branch-decomposition $(T,L)$ of~$f$ and an edge~$e$ of~$T$, \nwe define the \\emph{width} of~$e$ as $f(L^{-1}(A_e))$, \nwhere $(A_e,B_e)$ is the partition of $V(T)$ induced by the components of $T-e$.\nThe \\emph{width} of a branch-decomposition $(T,L)$ is the maximum width of all edges of $T$.\nThe \\emph{branch-width} of~$f$, denoted by $\\bw(f)$, is the minimum width of all branch-decompositions of~$f$.\nIf $\\abs{V}\\le 1$, there is no branch-decomposition and yet we define the branch-width of $f$ to be $0$.\n\nHicks and Oum~\\cite{HO2011} wrote the following remark in 2011.\n\\begin{quote}\n For many applications on fixed-parameter tractable algorithms, it is desirable to have an algorithm which runs in time $O(g(k)n^c)$ for some function $g$ and a constant $c$ independent of $k$. Such\nan algorithm is called a \\emph{fixed-parameter tractable} algorithm with parameter $k$. It is still unknown\nwhether there is a fixed-parameter tractable algorithm to decide whether branch-width of $f$ is at\nmost $k$ when $f$ is an integer-valued symmetric submodular function given as an oracle.\n\\end{quote}\n\nHere is our main theorem, proving that there is a fixed-parameter tractable algorithm to find a branch-decomposition of width at most $k$ if one exists for general connectivity functions.", "context": "\\label{sec:intro}\nIn 2005, Hlin\\v{e}n\\'y~\\cite{Hlineny2002} asked the following questions for matroids.\n\n\\begin{quote}\n What is the parameterized complexity of the problem to determine the branch-width of a matroid $M$?\n \\begin{enumerate}[label=\\rm(\\arabic*)]\n \\item If $M=M(A)$ is given by a matrix representation over an infinite field?\n \\item If $M$ is given by a rank oracle?\n \\end{enumerate}\n\\end{quote}\nWe resolve this long-standing open problem completely, in a more general setting, by showing that the branch-width of connectivity functions is fixed-parameter tractable (FPT).\nA \\emph{connectivity function} is an integer-valued function $f$ defined on all subsets of a finite set $V$ such that \n\\begin{enumerate}[label=\\rm (\\roman*)]\n \\item (symmetric) $f(X)=f(V-X)$ for all sets $X$,\n \\item (submodular) $f(X)+f(Y)\\ge f(X\\cap Y)+f(X\\cup Y)$ for all sets~$X$ and $Y$, and \n \\item $f(\\emptyset)=0$.\n\\end{enumerate}\n\nBranch-width of graphs and matroids was introduced by Robertson and Seymour in their Graph Minors series~\\cite{RS1991}.\nIt was later generalized for connectivity functions by Oum and Seymour~\\cite{OS2004,OS2005}.\nLet us provide the general definition of branch-width of connectivity functions.\n\nA tree is \\emph{subcubic} if every vertex has degree $3$ or $1$.\nLet $f:2^V\\to\\mathbb Z$ be a connectivity function on a set $V$.\nA \\emph{branch-decomposition} of~$f$ is a pair $(T,L)$ of a subcubic tree $T$ and a bijection $L$ from the set of leaves of $T$ to $V$.\nFor a branch-decomposition $(T,L)$ of~$f$ and an edge~$e$ of~$T$, \nwe define the \\emph{width} of~$e$ as $f(L^{-1}(A_e))$, \nwhere $(A_e,B_e)$ is the partition of $V(T)$ induced by the components of $T-e$.\nThe \\emph{width} of a branch-decomposition $(T,L)$ is the maximum width of all edges of $T$.\nThe \\emph{branch-width} of~$f$, denoted by $\\bw(f)$, is the minimum width of all branch-decompositions of~$f$.\nIf $\\abs{V}\\le 1$, there is no branch-decomposition and yet we define the branch-width of $f$ to be $0$.\n\nHicks and Oum~\\cite{HO2011} wrote the following remark in 2011.\n\\begin{quote}\n For many applications on fixed-parameter tractable algorithms, it is desirable to have an algorithm which runs in time $O(g(k)n^c)$ for some function $g$ and a constant $c$ independent of $k$. Such\nan algorithm is called a \\emph{fixed-parameter tractable} algorithm with parameter $k$. It is still unknown\nwhether there is a fixed-parameter tractable algorithm to decide whether branch-width of $f$ is at\nmost $k$ when $f$ is an integer-valued symmetric submodular function given as an oracle.\n\\end{quote}\n\nHere is our main theorem, proving that there is a fixed-parameter tractable algorithm to find a branch-decomposition of width at most $k$ if one exists for general connectivity functions.", "full_context": "\\label{sec:intro}\nIn 2005, Hlin\\v{e}n\\'y~\\cite{Hlineny2002} asked the following questions for matroids.\n\n\\begin{quote}\n What is the parameterized complexity of the problem to determine the branch-width of a matroid $M$?\n \\begin{enumerate}[label=\\rm(\\arabic*)]\n \\item If $M=M(A)$ is given by a matrix representation over an infinite field?\n \\item If $M$ is given by a rank oracle?\n \\end{enumerate}\n\\end{quote}\nWe resolve this long-standing open problem completely, in a more general setting, by showing that the branch-width of connectivity functions is fixed-parameter tractable (FPT).\nA \\emph{connectivity function} is an integer-valued function $f$ defined on all subsets of a finite set $V$ such that \n\\begin{enumerate}[label=\\rm (\\roman*)]\n \\item (symmetric) $f(X)=f(V-X)$ for all sets $X$,\n \\item (submodular) $f(X)+f(Y)\\ge f(X\\cap Y)+f(X\\cup Y)$ for all sets~$X$ and $Y$, and \n \\item $f(\\emptyset)=0$.\n\\end{enumerate}\n\nBranch-width of graphs and matroids was introduced by Robertson and Seymour in their Graph Minors series~\\cite{RS1991}.\nIt was later generalized for connectivity functions by Oum and Seymour~\\cite{OS2004,OS2005}.\nLet us provide the general definition of branch-width of connectivity functions.\n\nA tree is \\emph{subcubic} if every vertex has degree $3$ or $1$.\nLet $f:2^V\\to\\mathbb Z$ be a connectivity function on a set $V$.\nA \\emph{branch-decomposition} of~$f$ is a pair $(T,L)$ of a subcubic tree $T$ and a bijection $L$ from the set of leaves of $T$ to $V$.\nFor a branch-decomposition $(T,L)$ of~$f$ and an edge~$e$ of~$T$, \nwe define the \\emph{width} of~$e$ as $f(L^{-1}(A_e))$, \nwhere $(A_e,B_e)$ is the partition of $V(T)$ induced by the components of $T-e$.\nThe \\emph{width} of a branch-decomposition $(T,L)$ is the maximum width of all edges of $T$.\nThe \\emph{branch-width} of~$f$, denoted by $\\bw(f)$, is the minimum width of all branch-decompositions of~$f$.\nIf $\\abs{V}\\le 1$, there is no branch-decomposition and yet we define the branch-width of $f$ to be $0$.\n\nHicks and Oum~\\cite{HO2011} wrote the following remark in 2011.\n\\begin{quote}\n For many applications on fixed-parameter tractable algorithms, it is desirable to have an algorithm which runs in time $O(g(k)n^c)$ for some function $g$ and a constant $c$ independent of $k$. Such\nan algorithm is called a \\emph{fixed-parameter tractable} algorithm with parameter $k$. It is still unknown\nwhether there is a fixed-parameter tractable algorithm to decide whether branch-width of $f$ is at\nmost $k$ when $f$ is an integer-valued symmetric submodular function given as an oracle.\n\\end{quote}\n\nHere is our main theorem, proving that there is a fixed-parameter tractable algorithm to find a branch-decomposition of width at most $k$ if one exists for general connectivity functions.\n\nHicks and Oum~\\cite{HO2011} wrote the following remark in 2011.\n\\begin{quote}\n For many applications on fixed-parameter tractable algorithms, it is desirable to have an algorithm which runs in time $O(g(k)n^c)$ for some function $g$ and a constant $c$ independent of $k$. Such\nan algorithm is called a \\emph{fixed-parameter tractable} algorithm with parameter $k$. It is still unknown\nwhether there is a fixed-parameter tractable algorithm to decide whether branch-width of $f$ is at\nmost $k$ when $f$ is an integer-valued symmetric submodular function given as an oracle.\n\\end{quote}\n\nThe branch-width of a matroid $M$ on a ground set $E(M)$ is defined as the branch-width of the connectivity function $\\lambda(X) = r(X) + r(E(M)-X) - r(E(M))$ of the matroid, where $r(X)$ denotes the rank of a set $X \\subseteq E(M)$.\nTherefore, \\zcref{thm:main-simplified} answers both of the questions of Hlin\\v{e}n\\'y affirmatively.\nFurthermore, we can improve its running time by using algorithms dedicated for matroids and obtain the following.\n\n\\begin{theorem}[label=thm:os,note={Oum and Seymour~\\cite{OS2005}}]\n Let $n$, $k$ be positive integers.\n Let $f:2^V\\to \\mathbb Z$ be a connectivity function on an $n$-element set~$V$.\n Let $\\gamma$ be the time to compute $f(X)$ for any subset $X$ of~$V$.\n There is an algorithm to \n find a branch-decomposition of~$f$ of width at most $k$ if one exists, \n in time $O(\\gamma n^{8k+9}\\log n)$.\n\\end{theorem}\n\n\\begin{proposition}[label=prop:split]\n Let $f:2^V\\to\\mathbb Z$ be a connectivity function of branch-width at most $k$ and denote $n = |V|$.\nThere exists a subset $A$ of~$V$ satisfying the following two conditions.\n \\begin{enumerate}[label=\\rm(\\roman*)]\n \\item $\\abs{A} \\ge \\frac{n}{3^{k+1}}$ and $\\abs{\\co{A}} \\ge \\frac{n}{3^{k+1}}$.\n \\item Both $A$ and $\\co{A}$ are titanic.\n \\end{enumerate}\n Moreover, if a branch-decomposition of width at most $k$ is given,\n then we can find such a set $A$ in time $O(\n 3^{3k}k n \\SUB(n,kn,\\gamma))$, where $\\gamma$ is the time to compute $f(X)$ for any set $X$.\n\\end{proposition}\n\n\\begin{proposition}[label=prop:approx-exact]\n Let $n$ be a positive integer.\n Let $f:2^V\\to \\mathbb Z$ be a connectivity function on an $n$-element set~$V$.\n Let us assume that $\\gamma$ is the time to compute $f(X)$ for any subset $X$ of~$V$.\n Suppose that we are given a branch-decomposition of~$f$ of width at most $\\ell$.\n In time \n$O(3^{4\\ell}\\ell n \\SUB(n,k n,\\gamma)\\log n\n + \\ell 3^{(\\ell+1)(8k+9)}\\gamma n)$, \n we can either find a branch-decomposition of~$f$ of width at most~$k$, \n or confirm that the branch-width of~$f$ is larger than~$k$. \n\\end{proposition}\n\\begin{proof}\n We proceed by induction on~$n$, and prove that there is such an algorithm with the running time $T(n)$.\n We may assume that $\\ell>k$.\n We may also assume that $f(\\{v\\})\\le k$ for all $v\\in V$, because otherwise the branch-width of $f$ is larger than $k$.\n Therefore $f(X)\\le kn$ for all sets~$X$ by submodularity.\n\n\\begin{theorem}[label=thm:main,store=main]\n Let $n$ be a positive integer.\n Let $f:2^V\\to \\mathbb Z$ be a connectivity function on an $n$-element set~$V$.\nLet $f^*:3^V\\to\\mathbb Z$ be an interpolation of $f$.\n Let $\\gamma'$ be the time to compute $f^*(X,Y)$ for any disjoint subsets $X$, $Y$ of~$V$.\n In time \n$O(3^{8k}k n^2 \\SUB(n,k n,\\gamma')\\log n\n + k 3^{(2k+1)(8k+9)}\\gamma'n^2)$\n we can either find a branch-decomposition of~$f$ of width at most~$k$, \n or confirm that the branch-width of~$f$ is larger than~$k$.\n\nInstead of using the iterative compression, we can use the existing fixed-parameter tractable approximation algorithm for branch-width, due to Oum and Seymour~\\cite{OS2004}. \nHicks and Oum~\\cite{HO2011} noted that the following theorem can be deduced from the argument of Oum and Seymour~\\cite{OS2004}, which was originally written only for the case that $f(\\{v\\})\\le 1$ for all $v\\in V$.\n\\begin{theorem}[note={Oum and Seymour~\\cite{OS2004}},label=thm:approx]\n Let $n$ be a positive integer.\n Let $f:2^V\\to \\mathbb Z$ be a connectivity function on an $n$-element set~$V$\n and let $c'=\\max\\{f(\\{v\\}):v\\in V\\}$. \n Let $\\gamma$ be the time to compute $f(X)$ for any set~$X$.\n Then in time $O(2^{3k+c'} n^2 \\SUB(n,kn,\\gamma))$, we can either find a branch-decomposition of $f$ of width at most $3k+c'$ \n or confirm that the branch-width of~$f$ is larger than $k$.\n\\end{theorem}\nBy combining with \\zcref{prop:approx-exact}, we deduce the following.\n\\begin{theorem}[label=thm:main2]\n Let $n$ be a positive integer.\n Let $f:2^V\\to \\mathbb Z$ be a connectivity function on an $n$-element set~$V$.\n Let us assume that $\\gamma$ is the time to compute $f(X)$ for any subset $X$ of~$V$.\n In time \n$O(2^{4k} n^2 \\SUB(n,kn,\\gamma)+3^{16k}k n \\SUB(n,kn,\\gamma)\\log n\n + k 3^{(4k+1)(8k+9)}\\gamma n)$, \n we can either find a branch-decomposition of~$f$ of width at most~$k$, \n or confirm that the branch-width of~$f$ is larger than~$k$.\n\n\\begin{theorem}[note={Chakrabarty, Lee, Sidford, Singla, and Wong~\\cite{CLSSW2019}},label=thm:matroidint]\n Let $M_1$, $M_2$ be two matroids on an $n$-element set given by rank oracles, and let $\\gamma$ be the time to compute the rank of any set.\n Then we can find a largest common independent set in time $O(\\gamma n \\sqrt{r}\\log n)$, where $r$ is the size of a largest common independent set.\n\\end{theorem}\nUsing the remark in \\cite[page 526]{OS2004} combined with a newer algorithm for the matroid intersection problem stated in \\zcref{thm:matroidint}, we deduce the following.\nFor completeness, we include the proof.\n\\begin{proposition}[note={Oum and Seymour~\\cite[page 526]{OS2004}},label=prop:matroidmin]\nThere is an algorithm that, with input an $n$-element matroid $M$ \n given by its rank oracle and two disjoint sets $X$ and $Y$, finds a \n set $Z$ with $X\\subseteq Z\\subseteq E(M)-Y$ that minimizes $\\lambda(Z)$ in time $O(\\gamma n^{1.5}\\log n)$, where $\\lambda(Z)=r(Z)+r(E(M)-Z)-r(E(M))$ is the connectivity function of $M$ and $\\gamma$ is the time to compute the rank of any set. \n\\end{proposition}\n\\begin{proof}\n Let $r$ be the rank function of $M$.\n Let $M_1=M/X\\setminus Y$ and $M_2=M\\setminus X/Y$. \n Let $r_1$, $r_2$ be the rank function of $M_1$ and $M_2$, respectively. \n Then, for $U\\subseteq E(M)-(X\\cup Y)$, we have \n \\[ r_1(U)=r(U\\cup X)-r(X) \\text{ and } r_2(U)=r(U\\cup Y)-r(Y).\\] \n The matroid intersection theorem, \\zcref{thm:edmonds}, allows us to find $U$ that minimizes\n \\begin{align*} \n r_1(U)+r_2(E(M_2)-U) &= r(U\\cup X)-r(X) + r(E(M)-(X\\cup U)) - r(Y)\\\\ \n &= \\lambda(U\\cup X)+ r(E(M))-r(X)-r(Y) .\n \\end{align*} \n By \\zcref{thm:matroidint}, such $U$ can be found in time $O(\\gamma n^{1.5}\\log n)$.\n (Their paper is stated for finding a maximum common independent set, but it is easy to output the dual optimum, the set $U$ that minimizes $r_1(U)+r_2(E(M_2)-U)$ from the algorithm.)\n\\end{proof}\n\\begin{corollary}[note={Oum and Seymour~\\cite[Corollary 7.2]{OS2004}},label=cor:matroid]\n For an integer $k$, there is an algorithm that, with input an $n$-element matroid, given by its rank oracle, either concludes that the branch-width is larger than $k$\n or outputs its branch-decomposition of width at most $3k+1$, in time $O(\\gamma 8^k n^{2.5}\\log n)$, where $\\gamma$ is the time to compute the rank of any set. \n\\end{corollary}\n\\begin{proof}\n The proof is identical to the proof in \\cite{OS2004}, except for the fact that we now use an improved running time from \\zcref{thm:matroidint,prop:matroidmin}.\n\\end{proof}", "post_theorem_intro_text_len": 5652, "post_theorem_intro_text": "The branch-width of a matroid $M$ on a ground set $E(M)$ is defined as the branch-width of the connectivity function $\\lambda(X) = r(X) + r(E(M)-X) - r(E(M))$ of the matroid, where $r(X)$ denotes the rank of a set $X \\subseteq E(M)$.\nTherefore, \\zcref{thm:main-simplified} answers both of the questions of Hlin\\v{e}n\\'y affirmatively.\nFurthermore, we can improve its running time by using algorithms dedicated for matroids and obtain the following.\n\n\\begin{theorem}[label=thm:matroidmain,store=matroidmain]\n There is an algorithm that, with input an $n$-element matroid $M$, given by its rank oracle, and an integer $k$, finds a branch-decomposition of width at most $k$, if one exists, in time $2^{O(k)} \\gamma n^{2.5}\\log^2 n + 2^{O(k^2)}\\gamma n$, where $\\gamma$ is the time to compute the rank of any set. \n\\end{theorem}\n\nThe previous best algorithm for branch-width of a general connectivity function was given by Oum and Seymour~\\cite{OS2005} in 2007.\nThey showed that the problem of computing branch-width of a connectivity function given by an oracle is slice-wise polynomial (XP) by giving an algorithm with running time $\\gamma n^{O(k)}$.\nSlightly earlier, Oum and Seymour~\\cite{OS2004} gave an FPT approximation algorithm for branch-width of connectivity functions.\nTheir algorithm runs in time $2^{O(k)} \\gamma n^{O(1)} $, and either determines that the branch-width is more than $k$, or returns a branch-decomposition of width at most $3k+c$, where $c \\le k$ is the maximum of $f(\\{v\\})$ for $v \\in V$.\nOum~\\cite{Oum2009} also presented an exponential-time algorithm to compute branch-width.\n\nFor branch-width of matroids represented over a fixed finite field, Hlin\\v{e}n\\'y and Oum~\\cite{HO2006} gave an FPT algorithm in~2008 (see also~\\cite{Hlineny2002,JKO2019}).\nFor more general settings of matroid branch-width, the problem of finding an FPT algorithm remained open before this work.\n\nIn addition to branch-width of matroids, the branch-width of connectivity functions captures several well-studied graph width parameters, including rank-width, carving-width, and branch-width of graphs.\nEach of these was known to be fixed-parameter tractable: Bodlaender and Thilikos~\\cite{BT1997} gave a linear FPT algorithm for branch-width of graphs, Thilikos, Serna, and Bodlaender~\\cite{TSB2000} gave a linear FPT algorithm for carving-width, and Courcelle and Oum~\\cite{DBLP:journals/jct/CourcelleO07} gave an FPT algorithm for rank-width.\nThe FPT algorithm for rank-width was later improved to almost-linear by Korhonen and Soko{\\l}owski~\\cite{KS2024} (see also~\\cite{HO2006,Oum2006,JKO2019,FK2024} for other works on computing rank-width).\n\nDespite the earlier FPT algorithms for the aforementioned special cases of branch-width of connectivity functions, we believe that the algorithm of \\zcref{thm:main-simplified} is interesting also in their context.\nIts running time dependency on the parameter $k$ is only $2^{O(k^2)}$, while for the algorithms for graph branch-width and carving-width~\\cite{BT1997,TSB2000} it is at least $2^{\\Omega(k^3)}$, and for the algorithms for rank-width and finite field matroid branch-width~\\cite{DBLP:journals/jct/CourcelleO07,HO2006,JKO2019,FK2024,KS2024} it is at least doubly exponential.\nFurthermore, in our opinion, the algorithm of \\zcref{thm:main-simplified} provides so far the simplest proof of fixed-parameter tractability for all of these width parameters.\nIt is an interesting direction of future research to give faster problem-specific implementations of the technique of \\zcref{thm:main-simplified} for these special cases.\n\n\\paragraph{Outline of the algorithm.}\nThe proof of \\zcref{thm:main-simplified} is not long, but let us still shortly outline it.\nWe say that a \\emph{cut} of a connectivity function $f \\colon 2^{V} \\to \\mathbb{Z}$ is a bipartition $(A,B)$ of $V$.\nA \\emph{safe cut} is a cut $(A,B)$ such that there exists an optimum-width branch-decomposition where $(A,B)$ corresponds to an edge of the decomposition.\n\nThe main idea of the algorithm of \\zcref{thm:main-simplified} is to repeatedly find safe cuts and greedily decompose $f$ along them.\nIn particular, if we find a safe cut $(A,B)$ with $|A|,|B| \\ge 2$, we can construct two smaller connectivity functions $f_A$ and $f_B$, recursively find optimum-width branch-decompositions of them, and combine them to an optimum-width branch-decomposition of $f$.\n\nThe surprising fact that makes the above idea work is that we can always find safe cuts, provided that the branch-width of $f$ is small enough compared to $|V|$.\nIn particular, we show that if we are given a branch-decomposition of $f$ of width $\\ell$, where $3^{\\ell+1} < |V|$, then we can find a safe cut $(A,B)$ with $|A|,|B| \\ge 2$ in time $2^{O(\\ell)}\\gamma n^{O(1)} $.\nTherefore, by using either iterative compression or the FPT approximation algorithm of Oum and Seymour~\\cite{OS2004}, we can decompose along safe cuts until $|V| \\le 2^{O(k)}$.\nOnce $|V| \\le 2^{O(k)}$, we can use the XP algorithm of Oum and Seymour~\\cite{OS2005} to find an optimum-width branch-decomposition in $2^{O(k^2)} \\gamma$ time.\n\n\\paragraph{Organization of the paper.}\nThe paper is organized as follows.\nIn \\zcref{sec:prelim} we review the necessary definitions and preliminary results.\nIn \\zcref{sec:search} we provide an algorithm for testing if a cut satisfies a certain condition that implies it is safe. \nIn \\zcref{sec:titanic} we provide an algorithm for turning cuts into safe cuts.\nIn \\zcref{sec:algorithm} we give our algorithm, and in \\zcref{sec:matroid} the optimized version for matroids.\nWe conclude with discussions and open questions in \\zcref{sec:conclusion}.", "sketch": "We say that a \\emph{cut} of a connectivity function $f \\colon 2^{V} \\to \\mathbb{Z}$ is a bipartition $(A,B)$ of $V$. A \\emph{safe cut} is a cut $(A,B)$ such that there exists an optimum-width branch-decomposition where $(A,B)$ corresponds to an edge of the decomposition.\n\nThe main idea of the algorithm of \\zcref{thm:main-simplified} is to \"repeatedly find safe cuts and greedily decompose $f$ along them.\" In particular, if we find a safe cut $(A,B)$ with $|A|,|B| \\ge 2$, then we \"construct two smaller connectivity functions $f_A$ and $f_B$, recursively find optimum-width branch-decompositions of them, and combine them\" into an optimum-width branch-decomposition of $f$.\n\nThe key fact used is that \"we can always find safe cuts, provided that the branch-width of $f$ is small enough compared to $|V|$\": if we are given a branch-decomposition of width $\\ell$ with $3^{\\ell+1} < |V|$, then we can find a safe cut $(A,B)$ with $|A|,|B| \\ge 2$ in time $2^{O(\\ell)}\\gamma n^{O(1)}$.\n\nThus, \"by using either iterative compression or the FPT approximation algorithm of Oum and Seymour~\\cite{OS2004}, we can decompose along safe cuts until $|V| \\le 2^{O(k)}$.\" Once $|V| \\le 2^{O(k)}$, we \"use the XP algorithm of Oum and Seymour~\\cite{OS2005} to find an optimum-width branch-decomposition\" in $2^{O(k^2)}\\gamma$ time.", "expanded_sketch": "We say that a \\emph{cut} of a connectivity function $f \\colon 2^{V} \\to \\mathbb{Z}$ is a bipartition $(A,B)$ of $V$. A \\emph{safe cut} is a cut $(A,B)$ such that there exists an optimum-width branch-decomposition where $(A,B)$ corresponds to an edge of the decomposition.\n\nThe main idea of the algorithm of \\zcref{thm:main-simplified} is to \"repeatedly find safe cuts and greedily decompose $f$ along them.\" In particular, if we find a safe cut $(A,B)$ with $|A|,|B| \\ge 2$, then we \"construct two smaller connectivity functions $f_A$ and $f_B$, recursively find optimum-width branch-decompositions of them, and combine them\" into an optimum-width branch-decomposition of $f$.\n\nThe key fact used is that \"we can always find safe cuts, provided that the branch-width of $f$ is small enough compared to $|V|$\": if we are given a branch-decomposition of width $\\ell$ with $3^{\\ell+1} < |V|$, then we can find a safe cut $(A,B)$ with $|A|,|B| \\ge 2$ in time $2^{O(\\ell)}\\gamma n^{O(1)}$.\n\nThus, \"by using either iterative compression or the FPT approximation algorithm of Oum and Seymour~\\cite{OS2004}, we can decompose along safe cuts until $|V| \\le 2^{O(k)}$.\" Once $|V| \\le 2^{O(k)}$, we \"use the XP algorithm of Oum and Seymour~\\cite{OS2005} to find an optimum-width branch-decomposition\" in $2^{O(k^2)}\\gamma$ time.,", "expanded_theorem": "\\begin{theorem}[Simplified main theorem]\\label{thm:main-simplified}\nLet $n>1$ be an integer and $f:2^V\\to \\mathbb Z$ a connectivity function on an $n$-element set~$V$.\nLet $\\gamma$ be the time to compute $f(X)$ for any subset $X$ of~$V$.\nIn time\n$2^{O(k)}\\gamma n^6 \\log n + 2^{O(k^2)} \\gamma n$,\nwe can either find a branch-decomposition of~$f$ of width at most~$k$, \nor confirm that the branch-width of~$f$ is larger than~$k$.\n\\end{theorem}", "theorem_type": ["Algorithmic or Constructive", "Implication"], "mcq": {"question": "Let $n>1$ and $k$ be integers, and let $f:2^V\\to\\mathbb Z$ be a connectivity function on an $n$-element set $V$, meaning that $f(\\emptyset)=0$, $f(X)=f(V\\setminus X)$ for all $X\\subseteq V$, and $f$ is submodular: $f(X)+f(Y)\\ge f(X\\cap Y)+f(X\\cup Y)$ for all $X,Y\\subseteq V$. A branch-decomposition of $f$ is a pair $(T,L)$ where $T$ is a subcubic tree and $L$ is a bijection from the leaves of $T$ to $V$; for each edge $e$ of $T$, if $T-e$ has components inducing a partition $(A_e,B_e)$ of $V(T)$, then the width of $e$ is $f(L^{-1}(A_e))$, the width of $(T,L)$ is the maximum width over all edges, and the branch-width $\\operatorname{bw}(f)$ is the minimum such width. If $\\gamma$ is the time needed to compute $f(X)$ for any subset $X\\subseteq V$, which statement holds?", "correct_choice": {"label": "A", "text": "There is an algorithm running in time $2^{O(k)}\\gamma n^6\\log n + 2^{O(k^2)}\\gamma n$ that either finds a branch-decomposition of $f$ of width at most $k$, or confirms that $\\operatorname{bw}(f)>k$."}, "choices": [{"label": "B", "text": "There is an algorithm running in time $2^{O(k)}\\gamma n^6\\log n + 2^{O(k^2)}\\gamma n$ that, whenever $\\operatorname{bw}(f)\\le k$, finds a branch-decomposition of $f$ of width exactly $\\operatorname{bw}(f)$, and otherwise confirms that $\\operatorname{bw}(f)>k$."}, {"label": "C", "text": "There is an algorithm running in time $2^{O(k)}\\gamma n^6\\log n + 2^{O(k^2)}\\gamma n$ that either finds a branch-decomposition of $f$ of width at most $k$, or outputs no such decomposition."}, {"label": "D", "text": "For every fixed integer $k$, there is an algorithm running in time $2^{O(k)}\\gamma n^6\\log n$ that either finds a branch-decomposition of $f$ of width at most $k$, or confirms that $\\operatorname{bw}(f)>k$."}, {"label": "E", "text": "There is an algorithm running in time $2^{O(k)}\\gamma n^6\\log n + 2^{O(k^2)}\\gamma n$ that, provided $3^{k+1}k$."}], "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": "target-output upgraded from width-at-most-k to optimum-width decomposition", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the certification clause confirming that branch-width is larger than k", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "computational_check", "tampered_component": "removed the terminal XP stage on instances of size at most $2^{O(k)}$", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "promoted the safe-cut threshold condition $3^{\\ell+1}<|V|$ from an internal case split to a theorem hypothesis", "template_used": "wildcard"}]}} +{"id": "2601.05121v1", "paper_link": "http://arxiv.org/abs/2601.05121v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{theorem1.1}\nWhen $k\\geqslant 2$, one has\n\\[\nV_k(P)-L_k(P)\\ll P^{\\alpha_k+\\varepsilon}.\n\\]\nIn particular, one has\n\\[\nV_k(P)-L_k(P)\\ll P^{\\sqrt{8k+9}-1+\\varepsilon}.\n\\]", "start_pos": 8031, "end_pos": 8227, "label": "theorem1.1"}, "ref_dict": {"1.4": "\\begin{equation}\\label{1.4}\n\\sum_{i=1}^{k+1}x_i^j=\\sum_{i=1}^{k+1}y_i^j\\quad (1\\le j\\le k).\n\\end{equation}", "theorem1.1": "\\begin{theorem}\\label{theorem1.1}\nWhen $k\\ge 2$, one has\n\\[\nV_k(P)-L_k(P)\\ll P^{\\alpha_k+\\varepsilon}.\n\\]\nIn particular, one has\n\\[\nV_k(P)-L_k(P)\\ll P^{\\sqrt{8k+9}-1+\\varepsilon}.\n\\]\n\\end{theorem}", "1.3": "\\begin{equation}\\label{1.3}\n\\sum_{i=1}^{k+1}x_i^{2j-1}=\\sum_{i=1}^{k+1}y_i^{2j-1}\\quad (1\\le j\\le k),\n\\end{equation}", "1.1": "\\begin{equation}\\label{1.1}\n\\sum_{i=1}^{2k+2}z_i^{2j-1}=0\\quad (1\\le j\\le k).\n\\end{equation}", "theorem1.2": "\\begin{theorem}\\label{theorem1.2}\nWhen $k\\ge 2$, one has\n\\[\nV_k^*(P)-L_k^*(P)\\ll P^{\\alpha_k+\\varepsilon}.\n\\]\nIn particular, one has\n\\[\nV_k^*(P)-L_k^*(P)\\ll P^{\\sqrt{8k+9}-1+\\varepsilon}.\n\\]\n\\end{theorem}"}, "pre_theorem_intro_text_len": 1722, "pre_theorem_intro_text": "This memoir is devoted to the system of simultaneous Diophantine equations\n\\begin{equation}\\label{1.1}\n\\sum_{i=1}^{2k+2}z_i^{2j-1}=0\\quad (1\\leqslant j\\leqslant k).\n\\end{equation}\nWhen $k$ is a natural number and $P$ is a large real number, we denote by $V_k(P)$ the number of integral \nsolutions of the system \\eqref{1.1} with $|z_i|\\leqslant P$ $(1\\leqslant i\\leqslant 2k+2)$. The situation with $k=1$ being too mundane \nto attract our attention, we focus instead on those scenarios in which $k\\geqslant 2$. The system \\eqref{1.1} possesses \nlinear spaces of solutions having affine dimension $k+1$ typified by the one defined by the equations\n\\[\nz_{2i-1}+z_{2i}=0\\quad (1\\leqslant i\\leqslant k+1).\n\\]\nOn considering appropriate permutations of the underlying indices $\\{1,2,\\ldots ,2k+2\\}$, a moment of reflection \nreveals that the number of such linear spaces is equal to\n\\[\nc_k=2^{-k-1}\\frac{(2k+2)!}{(k+1)!}=\\prod_{j=1}^{k+1}(2j-1).\n\\]\nWe denote by $L_k(P)$ the number of integral solutions of the system \\eqref{1.1} lying on the collection of all such \nlinear spaces with $|z_i|\\leqslant P$ $(1\\leqslant i\\leqslant 2k+2)$. Thus, we have\n\\[\nL_k(P)=c_k(2P+1)^{k+1}+O(P^k).\n\\]\nWe think of these solutions lying on such linear spaces as being {\\it trivial}, and refer to any integral solution not \nof this type as being {\\it non-trivial}. Our goal in this paper is to show that, when $k\\geqslant 2$, there is a paucity of \nnon-trivial solutions to the system \\eqref{1.1} in a particularly strong sense.\\par\n\nIn order to describe our main conclusion, we define the exponent\n\\begin{equation}\\label{1.2}\n\\alpha_k=\\min_{\\substack{r\\in \\mathbb N\\\\ 2\\leqslant r\\leqslant 2k+1}}\\Bigl( r-1+\\frac{2k+2}{r}\\Bigr) .\n\\end{equation}", "context": "This memoir is devoted to the system of simultaneous Diophantine equations\n\\begin{equation}\\label{1.1}\n\\sum_{i=1}^{2k+2}z_i^{2j-1}=0\\quad (1\\leqslant j\\leqslant k).\n\\end{equation}\nWhen $k$ is a natural number and $P$ is a large real number, we denote by $V_k(P)$ the number of integral \nsolutions of the system \\eqref{1.1} with $|z_i|\\leqslant P$ $(1\\leqslant i\\leqslant 2k+2)$. The situation with $k=1$ being too mundane \nto attract our attention, we focus instead on those scenarios in which $k\\geqslant 2$. The system \\eqref{1.1} possesses \nlinear spaces of solutions having affine dimension $k+1$ typified by the one defined by the equations\n\\[\nz_{2i-1}+z_{2i}=0\\quad (1\\leqslant i\\leqslant k+1).\n\\]\nOn considering appropriate permutations of the underlying indices $\\{1,2,\\ldots ,2k+2\\}$, a moment of reflection \nreveals that the number of such linear spaces is equal to\n\\[\nc_k=2^{-k-1}\\frac{(2k+2)!}{(k+1)!}=\\prod_{j=1}^{k+1}(2j-1).\n\\]\nWe denote by $L_k(P)$ the number of integral solutions of the system \\eqref{1.1} lying on the collection of all such \nlinear spaces with $|z_i|\\leqslant P$ $(1\\leqslant i\\leqslant 2k+2)$. Thus, we have\n\\[\nL_k(P)=c_k(2P+1)^{k+1}+O(P^k).\n\\]\nWe think of these solutions lying on such linear spaces as being {\\it trivial}, and refer to any integral solution not \nof this type as being {\\it non-trivial}. Our goal in this paper is to show that, when $k\\geqslant 2$, there is a paucity of \nnon-trivial solutions to the system \\eqref{1.1} in a particularly strong sense.\\par\n\nIn order to describe our main conclusion, we define the exponent\n\\begin{equation}\\label{1.2}\n\\alpha_k=\\min_{\\substack{r\\in \\mathbb N\\\\ 2\\leqslant r\\leqslant 2k+1}}\\Bigl( r-1+\\frac{2k+2}{r}\\Bigr) .\n\\end{equation}\n\n\\begin{equation}\\label{1.1}\n\\sum_{i=1}^{2k+2}z_i^{2j-1}=0\\quad (1\\le j\\le k).\n\\end{equation}", "full_context": "This memoir is devoted to the system of simultaneous Diophantine equations\n\\begin{equation}\\label{1.1}\n\\sum_{i=1}^{2k+2}z_i^{2j-1}=0\\quad (1\\leqslant j\\leqslant k).\n\\end{equation}\nWhen $k$ is a natural number and $P$ is a large real number, we denote by $V_k(P)$ the number of integral \nsolutions of the system \\eqref{1.1} with $|z_i|\\leqslant P$ $(1\\leqslant i\\leqslant 2k+2)$. The situation with $k=1$ being too mundane \nto attract our attention, we focus instead on those scenarios in which $k\\geqslant 2$. The system \\eqref{1.1} possesses \nlinear spaces of solutions having affine dimension $k+1$ typified by the one defined by the equations\n\\[\nz_{2i-1}+z_{2i}=0\\quad (1\\leqslant i\\leqslant k+1).\n\\]\nOn considering appropriate permutations of the underlying indices $\\{1,2,\\ldots ,2k+2\\}$, a moment of reflection \nreveals that the number of such linear spaces is equal to\n\\[\nc_k=2^{-k-1}\\frac{(2k+2)!}{(k+1)!}=\\prod_{j=1}^{k+1}(2j-1).\n\\]\nWe denote by $L_k(P)$ the number of integral solutions of the system \\eqref{1.1} lying on the collection of all such \nlinear spaces with $|z_i|\\leqslant P$ $(1\\leqslant i\\leqslant 2k+2)$. Thus, we have\n\\[\nL_k(P)=c_k(2P+1)^{k+1}+O(P^k).\n\\]\nWe think of these solutions lying on such linear spaces as being {\\it trivial}, and refer to any integral solution not \nof this type as being {\\it non-trivial}. Our goal in this paper is to show that, when $k\\geqslant 2$, there is a paucity of \nnon-trivial solutions to the system \\eqref{1.1} in a particularly strong sense.\\par\n\nIn order to describe our main conclusion, we define the exponent\n\\begin{equation}\\label{1.2}\n\\alpha_k=\\min_{\\substack{r\\in \\mathbb N\\\\ 2\\leqslant r\\leqslant 2k+1}}\\Bigl( r-1+\\frac{2k+2}{r}\\Bigr) .\n\\end{equation}\n\n\\begin{equation}\\label{1.1}\n\\sum_{i=1}^{2k+2}z_i^{2j-1}=0\\quad (1\\le j\\le k).\n\\end{equation}\n\nIn order to describe our main conclusion, we define the exponent\n\\begin{equation}\\label{1.2}\n\\alpha_k=\\min_{\\substack{r\\in \\mathbb N\\\\ 2\\le r\\le 2k+1}}\\Bigl( r-1+\\frac{2k+2}{r}\\Bigr) .\n\\end{equation}\n\nThe conclusion of Theorem \\ref{theorem1.1} shows that as $P\\rightarrow \\infty$, one has the asymptotic formula\n\\[\nV_k(P)\\sim c_k(2P+1)^k,\n\\]\nwhenever $k\\ge 6$. This conclusion was obtained more than a decade ago by Br\\\"udern and Robert \\cite{BR2012} \n(see the antepenultimate paragraph of the introduction of \\cite{BR2012}), although the main focus of the latter \nauthors was an analogue of Theorem \\ref{theorem1.1} addressing the system \\eqref{1.3} below in which the \nunderlying variables are constrained to be positive. They show, in fact, that\n\\[\nV_k(P)-L_k(P)\\ll P^{\\lambda_k+\\varepsilon},\n\\]\nwhere $\\lambda_ki_p\\ge 1\\, (l\\ne p)}}\\alpha_{\\mathbf i}.\n\\]\nThen we have\n\\[\n\\biggl| \\prod_{p=1}^rA_p\\biggr|\\le \\prod_{\\substack{\\mathbf i\\in \\mathcal I\\\\ i_m\\ge 1\\, (1\\le m\\le r)}}\n|\\alpha_{\\mathbf i}|\\le \\prod_{l=1}^\\kappa |{\\widetilde \\alpha}_{l1}^\\pm|\\le (2P)^\\kappa .\n\\] \nConsequently, in any solution ${\\boldsymbol \\alpha}$ counted by $X_r(P)$, there exists a choice for the index $p$ \nwith $1\\le p\\le r$ for which one has\n\\[\n1\\le |A_p|\\le (2P)^{\\kappa/r}.\n\\]\nFurthermore, given an index $m$ with $1\\le m\\le r$, it follows from \\eqref{3.2} that, for each solution \n${\\boldsymbol \\alpha}$ counted by $X_r(P)$, when $1\\le j\\le r$ and $j\\ne m$, one has\n\\[\n{\\widetilde \\alpha}_{im}^\\pm -{\\widetilde \\alpha}_{ij}^\\pm=z_m-z_j\\quad (1\\le i\\le \\kappa).\n\\]\nBy relabelling variables, if necessary, it follows that $X_r(P)\\ll Y_r(P)$, where $Y_r(P)$ denotes the number of solutions \nof the system of equations \n\\begin{equation}\\label{3.5}\n{\\widetilde \\alpha}_{im}^\\pm -{\\widetilde \\alpha}_{i1}^\\pm=z_m-z_1\\quad (2\\le m\\le r,\\, 1\\le i\\le \\kappa),\n\\end{equation}\nwith $z_m-z_1$ fixed and non-zero with $|z_m-z_1|\\le 2P$, and with the $\\alpha_{\\mathbf i}$ satisfying \\eqref{3.3} \nand \nthe inequality\n\\begin{equation}\\label{3.6}\n1\\le |A_1|\\le (2P)^{\\kappa/r}.\n\\end{equation}\nFurthermore, by \\eqref{3.4}, we have\n\\begin{equation}\\label{3.7}\nV_k(P)-L_k(P)\\ll P^{r-1}+P^{r+\\varepsilon}Y_r(P).\n\\end{equation}\n\n\\par We claim that when the variables $\\alpha_{\\mathbf i}$ with\n\\begin{equation}\\label{3.8}\n\\mathbf i\\in \\mathcal I\\qquad \\text{and}\\qquad i_m>i_1\\quad (2\\le m\\le r)\n\\end{equation}\nare fixed, then there are $O(P^\\varepsilon)$ possible choices for the variables $\\alpha_{\\mathbf i}$ satisfying \n\\eqref{3.3} and \\eqref{3.5}. Supposing temporarily such to be the case, the combination of \\eqref{3.6} and \\eqref{3.7}, \ntogether with a standard estimate for the divisor function, shows that\n\\begin{equation}\\label{3.9}\nV_k(P)-L_k(P)\\ll P^{r-1}+P^{r+\\kappa/r+\\varepsilon}.\n\\end{equation}\nOn recalling from \\eqref{2.10} that $\\kappa=2k+2-r$, the first conclusion of Theorem \\ref{theorem1.1} follows by \nreference to the definition \\eqref{1.2}.\\par\n\n\\section{A generalisation of the Br\\\"udern-Robert system}\nIn this section we consider the generalisation of the system \\eqref{1.3} given by the simultaneous equations\n\\begin{equation}\\label{5.1}\n\\sum_{i=1}^{k+1} x_i^{(2j-1)d}=\\sum_{i=1}^{k+1}y_i^{(2j-1)d}\\quad (1\\le j\\le k),\n\\end{equation}\nin which $d$ is a fixed natural number. We denote by $V_{k,d}^*(P)$ the number of integral solutions of the system \n\\eqref{5.1} with $1\\le x_i,y_i\\le P$ $(1\\le i\\le k+1)$, and again denote by $L_k^*(P)$ the corresponding number of \ndiagonal solutions defined as in the preamble to the statement of Theorem \\ref{theorem1.2}. Of course, the system\n\\eqref{5.1} is obtained from \\eqref{1.3} by replacing $x_i$ and $y_i$ by $x_i^d$ and $y_i^d$ throughout. It is \ntherefore immediate from Theorem \\ref{theorem1.2} that one has\n\\[\nV_{k,d}^*(P)-L_k^*(P)\\ll (P^d)^{\\alpha_k+\\varepsilon}\\ll P^{d\\sqrt{8k+9}-d+d\\varepsilon}.\n\\]\nThus, the asymptotic formula $V_{k,d}^*(P)\\sim (k+1)!P^{k+1}$ holds whenever $1\\le d\\le \\sqrt{k/8}$. By a more \ncareful analysis along the lines of \\S\\S2 and 3, a sharper conclusion may be obtained.\n\n\\begin{theorem}\\label{theorem5.1}\nWhen $k\\ge 2$ and $d\\ge 1$, one has\n\\[\nV_{k,d}^*(P)-L_k^*(P)\\ll P^{\\alpha_{k,d}+\\varepsilon},\n\\]\nwhere\n\\[\n\\alpha_{k,d}=\\min_{\\substack{r\\in \\mathbb N\\\\ 2\\le r\\le 2k+1}}\\Bigl( r-d+\\frac{(2k+2)d}{r}\\Bigr) .\n\\]\nIn particular, one has\n\\[\nV_{k,d}^*(P)-L_k^*(P)\\ll P^{\\sqrt{8d(k+1)+1}-d+\\varepsilon}.\n\\]\n\\end{theorem}\n\n\\begin{equation}\\label{1.1}\n\\sum_{i=1}^{2k+2}z_i^{2j-1}=0\\quad (1\\le j\\le k).\n\\end{equation}\n\n\\begin{equation}\\label{1.3}\n\\sum_{i=1}^{k+1}x_i^{2j-1}=\\sum_{i=1}^{k+1}y_i^{2j-1}\\quad (1\\le j\\le k),\n\\end{equation}\n\n\\begin{theorem}\\label{theorem1.1}\nWhen $k\\ge 2$, one has\n\\[\nV_k(P)-L_k(P)\\ll P^{\\alpha_k+\\varepsilon}.\n\\]\nIn particular, one has\n\\[\nV_k(P)-L_k(P)\\ll P^{\\sqrt{8k+9}-1+\\varepsilon}.\n\\]\n\\end{theorem}", "post_theorem_intro_text_len": 5629, "post_theorem_intro_text": "The conclusion of Theorem \\ref{theorem1.1} shows that as $P\\rightarrow \\infty$, one has the asymptotic formula\n\\[\nV_k(P)\\sim c_k(2P+1)^k,\n\\]\nwhenever $k\\geqslant 6$. This conclusion was obtained more than a decade ago by Br\\\"udern and Robert \\cite{BR2012} \n(see the antepenultimate paragraph of the introduction of \\cite{BR2012}), although the main focus of the latter \nauthors was an analogue of Theorem \\ref{theorem1.1} addressing the system \\eqref{1.3} below in which the \nunderlying variables are constrained to be positive. They show, in fact, that\n\\[\nV_k(P)-L_k(P)\\ll P^{\\lambda_k+\\varepsilon},\n\\]\nwhere $\\lambda_k0$. Throughout, the symbols \n$\\ll $ and $\\gg $ denote Vinogradov's well-known notation. Implicit constants in both the notations of Vinogradov \nand Landau will depend at most on $\\varepsilon$, $k$ and $r$. We make frequent use of vector notation in the form \n$\\mathbf x=(x_1,\\ldots ,x_r)$. Here, the dimension $r$ depends on the course of the argument.\\par\n\nWork on this paper was conducted while the author was supported by NSF grant DMS-2502625 and Simons \nFellowship in Mathematics SFM-00011955. The author is grateful to the Institute for Advanced Study, Princeton, for \nhosting his sabbatical, during which period this paper was completed.\\par\n\n\\noindent{\\bf Historical note:} The author's interest in and work on the topic of this memoir can be traced back to his \nPh.D.~studies, at which time the author came across the paper of Roger Heath-Brown \\cite{HB1988} making use of \nquasi-paucity estimates for the Diophantine system\n\\[\n\\left .\\begin{aligned}x_1^3+x_2^3+x_3^3&=y_1^3+y_2^3+y_3^3\\\\\nx_1+x_2+x_3&=y_1+y_2+y_3\\end{aligned}\\right\\}.\n\\]\nThis inspired work in the author's thesis \\cite[Theorem 1.3 of Chapter 3]{Woo1990}, completed under the supervision \nof Bob Vaughan in 1990, providing analogous conclusions concerning pairs of equations having arbitrary degrees. The \nauthor is grateful to Roger for this, and many other contributions, that have provided such inspiration for nearly four \ndecades.", "sketch": "Our approach to the proof of Theorem \\ref{theorem1.1} is based on that employed in our work joint with Vaughan \\cite{VW1997} concerning the corresponding Vinogradov system \\eqref{1.4}. In that setting, the bound is obtained by \"showing that the solutions of the system \\eqref{1.4} satisfy auxiliary equations exhibiting copious multiplicative structure\", which \"fosters a parameterisation of solutions\" allowing \"considerable control\" on the number of non-trivial solutions. The introduction states that \"a similar approach is possible for the system \\eqref{1.1}, as will be apparent from the discussion of \\S2\"; then \"the exploitation of this parameterisation of the solutions will be discussed in \\S3, where we complete the proofs of Theorems \\ref{theorem1.1} and \\ref{theorem1.2}.\" It is also noted that in \\S6 an \"elementary discrete inequality\" is recorded, \"applied in proving the second assertion of Theorem \\ref{theorem1.1}.\"", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{theorem1.1}\nWhen $k\\geqslant 2$, one has\n\\[\nV_k(P)-L_k(P)\\ll P^{\\alpha_k+\\varepsilon}.\n\\]\nIn particular, one has\n\\[\nV_k(P)-L_k(P)\\ll P^{\\sqrt{8k+9}-1+\\varepsilon}.\n\\],", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "For an integer $k\\ge 2$ and a large real number $P$, let $V_k(P)$ denote the number of integer solutions $(z_1,\\dots,z_{2k+2})$ with $|z_i|\\le P$ $(1\\le i\\le 2k+2)$ to the system\n\\[\n\\sum_{i=1}^{2k+2} z_i^{2j-1}=0\\qquad (1\\le j\\le k).\n\\]\nLet $L_k(P)$ denote the number of such solutions that lie on one of the “trivial” affine linear spaces obtained by permuting coordinates in the family\n\\[\nz_{2i-1}+z_{2i}=0\\qquad (1\\le i\\le k+1).\n\\]\nDefine\n\\[\n\\alpha_k=\\min_{\\substack{r\\in\\mathbb N\\\\ 2\\le r\\le 2k+1}}\\left(r-1+\\frac{2k+2}{r}\\right).\n\\]\nWhich statement holds for every integer $k\\ge 2$?", "correct_choice": {"label": "A", "text": "For every $\\varepsilon>0$, one has\n\\[\nV_k(P)-L_k(P)\\ll P^{\\alpha_k+\\varepsilon},\n\\]\nand in particular\n\\[\nV_k(P)-L_k(P)\\ll P^{\\sqrt{8k+9}-1+\\varepsilon}.\n\\]"}, "choices": [{"label": "B", "text": "For every $\\varepsilon>0$, one has\n\\[\nV_k(P)-L_k(P)\\ll P^{\\alpha_k+\\varepsilon},\n\\]\nand in particular\n\\[\nV_k(P)-L_k(P)\\ll P^{\\sqrt{8k+9}+\\varepsilon}.\n\\]"}, {"label": "C", "text": "For every $\\varepsilon>0$, one has\n\\[\nV_k(P)-L_k(P)\\ll P^{\\sqrt{8k+9}-1+\\varepsilon}.\n\\]"}, {"label": "D", "text": "For every $\\varepsilon>0$, one has\n\\[\nV_k(P)-L_k(P)\\ll P^{\\alpha_k+\\varepsilon},\n\\]\nwhere\n\\[\n\\alpha_k=\\min_{\\substack{r\\in\\mathbb N\\\\ 1\\le r\\le 2k+2}}\\left(r-1+\\frac{2k+2}{r}\\right).\n\\]"}, {"label": "E", "text": "There exists $\\varepsilon>0$ such that for every integer $k\\ge 2$, one has\n\\[\nV_k(P)-L_k(P)\\ll P^{\\alpha_k+\\varepsilon},\n\\]\nand in particular\n\\[\nV_k(P)-L_k(P)\\ll P^{\\sqrt{8k+9}-1+\\varepsilon}.\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": "derived explicit exponent loses the subtractive constant", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "drops the sharper bound involving the minimised exponent \\alpha_k", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "range of the minimisation defining \\alpha_k", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "quantifier order on \\varepsilon", "template_used": "quantifier_dependence"}]}} +{"id": "2601.05915v1", "paper_link": "http://arxiv.org/abs/2601.05915v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{PrincipalSinCalculoExplicito}\n The Lyapunov spectrum $L_{p}$ is real analytic on $[\\log(p),\\infty)$. For each $\\alpha\\geq \\log p$\n \\begin{align}\\label{EcuSInCalculoExplicito}\n L_{p}(\\alpha)=\\dfrac{1}{\\alpha}\\inf\\left\\{P(-t\\log \\psi)+t\\alpha:t>0\\right\\},\n \\end{align}where $P(-t\\log\\psi)$ is the topological pressure of $-t\\log\\psi$ with respect to $T_p$. The infimum is attained at a unique $t_{\\alpha}>0$ such that $\\frac{d}{dt}P(-t\\log \\psi)|_{t=t_{\\alpha}}=-\\alpha$. Moreover, there exists a unique equilibrium state $\\mu_{t_{\\alpha}}$ for $-t_{\\alpha}\\log \\psi$ such that $\\mu_{t_{\\alpha}}(J_{p}(\\alpha))=1$.", "start_pos": 21909, "end_pos": 22576, "label": "PrincipalSinCalculoExplicito"}, "ref_dict": {"ApproximationSpectrumTHM": "\\begin{corollary}\\label{ApproximationSpectrumTHM}\n For each $\\alpha\\geq \\log p$, the set \n \\begin{align*}\n \\left\\{x\\in p\\Z_{p}\\setminus F:-\\lim_{n\\rightarrow \\infty}\\dfrac{1}{n}\\log \\left|x-\\dfrac{p_{n}(x)}{q_{n}(x)}\\right|_{p}=\\alpha\\right\\} \n \\end{align*}\n has Hausdorff dimension\n \\begin{align*}\n \\dfrac{\\log(p-1)+\\log(\\alpha-\\log p)-\\log\\log p+\\alpha\\log_{p}\\alpha-\\log_{p}(\\alpha-\\log p)}{\\alpha}.\n \\end{align*}\n \\end{corollary}", "continued": "\\begin{align}\\label{continued}\n x= \\cfrac{1}{a_{1}+\\cfrac{1}{a_{2}+\\cfrac{1}{a_{3}+\\cdots }}}=[a_{1},a_{2},a_{3},...],\n \\end{align}", "EcuLyaAsRationalApprox": "\\begin{align}\\label{EcuLyaAsRationalApprox}\n \\lambda(x)=-\\lim_{n\\rightarrow \\infty}\\dfrac{1}{n}\\log \\left|x-\\dfrac{P_{n}}{Q_{n}}\\right|.\n \\end{align}"}, "pre_theorem_intro_text_len": 9874, "pre_theorem_intro_text": "A central problem in multifractal analysis concerns the fractal decomposition of level sets of dynamically defined functions, typically quantified via their Hausdorff dimension. Let $f:M\\to M$ be a $C^{1}$ map on a compact manifold $M$. The Lyapunov exponent of $f$ at a point $x\\in M$ is defined by \n\\begin{align}\\label{Lyapunov exponent definition}\n \\lambda(x)=\\lim_{n\\rightarrow{\\infty}}\\dfrac{1}{n}\\log \\|Df^{n}(x)\\|,\n\\end{align} whenever this limit exists. This quantity measures the exponential rate at which nearby orbits diverge. For each real number $\\alpha$, we define\n\n$$J_{\\alpha}=\\{x\\in M: \\lambda(x)=\\alpha\\}.$$ \nBy the chain rule, $\\lambda(x)$ can be expressed as the pointwise limit of Birkhoff averages of $\\log \\|Df\\|$. Consequently, for any ergodic measure, the Lyapunov exponent is constant almost everywhere. In particular, a single ergodic measure cannot capture the geometric properties of different level sets $J_{\\alpha_1}$ and $J_{\\alpha_2}$. This motivates the study of these sets through their Hausdorff dimension. The Lyapunov spectrum of $\\alpha$ is defined by\n\\begin{align*}\n L(\\alpha)=\\dim_{\\mathrm{H}}\\left(J_{\\alpha}\\right),\n\\end{align*} whenever $J_{\\alpha}\\neq \\emptyset$, where $\\dim_{\\mathrm{H}}(\\cdot)$ denotes the Hausdorff dimension. \n\nFor conformal expanding maps on smooth manifolds with H\\\"older continuous derivative, this problem dates back to the work of Weiss \\cite{we}. Using tools such as Markov partitions and Gibbs measures, Weiss proved that $L$ is defined on a bounded interval $\\left[\\alpha_{min},\\alpha_{max}\\right]$, is real analytic, and satisfies\n\\begin{align}\\label{EcuWeiss}\n L(\\alpha)=\\dfrac{1}{\\alpha}\\inf\\left\\{P\\left(-t\\log\\|Df\\|\\right)+t\\alpha:t\\in \\mathbb{R}\\right\\},\n\\end{align} \nwhere $P(-t\\log\\|Df\\|)$ denotes the topological pressure associated with the potential $-t\\log\\|Df\\|$.\nFor interval maps, Pollicott and Weiss \\cite{mh}, and later Kesseb\\\"ohmer and Stratmann \\cite{kessestrat}, extended this result to the Gauss map $G$. This map has been extensively studied due to its intimate connection with the continued fraction expansion of real numbers.\n\nThe Gauss map $G: (0,1]\\rightarrow [0,1]$ is defined by $$G(x)=\\dfrac{1}{x}-\\left\\lfloor \\dfrac{1}{x}\\right\\rfloor,$$ where $\\lfloor \\cdot \\rfloor$ denotes the integer part.\nEvery irrational $x\\in (0,1)$ admits a unique continued fraction expansion of the form\n \\begin{align}\\label{continued}\n x= \\cfrac{1}{a_{1}+\\cfrac{1}{a_{2}+\\cfrac{1}{a_{3}+\\cdots }}}=[a_{1},a_{2},a_{3},...],\n \\end{align} where each $a_{i}$ is a positive integer and $a_{i}=\\left\\lfloor \\frac{1}{G^{i-1}x}\\right\\rfloor$. \nThe $n$-th rational approximation of $x$ is given by\n \\begin{align*}\n \\dfrac{P_n}{Q_{n}}=\\cfrac{1}{a_{1}+\\cfrac{1}{a_{2}+\\cfrac{1}{a_{3}+\\dfrac{1}{\\ddots +\\frac{1}{a_{n}}} }}}=[a_{1},a_{2},...,a_{n}].\n \\end{align*} \nThe definition of the Lyapunov exponent extends naturally to the Gauss map. For $x\\in (0,1]$ we define $\\lambda(x)=\\lim_{n\\to \\infty}\\frac{1}{n}\\log |(G^{n})'(x)|$, whenever this limit exists. Moreover, $\\lambda(x)$ can be written as (see \\cite[Page 160]{mh})\n \\begin{align}\\label{EcuLyaAsRationalApprox}\n \\lambda(x)=-\\lim_{n\\rightarrow \\infty}\\dfrac{1}{n}\\log \\left|x-\\dfrac{P_{n}}{Q_{n}}\\right|.\n \\end{align} \nBy equation \\eqref{EcuLyaAsRationalApprox}, $\\lambda(x)$ quantifies the exponential rate at which the rational approximations $P_n/Q_n$ converge to \n$x$, linking Lyapunov exponents to classical Diophantine approximation theory. For Lebesgue-almost every $x\\in (0,1)$, one has $\\lambda(x)=\\frac{\\pi^{2}}{6\\log 2}$ since $\\lambda\n(x)$ is a pointwise limit of Birkhoff averages. The domain of the Lyapunov spectrum $L$ for the Gauss map is $\\left(2\\log\\left(\\frac{\\sqrt{5}-1}{2}\\right),\\infty\\right)$. since $(0,1)\\setminus \\mathbb{Q}$ is a non-compact space, this leads to a different situation from the one considered by Weiss. Nevertheless, $L$ is a real analytic function, and for each $\\alpha$ in this domain, there exists a unique ergodic measure supported on $J_{\\alpha}$.\n\nIn this paper, we extend the study of Lyapunov spectra and their Diophantine implications to the Schneider map on $p\\mathbb{Z}_p$. \n\nLet $p$ be a prime number and let $v_p$ denote the $p$-adic valuation on $\\mathbb{Q}$, defined by the property that for each $x\\in \\mathbb{Q}$, there exists a unique integer\n$v_p(x)$ such that $x=p^{v_p(x)}\\frac{m}{n}$ with $(p,mn)=1$. The $p$-adic absolute value on $\\mathbb{Q}$ is defined by $|0|_p=0$, and $|x|_p=p^{-v_p(x)}$ if $x\\neq 0$. The field $\\Q_p$ of $p$-adic numbers is the completion of $\\mathbb{Q}$ with respect to this norm. The set of all elements $x\\in \\Q_p$ such that $|x|_p\\leq 1$ is denoted by $\\Z_p$, and the subset $p\\Z_p$ consists of those with $|x|_p<1$. Explicitly, any element in $\\Q_p$ can be expressed as $\\sum_{n\\geq n_0}c_np^{n}$, where $n_0$ is an integer depending only on $x$ and $c_n\\in \\{0,1,...,p-1\\}$. An element $x$ in $\\Z_p$ can be written as $\\sum_{n\\geq0} c_np^n$. Thus, $p\\mathbb{Z}_p$ consists of all elements in $\\Z_p$ with $c_0=0.$ \nSince $\\Q_p$ is locally compact, there exists a Haar measure $\\mu_p$ normalized by $\\mu_{p}(p\\Z_p)=1$. A detailed exposition on Haar measures can be found in \\cite{folland}.\n\nThe Schneider map $T_{p}:p\\Z_{p}\\rightarrow p\\Z_{p}$ is defined by $T_{p}(0)=0$, and for $x\\neq 0$\n \\begin{align*}\n T_{p}(x)=\\dfrac{p^{a_{1}(x)}}{x}-b_{1}(x),\n \\end{align*} where $a_{1}(x)=v_{p}(x)$ and $b_1(x)\\in\\{1,2,...,p-1\\}$ is uniquely determined by the congruence $b_{1}(x)\\equiv p^{a_{1}(x)}/x$ (mod $p$). Outside a countable set, every element of $p\\Z_p$ admits a continued fraction expansion analogous to \\eqref{continued}. More precisely, for $x\\in p\\Z_{p}\\setminus \\bigcup_{k\\in \\mathbb{N}} T_p^{-k}(0)$ and for each $n\\in \\mathbb{N}$, one can write\n \\begin{align}\\label{SchneiderContinued}\n x= \\cfrac{p^{a_{1}(x)}}{b_{1}(x)+\\cfrac{p^{a_{2}(x)}}{b_{2}(x)+\\cfrac{p^{a_{3}(x)}}{\\ddots + \\cfrac{p^{a_{n}(x)}}{b_{n}(x)+T_{p}^n(x)}}}}.\n \\end{align}\nwhere $a_{i}(x)=a_{1}(T^{i-1}_{p}x)$ and $b_{i}(x)=b_{1}(T^{i-1}_{p}x)$. Note that a point $x\\in p\\mathbb{Z}_p$ has a finite continued fraction expansion if and only if $T^{n}_{p}(x)=0$ for some $n\\in \\mathbb{N}$. We denote by $F$ the set of points in $p\\Z_p$ with finite continued fraction expansion. By the preceding discussion, $F=\\bigcup_{k=0}^\\infty T_p^{-k}(0)$. This situation is analogous to the classical Gauss map, where the points with finite continued fraction expansion are $\\bigcup^{\\infty}_{n=0}G^{-n}(0)=\\mathbb{Q}$. In contrast, if $x$ is a rational number with a infinite Schneider continued fraction expansion, then there exists $N\\in \\mathbb{N}$ such that $a_{n}(x)=1$ and $b_{n}(x)=p-1$ for all $n\\geq N$ (see \\cite[Theorem 1]{hw}). \n The Schneider map is ergodic with respect to the Haar measure $\\mu_p$ on $p\\Z_p$ (see \\cite[Lemma 2]{hw}) and its measure theoretic entropy is $\\frac{p}{p-1}\\log p $ (see \\cite{hn}).\n\n From a fractal geometric perspective, Hu, Yu and Zhao (\\cite{hyz}) studied multifractal decomposition determined by the asymptotic arithmetic mean of the digits appearing in the continued fraction expansion. More recently, Song, Wu, Yu and Zeng \\cite{swyz} investigated the Hausdorff dimension of related level sets. We continue this line of research by studying the fractal decomposition induced by rational approximations associated with the Schneider map.\n\nFor each pair of integers $a\\geq 1$ and $1\\leq b\\leq p-1$, we define the cylinder set \n $$I_{(a,b)}:=\\{x\\in p\\Z_p:a_{1}(x)=a \\text{ and $b_{1}(x)=b$}\\},$$\n and introduce the potential function\n\\begin{align*}\n \\psi(x)=\\dfrac{1}{\\left|I_{(a_{1}(x),b_{1}(x))}\\right|_{p}}=p^{a_{1}(x)}.\n\\end{align*} \nAs will be shown, this potential plays a role analogous to $\\log\\|Df\\|$ in the classical setting, in the sense that $\\log \\psi(x)$ measures the local expansion rate of $T_p$. \n\nFor $x\\in p\\Z_p\\setminus F$, we define the Lyapunov exponent by\n\n\\begin{align*}\n \\lambda_{p}(x)=\\lim_{n\\rightarrow{\\infty}}\\dfrac{1}{n}S_{n}\\log\\psi (x),\n\\end{align*} \nwhenever this limit exists, where $S_n\\log\\psi=\\sum^{n-1}_{k=0}\\log\\psi\\circ T^{k}_{p}$ denotes the $n$-th Birkhoff sum of $\\log\\psi$ with respect to $T_{p}$. It follows that\n\\begin{align}\n \\lambda_{p}(x)=\\lim_{n\\rightarrow{\\infty}}\\log p \\cdot \\dfrac{a_{1}(x)+a_{2}(x)+\\cdots +a_{n}(x)}{n},\n\\end{align} \nso that, up to a multiplicative constant, the Lyapunov exponent coincides with the asymptotic arithmetic mean of the digits in the continued fraction expansion. The associated level sets are $$J_{p}(\\alpha)=\\{x\\in p\\Z_{p}\\setminus F: \\lambda_{p}(x)=\\alpha\\},$$ and the Lyapunov spectrum is defined by $$L_{p}(\\alpha)=\\mathrm{dim}_{\\mathrm{H}}\\,J_{p}(\\alpha),$$ where $\\mathrm{dim}_{\\mathrm{H}}$ now denotes the Hausdorff dimension with respect to the $p$-adic norm. In the case of the Gauss map, the Lyapunov spectrum is computed via the topological pressure of the geometric potential $-t\\log|G'|$, which allows one to estimate the diameters of cylinder sets using the Mean Value Theorem. However, since the Gauss map acts on a non-compact space, the thermodynamic formalism in this setting requires additional assumptions, such as the Big Images Property and H\\\"older continuity of the potential. Our main difficulty in developing a thermodynamic formalism for $T_p$ stems from two facts. First, the set of points in $p\\Z_p$ with infinite continued fraction expansions is non-compact. Second, continuous functions on $p\\Z_p$ do not, in general, satisfy a Mean Value Theorem. These considerations motivate our definition of the Lyapunov exponent $\\lambda_p$, which directly captures the diameter of cylinder sets and links it to Hausdorff dimension.\n\nOur main theorems extends to the $p$-adic setting the results of Weiss \\cite{we} and Pollicott-Weiss \\cite{mh} (see also \\cite{io,kms,an}).", "context": "The Gauss map $G: (0,1]\\rightarrow [0,1]$ is defined by $$G(x)=\\dfrac{1}{x}-\\left\\lfloor \\dfrac{1}{x}\\right\\rfloor,$$ where $\\lfloor \\cdot \\rfloor$ denotes the integer part.\nEvery irrational $x\\in (0,1)$ admits a unique continued fraction expansion of the form\n \\begin{align}\\label{continued}\n x= \\cfrac{1}{a_{1}+\\cfrac{1}{a_{2}+\\cfrac{1}{a_{3}+\\cdots }}}=[a_{1},a_{2},a_{3},...],\n \\end{align} where each $a_{i}$ is a positive integer and $a_{i}=\\left\\lfloor \\frac{1}{G^{i-1}x}\\right\\rfloor$. \nThe $n$-th rational approximation of $x$ is given by\n \\begin{align*}\n \\dfrac{P_n}{Q_{n}}=\\cfrac{1}{a_{1}+\\cfrac{1}{a_{2}+\\cfrac{1}{a_{3}+\\dfrac{1}{\\ddots +\\frac{1}{a_{n}}} }}}=[a_{1},a_{2},...,a_{n}].\n \\end{align*} \nThe definition of the Lyapunov exponent extends naturally to the Gauss map. For $x\\in (0,1]$ we define $\\lambda(x)=\\lim_{n\\to \\infty}\\frac{1}{n}\\log |(G^{n})'(x)|$, whenever this limit exists. Moreover, $\\lambda(x)$ can be written as (see \\cite[Page 160]{mh})\n \\begin{align}\\label{EcuLyaAsRationalApprox}\n \\lambda(x)=-\\lim_{n\\rightarrow \\infty}\\dfrac{1}{n}\\log \\left|x-\\dfrac{P_{n}}{Q_{n}}\\right|.\n \\end{align} \nBy equation \\eqref{EcuLyaAsRationalApprox}, $\\lambda(x)$ quantifies the exponential rate at which the rational approximations $P_n/Q_n$ converge to \n$x$, linking Lyapunov exponents to classical Diophantine approximation theory. For Lebesgue-almost every $x\\in (0,1)$, one has $\\lambda(x)=\\frac{\\pi^{2}}{6\\log 2}$ since $\\lambda\n(x)$ is a pointwise limit of Birkhoff averages. The domain of the Lyapunov spectrum $L$ for the Gauss map is $\\left(2\\log\\left(\\frac{\\sqrt{5}-1}{2}\\right),\\infty\\right)$. since $(0,1)\\setminus \\mathbb{Q}$ is a non-compact space, this leads to a different situation from the one considered by Weiss. Nevertheless, $L$ is a real analytic function, and for each $\\alpha$ in this domain, there exists a unique ergodic measure supported on $J_{\\alpha}$.\n\nThe Schneider map $T_{p}:p\\Z_{p}\\rightarrow p\\Z_{p}$ is defined by $T_{p}(0)=0$, and for $x\\neq 0$\n \\begin{align*}\n T_{p}(x)=\\dfrac{p^{a_{1}(x)}}{x}-b_{1}(x),\n \\end{align*} where $a_{1}(x)=v_{p}(x)$ and $b_1(x)\\in\\{1,2,...,p-1\\}$ is uniquely determined by the congruence $b_{1}(x)\\equiv p^{a_{1}(x)}/x$ (mod $p$). Outside a countable set, every element of $p\\Z_p$ admits a continued fraction expansion analogous to \\eqref{continued}. More precisely, for $x\\in p\\Z_{p}\\setminus \\bigcup_{k\\in \\mathbb{N}} T_p^{-k}(0)$ and for each $n\\in \\mathbb{N}$, one can write\n \\begin{align}\\label{SchneiderContinued}\n x= \\cfrac{p^{a_{1}(x)}}{b_{1}(x)+\\cfrac{p^{a_{2}(x)}}{b_{2}(x)+\\cfrac{p^{a_{3}(x)}}{\\ddots + \\cfrac{p^{a_{n}(x)}}{b_{n}(x)+T_{p}^n(x)}}}}.\n \\end{align}\nwhere $a_{i}(x)=a_{1}(T^{i-1}_{p}x)$ and $b_{i}(x)=b_{1}(T^{i-1}_{p}x)$. Note that a point $x\\in p\\mathbb{Z}_p$ has a finite continued fraction expansion if and only if $T^{n}_{p}(x)=0$ for some $n\\in \\mathbb{N}$. We denote by $F$ the set of points in $p\\Z_p$ with finite continued fraction expansion. By the preceding discussion, $F=\\bigcup_{k=0}^\\infty T_p^{-k}(0)$. This situation is analogous to the classical Gauss map, where the points with finite continued fraction expansion are $\\bigcup^{\\infty}_{n=0}G^{-n}(0)=\\mathbb{Q}$. In contrast, if $x$ is a rational number with a infinite Schneider continued fraction expansion, then there exists $N\\in \\mathbb{N}$ such that $a_{n}(x)=1$ and $b_{n}(x)=p-1$ for all $n\\geq N$ (see \\cite[Theorem 1]{hw}). \n The Schneider map is ergodic with respect to the Haar measure $\\mu_p$ on $p\\Z_p$ (see \\cite[Lemma 2]{hw}) and its measure theoretic entropy is $\\frac{p}{p-1}\\log p $ (see \\cite{hn}).\n\nFor each pair of integers $a\\geq 1$ and $1\\leq b\\leq p-1$, we define the cylinder set \n $$I_{(a,b)}:=\\{x\\in p\\Z_p:a_{1}(x)=a \\text{ and $b_{1}(x)=b$}\\},$$\n and introduce the potential function\n\\begin{align*}\n \\psi(x)=\\dfrac{1}{\\left|I_{(a_{1}(x),b_{1}(x))}\\right|_{p}}=p^{a_{1}(x)}.\n\\end{align*} \nAs will be shown, this potential plays a role analogous to $\\log\\|Df\\|$ in the classical setting, in the sense that $\\log \\psi(x)$ measures the local expansion rate of $T_p$.\n\n\\begin{align*}\n \\lambda_{p}(x)=\\lim_{n\\rightarrow{\\infty}}\\dfrac{1}{n}S_{n}\\log\\psi (x),\n\\end{align*} \nwhenever this limit exists, where $S_n\\log\\psi=\\sum^{n-1}_{k=0}\\log\\psi\\circ T^{k}_{p}$ denotes the $n$-th Birkhoff sum of $\\log\\psi$ with respect to $T_{p}$. It follows that\n\\begin{align}\n \\lambda_{p}(x)=\\lim_{n\\rightarrow{\\infty}}\\log p \\cdot \\dfrac{a_{1}(x)+a_{2}(x)+\\cdots +a_{n}(x)}{n},\n\\end{align} \nso that, up to a multiplicative constant, the Lyapunov exponent coincides with the asymptotic arithmetic mean of the digits in the continued fraction expansion. The associated level sets are $$J_{p}(\\alpha)=\\{x\\in p\\Z_{p}\\setminus F: \\lambda_{p}(x)=\\alpha\\},$$ and the Lyapunov spectrum is defined by $$L_{p}(\\alpha)=\\mathrm{dim}_{\\mathrm{H}}\\,J_{p}(\\alpha),$$ where $\\mathrm{dim}_{\\mathrm{H}}$ now denotes the Hausdorff dimension with respect to the $p$-adic norm. In the case of the Gauss map, the Lyapunov spectrum is computed via the topological pressure of the geometric potential $-t\\log|G'|$, which allows one to estimate the diameters of cylinder sets using the Mean Value Theorem. However, since the Gauss map acts on a non-compact space, the thermodynamic formalism in this setting requires additional assumptions, such as the Big Images Property and H\\\"older continuity of the potential. Our main difficulty in developing a thermodynamic formalism for $T_p$ stems from two facts. First, the set of points in $p\\Z_p$ with infinite continued fraction expansions is non-compact. Second, continuous functions on $p\\Z_p$ do not, in general, satisfy a Mean Value Theorem. These considerations motivate our definition of the Lyapunov exponent $\\lambda_p$, which directly captures the diameter of cylinder sets and links it to Hausdorff dimension.\n\nOur main theorems extends to the $p$-adic setting the results of Weiss \\cite{we} and Pollicott-Weiss \\cite{mh} (see also \\cite{io,kms,an}).", "full_context": "The Gauss map $G: (0,1]\\rightarrow [0,1]$ is defined by $$G(x)=\\dfrac{1}{x}-\\left\\lfloor \\dfrac{1}{x}\\right\\rfloor,$$ where $\\lfloor \\cdot \\rfloor$ denotes the integer part.\nEvery irrational $x\\in (0,1)$ admits a unique continued fraction expansion of the form\n \\begin{align}\\label{continued}\n x= \\cfrac{1}{a_{1}+\\cfrac{1}{a_{2}+\\cfrac{1}{a_{3}+\\cdots }}}=[a_{1},a_{2},a_{3},...],\n \\end{align} where each $a_{i}$ is a positive integer and $a_{i}=\\left\\lfloor \\frac{1}{G^{i-1}x}\\right\\rfloor$. \nThe $n$-th rational approximation of $x$ is given by\n \\begin{align*}\n \\dfrac{P_n}{Q_{n}}=\\cfrac{1}{a_{1}+\\cfrac{1}{a_{2}+\\cfrac{1}{a_{3}+\\dfrac{1}{\\ddots +\\frac{1}{a_{n}}} }}}=[a_{1},a_{2},...,a_{n}].\n \\end{align*} \nThe definition of the Lyapunov exponent extends naturally to the Gauss map. For $x\\in (0,1]$ we define $\\lambda(x)=\\lim_{n\\to \\infty}\\frac{1}{n}\\log |(G^{n})'(x)|$, whenever this limit exists. Moreover, $\\lambda(x)$ can be written as (see \\cite[Page 160]{mh})\n \\begin{align}\\label{EcuLyaAsRationalApprox}\n \\lambda(x)=-\\lim_{n\\rightarrow \\infty}\\dfrac{1}{n}\\log \\left|x-\\dfrac{P_{n}}{Q_{n}}\\right|.\n \\end{align} \nBy equation \\eqref{EcuLyaAsRationalApprox}, $\\lambda(x)$ quantifies the exponential rate at which the rational approximations $P_n/Q_n$ converge to \n$x$, linking Lyapunov exponents to classical Diophantine approximation theory. For Lebesgue-almost every $x\\in (0,1)$, one has $\\lambda(x)=\\frac{\\pi^{2}}{6\\log 2}$ since $\\lambda\n(x)$ is a pointwise limit of Birkhoff averages. The domain of the Lyapunov spectrum $L$ for the Gauss map is $\\left(2\\log\\left(\\frac{\\sqrt{5}-1}{2}\\right),\\infty\\right)$. since $(0,1)\\setminus \\mathbb{Q}$ is a non-compact space, this leads to a different situation from the one considered by Weiss. Nevertheless, $L$ is a real analytic function, and for each $\\alpha$ in this domain, there exists a unique ergodic measure supported on $J_{\\alpha}$.\n\nThe Schneider map $T_{p}:p\\Z_{p}\\rightarrow p\\Z_{p}$ is defined by $T_{p}(0)=0$, and for $x\\neq 0$\n \\begin{align*}\n T_{p}(x)=\\dfrac{p^{a_{1}(x)}}{x}-b_{1}(x),\n \\end{align*} where $a_{1}(x)=v_{p}(x)$ and $b_1(x)\\in\\{1,2,...,p-1\\}$ is uniquely determined by the congruence $b_{1}(x)\\equiv p^{a_{1}(x)}/x$ (mod $p$). Outside a countable set, every element of $p\\Z_p$ admits a continued fraction expansion analogous to \\eqref{continued}. More precisely, for $x\\in p\\Z_{p}\\setminus \\bigcup_{k\\in \\mathbb{N}} T_p^{-k}(0)$ and for each $n\\in \\mathbb{N}$, one can write\n \\begin{align}\\label{SchneiderContinued}\n x= \\cfrac{p^{a_{1}(x)}}{b_{1}(x)+\\cfrac{p^{a_{2}(x)}}{b_{2}(x)+\\cfrac{p^{a_{3}(x)}}{\\ddots + \\cfrac{p^{a_{n}(x)}}{b_{n}(x)+T_{p}^n(x)}}}}.\n \\end{align}\nwhere $a_{i}(x)=a_{1}(T^{i-1}_{p}x)$ and $b_{i}(x)=b_{1}(T^{i-1}_{p}x)$. Note that a point $x\\in p\\mathbb{Z}_p$ has a finite continued fraction expansion if and only if $T^{n}_{p}(x)=0$ for some $n\\in \\mathbb{N}$. We denote by $F$ the set of points in $p\\Z_p$ with finite continued fraction expansion. By the preceding discussion, $F=\\bigcup_{k=0}^\\infty T_p^{-k}(0)$. This situation is analogous to the classical Gauss map, where the points with finite continued fraction expansion are $\\bigcup^{\\infty}_{n=0}G^{-n}(0)=\\mathbb{Q}$. In contrast, if $x$ is a rational number with a infinite Schneider continued fraction expansion, then there exists $N\\in \\mathbb{N}$ such that $a_{n}(x)=1$ and $b_{n}(x)=p-1$ for all $n\\geq N$ (see \\cite[Theorem 1]{hw}). \n The Schneider map is ergodic with respect to the Haar measure $\\mu_p$ on $p\\Z_p$ (see \\cite[Lemma 2]{hw}) and its measure theoretic entropy is $\\frac{p}{p-1}\\log p $ (see \\cite{hn}).\n\nFor each pair of integers $a\\geq 1$ and $1\\leq b\\leq p-1$, we define the cylinder set \n $$I_{(a,b)}:=\\{x\\in p\\Z_p:a_{1}(x)=a \\text{ and $b_{1}(x)=b$}\\},$$\n and introduce the potential function\n\\begin{align*}\n \\psi(x)=\\dfrac{1}{\\left|I_{(a_{1}(x),b_{1}(x))}\\right|_{p}}=p^{a_{1}(x)}.\n\\end{align*} \nAs will be shown, this potential plays a role analogous to $\\log\\|Df\\|$ in the classical setting, in the sense that $\\log \\psi(x)$ measures the local expansion rate of $T_p$.\n\n\\begin{align*}\n \\lambda_{p}(x)=\\lim_{n\\rightarrow{\\infty}}\\dfrac{1}{n}S_{n}\\log\\psi (x),\n\\end{align*} \nwhenever this limit exists, where $S_n\\log\\psi=\\sum^{n-1}_{k=0}\\log\\psi\\circ T^{k}_{p}$ denotes the $n$-th Birkhoff sum of $\\log\\psi$ with respect to $T_{p}$. It follows that\n\\begin{align}\n \\lambda_{p}(x)=\\lim_{n\\rightarrow{\\infty}}\\log p \\cdot \\dfrac{a_{1}(x)+a_{2}(x)+\\cdots +a_{n}(x)}{n},\n\\end{align} \nso that, up to a multiplicative constant, the Lyapunov exponent coincides with the asymptotic arithmetic mean of the digits in the continued fraction expansion. The associated level sets are $$J_{p}(\\alpha)=\\{x\\in p\\Z_{p}\\setminus F: \\lambda_{p}(x)=\\alpha\\},$$ and the Lyapunov spectrum is defined by $$L_{p}(\\alpha)=\\mathrm{dim}_{\\mathrm{H}}\\,J_{p}(\\alpha),$$ where $\\mathrm{dim}_{\\mathrm{H}}$ now denotes the Hausdorff dimension with respect to the $p$-adic norm. In the case of the Gauss map, the Lyapunov spectrum is computed via the topological pressure of the geometric potential $-t\\log|G'|$, which allows one to estimate the diameters of cylinder sets using the Mean Value Theorem. However, since the Gauss map acts on a non-compact space, the thermodynamic formalism in this setting requires additional assumptions, such as the Big Images Property and H\\\"older continuity of the potential. Our main difficulty in developing a thermodynamic formalism for $T_p$ stems from two facts. First, the set of points in $p\\Z_p$ with infinite continued fraction expansions is non-compact. Second, continuous functions on $p\\Z_p$ do not, in general, satisfy a Mean Value Theorem. These considerations motivate our definition of the Lyapunov exponent $\\lambda_p$, which directly captures the diameter of cylinder sets and links it to Hausdorff dimension.\n\nOur main theorems extends to the $p$-adic setting the results of Weiss \\cite{we} and Pollicott-Weiss \\cite{mh} (see also \\cite{io,kms,an}).\n\nOur main theorems extends to the $p$-adic setting the results of Weiss \\cite{we} and Pollicott-Weiss \\cite{mh} (see also \\cite{io,kms,an}).\n\n\\begin{theorem}\\label{PrincipalTheorem}\n For each $\\alpha\\geq \\log p,$ \n \\begin{align}\\label{Ecu2PrincipalTheorem}\n L_p(\\alpha)=\\dfrac{\\log(p-1)+\\log(\\alpha-\\log p)-\\log\\log p+\\alpha\\log_{p} \\alpha-\\alpha\\log_{p}(\\alpha-\\log p)}{\\alpha}.\n \\end{align}\n\\end{theorem}\n\n\\begin{theorem}\nFor all $t\\in \\R$\n\\begin{align}\\label{TopPressure}\n P(-t\\log \\psi)=\\begin{cases}\n \\log\\left(\\dfrac{p-1}{p^{t}-1}\\right) & \\text{if $t>0$}\\\\\n \\infty & \\text{if $t\\leq 0$},\n \\end{cases}\n\\end{align} and for each $t>0$ there exists a unique equilibrium state $\\nu_t$ of $-t\\log\\psi$. If $\\alpha=-\\frac{d}{dt}P(-t\\log \\psi)|_{t=t_{\\alpha}}$, then $\\nu_{t_\\alpha}$ is a weak-* accumulation point of the sequence $\\{\\nu_{\\alpha,n}\\}_{n\\in \\N}$.\n\\end{theorem}\n\\begin{remark}\\label{Remark1}\nIn this setting, the equilibrium state for $-\\log\\psi$ ( when $t=1$) coincides with the Haar measure. Indeed, $P(-\\log \\psi)=0$ and $\\mu_{p}$ attains the supremum in equation \\eqref{DefPressure} since\n\\begin{align}\n h_{\\mu_{p}}-\\int \\log\\psi \\ d\\mu_{p}= \\frac{p}{p-1}\\log p-\\frac{p}{p-1}\\log p=0,\n\\end{align}where $h_{\\mu_{p}}=\\frac{p}{p-1}\\log p$ (\\cite[Theorem 1.1]{hn}) and $\\int \\log\\psi \\ d\\mu_{p}=\\frac{p}{p-1}\\log p$ (\\cite[Theorem 3]{hw}). Furthermore, $L_{p}$ achieves its maximum value $\\dim_{\\mathrm{H}}(p\\Zp\\setminus F)=1$ at $\\alpha=\\frac{p\\log p}{p-1}$.\n\\end{remark}\n\n\\begin{proof}Observe that\n \\begin{align*}\n \\dfrac{\\psi_{2}(T_p(x))}{\\psi_{2}(x)} &=\\dfrac{p^{-a_1\\left(T_p(x)\\right)}\\left| T_p(x)-\\dfrac{p^{a_1(T_p(x))}}{b_1(T_p(x))}\\right|_p^{-1}}{p^{-a_1(x)}\\left| x-\\dfrac{p^{a_1(x)}}{b_1(x)} \\right|_p^{-1}}\\\\\n &= p^{a_1(x)-a_1(T_p(x))} \\dfrac{\\left|\\dfrac{b_1(T_p(x))T_p(x)-p^{a_1(T_p(x))}}{b_1(T_p(x))}\\right|^{-1}_p}{\\left|\\dfrac{xb_1(x)-p^{a_1(x)}}{b_1(x)}\\right|^{-1}_p}\\\\\n &=p^{a_1(x)-a_1(T_p(x))}\\left|\\dfrac{b_1(T_p(x))T_p(x)-p^{a_1(T_p(x))}}{xb_1(x)-p^{a_1(x)}} \\right|_p^{-1}.\n\\end{align*} Using the definition of $T_{p}$, one finds $\\left|\\frac{p^{a_{1}(x)}}{b_{1}(x)+T_{p}(x)}b_{1}(x)-p^{a_{1}(x)}\\right|_{p}=|T_{p}(x)|_{p}|x|_{p}$, so that\n\\begin{align*}\n \\dfrac{\\psi_{2}(T_p(x))}{\\psi_{2}(x)} &=p^{a_1(x)-a_2(x)} \\left|\\dfrac{T_p(x)T_p^2(x)}{xT_p(x)} \\right|_p^{-1}\\\\\n &=p^{a_1(x)-a_2(x)}\\cdot p^{v_p(T_p^2(x))-v_p(x)}\\\\\n &=p^{v_p(T_p^2(x))-v_p(T_p(x))}.\n\\end{align*}\nTherefore, if $x$ and $y$ belong to the same cylinder of length 3 (that is, $a_{i}(x)=a_{i}(y)$ and $b_{i}(x)=b_{i}(y)$ for $1\\leq i \\leq 3$), then\n \\begin{align*}\n \\left|\\log\\varphi(x)-\\log\\varphi(y) \\right|&=\\left|\\log\\left(p^{a_2(x)}\\dfrac{\\psi_{2}(T_p(x))}{\\psi_{2}(x)} \\right) -\\log\\left(p^{a_2(y)}\\dfrac{\\psi_{2}(T_p(y))}{\\psi_{2}(y)} \\right)\\right|\\\\\n &=|a_2(x)+v_p(T_p^2(x))-v_p(T_p(x))-a_2(y)-v_p(T_p^2(y))+v_p(T_p(x))|\\\\\n &=|v_p(T_p^2(x))-v_p(T_p^2(y))|\\\\\n &=|a_{3}(x)-a_{3}(y)|=0.\n \\end{align*}\nSo $\\log \\varphi$ is locally constant on such cylinders, and $\\log\\varphi (x)=a_{3}(x)\\log p$ for all $x\\in p\\Z_{p}\\setminus F$. \n\\end{proof}\nLet $x\\in J_{p}(\\alpha)$. Since $J_{p}(\\alpha)$ is a $T_{p}$-invariant set, we have $\\alpha=\\lambda_{p}(x)=\\lambda_{p}(T_{p}^{2}(x))$. Then, by Lemma \\ref{Lemma1} and Proposition \\ref{Proposition 52}, \n\\begin{align*}\n \\alpha =\\lim_{n\\to \\infty}\\log p\\cdot \\dfrac{a_{1}(T_{p}^{2}(x))+\\cdots+a_{n}(T_{p}^{2}(x))}{n}=\\lim_{n\\to \\infty}\\log p\\cdot \\dfrac{a_{3}(x)+\\cdots+a_{3}(T_{p}^{n-1}(x))}{n}.\n\\end{align*} Hence\n\\begin{align*}\n \\lambda_{p}(x)=\\lim_{n\\to \\infty}\\dfrac{1}{n}\\sum^{n-1}_{k=0}\\log\\varphi(T_{p}^{k}(x))=-\\lim_{n\\to \\infty}\\dfrac{1}{n}\\log \\left|x-\\dfrac{p_{n}(x)}{q_{n}(x)}\\right|_{p}.\n\\end{align*}Thus, the Lyapunov exponent of $x$ coincides with the mean exponential rate at which $p$-adic rational approximations converge to $x$. Consequently, Theorem \\ref{PrincipalTheorem} implies Corollary \\ref{ApproximationSpectrumTHM}. Indeed, Theorem \\ref{LyapunovSpectrumCompactAproximation} yields an analogous statement for each subsystem $(T_{p,n},p\\Z_{p_,n}\\setminus F)$. This allows us to characterize, in terms of dimension theory, sets of $p$-adic integers in $p\\Z_{p,n}$ according to the rate at which their rational approximations converge.\n\\begin{theorem}\n Let $n\\in \\N$. For each $\\alpha\\in [\\log p,n\\log p]$ the Hausdorff dimension of the set of points $x$ in $p\\Z_{p,n}\\setminus F$ satisfying\n \\begin{align*}\n -\\lim_{m\\to \\infty}\\dfrac{1}{m}\\log \\left|x-\\dfrac{p_{m}(x)}{q_{m}(x)}\\right|_{p}=\\alpha,\n \\end{align*} equals $\\frac{1}{\\alpha}P_{n}(-t_{\\alpha}\\log\\psi)+t_{\\alpha,n},$ where $P_{n}(-t_{\\alpha,n}\\log \\psi)$ denotes the topological pressure of the potential $-t_{\\alpha,n}\\log \\psi$ and $t_{\\alpha,n}$ is the unique real number $t$ satisfying \n \\begin{align*}\n -\\alpha=\\log p\\cdot \\left(\\dfrac{n}{p^{tn}-1}-\\dfrac{p^{t}}{p^{t}-1}\\right).\n \\end{align*}\n\\end{theorem}\n\nLet $00$, for each $w\\in \\tilde{K}_{\\alpha(q)}$ there exists a positive integer $N(w)$ such that for all $n>N(w)$\n\\begin{align}\\label{Ecu8}\n \\left|\\dfrac{\\sum^{n-1}_{k=0}\\log(\\vartheta\\circ \\sigma^{k}(w))}{\\sum^{n-1}_{k=0}\\log(\\varphi \\circ \\pi_{n}(\\sigma^{k}(w)))}-\\alpha(q)\\right|\\leq \\varepsilon.\n\\end{align} We truncate elements in $\\tilde{K}_{\\alpha(q)}$ depending on $N(w)$. Denote $Q_{l}=\\{w\\in \\tilde{K}_{\\alpha(q)}:N(w)\\leq l \\}$ for each $l>0$. These sets are concatenated by inclusion $Q_{l}\\subset Q_{l+1}$ and $\\tilde{K}_{\\alpha(q)}=\\bigcup^{\\infty}_{l=1}Q_{l}$. Therefore, there exists some $l_{0}>0$ such that $\\mu_{q}( Q_{l})>0$ for all $l \\geq l_{0}$.", "post_theorem_intro_text_len": 3176, "post_theorem_intro_text": "\\begin{theorem}\\label{PrincipalTheorem}\n For each $\\alpha\\geq \\log p,$ \n \\begin{align}\\label{Ecu2PrincipalTheorem}\n L_p(\\alpha)=\\dfrac{\\log(p-1)+\\log(\\alpha-\\log p)-\\log\\log p+\\alpha\\log_{p} \\alpha-\\alpha\\log_{p}(\\alpha-\\log p)}{\\alpha}.\n \\end{align}\n\\end{theorem}\n\nThis explicit formula for the Lyapunov spectrum is remarkable. Such closed expressions are seldom available outside the context of affine iterated function systems on $\\mathbb{R}^n$.\n\nWe conclude by highlighting some arithmetic consequences. \nSetting $\\hat{\\alpha}=\\alpha/\\log p$ we recover and refine previous results concerning digit means (see \\cite[Theorem 2.2]{hyz}) : the set of points $x$ such that the digits $\\{a_{1}(x),a_{2}(x),...\\}$ have asymptotic mean $\\hat{\\alpha}$ has Hausdorff dimension \n\n $$\\dfrac{\\widehat{\\alpha}\\log\\widehat{\\alpha}-(\\widehat{\\alpha}-1)\\log(\\widehat{\\alpha}-1)+\\log(p-1)}{\\widehat{\\alpha} \\log p}.$$\n\nWe also establish a precise relationship between the Lyapunov exponent and the rate of convergence of the rational approximations arising from Schneider continued fraction, obtaining a $p$-adic analogue of classical Diophantine results for the Gauss map. This is\n$$\\lambda_p(x)=-\\lim_{n\\to \\infty}\\dfrac{1}{n}\\log \\left|x-\\dfrac{p_n(x)}{q_n(x)}\\right|_p,$$\nwhere \n\\begin{align*}\n \\dfrac{p_{n}(x)}{q_{n}(x)}= \\cfrac{p^{a_{1}(x)}}{b_{1}(x)+\\cfrac{p^{a_{2}(x)}}{b_{2}(x)+\\cfrac{p^{a_{3}(x)}}{b_{3}(x)+\\cdots \\dfrac{p^{a_{n}(x)}}{b_{n}(x)}}}}.\n\\end{align*}As a consequence, we obtain the following corollary.\n\n\\begin{corollary}\\label{ApproximationSpectrumTHM}\n For each $\\alpha\\geq \\log p$, the set \n \\begin{align*}\n \\left\\{x\\in p\\Z_{p}\\setminus F:-\\lim_{n\\rightarrow \\infty}\\dfrac{1}{n}\\log \\left|x-\\dfrac{p_{n}(x)}{q_{n}(x)}\\right|_{p}=\\alpha\\right\\} \n \\end{align*}\n has Hausdorff dimension\n \\begin{align*}\n \\dfrac{\\log(p-1)+\\log(\\alpha-\\log p)-\\log\\log p+\\alpha\\log_{p}\\alpha-\\log_{p}(\\alpha-\\log p)}{\\alpha}.\n \\end{align*}\n \\end{corollary}\n\nThe proof of our main theorems combine several key ingredients.\nFirst, we employ a compact approximation scheme by analyzing truncated subsystems where points have bounded digits.\nSecond, we apply the thermodynamic formalism to establish the existence and uniqueness of equilibrium states for the relevant family of geometric potentials.\nFinally, we use Gibbs measures and multifractal analysis to derive dimension formulas via pointwise dimensions and Legendre transforms, and then extend the results to the full system.\n\n\\subsection*{Overview of the article} In Section 2, we collect preliminaries from symbolic dynamics, the Schneider map, and thermodynamic formalism.\nIn section 3, we study compact approximations of $(T_{p},p\\mathbb{Z}_p)$ and derive the multifractal spectra for these subsystems. Section 4 is devoted to the thermodynamic formalism of the Schneider map, where we establish the main results of the article. In Section 5, we relate the geometric potential to rational approximations and prove Corollary \\ref{ApproximationSpectrumTHM}. Finally, in order to keep the presentation clear and focused, the technical lemmas are deferred to section 6.", "sketch": "The post-theorem text does not sketch a proof of Theorem~\\ref{PrincipalSinCalculoExplicito} directly, but it does give a general proof outline for the paper’s main theorems (including the explicit spectrum formula and related results). The stated key ingredients are:\n\n1. “First, we employ a compact approximation scheme by analyzing truncated subsystems where points have bounded digits.”\n\n2. “Second, we apply the thermodynamic formalism to establish the existence and uniqueness of equilibrium states for the relevant family of geometric potentials.”\n\n3. “Finally, we use Gibbs measures and multifractal analysis to derive dimension formulas via pointwise dimensions and Legendre transforms, and then extend the results to the full system.”\n\nIt also indicates where these components appear: Section 3 treats “compact approximations… and derive[s] the multifractal spectra for these subsystems,” Section 4 develops “the thermodynamic formalism… where we establish the main results,” and Section 5 relates “the geometric potential to rational approximations and prove[s] Corollary \\ref{ApproximationSpectrumTHM}.”", "expanded_sketch": "The post-theorem text does not sketch a proof of the main theorem directly, but it does give a general proof outline for the paper’s main theorems (including the explicit spectrum formula and related results). The stated key ingredients are:\n\n1. “First, we employ a compact approximation scheme by analyzing truncated subsystems where points have bounded digits.”\n\n2. “Second, we apply the thermodynamic formalism to establish the existence and uniqueness of equilibrium states for the relevant family of geometric potentials.”\n\n3. “Finally, we use Gibbs measures and multifractal analysis to derive dimension formulas via pointwise dimensions and Legendre transforms, and then extend the results to the full system.”\n\nIt also indicates where these components appear: next it treats “compact approximations… and derive[s] the multifractal spectra for these subsystems,” then it develops “the thermodynamic formalism… where we establish the main results,” and finally it relates “the geometric potential to rational approximations” and proves the following corollary.\n\n\\begin{corollary}\\label{ApproximationSpectrumTHM}\n For each $\\alpha\\geq \\log p$, the set \n \\begin{align*}\n \\left\\{x\\in p\\Z_{p}\\setminus F:-\\lim_{n\\rightarrow \\infty}\\dfrac{1}{n}\\log \\left|x-\\dfrac{p_{n}(x)}{q_{n}(x)}\\right|_{p}=\\alpha\\right\\} \n \\end{align*}\n has Hausdorff dimension\n \\begin{align*}\n \\dfrac{\\log(p-1)+\\log(\\alpha-\\log p)-\\log\\log p+\\alpha\\log_{p}\\alpha-\\log_{p}(\\alpha-\\log p)}{\\alpha}.\n \\end{align*}\n \\end{corollary}", "expanded_theorem": "\\label{PrincipalSinCalculoExplicito}\n The Lyapunov spectrum $L_{p}$ is real analytic on $[\\log(p),\\infty)$. For each $\\alpha\\geq \\log p$\n \\begin{align}\\label{EcuSInCalculoExplicito}\n L_{p}(\\alpha)=\\dfrac{1}{\\alpha}\\inf\\left\\{P(-t\\log \\psi)+t\\alpha:t>0\\right\\},\n \\end{align}where $P(-t\\log\\psi)$ is the topological pressure of $-t\\log\\psi$ with respect to $T_p$. The infimum is attained at a unique $t_{\\alpha}>0$ such that $\\frac{d}{dt}P(-t\\log \\psi)|_{t=t_{\\alpha}}=-\\alpha$. Moreover, there exists a unique equilibrium state $\\mu_{t_{\\alpha}}$ for $-t_{\\alpha}\\log \\psi$ such that $\\mu_{t_{\\alpha}}(J_{p}(\\alpha))=1$.", "theorem_type": ["Classification or Bijection", "Uniqueness"], "mcq": {"question": "Let $T_p:p\\mathbb Z_p\\to p\\mathbb Z_p$ be the Schneider map, let $\\psi(x)=p^{a_1(x)}$, and for $x\\in p\\mathbb Z_p\\setminus F$ define the Lyapunov exponent by $\\lambda_p(x)=\\lim_{n\\to\\infty}\\frac1n S_n\\log\\psi(x)$ when the limit exists. For $\\alpha\\ge \\log p$, set $J_p(\\alpha)=\\{x\\in p\\mathbb Z_p\\setminus F: \\lambda_p(x)=\\alpha\\}$ and $L_p(\\alpha)=\\dim_{\\mathrm H} J_p(\\alpha)$. Which statement gives the correct complete description of the Lyapunov spectrum and the object uniquely associated to each level $\\alpha\\ge \\log p$?", "correct_choice": {"label": "A", "text": "The function $L_p$ is real analytic on $[\\log p,\\infty)$. For every $\\alpha\\ge \\log p$,\\n\\n$$L_p(\\alpha)=\\frac1\\alpha\\inf\\{P(-t\\log\\psi)+t\\alpha:t>0\\},$$\\n\\nwhere $P(-t\\log\\psi)$ is the topological pressure of $-t\\log\\psi$ for $T_p$. This infimum is attained at a unique number $t_\\alpha>0$ satisfying\\n\\n$$\\left.\\frac{d}{dt}P(-t\\log\\psi)\\right|_{t=t_\\alpha}=-\\alpha,$$\\n\\nand there is a unique equilibrium state $\\mu_{t_\\alpha}$ for the potential $-t_\\alpha\\log\\psi$ such that $\\mu_{t_\\alpha}(J_p(\\alpha))=1$."}, "choices": [{"label": "B", "text": "The function $L_p$ is real analytic on $(\\log p,\\infty)$. For every $\\alpha>\\log p$, \n\n$$L_p(\\alpha)=\\frac1\\alpha\\inf\\{P(-t\\log\\psi)+t\\alpha:t\\ge 0\\},$$\n\nwhere $P(-t\\log\\psi)$ is the topological pressure of $-t\\log\\psi$ for $T_p$. This infimum is attained at a unique number $t_\\alpha\\ge 0$ satisfying\n\n$$\\left.\\frac{d}{dt}P(-t\\log\\psi)\\right|_{t=t_\\alpha}=-\\alpha,$$\n\nand there is a unique equilibrium state $\\mu_{t_\\alpha}$ for the potential $-t_\\alpha\\log\\psi$ such that $\\mu_{t_\\alpha}(J_p(\\alpha))=1$."}, {"label": "C", "text": "For every $\\alpha\\ge \\log p$,\n\n$$L_p(\\alpha)=\\frac1\\alpha\\inf\\{P(-t\\log\\psi)+t\\alpha:t>0\\},$$\n\nwhere $P(-t\\log\\psi)$ is the topological pressure of $-t\\log\\psi$ for $T_p$. Moreover, for each such $\\alpha$ there exists an equilibrium state $\\mu_{t_\\alpha}$ for the potential $-t_\\alpha\\log\\psi$ such that $\\mu_{t_\\alpha}(J_p(\\alpha))=1$."}, {"label": "D", "text": "The function $L_p$ is real analytic on $[\\log p,\\infty)$. For every $\\alpha\\ge \\log p$,\n\n$$L_p(\\alpha)=\\frac1\\alpha\\inf\\{P(-t\\log\\psi)+t\\alpha:t>0\\},$$\n\nwhere $P(-t\\log\\psi)$ is the topological pressure of $-t\\log\\psi$ for $T_p$. This infimum is attained at a unique number $t_\\alpha>0$ satisfying\n\n$$\\left.\\frac{d}{dt}P(-t\\log\\psi)\\right|_{t=t_\\alpha}=\\alpha,$$\n\nand there is a unique equilibrium state $\\mu_{t_\\alpha}$ for the potential $-t_\\alpha\\log\\psi$ such that $\\mu_{t_\\alpha}(J_p(\\alpha))=1$."}, {"label": "E", "text": "The function $L_p$ is real analytic on $[\\log p,\\infty)$. For every $\\alpha\\ge \\log p$,\n\n$$L_p(\\alpha)=\\frac1\\alpha\\sup\\{P(-t\\log\\psi)+t\\alpha:t>0\\},$$\n\nwhere $P(-t\\log\\psi)$ is the topological pressure of $-t\\log\\psi$ for $T_p$. This extremum is attained at a unique number $t_\\alpha>0$ satisfying\n\n$$\\left.\\frac{d}{dt}P(-t\\log\\psi)\\right|_{t=t_\\alpha}=-\\alpha,$$\n\nand there is a unique equilibrium state $\\mu_{t_\\alpha}$ for the potential $-t_\\alpha\\log\\psi$ such that $\\mu_{t_\\alpha}(J_p(\\alpha))=1$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "compact approximation scheme by analyzing truncated subsystems where points have bounded digits. The theorem in the full system explicitly includes the boundary level \u0007lpha=\\log p and requires the full domain $[\\log p,\\infty)$ with $t_\\alpha>0$; replacing this by the open range $(\\log p,\\infty)$ and allowing $t=0$ mimics what often happens on compact approximants but is not the stated full-system result. Weaker-true option C drops the uniqueness/attainment/analyticity parts while preserving the variational formula and existence of a supporting equilibrium state. False options D and E tamper with the Legendre-transform step from multifractal analysis: D flips the sign in the derivative condition, while E replaces the infimum by a supremum, both highly plausible if one misremembers the pressure formalism.", "tampered_component": "endpoint domain and positivity of minimizer", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "drops real analyticity and uniqueness/derivative characterization while keeping the variational formula and existence of an equilibrium state", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "sign in Euler-Lagrange/pressure derivative condition", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "Legendre-transform extremum direction", "template_used": "stronger_trap"}]}} +{"id": "2601.06350v1", "paper_link": "http://arxiv.org/abs/2601.06350v1", "theorems_cnt": 2, "theorem": {"env_name": "cor", "content": "\\label{cor}\nLet $G=\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z})$. Then $L(\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})$ is the unique maximal Haagerup $G$-invariant von Neumann subalgebra in $L(G)$, where $I_2$ denotes the identity matrix in $SL_2(\\mathbb{Z})$.", "start_pos": 12383, "end_pos": 12645, "label": "cor"}, "ref_dict": {"thm": "\\begin{thm}\\label{thm}\nLet $G=\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z})$. Then a von Neumann subalgebra $P\\subseteq L(G)$ is $G$-invariant if and only if either $P=L(H)$ for some normal subgroup $H\\subseteq G$ or $P=A_n$ for some $n\\geq 0$, where $A_n:=\\{x\\in L(n\\mathbb{Z}^2):~\\tau(xu_g)=\\tau(xu_{g^{-1}}), ~\\forall~g\\in G\\}$, where $\\tau$ denotes the canonical trace on $L(G)$ defined by $\\tau(x)=\\langle x\\delta_e, \\delta_e\\rangle$ for any $x\\in L(G)\\subseteq B(\\ell^2(G))$. \n\\end{thm}", "cor": "\\begin{cor}\\label{cor}\nLet $G=\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z})$. Then $L(\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})$ is the unique maximal Haagerup $G$-invariant von Neumann subalgebra in $L(G)$, where $I_2$ denotes the identity matrix in $SL_2(\\mathbb{Z})$.\n\\end{cor}"}, "pre_theorem_intro_text_len": 7020, "pre_theorem_intro_text": "Let $G$ be a countable discrete group and $L(G)$ be the corresponding group von Neumann algebra. Note that $G$ acts on $L(G)$ naturally by conjugation.\nAfter the initial work by Alekseev-Brugger \\cite{ab}, the problem of classifying $G$-invariant von Neumann subalgebras in $L(G)$, i.e., those which are globally invariant as a subset under the $G$-conjugation action, has quickly received quite much attention recently \\cites{cd,kp,aj,cds,aho,jz}. There are at least two potential applications behind this line of research. One is that every $G$-invariant von Neumann subalgebra in $L(G)$ is clearly regular, i.e., its normalizers generate the whole ambient von Neumann algebra $L(G)$. Thus, maximal abelian $G$-invariant von Neumann subalgebras are automatically Cartan subalgebras in $L(G)$. Hence the study of classifying $G$-invariant von Neumann subalgebras in $L(G)$ may shed light on the famous open question on the existence of Cartan subalgebras in $L(G)$ for certain classes of groups \\cite[Problem V]{ioana_icm}, e.g., the class of icc groups with positive first $L^2$-Betti numbers \\cite[Corollary 4.4]{aj} or lattices in higher rank simple Lie groups, e.g., $G=SL_3(\\mathbb{Z})$ \\cite[Corollary 2.8]{kp}. The other one comes from the problem of classifying the so-called invariant random von Neumann subalgebras, a new concept as introduced in \\cite{aho}. This notion may be considered as the von Neumann algebra counterpart for the quite hot topic of invariant random subgroups \\cites{agv,gel}. In fact, it is \nnot hard to see, and has been partially indicated in \\cite{aho,jz} that the concept of invariant random von Neumann subalgebras serves as a middle link between invariant random subgroups and characters on groups \\cite{bd}. Moreover, invariant von Neumann subalgebras are the simplest, i.e., the Dirac type invariant random von Neumann subalgebras.\n\nFor the above classification problem, an extreme situation is that every $G$-invariant von Neumann subalgebra $P$ in $L(G)$ satisfies that $P=L(H)$ for some normal subgroup $H\\lhd G$. Once this happens, we say that $G$ has the invariant von Neumann subalgebras rigidity property (ISR property, for short) following Amrutam-Jiang in \\cite{aj}. It is clear that by Pontryagin duality, infinite abelian groups do not have the ISR property. In fact, it was proved in \\cite[Proposition 3.1]{aj} that if an infinite group $G$ is not icc (i.e., does not satisfy the infinite-conjugacy-class condition), then $G$ does not have the ISR property. On the other hand, many icc groups with trivial amenable radical are known to have the ISR property, including irreducible lattices in higher rank simple Lie groups \\cite{kp}, \n non-abelian free groups and a finite direct sum of them \\cite{aj}, all acylindrically hyperbolic groups with trivial amenable radical \\cite{cds}, etc. Recently, the (amenable) finitary permutation group\n $S_{\\mathbb{N}}$ was proved to have this ISR property in \\cite{jz}. Besides this extreme situation, it seems nothing is known on classifying invariant von Neumann subalgebras inside non-abelian ambient von Neumann algebras. Therefore,\nit is natural to ask whether we can classify all $G$-invariant von Neumann subalgebras in $L(G)$ if $G$ does not have the ISR property.\n\nRecall that in \\cite[Example 3.5]{aj}, Amrutam and the first named author presented an example showing that i.c.c. condition is not yet sufficient for deducing the ISR property. More precisely, it was proved that for $G=\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z})$, there is a $G$-invariant von Neumann subalgebra $M\\subsetneq L(\\mathbb{Z}^2)$ such that $M$ is not equal to $L(H)$ for any normal subgroup $H$ in $G$. This ambient group $G$ and the associated action $SL_2(\\mathbb{Z})\\curvearrowright \\mathbb{T}^2=\\widehat{\\mathbb{Z}^2}$ have played prominent roles in the modern development of von Neumann algebras and measurable group theory. In \\cite{popa_betti}, Popa proved that $L(G)$ has trivial fundamental group, hence solving a long standing problem proposed by R. V. Kadison. This was also one of the early achievements of Popa's highly influential deformation/rigidity technique, see \\cites{popa_icm, ioana_icm, vaes_icm} for an overview. Ozawa proved that $L(G)$ is solid \\cite{ozawa_hokkaido} and hence it is prime in the sense that it can not be decomposed as a tensor product of two II$_1$ factors. In \\cite{ioana_subequivalence}, Ioana established an alternative principle for all ergodic subequivalence relations inside the equivalence relation defined by $SL_2(\\mathbb{Z})\\curvearrowright \\mathbb{T}^2$. He also \nproved that $L(G)$ has a unique group measure space Cartan subalgebra in \\cite{ioana_gafa}. Shortly, this result was improved to be a unique Cartan by Popa and Vaes in \\cite{pv_crelle}. Recently, as a typical case of studying the notion of maximal Haagerup property as initiated in \\cite{jiangskalski}, the first named author showed that $L(SL_2(\\mathbb{Z}))$ is a maximal Haagerup von Neumann subalgebra in $L(G)$. For more recent study of the maximal Haagerup aspects and generalizations of the above results, see \\cites{val, jv}.\n\nConcerning the importance of studying various structure properties of $L(\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z}))$, it is natural to wonder whether we can classify all invariant von Neumann subalgebras in it. In this paper, we answer this question positively.\n\n\\begin{thm}\\label{thm}\nLet $G=\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z})$. Then a von Neumann subalgebra $P\\subseteq L(G)$ is $G$-invariant if and only if either $P=L(H)$ for some normal subgroup $H\\subseteq G$ or $P=A_n$ for some $n\\geq 0$, where $A_n:=\\{x\\in L(n\\mathbb{Z}^2):~\\tau(xu_g)=\\tau(xu_{g^{-1}}), ~\\forall~g\\in G\\}$, where $\\tau$ denotes the canonical trace on $L(G)$ defined by $\\tau(x)=\\langle x\\delta_e, \\delta_e\\rangle$ for any $x\\in L(G)\\subseteq B(\\ell^2(G))$. \n\\end{thm}\nIn \\cite{aho}, the authors proved the striking result that for any countable discrete group $G$, $L(G_a)$ is the maximal amenable $G$-invariant von Neumann subalgebra in $L(G)$, where $G_a$ denotes the amenable radical in $G$, i.e., the unique maximal amenable normal subgroup in $G$.\nAs a natural generalization of amenable radicals for groups, the notion of Haagerup radical was considered in \\cite{jiangskalski}. However, it is still unclear whether every countable group admits the Haagerup radical (see e.g., \\cite[Question 2.11]{jiangskalski}), although this radical does exist for several classes of groups, see e.g., \\cite[Proposition 2.10]{jiangskalski} and \\cite[Proposition 4.1]{val}. Similarly, as suggested by Amrutam, one may also consider the notion of maximal Haagerup $G$-invariant von Neumann subalgebras in $L(G)$. Although the existence of such subalgebras in $L(G)$ (for $G$ without the Haagerup property) is still unclear in general, we deduce the following parallel result to \\cite[Theorem A]{aho} but for a particular ambient group. To the best knowledge of the authors, this is the first known result along this direction.", "context": "For the above classification problem, an extreme situation is that every $G$-invariant von Neumann subalgebra $P$ in $L(G)$ satisfies that $P=L(H)$ for some normal subgroup $H\\lhd G$. Once this happens, we say that $G$ has the invariant von Neumann subalgebras rigidity property (ISR property, for short) following Amrutam-Jiang in \\cite{aj}. It is clear that by Pontryagin duality, infinite abelian groups do not have the ISR property. In fact, it was proved in \\cite[Proposition 3.1]{aj} that if an infinite group $G$ is not icc (i.e., does not satisfy the infinite-conjugacy-class condition), then $G$ does not have the ISR property. On the other hand, many icc groups with trivial amenable radical are known to have the ISR property, including irreducible lattices in higher rank simple Lie groups \\cite{kp}, \n non-abelian free groups and a finite direct sum of them \\cite{aj}, all acylindrically hyperbolic groups with trivial amenable radical \\cite{cds}, etc. Recently, the (amenable) finitary permutation group\n $S_{\\mathbb{N}}$ was proved to have this ISR property in \\cite{jz}. Besides this extreme situation, it seems nothing is known on classifying invariant von Neumann subalgebras inside non-abelian ambient von Neumann algebras. Therefore,\nit is natural to ask whether we can classify all $G$-invariant von Neumann subalgebras in $L(G)$ if $G$ does not have the ISR property.\n\nRecall that in \\cite[Example 3.5]{aj}, Amrutam and the first named author presented an example showing that i.c.c. condition is not yet sufficient for deducing the ISR property. More precisely, it was proved that for $G=\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z})$, there is a $G$-invariant von Neumann subalgebra $M\\subsetneq L(\\mathbb{Z}^2)$ such that $M$ is not equal to $L(H)$ for any normal subgroup $H$ in $G$. This ambient group $G$ and the associated action $SL_2(\\mathbb{Z})\\curvearrowright \\mathbb{T}^2=\\widehat{\\mathbb{Z}^2}$ have played prominent roles in the modern development of von Neumann algebras and measurable group theory. In \\cite{popa_betti}, Popa proved that $L(G)$ has trivial fundamental group, hence solving a long standing problem proposed by R. V. Kadison. This was also one of the early achievements of Popa's highly influential deformation/rigidity technique, see \\cites{popa_icm, ioana_icm, vaes_icm} for an overview. Ozawa proved that $L(G)$ is solid \\cite{ozawa_hokkaido} and hence it is prime in the sense that it can not be decomposed as a tensor product of two II$_1$ factors. In \\cite{ioana_subequivalence}, Ioana established an alternative principle for all ergodic subequivalence relations inside the equivalence relation defined by $SL_2(\\mathbb{Z})\\curvearrowright \\mathbb{T}^2$. He also \nproved that $L(G)$ has a unique group measure space Cartan subalgebra in \\cite{ioana_gafa}. Shortly, this result was improved to be a unique Cartan by Popa and Vaes in \\cite{pv_crelle}. Recently, as a typical case of studying the notion of maximal Haagerup property as initiated in \\cite{jiangskalski}, the first named author showed that $L(SL_2(\\mathbb{Z}))$ is a maximal Haagerup von Neumann subalgebra in $L(G)$. For more recent study of the maximal Haagerup aspects and generalizations of the above results, see \\cites{val, jv}.\n\nConcerning the importance of studying various structure properties of $L(\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z}))$, it is natural to wonder whether we can classify all invariant von Neumann subalgebras in it. In this paper, we answer this question positively.\n\n\\begin{thm}\\label{thm}\nLet $G=\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z})$. Then a von Neumann subalgebra $P\\subseteq L(G)$ is $G$-invariant if and only if either $P=L(H)$ for some normal subgroup $H\\subseteq G$ or $P=A_n$ for some $n\\geq 0$, where $A_n:=\\{x\\in L(n\\mathbb{Z}^2):~\\tau(xu_g)=\\tau(xu_{g^{-1}}), ~\\forall~g\\in G\\}$, where $\\tau$ denotes the canonical trace on $L(G)$ defined by $\\tau(x)=\\langle x\\delta_e, \\delta_e\\rangle$ for any $x\\in L(G)\\subseteq B(\\ell^2(G))$. \n\\end{thm}\nIn \\cite{aho}, the authors proved the striking result that for any countable discrete group $G$, $L(G_a)$ is the maximal amenable $G$-invariant von Neumann subalgebra in $L(G)$, where $G_a$ denotes the amenable radical in $G$, i.e., the unique maximal amenable normal subgroup in $G$.\nAs a natural generalization of amenable radicals for groups, the notion of Haagerup radical was considered in \\cite{jiangskalski}. However, it is still unclear whether every countable group admits the Haagerup radical (see e.g., \\cite[Question 2.11]{jiangskalski}), although this radical does exist for several classes of groups, see e.g., \\cite[Proposition 2.10]{jiangskalski} and \\cite[Proposition 4.1]{val}. Similarly, as suggested by Amrutam, one may also consider the notion of maximal Haagerup $G$-invariant von Neumann subalgebras in $L(G)$. Although the existence of such subalgebras in $L(G)$ (for $G$ without the Haagerup property) is still unclear in general, we deduce the following parallel result to \\cite[Theorem A]{aho} but for a particular ambient group. To the best knowledge of the authors, this is the first known result along this direction.", "full_context": "For the above classification problem, an extreme situation is that every $G$-invariant von Neumann subalgebra $P$ in $L(G)$ satisfies that $P=L(H)$ for some normal subgroup $H\\lhd G$. Once this happens, we say that $G$ has the invariant von Neumann subalgebras rigidity property (ISR property, for short) following Amrutam-Jiang in \\cite{aj}. It is clear that by Pontryagin duality, infinite abelian groups do not have the ISR property. In fact, it was proved in \\cite[Proposition 3.1]{aj} that if an infinite group $G$ is not icc (i.e., does not satisfy the infinite-conjugacy-class condition), then $G$ does not have the ISR property. On the other hand, many icc groups with trivial amenable radical are known to have the ISR property, including irreducible lattices in higher rank simple Lie groups \\cite{kp}, \n non-abelian free groups and a finite direct sum of them \\cite{aj}, all acylindrically hyperbolic groups with trivial amenable radical \\cite{cds}, etc. Recently, the (amenable) finitary permutation group\n $S_{\\mathbb{N}}$ was proved to have this ISR property in \\cite{jz}. Besides this extreme situation, it seems nothing is known on classifying invariant von Neumann subalgebras inside non-abelian ambient von Neumann algebras. Therefore,\nit is natural to ask whether we can classify all $G$-invariant von Neumann subalgebras in $L(G)$ if $G$ does not have the ISR property.\n\nRecall that in \\cite[Example 3.5]{aj}, Amrutam and the first named author presented an example showing that i.c.c. condition is not yet sufficient for deducing the ISR property. More precisely, it was proved that for $G=\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z})$, there is a $G$-invariant von Neumann subalgebra $M\\subsetneq L(\\mathbb{Z}^2)$ such that $M$ is not equal to $L(H)$ for any normal subgroup $H$ in $G$. This ambient group $G$ and the associated action $SL_2(\\mathbb{Z})\\curvearrowright \\mathbb{T}^2=\\widehat{\\mathbb{Z}^2}$ have played prominent roles in the modern development of von Neumann algebras and measurable group theory. In \\cite{popa_betti}, Popa proved that $L(G)$ has trivial fundamental group, hence solving a long standing problem proposed by R. V. Kadison. This was also one of the early achievements of Popa's highly influential deformation/rigidity technique, see \\cites{popa_icm, ioana_icm, vaes_icm} for an overview. Ozawa proved that $L(G)$ is solid \\cite{ozawa_hokkaido} and hence it is prime in the sense that it can not be decomposed as a tensor product of two II$_1$ factors. In \\cite{ioana_subequivalence}, Ioana established an alternative principle for all ergodic subequivalence relations inside the equivalence relation defined by $SL_2(\\mathbb{Z})\\curvearrowright \\mathbb{T}^2$. He also \nproved that $L(G)$ has a unique group measure space Cartan subalgebra in \\cite{ioana_gafa}. Shortly, this result was improved to be a unique Cartan by Popa and Vaes in \\cite{pv_crelle}. Recently, as a typical case of studying the notion of maximal Haagerup property as initiated in \\cite{jiangskalski}, the first named author showed that $L(SL_2(\\mathbb{Z}))$ is a maximal Haagerup von Neumann subalgebra in $L(G)$. For more recent study of the maximal Haagerup aspects and generalizations of the above results, see \\cites{val, jv}.\n\nConcerning the importance of studying various structure properties of $L(\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z}))$, it is natural to wonder whether we can classify all invariant von Neumann subalgebras in it. In this paper, we answer this question positively.\n\n\\begin{thm}\\label{thm}\nLet $G=\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z})$. Then a von Neumann subalgebra $P\\subseteq L(G)$ is $G$-invariant if and only if either $P=L(H)$ for some normal subgroup $H\\subseteq G$ or $P=A_n$ for some $n\\geq 0$, where $A_n:=\\{x\\in L(n\\mathbb{Z}^2):~\\tau(xu_g)=\\tau(xu_{g^{-1}}), ~\\forall~g\\in G\\}$, where $\\tau$ denotes the canonical trace on $L(G)$ defined by $\\tau(x)=\\langle x\\delta_e, \\delta_e\\rangle$ for any $x\\in L(G)\\subseteq B(\\ell^2(G))$. \n\\end{thm}\nIn \\cite{aho}, the authors proved the striking result that for any countable discrete group $G$, $L(G_a)$ is the maximal amenable $G$-invariant von Neumann subalgebra in $L(G)$, where $G_a$ denotes the amenable radical in $G$, i.e., the unique maximal amenable normal subgroup in $G$.\nAs a natural generalization of amenable radicals for groups, the notion of Haagerup radical was considered in \\cite{jiangskalski}. However, it is still unclear whether every countable group admits the Haagerup radical (see e.g., \\cite[Question 2.11]{jiangskalski}), although this radical does exist for several classes of groups, see e.g., \\cite[Proposition 2.10]{jiangskalski} and \\cite[Proposition 4.1]{val}. Similarly, as suggested by Amrutam, one may also consider the notion of maximal Haagerup $G$-invariant von Neumann subalgebras in $L(G)$. Although the existence of such subalgebras in $L(G)$ (for $G$ without the Haagerup property) is still unclear in general, we deduce the following parallel result to \\cite[Theorem A]{aho} but for a particular ambient group. To the best knowledge of the authors, this is the first known result along this direction.\n\n\\begin{abstract}\nWe classify all $G$-invariant von Neumann subalgebras in $L(G)$ for $G=\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z})$. This is the first result on classifying $G$-invariant von Neumann subalgebras in $L(G)$ for i.c.c. groups $G$ without the invariant von Neumann subalgebras rigidity property (ISR property for short) as introduced in Amrutam-Jiang's work. As a corollary, we show that $L(\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})$ is the unique maximal Haagerup $G$-invariant von Neumann subalgebra in $L(G)$, where $I_2$ denotes the identity matrix in $SL_2(\\mathbb{Z})$.\n\\end{abstract}\n\nRecall that in \\cite[Example 3.5]{aj}, Amrutam and the first named author presented an example showing that i.c.c. condition is not yet sufficient for deducing the ISR property. More precisely, it was proved that for $G=\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z})$, there is a $G$-invariant von Neumann subalgebra $M\\subsetneq L(\\mathbb{Z}^2)$ such that $M$ is not equal to $L(H)$ for any normal subgroup $H$ in $G$. This ambient group $G$ and the associated action $SL_2(\\mathbb{Z})\\curvearrowright \\mathbb{T}^2=\\widehat{\\mathbb{Z}^2}$ have played prominent roles in the modern development of von Neumann algebras and measurable group theory. In \\cite{popa_betti}, Popa proved that $L(G)$ has trivial fundamental group, hence solving a long standing problem proposed by R. V. Kadison. This was also one of the early achievements of Popa's highly influential deformation/rigidity technique, see \\cites{popa_icm, ioana_icm, vaes_icm} for an overview. Ozawa proved that $L(G)$ is solid \\cite{ozawa_hokkaido} and hence it is prime in the sense that it can not be decomposed as a tensor product of two II$_1$ factors. In \\cite{ioana_subequivalence}, Ioana established an alternative principle for all ergodic subequivalence relations inside the equivalence relation defined by $SL_2(\\mathbb{Z})\\curvearrowright \\mathbb{T}^2$. He also \nproved that $L(G)$ has a unique group measure space Cartan subalgebra in \\cite{ioana_gafa}. Shortly, this result was improved to be a unique Cartan by Popa and Vaes in \\cite{pv_crelle}. Recently, as a typical case of studying the notion of maximal Haagerup property as initiated in \\cite{jiangskalski}, the first named author showed that $L(SL_2(\\mathbb{Z}))$ is a maximal Haagerup von Neumann subalgebra in $L(G)$. For more recent study of the maximal Haagerup aspects and generalizations of the above results, see \\cites{val, jv}.\n\nConcerning the importance of studying various structure properties of $L(\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z}))$, it is natural to wonder whether we can classify all invariant von Neumann subalgebras in it. In this paper, we answer this question positively.\n\nNext, let us comment on the proof of Theorem \\ref{thm}.\nFor the proof, we combine \\cite[Theorem 5.1]{cds} with \\cite[Theorem A]{aho} to reduce the study to the case $P\\subseteq L(\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})$. Then by applying techniques as introduced in \\cite{aj}, a considerable amount of effort has to be devoted to showing that $P$ actually lies in $L(\\mathbb{Z}^2)$ unless $P=L(d\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})$, where $d\\in\\{1, 2\\}$.\nWe remark that a similar strategy can also be applied to the wreath product group $\\mathbb{Z}\\wr F_2$, see Section \\ref{section: last remark}.\n\nLet $(A,\\phi)$ be a tracial von Neumann subalgebra equipped with a trace $\\phi$ and $H\\curvearrowright (A,\\phi)$ be a $\\phi$-preserving action. Then we may form the crossed product von Neumann algebra $A\\rtimes H$ \\cite[\\S~5.2]{ap}, which is equipped with a trace $\\tau$ defined by \n\\[\\tau(\\sum_ha_hu_h)=\\phi(a_e),~\\text{where}~ \\sum_ha_hu_h\\in A\\rtimes H.\\]\nIn this paper, we are mainly interested in the case $A=L(\\mathbb{Z}^2)$ equipped with the canonical trace and $H=SL_2(\\mathbb{Z})$. By Pontrygain duality, $A\\cong L^{\\infty}(\\mathbb{T}^2,\\mu)$, where $\\mu$ denotes the Haar measure on the 2-torus $\\mathbb{T}^2$. The action $G\\curvearrowright A$ is induced by the canonical left matrix multiplication on column vectors: $SL_2(\\mathbb{Z})\\curvearrowright \\mathbb{Z}^2$. It is easy to check that $L(\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z}))\\cong L^{\\infty}(\\mathbb{T}^2,\\mu)\\rtimes SL_2(\\mathbb{Z})$.\n\n\\begin{prop}\\label{prop: version of thm}\nLet $G=\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z})$. Let $P$ be a $G$-invariant von Neumann subalgebra in $(L(G),\\tau)$. Then \n\\begin{itemize}\n\\item either $P=L(H)$ for some normal subgroup $H\\lhd G$; or,\n\\item $P=A_n$ for some $n\\geq 0$, where $A_n\\subset L(n\\mathbb{Z}^2)$ is defined by $A_n=\\{x\\in L(n\\mathbb{Z}^2): \\tau(xs)=\\tau(xs^{-1}), \\forall s\\in n\\mathbb{Z}^2\\}$.\n\\end{itemize}\n\\end{prop}\n\\begin{proof}\nFirst, note that $G$ satisfies condition 2) in \\cite[Theorem 5.1]{cds} as explained in Proposition \\ref{prop: G as in chifan-das-sun}. Therefore, we may apply \\cite[Theorem 5.1]{cds} to deduce that if $P$ is non-amenable, then $P=L(H)$ for some normal subgroup $H\\lhd G$.\n\nFinally, let us prove Corollary \\ref{cor}.\n\\begin{proof}[Proof of Corollary \\ref{cor}]\nLet $P$ be a $G$-invariant von Neumann subalgebra in $L(G)$ with the Haagerup property. Then $P=A_n$ for some $n\\geq 0$ or $L(H)$ for some normal subgroup $H\\lhd G$ with the Haagerup property by Theorem \\ref{thm}. By \\cite[Proposition 2.10]{jiangskalski}, we know that $H\\subseteq \\mathbb{Z}^2\\rtimes \\{\\pm I_2\\}$, where $I_2$ denotes the identity matrix in $SL_2(\\mathbb{Z})$. Therefore, $P\\subseteq L(\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})$ in both cases.\nNotice that $L(\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})$ is clearly $G$-invariant and has Haagerup property and hence it is the maximal one with these properties.\n\\end{proof}\n\n\\begin{question}\nLet $M=L^{\\infty}(X,\\mu)\\rtimes G$, where $G:=PSL_2(\\mathbb{Z})\\curvearrowright (X,\\mu)$ (as studied in \\cite{jiang_jot}) denotes the quotient of $SL_2(\\mathbb{Z})\\curvearrowright \\mathbb{T}^2$ by modding out the central subgroup action $\\{\\pm id\\}\\curvearrowright \\mathbb{T}^2$ and then the kernel of the action. Is every $G$-invariant von Neumann subalgebra in $M$ of the form $L^{\\infty}(Y,\\nu)\\rtimes H$ for some normal subgroup $H\\lhd PSL_2(\\mathbb{Z})$ and a quotient action $PSL_2(\\mathbb{Z})\\curvearrowright (Y,\\nu)$ of the action $PSL_2(\\mathbb{Z})\\curvearrowright (X,\\mu)$?\n\\end{question}\n\n\\begin{cor}\\label{cor}\nLet $G=\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z})$. Then $L(\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})$ is the unique maximal Haagerup $G$-invariant von Neumann subalgebra in $L(G)$, where $I_2$ denotes the identity matrix in $SL_2(\\mathbb{Z})$.\n\\end{cor}\n\n\\begin{thm}\\label{thm}\nLet $G=\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z})$. Then a von Neumann subalgebra $P\\subseteq L(G)$ is $G$-invariant if and only if either $P=L(H)$ for some normal subgroup $H\\subseteq G$ or $P=A_n$ for some $n\\geq 0$, where $A_n:=\\{x\\in L(n\\mathbb{Z}^2):~\\tau(xu_g)=\\tau(xu_{g^{-1}}), ~\\forall~g\\in G\\}$, where $\\tau$ denotes the canonical trace on $L(G)$ defined by $\\tau(x)=\\langle x\\delta_e, \\delta_e\\rangle$ for any $x\\in L(G)\\subseteq B(\\ell^2(G))$. \n\\end{thm}", "post_theorem_intro_text_len": 1671, "post_theorem_intro_text": "Next, let us comment on the proof of Theorem \\ref{thm}.\nFor the proof, we combine \\cite[Theorem 5.1]{cds} with \\cite[Theorem A]{aho} to reduce the study to the case $P\\subseteq L(\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})$. Then by applying techniques as introduced in \\cite{aj}, a considerable amount of effort has to be devoted to showing that $P$ actually lies in $L(\\mathbb{Z}^2)$ unless $P=L(d\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})$, where $d\\in\\{1, 2\\}$.\nWe remark that a similar strategy can also be applied to the wreath product group $\\mathbb{Z}\\wr F_2$, see Section \\ref{section: last remark}.\n\nThis paper is organized as follows: after recalling relevant techniques in Section \\ref{section: preliminaries}, we present the proof of Theorem \\ref{thm} and Corollary \\ref{cor} in Section \\ref{section: proof}. Finally, in Section \\ref{section: last remark}, we collect some remarks and open questions related to this work. \n\nIn this paper, we usually use $\\tau$ to mean a trace on a finite von Neumann algebra, e.g., a crossed product or group von Neumann algebra, which will be clear from the context.\n\\medskip\n\n\\paragraph{\\textbf{Acknowledgements}} \n\nThis work is partially supported by National Natural Science Foundation of China (Grant No. 12001081, No. 12271074) and the Fundamental Research Funds for the Central Universities (Grant No. DUT19RC(3)075). Part of this work was done during the visiting of Y. J. to the Institute for Advances Study in Mathematics of HIT in October, 2023. Y. J. is grateful to Prof. Simeng Wang for his invitation and hospitality during this visiting and to Prof. Adam Skalski and Dr. Amrutam Tattwamasi for helpful discussions on this paper.", "sketch": "For the proof (commented on for Theorem~\\ref{thm}), the argument \"combine[s] \\cite[Theorem 5.1]{cds} with \\cite[Theorem A]{aho} to reduce the study to the case $P\\subseteq L(\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})$.\" Then, \"by applying techniques as introduced in \\cite{aj},\" one devotes \"a considerable amount of effort\" to show that $P$ \"actually lies in $L(\\mathbb{Z}^2)$ unless $P=L(d\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})$, where $d\\in\\{1, 2\\}$.\"", "expanded_sketch": "For the proof (commented on for Theorem~\\ref{thm}), the argument “combine[s] \\cite[Theorem 5.1]{cds} with \\cite[Theorem A]{aho} to reduce the study to the case $P\\subseteq L(\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})$.” Then, “by applying techniques as introduced in \\cite{aj},” one devotes “a considerable amount of effort” to show that $P$ “actually lies in $L(\\mathbb{Z}^2)$ unless $P=L(d\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})$, where $d\\in\\{1, 2\\}$.”", "expanded_theorem": "\\label{cor}\nLet $G=\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z})$. Then $L(\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})$ is the unique maximal Haagerup $G$-invariant von Neumann subalgebra in $L(G)$, where $I_2$ denotes the identity matrix in $SL_2(\\mathbb{Z})$.", "theorem_type": ["Uniqueness", "Existence"], "mcq": {"question": "Let \n\\(G=\\mathbb{Z}^2\\rtimes SL_2(\\mathbb{Z})\\), and let \\(L(G)\\) denote the group von Neumann algebra of \\(G\\). For a von Neumann subalgebra \\(P\\subseteq L(G)\\), say that \\(P\\) is \\(G\\)-invariant if \\(u_gPu_g^*=P\\) for every \\(g\\in G\\), where \\(u_g\\) is the canonical group unitary in \\(L(G)\\). Which statement holds about the Haagerup \\(G\\)-invariant von Neumann subalgebras of \\(L(G)\\)?", "correct_choice": {"label": "A", "text": "If \\(I_2\\) denotes the identity matrix in \\(SL_2(\\mathbb{Z})\\), then \\(L(\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})\\) is the unique maximal Haagerup \\(G\\)-invariant von Neumann subalgebra of \\(L(G)\\); equivalently, it is \\(G\\)-invariant, has the Haagerup property, and every \\(G\\)-invariant von Neumann subalgebra of \\(L(G)\\) with the Haagerup property is contained in \\(L(\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})\\)."}, "choices": [{"label": "B", "text": "If \\(I_2\\) denotes the identity matrix in \\(SL_2(\\mathbb{Z})\\), then \\(L(\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})\\) is the unique maximal Haagerup \\(G\\)-invariant von Neumann subalgebra of \\(L(G)\\); equivalently, it is \\(G\\)-invariant, has the Haagerup property, and every \\(G\\)-invariant amenable von Neumann subalgebra of \\(L(G)\\) is contained in \\(L(\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})\\)."}, {"label": "C", "text": "If \\(I_2\\) denotes the identity matrix in \\(SL_2(\\mathbb{Z})\\), then \\(L(\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})\\) is a maximal Haagerup \\(G\\)-invariant von Neumann subalgebra of \\(L(G)\\); in particular, it is \\(G\\)-invariant and has the Haagerup property."}, {"label": "D", "text": "If \\(I_2\\) denotes the identity matrix in \\(SL_2(\\mathbb{Z})\\), then every \\(G\\)-invariant von Neumann subalgebra of \\(L(G)\\) with the Haagerup property is equal either to \\(L(\\mathbb{Z}^2)\\) or to \\(L(\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})\\)."}, {"label": "E", "text": "If \\(I_2\\) denotes the identity matrix in \\(SL_2(\\mathbb{Z})\\), then \\(L(\\mathbb{Z}^2\\rtimes \\{\\pm I_2\\})\\) is the unique maximal Haagerup von Neumann subalgebra of \\(L(G)\\); equivalently, every Haagerup von Neumann subalgebra of \\(L(G)\\) is contained in \\(L(\\mathbb{Z}^2\\rtimes \\{\\pm I_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": "Haagerup-versus-amenable conclusion after reduction to \\(L(\\mathbb{Z}^2\\rtimes\\{\\pm I_2\\})\\)", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped uniqueness/containment of all Haagerup \\(G\\)-invariant subalgebras", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "ignored the additional Haagerup \\(G\\)-invariant subalgebras inside \\(L(\\mathbb{Z}^2)\\)", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "removed the \\(G\\)-invariant hypothesis from the maximality statement", "template_used": "quantifier_dependence"}]}} +{"id": "2601.07642v1", "paper_link": "http://arxiv.org/abs/2601.07642v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "[Zibulski-Zeevi characterization]\n \\label{thm:ZZ_singular_values}\n Let $A,B>0$, and let $g \\in L^2(\\mathbb{R})$. Suppose $\\mathcal{G}(g,a,b)$\n is rationally oversampled Gabor system. Then the following\n assertions are equivalent:\n \\begin{enumerate}[label=(\\roman*)]\n \\item $\\mathcal{G}(g,a,b)$ is a Gabor frame for $L^2(\\mathbb{R})$ with bounds $A$ and $B$,\n \\item $\\set{\\phi^g_\\ell(x,\\gamma)}_{\\ell=0}^{q-1}$ is a frame for $\\mathbb{C}^p$ with\n uniform bounds $A$ and $B$ for a.e. $(x,\\gamma) \\in\n \\itvco{0, 1}^2$.\n \\end{enumerate}", "start_pos": 10705, "end_pos": 11276, "label": "thm:ZZ_singular_values"}, "ref_dict": {"thm:new-hyperbolas-frame-set-Bn": "\\begin{theorem}\n \\label{TEST}\n \\label{thm:new-hyperbolas-frame-set-Bn}\n Let $n \\in\\N$. For $(a_0,b_0) \\in P$ we set $a_0 b_0=p/q$ and $q=r\\mu$ with $p,q,r,\\mu$ as in \\Cref{thm:point-obstructions-grochenig}. Define the set\n \\begin{align}\n H & = \\bigcup_{(a_0,b_0) \\in P} H_{(a_0,b_0)}, \\label{eq:hyperbolic-obstruction-set} \\\\\n \\intertext{where}\n H_{(a_0,b_0)} & = \\set*{(a,b) \\in \\R_+^2 \\given ab=\\frac{p}{q} \\text{ and } \\;\\abs{b - b_0} \\le \\frac{1}{nq}\\bigl(\\mu-(q-p+1) \\bigr)}\n \\label{eq:hyperbolic-obstruction-b-interval}\n \\end{align}\n If $(a,b) \\in H$, then $\\cG (N_n,a,b)$ is \\emph{not} a frame for $L^2(\\R)$. In other words, the complement of the frame set $\\R_+^2 \\setminus \\cF(N_n)$ contains the set $H$.\n\\end{theorem}", "fig:point-set-P": "\\begin{figure}\n \\centering\n \\includegraphics[width=0.6\\textwidth]{point-set-P-with-box.pdf}\n \\caption{The point obstruction set $P$ defined in \\Cref{thm:point-obstructions-grochenig} plotted for $b \\le 15$. In each vertical band $a_0 = 1/\\mu$, the points become denser and denser as $r$ increases with accumulation point $(1/\\mu, \\mu)$, $\\mu \\ge 3$. The colorbar indicates the value of $ab \\in \\itvoo{1/2,1}$.\n }\n \\label{fig:point-set-P}\n\\end{figure}", "fig:hyperbola-set": "\\begin{figure}\n \\centering\n \\includegraphics[width=0.6\\textwidth]{hyperbolas_b_2-8.pdf}\n \\caption{The new hyperbolic obstruction set $H$ for $N_n$, $n=2$, from \\Cref{thm:new-hyperbolas-frame-set-Bn}. The hyperbolic segments, defined by \\eqref{eq:hyperbolic-obstruction-b-interval}, are colored by their $ab$ values.}\n \\label{fig:hyperbola-set}\n\\end{figure}", "eq:con_newNonFrame": "\\begin{equation}\n \\label{eq:con_newNonFrame}\n a_0= \\frac{1}{2m+1}, \\ b_0=\\frac{2k+1}{2}, \\ k,m\\in\\mathbb{N}, \\ k>m, \\ a_0 b_0 < 1,\n\\end{equation}", "eq:hyperbolic-obstruction-b-interval": "\\begin{align}\n H & = \\bigcup_{(a_0,b_0) \\in P} H_{(a_0,b_0)}, \\label{eq:hyperbolic-obstruction-set} \\\\\n \\intertext{where}\n H_{(a_0,b_0)} & = \\set*{(a,b) \\in \\R_+^2 \\given ab=\\frac{p}{q} \\text{ and } \\;\\abs{b - b_0} \\le \\frac{1}{nq}\\bigl(\\mu-(q-p+1) \\bigr)}\n \\label{eq:hyperbolic-obstruction-b-interval}\n \\end{align}", "thm:point-obstructions-grochenig": "\\begin{theorem}[Gr\\\"ochenig \\cite{GrochenigPartitions2015}]\n \\label{thm:point-obstructions-grochenig}\n Let $\\mu,\\nu,r \\in \\N$ with $r \\ge 2$. Let $p=r\\nu+j$ and $q=r\\mu$, where $j=1,\\dots,r-1$. Define the set $P$ of points $(a_0,b_0) \\in \\R_+^2$ where $a_0=\\frac{r}{q}=\\frac{1}{\\mu}$ and $b_0=\\frac{p}{r} = \\nu+\\frac{j}{r}$ subject to the constraints that $p$ and $q$ are relatively prime with $q-\\mu +1< p0$, is defined as\n\\[\n \\mathcal{G}(g,a,b) = \\set*{\\exponential^{2\\pi i b m \\cdot} g(\\cdot - a k) \\given k,m \\in \\mathbb{Z}}.\n\\]\nThe system $\\mathcal{G}(g,a,b)$ is called a \\emph{Gabor frame} if there exist constants $A,B > 0$ such that\n\\[\n A \\norm{f}^2 \\le \\sum_{k,m \\in \\mathbb{Z}} \\abs{\\ip{f,\\exponential^{2\\pi i b m \\cdot} g(\\cdot - a k)}}^2\n \\le B \\norm{f}^2 \\quad \\text{for all } f \\in L^2(\\mathbb{R}).\n\\]\nOne of the fundamental problems in Gabor analysis asks, given $g \\in L^2(\\mathbb{R})$, for the determination of the \\emph{frame set} $\\mathcal{F}(g)$, which consists of the parameter values $(a,b)\\in\\mathbb{R}_+^2$ for which $\\mathcal{G}(g,a,b)$ is a frame.\nThe modern formulation is due to Gr\\\"ochenig~\\cite{MR3232589}, who emphasized it as an important problem in time-frequency analysis; the question itself dates back to the 1990s, see \\cite{MR2031050,MR1066587,MR1955931}.\n\nIn this paper, we consider the frame set of the cardinal B-splines $N_n$ of order $n \\ge 2$, $n \\in \\mathbb{N}$. The cardinal B-splines are defined as the $n$-fold convolution of the indicator function of the unit interval $\\itvcc{0,1}$, i.e.,\n\\begin{equation*}\n N_1 = \\chi_{\\itvcc{0, 1}}, \\quad \\text{and} \\quad\n N_{n+1} = N_{n} \\ast N_{1}, \\quad \\text{for } n \\in \\mathbb{N}.\n\\end{equation*}\nDue to its many desirable properties such as compact support, smoothness, and partition of unity property, these B-splines are widely used as Gabor system generators. The characterization of $\\mathcal{F}(N_n)$ was one of six open problems in Gabor analysis posed by Christensen in \\cite{ChristensenNew2014}. Our main result, \\Cref{thm:new-hyperbolas-frame-set-Bn}, proves the existence of a new infinite family of hyperbolic obstructions to the frame set $\\mathcal{F}(N_n)$.\n\nThe geometric complexity of the frame set for the indicator function $N_1$ is well-documented, famously described by the 'Janssen tie' \\cite{MR1955931}, and culminating in the complete characterization of $\\mathcal{F}(N_1)$ by Dai and Sun \\cite{MR3545108}. Historically, it was unclear whether the complexity of $\\mathcal{F}(N_1)$ was an artifact of the discontinuity of $N_1$.\nHowever, the results in \\cite{MR3572909} showed that the frame set of B-spline windows of all orders must have a very complicated structure, sharing several similarities with $\\mathcal{F}(N_1)$. In the present paper, we strengthen this perspective: we argue that certain number-theoretic constraints on $a$ and $b$ determine frame obstructions, and that (at least parts of) the hyperbolic geometry in the Janssen tie is not unique to $N_1$, but rather a general phenomenon for all cardinal B-splines $N_n$.\n\nOur analysis is based on the Zak transform and the Zibulski-Zeevi representation of Gabor systems \\cite{MR1448221}. Hence, let us briefly summarize the main idea here. The Zak transform of a function $f \\in L^2(\\mathbb{R})$ is defined as\n\\begin{equation}\\label{eq:zakTransform}\n \\left(Z_{\\lambda}f\\right)(x,\\gamma)\n = \\sqrt{\\lambda}\\sum_{k\\in\\mathbb{Z}} f(\\lambda(x-\n k))\\exponential^{2\\pi i k \\gamma}, \\quad a.e.\\ x, \\gamma \\in \\mathbb{R},\n\\end{equation}\nwith convergence in $L^2_\\mathrm{loc}(\\mathbb{R})$. We only consider rationally oversampled Gabor systems, i.e., $\\mathcal{G}(g,a,b)$ with\n\\[\n ab \\in \\mathbb{Q}, \\quad ab=\\frac{p}{q} \\quad \\gcd(p,q)=1.\n\\]\nFor $g\\in L^2(\\mathbb{R})$, we define column vectors $\\phi^g_\\ell(x,\\gamma) \\in \\mathbb{C}^p$ for $\\ell\n \\in \\set{0,1, \\dots, q-1}$ by\n\\[\n \\phi^g_\\ell(x,\\gamma) = \\left(p^{-\\frac{1}{2}} (Z_{\\frac{1}{b}}g)(x-\\ell\n \\frac{p}{q},\\gamma+\\frac{k}{p})\\right)_{k=0}^{p-1} \\ a.e. \\ x,\\gamma \\in \\mathbb{R}.\n\\]\nThe $p\\times q$ matrix defined by $\\Phi^g(x,\\gamma)=[\\phi^g_\\ell(x,\\gamma)]_{\\ell=0}^{q-1}$ is\nthe so-called Zibulski-Zeevi matrix from which we get the following characterization of the frame property of rationally oversampled Gabor systems:", "context": "\\label{sec:intro}\n\nThe Gabor system generated by $g \\in L^2(\\mathbb{R})$ with time-frequency shifts along the lattice $a\\mathbb{Z} \\times b\\mathbb{Z}$, $a,b>0$, is defined as\n\\[\n \\mathcal{G}(g,a,b) = \\set*{\\exponential^{2\\pi i b m \\cdot} g(\\cdot - a k) \\given k,m \\in \\mathbb{Z}}.\n\\]\nThe system $\\mathcal{G}(g,a,b)$ is called a \\emph{Gabor frame} if there exist constants $A,B > 0$ such that\n\\[\n A \\norm{f}^2 \\le \\sum_{k,m \\in \\mathbb{Z}} \\abs{\\ip{f,\\exponential^{2\\pi i b m \\cdot} g(\\cdot - a k)}}^2\n \\le B \\norm{f}^2 \\quad \\text{for all } f \\in L^2(\\mathbb{R}).\n\\]\nOne of the fundamental problems in Gabor analysis asks, given $g \\in L^2(\\mathbb{R})$, for the determination of the \\emph{frame set} $\\mathcal{F}(g)$, which consists of the parameter values $(a,b)\\in\\mathbb{R}_+^2$ for which $\\mathcal{G}(g,a,b)$ is a frame.\nThe modern formulation is due to Gr\\\"ochenig~\\cite{MR3232589}, who emphasized it as an important problem in time-frequency analysis; the question itself dates back to the 1990s, see \\cite{MR2031050,MR1066587,MR1955931}.\n\nIn this paper, we consider the frame set of the cardinal B-splines $N_n$ of order $n \\ge 2$, $n \\in \\mathbb{N}$. The cardinal B-splines are defined as the $n$-fold convolution of the indicator function of the unit interval $\\itvcc{0,1}$, i.e.,\n\\begin{equation*}\n N_1 = \\chi_{\\itvcc{0, 1}}, \\quad \\text{and} \\quad\n N_{n+1} = N_{n} \\ast N_{1}, \\quad \\text{for } n \\in \\mathbb{N}.\n\\end{equation*}\nDue to its many desirable properties such as compact support, smoothness, and partition of unity property, these B-splines are widely used as Gabor system generators. The characterization of $\\mathcal{F}(N_n)$ was one of six open problems in Gabor analysis posed by Christensen in \\cite{ChristensenNew2014}. Our main result, \\Cref{thm:new-hyperbolas-frame-set-Bn}, proves the existence of a new infinite family of hyperbolic obstructions to the frame set $\\mathcal{F}(N_n)$.\n\nThe geometric complexity of the frame set for the indicator function $N_1$ is well-documented, famously described by the 'Janssen tie' \\cite{MR1955931}, and culminating in the complete characterization of $\\mathcal{F}(N_1)$ by Dai and Sun \\cite{MR3545108}. Historically, it was unclear whether the complexity of $\\mathcal{F}(N_1)$ was an artifact of the discontinuity of $N_1$.\nHowever, the results in \\cite{MR3572909} showed that the frame set of B-spline windows of all orders must have a very complicated structure, sharing several similarities with $\\mathcal{F}(N_1)$. In the present paper, we strengthen this perspective: we argue that certain number-theoretic constraints on $a$ and $b$ determine frame obstructions, and that (at least parts of) the hyperbolic geometry in the Janssen tie is not unique to $N_1$, but rather a general phenomenon for all cardinal B-splines $N_n$.\n\nOur analysis is based on the Zak transform and the Zibulski-Zeevi representation of Gabor systems \\cite{MR1448221}. Hence, let us briefly summarize the main idea here. The Zak transform of a function $f \\in L^2(\\mathbb{R})$ is defined as\n\\begin{equation}\\label{eq:zakTransform}\n \\left(Z_{\\lambda}f\\right)(x,\\gamma)\n = \\sqrt{\\lambda}\\sum_{k\\in\\mathbb{Z}} f(\\lambda(x-\n k))\\exponential^{2\\pi i k \\gamma}, \\quad a.e.\\ x, \\gamma \\in \\mathbb{R},\n\\end{equation}\nwith convergence in $L^2_\\mathrm{loc}(\\mathbb{R})$. We only consider rationally oversampled Gabor systems, i.e., $\\mathcal{G}(g,a,b)$ with\n\\[\n ab \\in \\mathbb{Q}, \\quad ab=\\frac{p}{q} \\quad \\gcd(p,q)=1.\n\\]\nFor $g\\in L^2(\\mathbb{R})$, we define column vectors $\\phi^g_\\ell(x,\\gamma) \\in \\mathbb{C}^p$ for $\\ell\n \\in \\set{0,1, \\dots, q-1}$ by\n\\[\n \\phi^g_\\ell(x,\\gamma) = \\left(p^{-\\frac{1}{2}} (Z_{\\frac{1}{b}}g)(x-\\ell\n \\frac{p}{q},\\gamma+\\frac{k}{p})\\right)_{k=0}^{p-1} \\ a.e. \\ x,\\gamma \\in \\mathbb{R}.\n\\]\nThe $p\\times q$ matrix defined by $\\Phi^g(x,\\gamma)=[\\phi^g_\\ell(x,\\gamma)]_{\\ell=0}^{q-1}$ is\nthe so-called Zibulski-Zeevi matrix from which we get the following characterization of the frame property of rationally oversampled Gabor systems:\n\n\\begin{theorem}\n \\label{TEST}\n \\label{thm:new-hyperbolas-frame-set-Bn}\n Let $n \\in\\N$. For $(a_0,b_0) \\in P$ we set $a_0 b_0=p/q$ and $q=r\\mu$ with $p,q,r,\\mu$ as in \\Cref{thm:point-obstructions-grochenig}. Define the set\n \\begin{align}\n H & = \\bigcup_{(a_0,b_0) \\in P} H_{(a_0,b_0)}, \\label{eq:hyperbolic-obstruction-set} \\\\\n \\intertext{where}\n H_{(a_0,b_0)} & = \\set*{(a,b) \\in \\R_+^2 \\given ab=\\frac{p}{q} \\text{ and } \\;\\abs{b - b_0} \\le \\frac{1}{nq}\\bigl(\\mu-(q-p+1) \\bigr)}\n \\label{eq:hyperbolic-obstruction-b-interval}\n \\end{align}\n If $(a,b) \\in H$, then $\\cG (N_n,a,b)$ is \\emph{not} a frame for $L^2(\\R)$. In other words, the complement of the frame set $\\R_+^2 \\setminus \\cF(N_n)$ contains the set $H$.\n\\end{theorem}", "full_context": "\\label{sec:intro}\n\nThe Gabor system generated by $g \\in L^2(\\mathbb{R})$ with time-frequency shifts along the lattice $a\\mathbb{Z} \\times b\\mathbb{Z}$, $a,b>0$, is defined as\n\\[\n \\mathcal{G}(g,a,b) = \\set*{\\exponential^{2\\pi i b m \\cdot} g(\\cdot - a k) \\given k,m \\in \\mathbb{Z}}.\n\\]\nThe system $\\mathcal{G}(g,a,b)$ is called a \\emph{Gabor frame} if there exist constants $A,B > 0$ such that\n\\[\n A \\norm{f}^2 \\le \\sum_{k,m \\in \\mathbb{Z}} \\abs{\\ip{f,\\exponential^{2\\pi i b m \\cdot} g(\\cdot - a k)}}^2\n \\le B \\norm{f}^2 \\quad \\text{for all } f \\in L^2(\\mathbb{R}).\n\\]\nOne of the fundamental problems in Gabor analysis asks, given $g \\in L^2(\\mathbb{R})$, for the determination of the \\emph{frame set} $\\mathcal{F}(g)$, which consists of the parameter values $(a,b)\\in\\mathbb{R}_+^2$ for which $\\mathcal{G}(g,a,b)$ is a frame.\nThe modern formulation is due to Gr\\\"ochenig~\\cite{MR3232589}, who emphasized it as an important problem in time-frequency analysis; the question itself dates back to the 1990s, see \\cite{MR2031050,MR1066587,MR1955931}.\n\nIn this paper, we consider the frame set of the cardinal B-splines $N_n$ of order $n \\ge 2$, $n \\in \\mathbb{N}$. The cardinal B-splines are defined as the $n$-fold convolution of the indicator function of the unit interval $\\itvcc{0,1}$, i.e.,\n\\begin{equation*}\n N_1 = \\chi_{\\itvcc{0, 1}}, \\quad \\text{and} \\quad\n N_{n+1} = N_{n} \\ast N_{1}, \\quad \\text{for } n \\in \\mathbb{N}.\n\\end{equation*}\nDue to its many desirable properties such as compact support, smoothness, and partition of unity property, these B-splines are widely used as Gabor system generators. The characterization of $\\mathcal{F}(N_n)$ was one of six open problems in Gabor analysis posed by Christensen in \\cite{ChristensenNew2014}. Our main result, \\Cref{thm:new-hyperbolas-frame-set-Bn}, proves the existence of a new infinite family of hyperbolic obstructions to the frame set $\\mathcal{F}(N_n)$.\n\nThe geometric complexity of the frame set for the indicator function $N_1$ is well-documented, famously described by the 'Janssen tie' \\cite{MR1955931}, and culminating in the complete characterization of $\\mathcal{F}(N_1)$ by Dai and Sun \\cite{MR3545108}. Historically, it was unclear whether the complexity of $\\mathcal{F}(N_1)$ was an artifact of the discontinuity of $N_1$.\nHowever, the results in \\cite{MR3572909} showed that the frame set of B-spline windows of all orders must have a very complicated structure, sharing several similarities with $\\mathcal{F}(N_1)$. In the present paper, we strengthen this perspective: we argue that certain number-theoretic constraints on $a$ and $b$ determine frame obstructions, and that (at least parts of) the hyperbolic geometry in the Janssen tie is not unique to $N_1$, but rather a general phenomenon for all cardinal B-splines $N_n$.\n\nOur analysis is based on the Zak transform and the Zibulski-Zeevi representation of Gabor systems \\cite{MR1448221}. Hence, let us briefly summarize the main idea here. The Zak transform of a function $f \\in L^2(\\mathbb{R})$ is defined as\n\\begin{equation}\\label{eq:zakTransform}\n \\left(Z_{\\lambda}f\\right)(x,\\gamma)\n = \\sqrt{\\lambda}\\sum_{k\\in\\mathbb{Z}} f(\\lambda(x-\n k))\\exponential^{2\\pi i k \\gamma}, \\quad a.e.\\ x, \\gamma \\in \\mathbb{R},\n\\end{equation}\nwith convergence in $L^2_\\mathrm{loc}(\\mathbb{R})$. We only consider rationally oversampled Gabor systems, i.e., $\\mathcal{G}(g,a,b)$ with\n\\[\n ab \\in \\mathbb{Q}, \\quad ab=\\frac{p}{q} \\quad \\gcd(p,q)=1.\n\\]\nFor $g\\in L^2(\\mathbb{R})$, we define column vectors $\\phi^g_\\ell(x,\\gamma) \\in \\mathbb{C}^p$ for $\\ell\n \\in \\set{0,1, \\dots, q-1}$ by\n\\[\n \\phi^g_\\ell(x,\\gamma) = \\left(p^{-\\frac{1}{2}} (Z_{\\frac{1}{b}}g)(x-\\ell\n \\frac{p}{q},\\gamma+\\frac{k}{p})\\right)_{k=0}^{p-1} \\ a.e. \\ x,\\gamma \\in \\mathbb{R}.\n\\]\nThe $p\\times q$ matrix defined by $\\Phi^g(x,\\gamma)=[\\phi^g_\\ell(x,\\gamma)]_{\\ell=0}^{q-1}$ is\nthe so-called Zibulski-Zeevi matrix from which we get the following characterization of the frame property of rationally oversampled Gabor systems:\n\n\\begin{theorem}\n \\label{TEST}\n \\label{thm:new-hyperbolas-frame-set-Bn}\n Let $n \\in\\N$. For $(a_0,b_0) \\in P$ we set $a_0 b_0=p/q$ and $q=r\\mu$ with $p,q,r,\\mu$ as in \\Cref{thm:point-obstructions-grochenig}. Define the set\n \\begin{align}\n H & = \\bigcup_{(a_0,b_0) \\in P} H_{(a_0,b_0)}, \\label{eq:hyperbolic-obstruction-set} \\\\\n \\intertext{where}\n H_{(a_0,b_0)} & = \\set*{(a,b) \\in \\R_+^2 \\given ab=\\frac{p}{q} \\text{ and } \\;\\abs{b - b_0} \\le \\frac{1}{nq}\\bigl(\\mu-(q-p+1) \\bigr)}\n \\label{eq:hyperbolic-obstruction-b-interval}\n \\end{align}\n If $(a,b) \\in H$, then $\\cG (N_n,a,b)$ is \\emph{not} a frame for $L^2(\\R)$. In other words, the complement of the frame set $\\R_+^2 \\setminus \\cF(N_n)$ contains the set $H$.\n\\end{theorem}\n\nThe geometric complexity of the frame set for the indicator function $N_1$ is well-documented, famously described by the 'Janssen tie' \\cite{MR1955931}, and culminating in the complete characterization of $\\cF(N_1)$ by Dai and Sun \\cite{MR3545108}. Historically, it was unclear whether the complexity of $\\cF(N_1)$ was an artifact of the discontinuity of $N_1$.\nHowever, the results in \\cite{MR3572909} showed that the frame set of B-spline windows of all orders must have a very complicated structure, sharing several similarities with $\\cF(N_1)$. In the present paper, we strengthen this perspective: we argue that certain number-theoretic constraints on $a$ and $b$ determine frame obstructions, and that (at least parts of) the hyperbolic geometry in the Janssen tie is not unique to $N_1$, but rather a general phenomenon for all cardinal B-splines $N_n$.\n\nFor windows in the Feichtinger algebra $M^1(\\R)$, the Zibulski-Zeevi matrix has continuous entries, and thus, to show that $\\gaborG{g}$ is not a frame, it suffices to find a single point $(x,\\gamma) \\in \\itvco{0, 1}^2$ such that the set $\\set{\\phi^g_\\ell(x,\\gamma)}_{\\ell=0}^{q-1}$ does not span $\\C^p$.\n\nFor $x\\in \\R$, we let $\\round{x}$ denote the round function to the nearest integer, i.e., $R(x)=\\floor{x+\\frac12}$, and we let $\\sfrac{x}=x-\\round{x} \\in\n \\itvoc{-\\frac12, \\frac12}$ denote the (signed) fractional part of $x$. The main result of \\cite{MR3572909} states that $\\gaborG{N_n}$ is not a frame for\n\\begin{equation}\n \\label{eq:old_hyperbolic_obstructions}\n ab=\\frac{p}{q}<1 \\quad \\text{for } \\abs{\\sfrac{b}} \\le \\frac{1}{nq} \\;\\text{ and }\\; b > \\frac{3}{2},\n\\end{equation}\nwhere $\\gcd(p,q)=1$. Based on a large number of computer-assisted symbolic calculations by Kamilla~H.~Nielsen, we realized that there are further hyperbolic obstructions for the frame property of the Gabor system $\\mathcal{G}(N_n,a,b)$ ``far'' away from integer values of $b=2,3,4, \\dots$. Indeed, we conjectured in \\cite{MR3572909} the following: $\\mathcal{G}(N_2,a_0,b_0)$ is not a frame for\n\\begin{equation}\n \\label{eq:con_newNonFrame}\n a_0= \\frac{1}{2m+1}, \\ b_0=\\frac{2k+1}{2}, \\ k,m\\in\\mathbb{N}, \\ k>m, \\ a_0 b_0 < 1,\n\\end{equation}\nand, furthermore, $\\mathcal{G}(N_2,a,b)$ is not a frame along the hyperbolas\n\\begin{equation}\n \\label{eq:con_hyperbelStykke}\n \\ ab=\\frac{2k+1}{2\\left(2m+1\\right)}, \\quad \\text{for } \\abs{b-b_0} \\le \\frac{k-m}{2(2m+1)},\n\\end{equation}\nfor every $a_0$ and $b_0$ defined by (\\ref{eq:con_newNonFrame}). We provided a proof of the case $m=1$ and $k=2$ in \\cite{MR3572909}. Gr\\\"ochenig proved the first part \\eqref{eq:con_newNonFrame} of this conjecture in the note \\cite{GrochenigPartitions2015}. The full conjecture remained open until Ghosh and Selvan \\cite{MR4917072} recently were able to verify it.\n\n\\begin{lemma}[\\!\\cite{MR3572909}]\n \\label{lem:partly-pou}\n Let $n \\in \\N$ and $c>0$. Assume that $\\abs{\\sfrac{c}} \\le\n \\frac{1}{n}$.\n \\begin{enumerate}[label=(\\roman*)]\n \\item If $\\sfrac{c}\\ge 0$, then\n \\begin{equation}\n \\label{eq:partly-part-of-unity-1}\n \\sum_{k \\in \\Z} N_n((x+k)/c) = \\mathrm{const} \\quad \\text{for }\n x \\in \\bigcup_{m\\in \\Z}\\itvcc{m+n\\sfrac{c}, m+1}\n \\end{equation}\n \\item If $\\sfrac{c}\\le 0$, then\n \\begin{equation}\n \\label{eq:partly-part-of-unity-2}\n \\sum_{k \\in \\Z} N_n((x+k)/c) = \\mathrm{const} \\quad \\text{for }\n x \\in \\bigcup_{m\\in \\Z}\\itvcc{m, m+1+n\\sfrac{c}}\n \\end{equation}\n \\end{enumerate}\n\\end{lemma}\n\n\\begin{lemma}\n \\label{lem:ZZ-structure}\n Let $n \\in\\N$. Let $\\mu,r, k \\in \\N$ be as in \\Cref{def:point-set-only-mu}. Set $p=r\\mu-k$, $q=r\\mu$, and $b_0 = \\mu - k/r$.\n Let $b>3/2$ be given such that $\\abs{b-b_0} \\le \\frac{1}{nr}$, i.e., $\\abs{\\sfrac{rb}} \\le \\frac1n$.\n \\begin{enumerate}[label=(\\roman*)]\n \\item If $\\sfrac{rb}\\ge 0$, there exists a\n constant $K$ such that \\[\n \\sum_{\\ell=0}^{r-1} \\phi_{\\ell \\mu}^{N_n}(\\frac{x}{r},0) = K e_0 \\in \\C^p\n \\qquad \\text{for $x \\in \\bigcup_{m \\in\n \\Z}\\itvcc{m+n\\sfrac{rb}, m+1}$.}\n \\]\n \\label{item:ZZ-cancellation-pos}\n \\item If $\\sfrac{rb}\\le 0$, there exists a\n constant $K$ such that \\[\n \\sum_{\\ell=0}^{r-1} \\phi_{\\ell \\mu}^{N_n}(\\frac{x}{r},0) = K e_0 \\in \\C^p\n \\qquad \\text{for } x \\in \\bigcup_{m \\in\n \\Z}\\itvcc{m, m+1+n\\sfrac{rb}}\n \\]\n \\label{item:ZZ-cancellation-neg}\n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n We will only prove \\ref{item:ZZ-cancellation-pos} as the proof of \\ref{item:ZZ-cancellation-neg} is similar. Note that $\\round{rb}=p$. Hence, $rb$ is ``close'' to the integer $p$ by $0 \\le \\sfrac{rb} \\le \\frac1n$.\n\n\\begin{remark}\n In the proof of \\Cref{thm:new-hyperbolas-frame-set-Bn}, we tacitly used that the Zak transform $Z_\\lambda N_n$ is continuous. This is not true for $n=1$. However, $Z_\\lambda N_1$ is continuous with respect to the second variable and piecewise continuous with respect to the first variable with finitely many jump discontinuities. It follows that we can still conclude that the minimum of the smallest singular value of the Zibulski-Zeevi matrix $\\Phi^{N_1}(x,\\gamma)$ over $(x,\\gamma)\\in \\itvco{0,1}^2$ is zero, which is the conclusion we need.\n\\end{remark}\n\\begin{remark}\n The obstructions in \\Cref{thm:new-hyperbolas-frame-set-Bn} have been known to the author for a number of years. They were originally presented at \\emph{The International Workshop on Operator Theory and Applications (IWOTA)} in 2016, but have not previously appeared in print.\n\\end{remark}\n\nWe end the paper with a general remark on the nature of the known general obstructions for B-splines and other functions that generate partitions of unity. For functions $g \\in M^1(\\R)$ that generate partitions of unity, there are two types of obstructions to the frame property:\n\\begin{enumerate}[label=(\\Roman*)]\n \\item For $b=2,3,\\dots$ and any $a>0$ the Gabor system $\\gaborG{g}$ is not a frame \\cite{MR2657413,MR2045812,MR1752589} \\label{item:integer-obstruction},\n \\item For $(a,b)$ in the point set $P$ the Gabor system $\\gaborG{g}$ is not a frame \\cite{GrochenigPartitions2015} \\label{item:point-obstruction}.\n\\end{enumerate}\nWhen viewed along hyperbolas $ab=p/q<1$, these two types of obstructions are both point obstructions. For B-splines, the hyperbolic obstructions in \\cite{MR4917072} are ``grown'' out of the first type of obstruction \\ref{item:integer-obstruction} along $ab=p/q<1$, whereas the obstructions in \\Cref{thm:new-hyperbolas-frame-set-Bn} are ``grown'' out of the second type of obstruction \\ref{item:point-obstruction}, also along $ab=p/q<1$. In the opposite direction, as the B-spline order $n$ increases, these hyperbolic obstructions contract, degenerating to the point obstructions \nas $n \\to \\infty$. To see why B-splines admit these hyperbolic extensions of point obstructions, note first that any function $g$ that generates a partition of unity $\\sum_{k \\in \\Z} g(\\cdot + k) = 1$ will, for integer $c$, also satisfy $\\sum_{k \\in \\Z} g((\\cdot+k)/c) = c$. B-splines further satisfy a partly partition of unity property in \nLemma~\\ref{lem:partly-pou} for $c$ close to an integer -- that is, $\\sum_{k \\in \\Z} N_n((\\cdot+k)/c)$ is constant on some subinterval of $\\itvco{0,1}$ -- which is the key additional property that allows the point obstructions to be extended into hyperbolic segments. In fact, any sufficiently nice function $g$ that generates a partition of unity and retains such a partly partition of unity under dilations $(1/c)\\Z$ for $c \\approx 1$ will have similar hyperbolic obstructions.", "post_theorem_intro_text_len": 6283, "post_theorem_intro_text": "For windows in the Feichtinger algebra $M^1(\\mathbb{R})$, the Zibulski-Zeevi matrix has continuous entries, and thus, to show that $\\mathcal{G}(g,a,b)$ is not a frame, it suffices to find a single point $(x,\\gamma) \\in \\itvco{0, 1}^2$ such that the set $\\set{\\phi^g_\\ell(x,\\gamma)}_{\\ell=0}^{q-1}$ does not span $\\mathbb{C}^p$.\n\nFor $x\\in \\mathbb{R}$, we let $R{(x)}$ denote the round function to the nearest integer, i.e., $R(x)=\\floor{x+\\frac12}$, and we let $F{(x)}=x-R{(x)} \\in\n \\itvoc{-\\frac12, \\frac12}$ denote the (signed) fractional part of $x$. The main result of \\cite{MR3572909} states that $\\mathcal{G}(N_n,a,b)$ is not a frame for\n\\begin{equation}\n \\label{eq:old_hyperbolic_obstructions}\n ab=\\frac{p}{q}<1 \\quad \\text{for } \\abs{F{(b)}} \\le \\frac{1}{nq} \\;\\text{ and }\\; b > \\frac{3}{2},\n\\end{equation}\nwhere $\\gcd(p,q)=1$. Based on a large number of computer-assisted symbolic calculations by Kamilla~H.~Nielsen, we realized that there are further hyperbolic obstructions for the frame property of the Gabor system $\\mathcal{G}(N_n,a,b)$ ``far'' away from integer values of $b=2,3,4, \\dots$. Indeed, we conjectured in \\cite{MR3572909} the following: $\\mathcal{G}(N_2,a_0,b_0)$ is not a frame for\n\\begin{equation}\n \\label{eq:con_newNonFrame}\n a_0= \\frac{1}{2m+1}, \\ b_0=\\frac{2k+1}{2}, \\ k,m\\in\\mathbb{N}, \\ k>m, \\ a_0 b_0 < 1,\n\\end{equation}\nand, furthermore, $\\mathcal{G}(N_2,a,b)$ is not a frame along the hyperbolas\n\\begin{equation}\n \\label{eq:con_hyperbelStykke}\n \\ ab=\\frac{2k+1}{2\\left(2m+1\\right)}, \\quad \\text{for } \\abs{b-b_0} \\le \\frac{k-m}{2(2m+1)},\n\\end{equation}\nfor every $a_0$ and $b_0$ defined by (\\ref{eq:con_newNonFrame}). We provided a proof of the case $m=1$ and $k=2$ in \\cite{MR3572909}. Gr\\\"ochenig proved the first part \\eqref{eq:con_newNonFrame} of this conjecture in the note \\cite{GrochenigPartitions2015}. The full conjecture remained open until Ghosh and Selvan \\cite{MR4917072} recently were able to verify it.\n\nImportant for the development of the present paper is that Gr\\\"ochenig~\\cite{GrochenigPartitions2015} realized that the pattern \\eqref{eq:con_newNonFrame} found in \\cite{MR3572909} was part of a much larger family of obstructions that holds for any Gabor system generator in the Feichtinger algebra $M^1(\\mathbb{R})$ satisfying the partition of unity property:\n\n\\begin{theorem}[Gr\\\"ochenig \\cite{GrochenigPartitions2015}]\n \\label{thm:point-obstructions-grochenig}\n Let $\\mu,\\nu,r \\in \\mathbb{N}$ with $r \\ge 2$. Let $p=r\\nu+j$ and $q=r\\mu$, where $j=1,\\dots,r-1$. Define the set $P$ of points $(a_0,b_0) \\in \\R_+^2$ where $a_0=\\frac{r}{q}=\\frac{1}{\\mu}$ and $b_0=\\frac{p}{r} = \\nu+\\frac{j}{r}$ subject to the constraints that $p$ and $q$ are relatively prime with $q-\\mu +1< p0$, and let $g \\in L^2(\\mathbb{R})$. Suppose $\\mathcal{G}(g,a,b)$\n is rationally oversampled Gabor system. Then the following\n assertions are equivalent:\n \\begin{enumerate}[label=(\\roman*)]\n \\item $\\mathcal{G}(g,a,b)$ is a Gabor frame for $L^2(\\mathbb{R})$ with bounds $A$ and $B$,\n \\item $\\set{\\phi^g_\\ell(x,\\gamma)}_{\\ell=0}^{q-1}$ is a frame for $\\mathbb{C}^p$ with\n uniform bounds $A$ and $B$ for a.e. $(x,\\gamma) \\in\n \\itvco{0, 1}^2$.\n \\end{enumerate}", "theorem_type": ["Biconditional or Equivalence", "Classification or Bijection"], "mcq": {"question": "Let $g\\in L^2(\\mathbb{R})$ and $A,B>0$. Assume the Gabor system\n\\[\n\\mathcal{G}(g,a,b)=\\{e^{2\\pi i bm\\,\\cdot}\\,g(\\cdot-ak):k,m\\in\\mathbb{Z}\\}\n\\]\nis rationally oversampled, meaning $ab=p/q\\in\\mathbb{Q}$ with $\\gcd(p,q)=1$. Define the Zak transform by\n\\[\n(Z_{\\lambda}f)(x,\\gamma)=\\sqrt{\\lambda}\\sum_{k\\in\\mathbb{Z}} f(\\lambda(x-k))e^{2\\pi i k\\gamma},\n\\]\nand for $\\ell=0,1,\\dots,q-1$ define vectors $\\phi^g_\\ell(x,\\gamma)\\in\\mathbb{C}^p$ by\n\\[\n\\phi^g_\\ell(x,\\gamma)=\\Big(p^{-1/2}(Z_{1/b}g)(x-\\ell p/q,\\,\\gamma+k/p)\\Big)_{k=0}^{p-1}.\n\\]\nWhich statement exactly characterizes the condition that $\\mathcal{G}(g,a,b)$ is a Gabor frame for $L^2(\\mathbb{R})$ with frame bounds $A$ and $B$?", "correct_choice": {"label": "A", "text": "For almost every $(x,\\gamma)\\in[0,1)^2$, the family $\\{\\phi^g_\\ell(x,\\gamma)\\}_{\\ell=0}^{q-1}$ is a frame for $\\mathbb{C}^p$ with the same uniform bounds $A$ and $B$; equivalently, for almost every $(x,\\gamma)\\in[0,1)^2$ and every $c\\in\\mathbb{C}^p$,\n\\[\nA\\|c\\|^2\\le \\sum_{\\ell=0}^{q-1}|\\langle c,\\phi^g_\\ell(x,\\gamma)\\rangle|^2\\le B\\|c\\|^2.\n\\]"}, "choices": [{"label": "B", "text": "For every $(x,\\gamma)\\in[0,1)^2$, the family $\\{\\phi^g_\\ell(x,\\gamma)\\}_{\\ell=0}^{q-1}$ is a frame for $\\mathbb{C}^p$ with the same uniform bounds $A$ and $B$; equivalently, for every $(x,\\gamma)\\in[0,1)^2$ and every $c\\in\\mathbb{C}^p$,\n\\[\nA\\|c\\|^2\\le \\sum_{\\ell=0}^{q-1}|\\langle c,\\phi^g_\\ell(x,\\gamma)\\rangle|^2\\le B\\|c\\|^2.\n\\]"}, {"label": "C", "text": "For almost every $(x,\\gamma)\\in[0,1)^2$, the family $\\{\\phi^g_\\ell(x,\\gamma)\\}_{\\ell=0}^{q-1}$ is a frame for $\\mathbb{C}^p$."}, {"label": "D", "text": "For almost every $(x,\\gamma)\\in[0,1)^2$, the family $\\{\\phi^g_\\ell(x,\\gamma)\\}_{\\ell=0}^{q-1}$ spans $\\mathbb{C}^p$; equivalently, for almost every $(x,\\gamma)\\in[0,1)^2$ there exist bounds $A_{x,\\gamma},B_{x,\\gamma}>0$ such that for every $c\\in\\mathbb{C}^p$,\n\\[\nA_{x,\\gamma}\\|c\\|^2\\le \\sum_{\\ell=0}^{q-1}|\\langle c,\\phi^g_\\ell(x,\\gamma)\\rangle|^2\\le B_{x,\\gamma}\\|c\\|^2.\n\\]"}, {"label": "E", "text": "For almost every $(x,\\gamma)\\in[0,1)^2$, the vectors $\\phi^g_0(x,\\gamma),\\dots,\\phi^g_{q-1}(x,\\gamma)$ form a basis of $\\mathbb{C}^p$ with basis constants bounded by $A$ and $B$ uniformly in $(x,\\gamma)$; equivalently, necessarily $q=p$ and for almost every $(x,\\gamma)\\in[0,1)^2$ and every $c\\in\\mathbb{C}^p$,\n\\[\nA\\|c\\|^2\\le \\sum_{\\ell=0}^{p-1}|\\langle c,\\phi^g_\\ell(x,\\gamma)\\rangle|^2\\le B\\|c\\|^2.\n\\]"}], "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": "a.e.-quantifier on $(x,\\gamma)$ replaced by pointwise-for-all", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped the requirement that the frame bounds are the same prescribed uniform bounds $A,B$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "uniform bounds $A,B$ replaced by non-uniform $(x,\\gamma)$-dependent bounds", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "frame of $q$ vectors in $\\mathbb{C}^p$ confused with a basis, forcing $q=p$", "template_used": "wildcard"}]}} +{"id": "2601.07642v1", "paper_link": "http://arxiv.org/abs/2601.07642v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "[Zibulski-Zeevi characterization]\n \\label{thm:ZZ_singular_values}\n Let $A,B>0$, and let $g \\in L^2(\\mathbb{R})$. Suppose $\\mathcal{G}(g,a,b)$\n is rationally oversampled Gabor system. Then the following\n assertions are equivalent:\n \\begin{enumerate}[label=(\\roman*)]\n \\item $\\mathcal{G}(g,a,b)$ is a Gabor frame for $L^2(\\mathbb{R})$ with bounds $A$ and $B$,\n \\item $\\set{\\phi^g_\\ell(x,\\gamma)}_{\\ell=0}^{q-1}$ is a frame for $\\mathbb{C}^p$ with\n uniform bounds $A$ and $B$ for a.e. $(x,\\gamma) \\in\n \\itvco{0, 1}^2$.\n \\end{enumerate}", "start_pos": 10705, "end_pos": 11276, "label": "thm:ZZ_singular_values"}, "ref_dict": {"thm:new-hyperbolas-frame-set-Bn": "\\begin{theorem}\n \\label{TEST}\n \\label{thm:new-hyperbolas-frame-set-Bn}\n Let $n \\in\\N$. For $(a_0,b_0) \\in P$ we set $a_0 b_0=p/q$ and $q=r\\mu$ with $p,q,r,\\mu$ as in \\Cref{thm:point-obstructions-grochenig}. Define the set\n \\begin{align}\n H & = \\bigcup_{(a_0,b_0) \\in P} H_{(a_0,b_0)}, \\label{eq:hyperbolic-obstruction-set} \\\\\n \\intertext{where}\n H_{(a_0,b_0)} & = \\set*{(a,b) \\in \\R_+^2 \\given ab=\\frac{p}{q} \\text{ and } \\;\\abs{b - b_0} \\le \\frac{1}{nq}\\bigl(\\mu-(q-p+1) \\bigr)}\n \\label{eq:hyperbolic-obstruction-b-interval}\n \\end{align}\n If $(a,b) \\in H$, then $\\cG (N_n,a,b)$ is \\emph{not} a frame for $L^2(\\R)$. In other words, the complement of the frame set $\\R_+^2 \\setminus \\cF(N_n)$ contains the set $H$.\n\\end{theorem}", "fig:point-set-P": "\\begin{figure}\n \\centering\n \\includegraphics[width=0.6\\textwidth]{point-set-P-with-box.pdf}\n \\caption{The point obstruction set $P$ defined in \\Cref{thm:point-obstructions-grochenig} plotted for $b \\le 15$. In each vertical band $a_0 = 1/\\mu$, the points become denser and denser as $r$ increases with accumulation point $(1/\\mu, \\mu)$, $\\mu \\ge 3$. The colorbar indicates the value of $ab \\in \\itvoo{1/2,1}$.\n }\n \\label{fig:point-set-P}\n\\end{figure}", "fig:hyperbola-set": "\\begin{figure}\n \\centering\n \\includegraphics[width=0.6\\textwidth]{hyperbolas_b_2-8.pdf}\n \\caption{The new hyperbolic obstruction set $H$ for $N_n$, $n=2$, from \\Cref{thm:new-hyperbolas-frame-set-Bn}. The hyperbolic segments, defined by \\eqref{eq:hyperbolic-obstruction-b-interval}, are colored by their $ab$ values.}\n \\label{fig:hyperbola-set}\n\\end{figure}", "eq:con_newNonFrame": "\\begin{equation}\n \\label{eq:con_newNonFrame}\n a_0= \\frac{1}{2m+1}, \\ b_0=\\frac{2k+1}{2}, \\ k,m\\in\\mathbb{N}, \\ k>m, \\ a_0 b_0 < 1,\n\\end{equation}", "eq:hyperbolic-obstruction-b-interval": "\\begin{align}\n H & = \\bigcup_{(a_0,b_0) \\in P} H_{(a_0,b_0)}, \\label{eq:hyperbolic-obstruction-set} \\\\\n \\intertext{where}\n H_{(a_0,b_0)} & = \\set*{(a,b) \\in \\R_+^2 \\given ab=\\frac{p}{q} \\text{ and } \\;\\abs{b - b_0} \\le \\frac{1}{nq}\\bigl(\\mu-(q-p+1) \\bigr)}\n \\label{eq:hyperbolic-obstruction-b-interval}\n \\end{align}", "thm:point-obstructions-grochenig": "\\begin{theorem}[Gr\\\"ochenig \\cite{GrochenigPartitions2015}]\n \\label{thm:point-obstructions-grochenig}\n Let $\\mu,\\nu,r \\in \\N$ with $r \\ge 2$. Let $p=r\\nu+j$ and $q=r\\mu$, where $j=1,\\dots,r-1$. Define the set $P$ of points $(a_0,b_0) \\in \\R_+^2$ where $a_0=\\frac{r}{q}=\\frac{1}{\\mu}$ and $b_0=\\frac{p}{r} = \\nu+\\frac{j}{r}$ subject to the constraints that $p$ and $q$ are relatively prime with $q-\\mu +1< p0$, is defined as\n\\[\n \\mathcal{G}(g,a,b) = \\set*{\\exponential^{2\\pi i b m \\cdot} g(\\cdot - a k) \\given k,m \\in \\mathbb{Z}}.\n\\]\nThe system $\\mathcal{G}(g,a,b)$ is called a \\emph{Gabor frame} if there exist constants $A,B > 0$ such that\n\\[\n A \\norm{f}^2 \\le \\sum_{k,m \\in \\mathbb{Z}} \\abs{\\ip{f,\\exponential^{2\\pi i b m \\cdot} g(\\cdot - a k)}}^2\n \\le B \\norm{f}^2 \\quad \\text{for all } f \\in L^2(\\mathbb{R}).\n\\]\nOne of the fundamental problems in Gabor analysis asks, given $g \\in L^2(\\mathbb{R})$, for the determination of the \\emph{frame set} $\\mathcal{F}(g)$, which consists of the parameter values $(a,b)\\in\\mathbb{R}_+^2$ for which $\\mathcal{G}(g,a,b)$ is a frame.\nThe modern formulation is due to Gr\\\"ochenig~\\cite{MR3232589}, who emphasized it as an important problem in time-frequency analysis; the question itself dates back to the 1990s, see \\cite{MR2031050,MR1066587,MR1955931}.\n\nIn this paper, we consider the frame set of the cardinal B-splines $N_n$ of order $n \\ge 2$, $n \\in \\mathbb{N}$. The cardinal B-splines are defined as the $n$-fold convolution of the indicator function of the unit interval $\\itvcc{0,1}$, i.e.,\n\\begin{equation*}\n N_1 = \\chi_{\\itvcc{0, 1}}, \\quad \\text{and} \\quad\n N_{n+1} = N_{n} \\ast N_{1}, \\quad \\text{for } n \\in \\mathbb{N}.\n\\end{equation*}\nDue to its many desirable properties such as compact support, smoothness, and partition of unity property, these B-splines are widely used as Gabor system generators. The characterization of $\\mathcal{F}(N_n)$ was one of six open problems in Gabor analysis posed by Christensen in \\cite{ChristensenNew2014}. Our main result, \\Cref{thm:new-hyperbolas-frame-set-Bn}, proves the existence of a new infinite family of hyperbolic obstructions to the frame set $\\mathcal{F}(N_n)$.\n\nThe geometric complexity of the frame set for the indicator function $N_1$ is well-documented, famously described by the 'Janssen tie' \\cite{MR1955931}, and culminating in the complete characterization of $\\mathcal{F}(N_1)$ by Dai and Sun \\cite{MR3545108}. Historically, it was unclear whether the complexity of $\\mathcal{F}(N_1)$ was an artifact of the discontinuity of $N_1$.\nHowever, the results in \\cite{MR3572909} showed that the frame set of B-spline windows of all orders must have a very complicated structure, sharing several similarities with $\\mathcal{F}(N_1)$. In the present paper, we strengthen this perspective: we argue that certain number-theoretic constraints on $a$ and $b$ determine frame obstructions, and that (at least parts of) the hyperbolic geometry in the Janssen tie is not unique to $N_1$, but rather a general phenomenon for all cardinal B-splines $N_n$.\n\nOur analysis is based on the Zak transform and the Zibulski-Zeevi representation of Gabor systems \\cite{MR1448221}. Hence, let us briefly summarize the main idea here. The Zak transform of a function $f \\in L^2(\\mathbb{R})$ is defined as\n\\begin{equation}\\label{eq:zakTransform}\n \\left(Z_{\\lambda}f\\right)(x,\\gamma)\n = \\sqrt{\\lambda}\\sum_{k\\in\\mathbb{Z}} f(\\lambda(x-\n k))\\exponential^{2\\pi i k \\gamma}, \\quad a.e.\\ x, \\gamma \\in \\mathbb{R},\n\\end{equation}\nwith convergence in $L^2_\\mathrm{loc}(\\mathbb{R})$. We only consider rationally oversampled Gabor systems, i.e., $\\mathcal{G}(g,a,b)$ with\n\\[\n ab \\in \\mathbb{Q}, \\quad ab=\\frac{p}{q} \\quad \\gcd(p,q)=1.\n\\]\nFor $g\\in L^2(\\mathbb{R})$, we define column vectors $\\phi^g_\\ell(x,\\gamma) \\in \\mathbb{C}^p$ for $\\ell\n \\in \\set{0,1, \\dots, q-1}$ by\n\\[\n \\phi^g_\\ell(x,\\gamma) = \\left(p^{-\\frac{1}{2}} (Z_{\\frac{1}{b}}g)(x-\\ell\n \\frac{p}{q},\\gamma+\\frac{k}{p})\\right)_{k=0}^{p-1} \\ a.e. \\ x,\\gamma \\in \\mathbb{R}.\n\\]\nThe $p\\times q$ matrix defined by $\\Phi^g(x,\\gamma)=[\\phi^g_\\ell(x,\\gamma)]_{\\ell=0}^{q-1}$ is\nthe so-called Zibulski-Zeevi matrix from which we get the following characterization of the frame property of rationally oversampled Gabor systems:", "context": "\\label{sec:intro}\n\nThe Gabor system generated by $g \\in L^2(\\mathbb{R})$ with time-frequency shifts along the lattice $a\\mathbb{Z} \\times b\\mathbb{Z}$, $a,b>0$, is defined as\n\\[\n \\mathcal{G}(g,a,b) = \\set*{\\exponential^{2\\pi i b m \\cdot} g(\\cdot - a k) \\given k,m \\in \\mathbb{Z}}.\n\\]\nThe system $\\mathcal{G}(g,a,b)$ is called a \\emph{Gabor frame} if there exist constants $A,B > 0$ such that\n\\[\n A \\norm{f}^2 \\le \\sum_{k,m \\in \\mathbb{Z}} \\abs{\\ip{f,\\exponential^{2\\pi i b m \\cdot} g(\\cdot - a k)}}^2\n \\le B \\norm{f}^2 \\quad \\text{for all } f \\in L^2(\\mathbb{R}).\n\\]\nOne of the fundamental problems in Gabor analysis asks, given $g \\in L^2(\\mathbb{R})$, for the determination of the \\emph{frame set} $\\mathcal{F}(g)$, which consists of the parameter values $(a,b)\\in\\mathbb{R}_+^2$ for which $\\mathcal{G}(g,a,b)$ is a frame.\nThe modern formulation is due to Gr\\\"ochenig~\\cite{MR3232589}, who emphasized it as an important problem in time-frequency analysis; the question itself dates back to the 1990s, see \\cite{MR2031050,MR1066587,MR1955931}.\n\nIn this paper, we consider the frame set of the cardinal B-splines $N_n$ of order $n \\ge 2$, $n \\in \\mathbb{N}$. The cardinal B-splines are defined as the $n$-fold convolution of the indicator function of the unit interval $\\itvcc{0,1}$, i.e.,\n\\begin{equation*}\n N_1 = \\chi_{\\itvcc{0, 1}}, \\quad \\text{and} \\quad\n N_{n+1} = N_{n} \\ast N_{1}, \\quad \\text{for } n \\in \\mathbb{N}.\n\\end{equation*}\nDue to its many desirable properties such as compact support, smoothness, and partition of unity property, these B-splines are widely used as Gabor system generators. The characterization of $\\mathcal{F}(N_n)$ was one of six open problems in Gabor analysis posed by Christensen in \\cite{ChristensenNew2014}. Our main result, \\Cref{thm:new-hyperbolas-frame-set-Bn}, proves the existence of a new infinite family of hyperbolic obstructions to the frame set $\\mathcal{F}(N_n)$.\n\nThe geometric complexity of the frame set for the indicator function $N_1$ is well-documented, famously described by the 'Janssen tie' \\cite{MR1955931}, and culminating in the complete characterization of $\\mathcal{F}(N_1)$ by Dai and Sun \\cite{MR3545108}. Historically, it was unclear whether the complexity of $\\mathcal{F}(N_1)$ was an artifact of the discontinuity of $N_1$.\nHowever, the results in \\cite{MR3572909} showed that the frame set of B-spline windows of all orders must have a very complicated structure, sharing several similarities with $\\mathcal{F}(N_1)$. In the present paper, we strengthen this perspective: we argue that certain number-theoretic constraints on $a$ and $b$ determine frame obstructions, and that (at least parts of) the hyperbolic geometry in the Janssen tie is not unique to $N_1$, but rather a general phenomenon for all cardinal B-splines $N_n$.\n\nOur analysis is based on the Zak transform and the Zibulski-Zeevi representation of Gabor systems \\cite{MR1448221}. Hence, let us briefly summarize the main idea here. The Zak transform of a function $f \\in L^2(\\mathbb{R})$ is defined as\n\\begin{equation}\\label{eq:zakTransform}\n \\left(Z_{\\lambda}f\\right)(x,\\gamma)\n = \\sqrt{\\lambda}\\sum_{k\\in\\mathbb{Z}} f(\\lambda(x-\n k))\\exponential^{2\\pi i k \\gamma}, \\quad a.e.\\ x, \\gamma \\in \\mathbb{R},\n\\end{equation}\nwith convergence in $L^2_\\mathrm{loc}(\\mathbb{R})$. We only consider rationally oversampled Gabor systems, i.e., $\\mathcal{G}(g,a,b)$ with\n\\[\n ab \\in \\mathbb{Q}, \\quad ab=\\frac{p}{q} \\quad \\gcd(p,q)=1.\n\\]\nFor $g\\in L^2(\\mathbb{R})$, we define column vectors $\\phi^g_\\ell(x,\\gamma) \\in \\mathbb{C}^p$ for $\\ell\n \\in \\set{0,1, \\dots, q-1}$ by\n\\[\n \\phi^g_\\ell(x,\\gamma) = \\left(p^{-\\frac{1}{2}} (Z_{\\frac{1}{b}}g)(x-\\ell\n \\frac{p}{q},\\gamma+\\frac{k}{p})\\right)_{k=0}^{p-1} \\ a.e. \\ x,\\gamma \\in \\mathbb{R}.\n\\]\nThe $p\\times q$ matrix defined by $\\Phi^g(x,\\gamma)=[\\phi^g_\\ell(x,\\gamma)]_{\\ell=0}^{q-1}$ is\nthe so-called Zibulski-Zeevi matrix from which we get the following characterization of the frame property of rationally oversampled Gabor systems:\n\n\\begin{theorem}\n \\label{TEST}\n \\label{thm:new-hyperbolas-frame-set-Bn}\n Let $n \\in\\N$. For $(a_0,b_0) \\in P$ we set $a_0 b_0=p/q$ and $q=r\\mu$ with $p,q,r,\\mu$ as in \\Cref{thm:point-obstructions-grochenig}. Define the set\n \\begin{align}\n H & = \\bigcup_{(a_0,b_0) \\in P} H_{(a_0,b_0)}, \\label{eq:hyperbolic-obstruction-set} \\\\\n \\intertext{where}\n H_{(a_0,b_0)} & = \\set*{(a,b) \\in \\R_+^2 \\given ab=\\frac{p}{q} \\text{ and } \\;\\abs{b - b_0} \\le \\frac{1}{nq}\\bigl(\\mu-(q-p+1) \\bigr)}\n \\label{eq:hyperbolic-obstruction-b-interval}\n \\end{align}\n If $(a,b) \\in H$, then $\\cG (N_n,a,b)$ is \\emph{not} a frame for $L^2(\\R)$. In other words, the complement of the frame set $\\R_+^2 \\setminus \\cF(N_n)$ contains the set $H$.\n\\end{theorem}", "full_context": "\\label{sec:intro}\n\nThe Gabor system generated by $g \\in L^2(\\mathbb{R})$ with time-frequency shifts along the lattice $a\\mathbb{Z} \\times b\\mathbb{Z}$, $a,b>0$, is defined as\n\\[\n \\mathcal{G}(g,a,b) = \\set*{\\exponential^{2\\pi i b m \\cdot} g(\\cdot - a k) \\given k,m \\in \\mathbb{Z}}.\n\\]\nThe system $\\mathcal{G}(g,a,b)$ is called a \\emph{Gabor frame} if there exist constants $A,B > 0$ such that\n\\[\n A \\norm{f}^2 \\le \\sum_{k,m \\in \\mathbb{Z}} \\abs{\\ip{f,\\exponential^{2\\pi i b m \\cdot} g(\\cdot - a k)}}^2\n \\le B \\norm{f}^2 \\quad \\text{for all } f \\in L^2(\\mathbb{R}).\n\\]\nOne of the fundamental problems in Gabor analysis asks, given $g \\in L^2(\\mathbb{R})$, for the determination of the \\emph{frame set} $\\mathcal{F}(g)$, which consists of the parameter values $(a,b)\\in\\mathbb{R}_+^2$ for which $\\mathcal{G}(g,a,b)$ is a frame.\nThe modern formulation is due to Gr\\\"ochenig~\\cite{MR3232589}, who emphasized it as an important problem in time-frequency analysis; the question itself dates back to the 1990s, see \\cite{MR2031050,MR1066587,MR1955931}.\n\nIn this paper, we consider the frame set of the cardinal B-splines $N_n$ of order $n \\ge 2$, $n \\in \\mathbb{N}$. The cardinal B-splines are defined as the $n$-fold convolution of the indicator function of the unit interval $\\itvcc{0,1}$, i.e.,\n\\begin{equation*}\n N_1 = \\chi_{\\itvcc{0, 1}}, \\quad \\text{and} \\quad\n N_{n+1} = N_{n} \\ast N_{1}, \\quad \\text{for } n \\in \\mathbb{N}.\n\\end{equation*}\nDue to its many desirable properties such as compact support, smoothness, and partition of unity property, these B-splines are widely used as Gabor system generators. The characterization of $\\mathcal{F}(N_n)$ was one of six open problems in Gabor analysis posed by Christensen in \\cite{ChristensenNew2014}. Our main result, \\Cref{thm:new-hyperbolas-frame-set-Bn}, proves the existence of a new infinite family of hyperbolic obstructions to the frame set $\\mathcal{F}(N_n)$.\n\nThe geometric complexity of the frame set for the indicator function $N_1$ is well-documented, famously described by the 'Janssen tie' \\cite{MR1955931}, and culminating in the complete characterization of $\\mathcal{F}(N_1)$ by Dai and Sun \\cite{MR3545108}. Historically, it was unclear whether the complexity of $\\mathcal{F}(N_1)$ was an artifact of the discontinuity of $N_1$.\nHowever, the results in \\cite{MR3572909} showed that the frame set of B-spline windows of all orders must have a very complicated structure, sharing several similarities with $\\mathcal{F}(N_1)$. In the present paper, we strengthen this perspective: we argue that certain number-theoretic constraints on $a$ and $b$ determine frame obstructions, and that (at least parts of) the hyperbolic geometry in the Janssen tie is not unique to $N_1$, but rather a general phenomenon for all cardinal B-splines $N_n$.\n\nOur analysis is based on the Zak transform and the Zibulski-Zeevi representation of Gabor systems \\cite{MR1448221}. Hence, let us briefly summarize the main idea here. The Zak transform of a function $f \\in L^2(\\mathbb{R})$ is defined as\n\\begin{equation}\\label{eq:zakTransform}\n \\left(Z_{\\lambda}f\\right)(x,\\gamma)\n = \\sqrt{\\lambda}\\sum_{k\\in\\mathbb{Z}} f(\\lambda(x-\n k))\\exponential^{2\\pi i k \\gamma}, \\quad a.e.\\ x, \\gamma \\in \\mathbb{R},\n\\end{equation}\nwith convergence in $L^2_\\mathrm{loc}(\\mathbb{R})$. We only consider rationally oversampled Gabor systems, i.e., $\\mathcal{G}(g,a,b)$ with\n\\[\n ab \\in \\mathbb{Q}, \\quad ab=\\frac{p}{q} \\quad \\gcd(p,q)=1.\n\\]\nFor $g\\in L^2(\\mathbb{R})$, we define column vectors $\\phi^g_\\ell(x,\\gamma) \\in \\mathbb{C}^p$ for $\\ell\n \\in \\set{0,1, \\dots, q-1}$ by\n\\[\n \\phi^g_\\ell(x,\\gamma) = \\left(p^{-\\frac{1}{2}} (Z_{\\frac{1}{b}}g)(x-\\ell\n \\frac{p}{q},\\gamma+\\frac{k}{p})\\right)_{k=0}^{p-1} \\ a.e. \\ x,\\gamma \\in \\mathbb{R}.\n\\]\nThe $p\\times q$ matrix defined by $\\Phi^g(x,\\gamma)=[\\phi^g_\\ell(x,\\gamma)]_{\\ell=0}^{q-1}$ is\nthe so-called Zibulski-Zeevi matrix from which we get the following characterization of the frame property of rationally oversampled Gabor systems:\n\n\\begin{theorem}\n \\label{TEST}\n \\label{thm:new-hyperbolas-frame-set-Bn}\n Let $n \\in\\N$. For $(a_0,b_0) \\in P$ we set $a_0 b_0=p/q$ and $q=r\\mu$ with $p,q,r,\\mu$ as in \\Cref{thm:point-obstructions-grochenig}. Define the set\n \\begin{align}\n H & = \\bigcup_{(a_0,b_0) \\in P} H_{(a_0,b_0)}, \\label{eq:hyperbolic-obstruction-set} \\\\\n \\intertext{where}\n H_{(a_0,b_0)} & = \\set*{(a,b) \\in \\R_+^2 \\given ab=\\frac{p}{q} \\text{ and } \\;\\abs{b - b_0} \\le \\frac{1}{nq}\\bigl(\\mu-(q-p+1) \\bigr)}\n \\label{eq:hyperbolic-obstruction-b-interval}\n \\end{align}\n If $(a,b) \\in H$, then $\\cG (N_n,a,b)$ is \\emph{not} a frame for $L^2(\\R)$. In other words, the complement of the frame set $\\R_+^2 \\setminus \\cF(N_n)$ contains the set $H$.\n\\end{theorem}\n\nThe geometric complexity of the frame set for the indicator function $N_1$ is well-documented, famously described by the 'Janssen tie' \\cite{MR1955931}, and culminating in the complete characterization of $\\cF(N_1)$ by Dai and Sun \\cite{MR3545108}. Historically, it was unclear whether the complexity of $\\cF(N_1)$ was an artifact of the discontinuity of $N_1$.\nHowever, the results in \\cite{MR3572909} showed that the frame set of B-spline windows of all orders must have a very complicated structure, sharing several similarities with $\\cF(N_1)$. In the present paper, we strengthen this perspective: we argue that certain number-theoretic constraints on $a$ and $b$ determine frame obstructions, and that (at least parts of) the hyperbolic geometry in the Janssen tie is not unique to $N_1$, but rather a general phenomenon for all cardinal B-splines $N_n$.\n\nFor windows in the Feichtinger algebra $M^1(\\R)$, the Zibulski-Zeevi matrix has continuous entries, and thus, to show that $\\gaborG{g}$ is not a frame, it suffices to find a single point $(x,\\gamma) \\in \\itvco{0, 1}^2$ such that the set $\\set{\\phi^g_\\ell(x,\\gamma)}_{\\ell=0}^{q-1}$ does not span $\\C^p$.\n\nFor $x\\in \\R$, we let $\\round{x}$ denote the round function to the nearest integer, i.e., $R(x)=\\floor{x+\\frac12}$, and we let $\\sfrac{x}=x-\\round{x} \\in\n \\itvoc{-\\frac12, \\frac12}$ denote the (signed) fractional part of $x$. The main result of \\cite{MR3572909} states that $\\gaborG{N_n}$ is not a frame for\n\\begin{equation}\n \\label{eq:old_hyperbolic_obstructions}\n ab=\\frac{p}{q}<1 \\quad \\text{for } \\abs{\\sfrac{b}} \\le \\frac{1}{nq} \\;\\text{ and }\\; b > \\frac{3}{2},\n\\end{equation}\nwhere $\\gcd(p,q)=1$. Based on a large number of computer-assisted symbolic calculations by Kamilla~H.~Nielsen, we realized that there are further hyperbolic obstructions for the frame property of the Gabor system $\\mathcal{G}(N_n,a,b)$ ``far'' away from integer values of $b=2,3,4, \\dots$. Indeed, we conjectured in \\cite{MR3572909} the following: $\\mathcal{G}(N_2,a_0,b_0)$ is not a frame for\n\\begin{equation}\n \\label{eq:con_newNonFrame}\n a_0= \\frac{1}{2m+1}, \\ b_0=\\frac{2k+1}{2}, \\ k,m\\in\\mathbb{N}, \\ k>m, \\ a_0 b_0 < 1,\n\\end{equation}\nand, furthermore, $\\mathcal{G}(N_2,a,b)$ is not a frame along the hyperbolas\n\\begin{equation}\n \\label{eq:con_hyperbelStykke}\n \\ ab=\\frac{2k+1}{2\\left(2m+1\\right)}, \\quad \\text{for } \\abs{b-b_0} \\le \\frac{k-m}{2(2m+1)},\n\\end{equation}\nfor every $a_0$ and $b_0$ defined by (\\ref{eq:con_newNonFrame}). We provided a proof of the case $m=1$ and $k=2$ in \\cite{MR3572909}. Gr\\\"ochenig proved the first part \\eqref{eq:con_newNonFrame} of this conjecture in the note \\cite{GrochenigPartitions2015}. The full conjecture remained open until Ghosh and Selvan \\cite{MR4917072} recently were able to verify it.\n\n\\begin{lemma}[\\!\\cite{MR3572909}]\n \\label{lem:partly-pou}\n Let $n \\in \\N$ and $c>0$. Assume that $\\abs{\\sfrac{c}} \\le\n \\frac{1}{n}$.\n \\begin{enumerate}[label=(\\roman*)]\n \\item If $\\sfrac{c}\\ge 0$, then\n \\begin{equation}\n \\label{eq:partly-part-of-unity-1}\n \\sum_{k \\in \\Z} N_n((x+k)/c) = \\mathrm{const} \\quad \\text{for }\n x \\in \\bigcup_{m\\in \\Z}\\itvcc{m+n\\sfrac{c}, m+1}\n \\end{equation}\n \\item If $\\sfrac{c}\\le 0$, then\n \\begin{equation}\n \\label{eq:partly-part-of-unity-2}\n \\sum_{k \\in \\Z} N_n((x+k)/c) = \\mathrm{const} \\quad \\text{for }\n x \\in \\bigcup_{m\\in \\Z}\\itvcc{m, m+1+n\\sfrac{c}}\n \\end{equation}\n \\end{enumerate}\n\\end{lemma}\n\n\\begin{lemma}\n \\label{lem:ZZ-structure}\n Let $n \\in\\N$. Let $\\mu,r, k \\in \\N$ be as in \\Cref{def:point-set-only-mu}. Set $p=r\\mu-k$, $q=r\\mu$, and $b_0 = \\mu - k/r$.\n Let $b>3/2$ be given such that $\\abs{b-b_0} \\le \\frac{1}{nr}$, i.e., $\\abs{\\sfrac{rb}} \\le \\frac1n$.\n \\begin{enumerate}[label=(\\roman*)]\n \\item If $\\sfrac{rb}\\ge 0$, there exists a\n constant $K$ such that \\[\n \\sum_{\\ell=0}^{r-1} \\phi_{\\ell \\mu}^{N_n}(\\frac{x}{r},0) = K e_0 \\in \\C^p\n \\qquad \\text{for $x \\in \\bigcup_{m \\in\n \\Z}\\itvcc{m+n\\sfrac{rb}, m+1}$.}\n \\]\n \\label{item:ZZ-cancellation-pos}\n \\item If $\\sfrac{rb}\\le 0$, there exists a\n constant $K$ such that \\[\n \\sum_{\\ell=0}^{r-1} \\phi_{\\ell \\mu}^{N_n}(\\frac{x}{r},0) = K e_0 \\in \\C^p\n \\qquad \\text{for } x \\in \\bigcup_{m \\in\n \\Z}\\itvcc{m, m+1+n\\sfrac{rb}}\n \\]\n \\label{item:ZZ-cancellation-neg}\n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n We will only prove \\ref{item:ZZ-cancellation-pos} as the proof of \\ref{item:ZZ-cancellation-neg} is similar. Note that $\\round{rb}=p$. Hence, $rb$ is ``close'' to the integer $p$ by $0 \\le \\sfrac{rb} \\le \\frac1n$.\n\n\\begin{remark}\n In the proof of \\Cref{thm:new-hyperbolas-frame-set-Bn}, we tacitly used that the Zak transform $Z_\\lambda N_n$ is continuous. This is not true for $n=1$. However, $Z_\\lambda N_1$ is continuous with respect to the second variable and piecewise continuous with respect to the first variable with finitely many jump discontinuities. It follows that we can still conclude that the minimum of the smallest singular value of the Zibulski-Zeevi matrix $\\Phi^{N_1}(x,\\gamma)$ over $(x,\\gamma)\\in \\itvco{0,1}^2$ is zero, which is the conclusion we need.\n\\end{remark}\n\\begin{remark}\n The obstructions in \\Cref{thm:new-hyperbolas-frame-set-Bn} have been known to the author for a number of years. They were originally presented at \\emph{The International Workshop on Operator Theory and Applications (IWOTA)} in 2016, but have not previously appeared in print.\n\\end{remark}\n\nWe end the paper with a general remark on the nature of the known general obstructions for B-splines and other functions that generate partitions of unity. For functions $g \\in M^1(\\R)$ that generate partitions of unity, there are two types of obstructions to the frame property:\n\\begin{enumerate}[label=(\\Roman*)]\n \\item For $b=2,3,\\dots$ and any $a>0$ the Gabor system $\\gaborG{g}$ is not a frame \\cite{MR2657413,MR2045812,MR1752589} \\label{item:integer-obstruction},\n \\item For $(a,b)$ in the point set $P$ the Gabor system $\\gaborG{g}$ is not a frame \\cite{GrochenigPartitions2015} \\label{item:point-obstruction}.\n\\end{enumerate}\nWhen viewed along hyperbolas $ab=p/q<1$, these two types of obstructions are both point obstructions. For B-splines, the hyperbolic obstructions in \\cite{MR4917072} are ``grown'' out of the first type of obstruction \\ref{item:integer-obstruction} along $ab=p/q<1$, whereas the obstructions in \\Cref{thm:new-hyperbolas-frame-set-Bn} are ``grown'' out of the second type of obstruction \\ref{item:point-obstruction}, also along $ab=p/q<1$. In the opposite direction, as the B-spline order $n$ increases, these hyperbolic obstructions contract, degenerating to the point obstructions \nas $n \\to \\infty$. To see why B-splines admit these hyperbolic extensions of point obstructions, note first that any function $g$ that generates a partition of unity $\\sum_{k \\in \\Z} g(\\cdot + k) = 1$ will, for integer $c$, also satisfy $\\sum_{k \\in \\Z} g((\\cdot+k)/c) = c$. B-splines further satisfy a partly partition of unity property in \nLemma~\\ref{lem:partly-pou} for $c$ close to an integer -- that is, $\\sum_{k \\in \\Z} N_n((\\cdot+k)/c)$ is constant on some subinterval of $\\itvco{0,1}$ -- which is the key additional property that allows the point obstructions to be extended into hyperbolic segments. In fact, any sufficiently nice function $g$ that generates a partition of unity and retains such a partly partition of unity under dilations $(1/c)\\Z$ for $c \\approx 1$ will have similar hyperbolic obstructions.", "post_theorem_intro_text_len": 6283, "post_theorem_intro_text": "For windows in the Feichtinger algebra $M^1(\\mathbb{R})$, the Zibulski-Zeevi matrix has continuous entries, and thus, to show that $\\mathcal{G}(g,a,b)$ is not a frame, it suffices to find a single point $(x,\\gamma) \\in \\itvco{0, 1}^2$ such that the set $\\set{\\phi^g_\\ell(x,\\gamma)}_{\\ell=0}^{q-1}$ does not span $\\mathbb{C}^p$.\n\nFor $x\\in \\mathbb{R}$, we let $R{(x)}$ denote the round function to the nearest integer, i.e., $R(x)=\\floor{x+\\frac12}$, and we let $F{(x)}=x-R{(x)} \\in\n \\itvoc{-\\frac12, \\frac12}$ denote the (signed) fractional part of $x$. The main result of \\cite{MR3572909} states that $\\mathcal{G}(N_n,a,b)$ is not a frame for\n\\begin{equation}\n \\label{eq:old_hyperbolic_obstructions}\n ab=\\frac{p}{q}<1 \\quad \\text{for } \\abs{F{(b)}} \\le \\frac{1}{nq} \\;\\text{ and }\\; b > \\frac{3}{2},\n\\end{equation}\nwhere $\\gcd(p,q)=1$. Based on a large number of computer-assisted symbolic calculations by Kamilla~H.~Nielsen, we realized that there are further hyperbolic obstructions for the frame property of the Gabor system $\\mathcal{G}(N_n,a,b)$ ``far'' away from integer values of $b=2,3,4, \\dots$. Indeed, we conjectured in \\cite{MR3572909} the following: $\\mathcal{G}(N_2,a_0,b_0)$ is not a frame for\n\\begin{equation}\n \\label{eq:con_newNonFrame}\n a_0= \\frac{1}{2m+1}, \\ b_0=\\frac{2k+1}{2}, \\ k,m\\in\\mathbb{N}, \\ k>m, \\ a_0 b_0 < 1,\n\\end{equation}\nand, furthermore, $\\mathcal{G}(N_2,a,b)$ is not a frame along the hyperbolas\n\\begin{equation}\n \\label{eq:con_hyperbelStykke}\n \\ ab=\\frac{2k+1}{2\\left(2m+1\\right)}, \\quad \\text{for } \\abs{b-b_0} \\le \\frac{k-m}{2(2m+1)},\n\\end{equation}\nfor every $a_0$ and $b_0$ defined by (\\ref{eq:con_newNonFrame}). We provided a proof of the case $m=1$ and $k=2$ in \\cite{MR3572909}. Gr\\\"ochenig proved the first part \\eqref{eq:con_newNonFrame} of this conjecture in the note \\cite{GrochenigPartitions2015}. The full conjecture remained open until Ghosh and Selvan \\cite{MR4917072} recently were able to verify it.\n\nImportant for the development of the present paper is that Gr\\\"ochenig~\\cite{GrochenigPartitions2015} realized that the pattern \\eqref{eq:con_newNonFrame} found in \\cite{MR3572909} was part of a much larger family of obstructions that holds for any Gabor system generator in the Feichtinger algebra $M^1(\\mathbb{R})$ satisfying the partition of unity property:\n\n\\begin{theorem}[Gr\\\"ochenig \\cite{GrochenigPartitions2015}]\n \\label{thm:point-obstructions-grochenig}\n Let $\\mu,\\nu,r \\in \\mathbb{N}$ with $r \\ge 2$. Let $p=r\\nu+j$ and $q=r\\mu$, where $j=1,\\dots,r-1$. Define the set $P$ of points $(a_0,b_0) \\in \\R_+^2$ where $a_0=\\frac{r}{q}=\\frac{1}{\\mu}$ and $b_0=\\frac{p}{r} = \\nu+\\frac{j}{r}$ subject to the constraints that $p$ and $q$ are relatively prime with $q-\\mu +1< p0$, and let $g \\in L^2(\\mathbb{R})$. Suppose $\\mathcal{G}(g,a,b)$\n is rationally oversampled Gabor system. Then the following\n assertions are equivalent:\n \\begin{enumerate}[label=(\\roman*)]\n \\item $\\mathcal{G}(g,a,b)$ is a Gabor frame for $L^2(\\mathbb{R})$ with bounds $A$ and $B$,\n \\item $\\set{\\phi^g_\\ell(x,\\gamma)}_{\\ell=0}^{q-1}$ is a frame for $\\mathbb{C}^p$ with\n uniform bounds $A$ and $B$ for a.e. $(x,\\gamma) \\in\n \\itvco{0, 1}^2$.\n \\end{enumerate}", "theorem_type": ["Biconditional or Equivalence", "Classification or Bijection"], "mcq": {"question": "Let $g\\in L^2(\\mathbb{R})$, let $A,B>0$, and suppose the Gabor system\n\\[\n\\mathcal G(g,a,b)=\\{e^{2\\pi i b m\\,\\cdot}\\, g(\\cdot-ak): k,m\\in\\mathbb Z\\}\n\\]\nis rationally oversampled, meaning $ab=p/q\\in\\mathbb Q$ with $\\gcd(p,q)=1$. Define the Zak transform by\n\\[\n(Z_\\lambda f)(x,\\gamma)=\\sqrt{\\lambda}\\sum_{m\\in\\mathbb Z} f(\\lambda(x-m))e^{2\\pi i m\\gamma},\n\\]\nand for $\\ell=0,1,\\dots,q-1$ define vectors $\\phi^g_\\ell(x,\\gamma)\\in\\mathbb C^p$ by\n\\[\n\\phi^g_\\ell(x,\\gamma)=\\left(p^{-1/2}(Z_{1/b}g)\\bigl(x-\\ell p/q,\\,\\gamma+k/p\\bigr)\\right)_{k=0}^{p-1}.\n\\]\nWhich statement is equivalent to saying that $\\mathcal G(g,a,b)$ is a Gabor frame for $L^2(\\mathbb R)$ with frame bounds $A$ and $B$?", "correct_choice": {"label": "A", "text": "For almost every $(x,\\gamma)\\in[0,1)^2$, the family $\\{\\phi^g_\\ell(x,\\gamma)\\}_{\\ell=0}^{q-1}$ is a frame for $\\mathbb C^p$ with the same uniform bounds $A$ and $B$; equivalently, for every $c\\in\\mathbb C^p$,\n\\[\nA\\|c\\|^2\\le \\sum_{\\ell=0}^{q-1}\\big|\\langle c,\\phi^g_\\ell(x,\\gamma)\\rangle\\big|^2\\le B\\|c\\|^2\n\\]\nfor almost every $(x,\\gamma)\\in[0,1)^2$."}, "choices": [{"label": "B", "text": "For every $(x,\\gamma)\\in[0,1)^2$, the family $\\{\\phi^g_\\ell(x,\\gamma)\\}_{\\ell=0}^{q-1}$ is a frame for $\\mathbb C^p$ with bounds $A$ and $B$; equivalently, for every $c\\in\\mathbb C^p$,\n\\[\nA\\|c\\|^2\\le \\sum_{\\ell=0}^{q-1}\\big|\\langle c,\\phi^g_\\ell(x,\\gamma)\\rangle\\big|^2\\le B\\|c\\|^2\n\\]\nfor every $(x,\\gamma)\\in[0,1)^2$."}, {"label": "C", "text": "For almost every $(x,\\gamma)\\in[0,1)^2$, the family $\\{\\phi^g_\\ell(x,\\gamma)\\}_{\\ell=0}^{q-1}$ spans $\\mathbb C^p$."}, {"label": "D", "text": "For almost every $(x,\\gamma)\\in[0,1)^2$, the family $\\{\\phi^g_\\ell(x,\\gamma)\\}_{\\ell=0}^{q-1}$ is a frame for $\\mathbb C^p$, and for each such $(x,\\gamma)$ there exist bounds $A(x,\\gamma),B(x,\\gamma)>0$ such that for every $c\\in\\mathbb C^p$,\n\\[\nA(x,\\gamma)\\|c\\|^2\\le \\sum_{\\ell=0}^{q-1}\\big|\\langle c,\\phi^g_\\ell(x,\\gamma)\\rangle\\big|^2\\le B(x,\\gamma)\\|c\\|^2.\n\\]"}, {"label": "E", "text": "There exists a single point $(x,\\gamma)\\in[0,1)^2$ such that the family $\\{\\phi^g_\\ell(x,\\gamma)\\}_{\\ell=0}^{q-1}$ is a frame for $\\mathbb C^p$ with bounds $A$ and $B$; equivalently, for every $c\\in\\mathbb C^p$,\n\\[\nA\\|c\\|^2\\le \\sum_{\\ell=0}^{q-1}\\big|\\langle c,\\phi^g_\\ell(x,\\gamma)\\rangle\\big|^2\\le B\\|c\\|^2.\n\\]"}], "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": "a.e.-in-(x,\\gamma) replaced by pointwise everywhere", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "uniform frame bounds A,B dropped, retaining only spanning/frame existence a.e.", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "same uniform bounds A,B replaced by bounds depending on (x,\\gamma)", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "a.e. fiberwise frame condition weakened to existence at one point", "template_used": "wildcard"}]}} +{"id": "2601.08195v1", "paper_link": "http://arxiv.org/abs/2601.08195v1", "theorems_cnt": 2, "theorem": {"env_name": "conj", "content": "\\label{conj}\nLet $\\Gamma$ be a finite subgroup of $SU(2)$. Let $X$ be the minimal resolution of $\\mathbb C^2/\\Gamma$. Let $S_1$,...$S_r$ be the components of $S^1$-fixed points of $X$ generating $H^2(X)$, and let $\\rho_j(t)=P_j(t)RP_j(t)^{-1}, 1\\leq j\\leq r$ be the $1$-parameter family of holonomy representations with the property that $$\\rho_j(t)=\\rho_{(\\alpha_j(t),\\beta_j(t))},$$ $$\\lim_{t\\to-\\infty}(\\alpha_j(t),\\beta_j(t))=(\\alpha_j,\\beta_j),$$ with $(\\alpha_j,\\beta_j)\\in S_j$. Then $$P_j^{\\lim}=\\lim_{t\\to\\infty}P_j(t)$$ is equal to the projector onto the nontrivial irreducible representation $\\rho_j$ of $\\Gamma$, and every nontrivial irreducible representation of $\\Gamma$ arises this way.", "start_pos": 12171, "end_pos": 12890, "label": "conj"}, "ref_dict": {"conj": "\\begin{conj}\\label{conj}\nLet $\\Gamma$ be a finite subgroup of $SU(2)$. Let $X$ be the minimal resolution of $\\C^2/\\Gamma$. Let $S_1$,...$S_r$ be the components of $S^1$-fixed points of $X$ generating $H^2(X)$, and let $\\rho_j(t)=P_j(t)RP_j(t)^{-1}, 1\\leq j\\leq r$ be the $1$-parameter family of holonomy representations with the property that $$\\rho_j(t)=\\rho_{(\\alpha_j(t),\\beta_j(t))},$$ $$\\lim_{t\\to-\\infty}(\\alpha_j(t),\\beta_j(t))=(\\alpha_j,\\beta_j),$$ with $(\\alpha_j,\\beta_j)\\in S_j$. Then $$P_j^{\\lim}=\\lim_{t\\to\\infty}P_j(t)$$ is equal to the projector onto the nontrivial irreducible representation $\\rho_j$ of $\\Gamma$, and every nontrivial irreducible representation of $\\Gamma$ arises this way. \n\\end{conj}"}, "pre_theorem_intro_text_len": 8557, "pre_theorem_intro_text": "The McKay correspondence is first stated in the seminal paper \\cite{mckay} by McKay where he observes that there is a one-to-one correspondence between the finite subgroups $\\Gamma \\subset \\text{SU}(2)$ and the simply laced Dynkin diagrams of types $A$, $D$, and $E$. McKay observes that decomposing the tensor product of the canonical representation with the irreducible representations of $\\Gamma$ yields an adjacency matrix exactly matching that of the affine Dynkin diagram. Initially a mysterious combinatorial coincidence, this correspondence inspires decades of subsequent research exploring the deeper conceptual explanation for such an occurance.\n\nThis link was given a geometric foundation by Gonzalez-Sprinberg and Verdier \\cite{GSV}, who lift the correspondence to the minimal resolution of the quotient singularity $X \\to \\mathbb{C}^2/\\Gamma$. They establish that the tautological bundles on the resolution $X$ provide a $K$-theoretic basis isomorphic to the representation ring $R(\\Gamma)$. Geometrically, this identified the nontrivial irreducible representations with a basis of $H^2(X, \\mathbb{Z})$. Furthermore, the intersection form of these cohomology cycles reproduces the negative Cartan matrix, thereby providing a geometric explanation for the appearance of the Dynkin diagram in the representation theory. A heuristic statement of the McKay correspondence is given below; standard references for the relevant basic topics include \\cite{fulton-harris}, \\cite{griffiths-harris}, and \\cite{G-S}.\n\n\\begin{statement}[McKay correspondence]\nLet $\\Gamma \\subset SU(2$) be a finite subgroup, and let $\\widetilde{ \\mathbb{C}^2/\\Gamma}$ be the minimal resolution of $\\mathbb C^2/\\Gamma$. There is a natural correspondence between the non-trivial irreducible representations of $\\Gamma$ and the irreducible exceptional curves in $\\widetilde{ \\mathbb{C}^2/\\Gamma}$, such that the intersection pairing of these curves is given by the Mckay quiver of $\\Gamma$.\n\n\\end{statement}\n\nSubsequently, more proofs of the McKay correspondence and its generalization to higher dimensions utilizing different, mainly algebriac tools and techniques, such as \\cite{B-D}, \\cite{BKR}, \\cite{D-L}, \\cite{I-N}, \\cite{I-R}, \\cite{kaledin}, appear in the years to come -- the list of references provided here is not nearly exhaustive. We direct the readers to \\cite{craw} and \\cite{reid} for further expositions. \n\nIn the meantime, Kronheimer in his PhD thesis \\cite{kronheimer}, \\cite{kronheimer2}, gives a complete construction of all the $4$-dimensional hyperkähler ALE spaces, showing that each such space is topologically equivalent to the minimal resolution $X$ of the corresponding quotient $\\mathbb{C}^2/\\Gamma$. In Kronheimer's construction, these hyperkähler manifolds are obtained via a finite-dimensional hyperkähler reduction. Kronheimer’s construction provided $X$ with a canonical metric structure, moving the correspondence from the realm of algebraic geometry into the domain of gauge theory and metric analysis. In a subsequent paper studying the moduli space of instantons on ALE spaces \\cite{kronheimer3}, joint with Nakajima, they provide a geometric interpretation of the McKay correspondence via APS-index theory, further linking the McKay correspondence with hyperkähler geometry and gauge theory. \n\nMore recently, the McKay correspondence is explored through the lens of symplectic and contact geometry in the works of McLean-Ritter \\cite{M-R} and Digiosia-Nelson \\cite{D-N}. These new perspectives use symplectic cohomology and cylindrical contact homology to establish the correspondence through the enumeration of pseudoholomorphic curves. One thing to note is that the symplectic and contact geometric interpretation of the McKay correspondence does not directly address irreducible representations of $\\Gamma$, and the correspondence is established between nontrivial conjugacy classes of $\\Gamma$ and the generators of $H^2(X, \\mathbb{Z})$.\n\nThe remarkable variety of tools, ranging from $K$-theory, Hilbert schemes, to symplectic and contact geometry, that have been used to study the McKay correspondence indicates a rich and deep link between representation theory, singularity theory and geometry. The strategy of all the existing approaches so far is to construct an isomorphism identifying bases (cohomology cycles to irreducible representations). While these celebrated results establish that an isomorphism exists, the nature of the correspondence remains largely static. A natural question, particularly in light of modern developments in gauge theory, Floer theory and mathematical physics, is whether this correspondence can be realized dynamically. Given the asymptotic metric structure of the ALE spaces constructed by Kronheimer \\cite{kronheimer}, one might ask if the correspondence can manifest through the evolution of gradient flow trajectories of a natural Morse-Bott function as the flow lines approach the boundary at infinity, bearing certain reminiscence to the construction of instanton Floery homology \\cite{floer}.\n\nIn this paper, we propose such a dynamical framework. The foundation of the framework lies upon a gauge-theoretic interpretation of Kronheimer's construction of the $4$-dimensional hyperkähler ALE spaces, developed in the thesis work of the author \\cite{yan}, in which each ALE space is realized as a moduli space of solutions to a system of equations for a pair consisting of a connection and a section of a vector bundle over an orbifold Riemann surface, modulo a gauge group action. We use this gauge-theoretic construction to seek a new gauge/Morse-theoretic interpretation of the McKay correspondence by studying an $S^1$-invariant Morse-Bott function on the ALE spaces and matching the gradient flow lines from its critical points with the irreducible representations of $\\Gamma$. \n\nThe setup is as follows: inspired by works of Atiyah-Bott \\cite{atiyah-bott}, Hitchin \\cite{hitchin} and Kirwan \\cite{kirwan}, we study the topology of the moduli space via an $S^1$-invariant Morse-Bott function induced by a Hamiltoninan $S^1$-action on the moduli space. Following Kronheimer's notations in \\cite{kronheimer}, a point in the minimal resolution $X$ can be expressed as a pair of matrices $(\\alpha,\\beta)$ with certain properties, with $\\alpha,\\beta \\in End(R)$ and $R$ being the regular representation of $\\Gamma$. The $S^1$-action on $X$ is simply given by $$e^{i\\varphi}\\cdot(\\alpha,\\beta)=(e^{i\\varphi}\\alpha,e^{i\\varphi}\\beta).$$ The induced $S^1$-invariant Morse-Bott function is equal to the norm square of $(\\alpha,\\beta)$ up to some constant, that is, $$\\Phi(\\alpha,\\beta)=k_0(\\|\\alpha\\|^2+\\|\\beta\\|^2),$$ for some fixed constant $k_0$. The critical points of this $S^1$-invariant Morse-Bott function are precisely the $S^1$-fixed point of the Hamiltoninan $S^1$-action. On the other hand, from the gauge-theoretic construction in \\cite{yan}, we realize each point $(\\alpha,\\beta)$ of an ALE space as a flat connection on a trivial bundle over $S^3/\\Gamma$ with fibers equal to $R$, the regular representation of $\\Gamma$. This yields a holonomy representation $\\rho_{(\\alpha,\\beta)}: \\Gamma \\to GL(R)$, which can be shown to be always isomorphic to $R$. More specifically, the flat connection is given by $$d + \\alpha dz_1 + \\beta dz_2,$$ where $z_1,z_2$ are coordinates in $\\mathbb C^2$. We set $[\\alpha, \\beta] = 0$ to ensure that the above connection is flat, with holonomy representation given by $$\\rho_{(\\alpha,\\beta)}(\\gamma)=R(\\gamma)\\exp((v_1-(uv_1-\\bar{v}v_2))\\alpha+(v_2-(\\bar{u}v_2+vv_1))\\beta),$$ where $(v_1,v_2)$ is the base point in $S^3/\\Gamma$, and $u,v$ are the coordinates of an element $\\gamma\\in\\Gamma$. We show that $\\rho_{(\\alpha,\\beta)}$ is always isomorphic to the regular representation $R$, for any point $(\\alpha,\\beta)$ in $X$. \n\nAlong a gradient flow line $(\\alpha(t),\\beta(t))$ from an index-$2$ critical point to infinity, we obtain a $1$-parameter family of representations $\\rho(t) = \\rho_{(\\alpha(t),\\beta(t))}$. Since $\\rho(t)$ is always isomorphic to the regular representation $R$, we obtain a $1$-parameter family of change of basis matrices $P(t)$ such that $\\rho(t) = P(t) R P(t)^{-1}$. As $H^2(X)$ is generated by the components of the $S^1$-fixed points, we propose the following conjecture, presenting an identification of the generators of $H^2(X)$ with the nontrivial irreducible representations of $\\Gamma$ via the evolution of gradient flow lines, creating an explicit dynamical framework for the McKay correspondence.", "context": "The McKay correspondence is first stated in the seminal paper \\cite{mckay} by McKay where he observes that there is a one-to-one correspondence between the finite subgroups $\\Gamma \\subset \\text{SU}(2)$ and the simply laced Dynkin diagrams of types $A$, $D$, and $E$. McKay observes that decomposing the tensor product of the canonical representation with the irreducible representations of $\\Gamma$ yields an adjacency matrix exactly matching that of the affine Dynkin diagram. Initially a mysterious combinatorial coincidence, this correspondence inspires decades of subsequent research exploring the deeper conceptual explanation for such an occurance.\n\nThis link was given a geometric foundation by Gonzalez-Sprinberg and Verdier \\cite{GSV}, who lift the correspondence to the minimal resolution of the quotient singularity $X \\to \\mathbb{C}^2/\\Gamma$. They establish that the tautological bundles on the resolution $X$ provide a $K$-theoretic basis isomorphic to the representation ring $R(\\Gamma)$. Geometrically, this identified the nontrivial irreducible representations with a basis of $H^2(X, \\mathbb{Z})$. Furthermore, the intersection form of these cohomology cycles reproduces the negative Cartan matrix, thereby providing a geometric explanation for the appearance of the Dynkin diagram in the representation theory. A heuristic statement of the McKay correspondence is given below; standard references for the relevant basic topics include \\cite{fulton-harris}, \\cite{griffiths-harris}, and \\cite{G-S}.\n\n\\begin{statement}[McKay correspondence]\nLet $\\Gamma \\subset SU(2$) be a finite subgroup, and let $\\widetilde{ \\mathbb{C}^2/\\Gamma}$ be the minimal resolution of $\\mathbb C^2/\\Gamma$. There is a natural correspondence between the non-trivial irreducible representations of $\\Gamma$ and the irreducible exceptional curves in $\\widetilde{ \\mathbb{C}^2/\\Gamma}$, such that the intersection pairing of these curves is given by the Mckay quiver of $\\Gamma$.\n\nIn the meantime, Kronheimer in his PhD thesis \\cite{kronheimer}, \\cite{kronheimer2}, gives a complete construction of all the $4$-dimensional hyperkähler ALE spaces, showing that each such space is topologically equivalent to the minimal resolution $X$ of the corresponding quotient $\\mathbb{C}^2/\\Gamma$. In Kronheimer's construction, these hyperkähler manifolds are obtained via a finite-dimensional hyperkähler reduction. Kronheimer’s construction provided $X$ with a canonical metric structure, moving the correspondence from the realm of algebraic geometry into the domain of gauge theory and metric analysis. In a subsequent paper studying the moduli space of instantons on ALE spaces \\cite{kronheimer3}, joint with Nakajima, they provide a geometric interpretation of the McKay correspondence via APS-index theory, further linking the McKay correspondence with hyperkähler geometry and gauge theory.\n\nThe setup is as follows: inspired by works of Atiyah-Bott \\cite{atiyah-bott}, Hitchin \\cite{hitchin} and Kirwan \\cite{kirwan}, we study the topology of the moduli space via an $S^1$-invariant Morse-Bott function induced by a Hamiltoninan $S^1$-action on the moduli space. Following Kronheimer's notations in \\cite{kronheimer}, a point in the minimal resolution $X$ can be expressed as a pair of matrices $(\\alpha,\\beta)$ with certain properties, with $\\alpha,\\beta \\in End(R)$ and $R$ being the regular representation of $\\Gamma$. The $S^1$-action on $X$ is simply given by $$e^{i\\varphi}\\cdot(\\alpha,\\beta)=(e^{i\\varphi}\\alpha,e^{i\\varphi}\\beta).$$ The induced $S^1$-invariant Morse-Bott function is equal to the norm square of $(\\alpha,\\beta)$ up to some constant, that is, $$\\Phi(\\alpha,\\beta)=k_0(\\|\\alpha\\|^2+\\|\\beta\\|^2),$$ for some fixed constant $k_0$. The critical points of this $S^1$-invariant Morse-Bott function are precisely the $S^1$-fixed point of the Hamiltoninan $S^1$-action. On the other hand, from the gauge-theoretic construction in \\cite{yan}, we realize each point $(\\alpha,\\beta)$ of an ALE space as a flat connection on a trivial bundle over $S^3/\\Gamma$ with fibers equal to $R$, the regular representation of $\\Gamma$. This yields a holonomy representation $\\rho_{(\\alpha,\\beta)}: \\Gamma \\to GL(R)$, which can be shown to be always isomorphic to $R$. More specifically, the flat connection is given by $$d + \\alpha dz_1 + \\beta dz_2,$$ where $z_1,z_2$ are coordinates in $\\mathbb C^2$. We set $[\\alpha, \\beta] = 0$ to ensure that the above connection is flat, with holonomy representation given by $$\\rho_{(\\alpha,\\beta)}(\\gamma)=R(\\gamma)\\exp((v_1-(uv_1-\\bar{v}v_2))\\alpha+(v_2-(\\bar{u}v_2+vv_1))\\beta),$$ where $(v_1,v_2)$ is the base point in $S^3/\\Gamma$, and $u,v$ are the coordinates of an element $\\gamma\\in\\Gamma$. We show that $\\rho_{(\\alpha,\\beta)}$ is always isomorphic to the regular representation $R$, for any point $(\\alpha,\\beta)$ in $X$.\n\nAlong a gradient flow line $(\\alpha(t),\\beta(t))$ from an index-$2$ critical point to infinity, we obtain a $1$-parameter family of representations $\\rho(t) = \\rho_{(\\alpha(t),\\beta(t))}$. Since $\\rho(t)$ is always isomorphic to the regular representation $R$, we obtain a $1$-parameter family of change of basis matrices $P(t)$ such that $\\rho(t) = P(t) R P(t)^{-1}$. As $H^2(X)$ is generated by the components of the $S^1$-fixed points, we propose the following conjecture, presenting an identification of the generators of $H^2(X)$ with the nontrivial irreducible representations of $\\Gamma$ via the evolution of gradient flow lines, creating an explicit dynamical framework for the McKay correspondence.", "full_context": "The McKay correspondence is first stated in the seminal paper \\cite{mckay} by McKay where he observes that there is a one-to-one correspondence between the finite subgroups $\\Gamma \\subset \\text{SU}(2)$ and the simply laced Dynkin diagrams of types $A$, $D$, and $E$. McKay observes that decomposing the tensor product of the canonical representation with the irreducible representations of $\\Gamma$ yields an adjacency matrix exactly matching that of the affine Dynkin diagram. Initially a mysterious combinatorial coincidence, this correspondence inspires decades of subsequent research exploring the deeper conceptual explanation for such an occurance.\n\nThis link was given a geometric foundation by Gonzalez-Sprinberg and Verdier \\cite{GSV}, who lift the correspondence to the minimal resolution of the quotient singularity $X \\to \\mathbb{C}^2/\\Gamma$. They establish that the tautological bundles on the resolution $X$ provide a $K$-theoretic basis isomorphic to the representation ring $R(\\Gamma)$. Geometrically, this identified the nontrivial irreducible representations with a basis of $H^2(X, \\mathbb{Z})$. Furthermore, the intersection form of these cohomology cycles reproduces the negative Cartan matrix, thereby providing a geometric explanation for the appearance of the Dynkin diagram in the representation theory. A heuristic statement of the McKay correspondence is given below; standard references for the relevant basic topics include \\cite{fulton-harris}, \\cite{griffiths-harris}, and \\cite{G-S}.\n\n\\begin{statement}[McKay correspondence]\nLet $\\Gamma \\subset SU(2$) be a finite subgroup, and let $\\widetilde{ \\mathbb{C}^2/\\Gamma}$ be the minimal resolution of $\\mathbb C^2/\\Gamma$. There is a natural correspondence between the non-trivial irreducible representations of $\\Gamma$ and the irreducible exceptional curves in $\\widetilde{ \\mathbb{C}^2/\\Gamma}$, such that the intersection pairing of these curves is given by the Mckay quiver of $\\Gamma$.\n\nIn the meantime, Kronheimer in his PhD thesis \\cite{kronheimer}, \\cite{kronheimer2}, gives a complete construction of all the $4$-dimensional hyperkähler ALE spaces, showing that each such space is topologically equivalent to the minimal resolution $X$ of the corresponding quotient $\\mathbb{C}^2/\\Gamma$. In Kronheimer's construction, these hyperkähler manifolds are obtained via a finite-dimensional hyperkähler reduction. Kronheimer’s construction provided $X$ with a canonical metric structure, moving the correspondence from the realm of algebraic geometry into the domain of gauge theory and metric analysis. In a subsequent paper studying the moduli space of instantons on ALE spaces \\cite{kronheimer3}, joint with Nakajima, they provide a geometric interpretation of the McKay correspondence via APS-index theory, further linking the McKay correspondence with hyperkähler geometry and gauge theory.\n\nThe setup is as follows: inspired by works of Atiyah-Bott \\cite{atiyah-bott}, Hitchin \\cite{hitchin} and Kirwan \\cite{kirwan}, we study the topology of the moduli space via an $S^1$-invariant Morse-Bott function induced by a Hamiltoninan $S^1$-action on the moduli space. Following Kronheimer's notations in \\cite{kronheimer}, a point in the minimal resolution $X$ can be expressed as a pair of matrices $(\\alpha,\\beta)$ with certain properties, with $\\alpha,\\beta \\in End(R)$ and $R$ being the regular representation of $\\Gamma$. The $S^1$-action on $X$ is simply given by $$e^{i\\varphi}\\cdot(\\alpha,\\beta)=(e^{i\\varphi}\\alpha,e^{i\\varphi}\\beta).$$ The induced $S^1$-invariant Morse-Bott function is equal to the norm square of $(\\alpha,\\beta)$ up to some constant, that is, $$\\Phi(\\alpha,\\beta)=k_0(\\|\\alpha\\|^2+\\|\\beta\\|^2),$$ for some fixed constant $k_0$. The critical points of this $S^1$-invariant Morse-Bott function are precisely the $S^1$-fixed point of the Hamiltoninan $S^1$-action. On the other hand, from the gauge-theoretic construction in \\cite{yan}, we realize each point $(\\alpha,\\beta)$ of an ALE space as a flat connection on a trivial bundle over $S^3/\\Gamma$ with fibers equal to $R$, the regular representation of $\\Gamma$. This yields a holonomy representation $\\rho_{(\\alpha,\\beta)}: \\Gamma \\to GL(R)$, which can be shown to be always isomorphic to $R$. More specifically, the flat connection is given by $$d + \\alpha dz_1 + \\beta dz_2,$$ where $z_1,z_2$ are coordinates in $\\mathbb C^2$. We set $[\\alpha, \\beta] = 0$ to ensure that the above connection is flat, with holonomy representation given by $$\\rho_{(\\alpha,\\beta)}(\\gamma)=R(\\gamma)\\exp((v_1-(uv_1-\\bar{v}v_2))\\alpha+(v_2-(\\bar{u}v_2+vv_1))\\beta),$$ where $(v_1,v_2)$ is the base point in $S^3/\\Gamma$, and $u,v$ are the coordinates of an element $\\gamma\\in\\Gamma$. We show that $\\rho_{(\\alpha,\\beta)}$ is always isomorphic to the regular representation $R$, for any point $(\\alpha,\\beta)$ in $X$.\n\nAlong a gradient flow line $(\\alpha(t),\\beta(t))$ from an index-$2$ critical point to infinity, we obtain a $1$-parameter family of representations $\\rho(t) = \\rho_{(\\alpha(t),\\beta(t))}$. Since $\\rho(t)$ is always isomorphic to the regular representation $R$, we obtain a $1$-parameter family of change of basis matrices $P(t)$ such that $\\rho(t) = P(t) R P(t)^{-1}$. As $H^2(X)$ is generated by the components of the $S^1$-fixed points, we propose the following conjecture, presenting an identification of the generators of $H^2(X)$ with the nontrivial irreducible representations of $\\Gamma$ via the evolution of gradient flow lines, creating an explicit dynamical framework for the McKay correspondence.\n\nAlong a gradient flow line $(\\alpha(t),\\beta(t))$ from an index-$2$ critical point to infinity, we obtain a $1$-parameter family of representations $\\rho(t) = \\rho_{(\\alpha(t),\\beta(t))}$. Since $\\rho(t)$ is always isomorphic to the regular representation $R$, we obtain a $1$-parameter family of change of basis matrices $P(t)$ such that $\\rho(t) = P(t) R P(t)^{-1}$. As $H^2(X)$ is generated by the components of the $S^1$-fixed points, we propose the following conjecture, presenting an identification of the generators of $H^2(X)$ with the nontrivial irreducible representations of $\\Gamma$ via the evolution of gradient flow lines, creating an explicit dynamical framework for the McKay correspondence.\n\nIn Section 6, we provide detailed verification of Conjecture \\ref{conj} in low rank cyclic cases $\\Z_2$, $\\Z_3$. The method is indicative of the general verification for all $A_n$-type ALE spaces, which will be contained in a subsequent version of this paper. The verification of Conjecture \\ref{conj} for nonabelian $\\Gamma$ requires more conceptual inputs as the computation for $S^1$-fixed points becomes cumbersome in those cases. A tentative approach for future work is to realize a gradient flow line as a connection on the ALE space or on $(S^3/\\Gamma) \\times \\mathbb{R}$ in temporal gauge, in light of the geometric interpretation of the McKay correspondence via APS-index theory \\cite{kronheimer3} and the construction of instanton Floer theory \\cite{floer}.\n\nFrom this point on, we fix $\\zeta=(\\zeta_1,0,0)$ to be an element in the good set. Note that this is always possible since the good set is the complement of a codimension-$3$ subset of $\\R^3\\otimes Z$, and by setting $\\zeta_2=\\zeta_3=0$, we can choose $\\zeta_1$ to lie in a codimension-$1$ subset of $Z$ to ensure $\\zeta=(\\zeta_1,0,0)$ lies in the good set. Continuing from the previous section, since $\\zeta_2=\\zeta_3=0$, a solution $\\Theta=(\\alpha,\\beta)$ gives rise to a flat connection $A_\\Theta$ and hence induces a holonomy representation $$\\rho_\\Theta: \\pi_1(S^3/\\Gamma)=\\Gamma \\to GL(R). $$\nWe now compute $\\rho_\\Theta$. First, fix $(v_1,v_2)\\in S^3\\subset \\C^2$ and $\\gamma\\in\\Gamma$, let $$\\tilde{\\gamma}:[0,1]\\to \\C^2$$ be the linear path connecting $(v_1,v_2)$ to $\\gamma\\cdot(v_1,v_2)$. More explicitly, we let $$\\gamma=\\begin{pmatrix}\nu & v \\\\\n-v^* & u^* \n\\end{pmatrix}\\in\\Gamma\\subset SU(2),$$ and we have $\\tilde{\\gamma}(t)=((1-t+tu)v_1-t\\bar{v}v_2,tvv_1+(1-t+t\\bar{u})v_2)$, with $\\gamma$ acting on $\\C^2$ on the right. Now we define the path $\\gamma(t)$ to be the projection of $\\tilde{\\gamma}(t)$ onto $S^3$, that is $$\\gamma(t)=\\frac{\\tilde{\\gamma}(t)}{\\Vert \\tilde{\\gamma}(t) \\Vert}.$$\n\nIt's not hard to see that combinatorially, such a pair $(\\alpha,\\beta)$ is represented by the following diagram: \\begin{center}\n\\includegraphics[width=0.5\\textwidth]{crtpt1}\n\\captionof{figure}{$(\\alpha,\\beta)=((\\alpha_1,...,\\alpha_n),(\\beta_1,...,\\beta_n))$}\n\\label{fig:myfigure}\n\\end{center} This diagram is also known as the extended $A_n$-type Dynkin diagram. Here, the vertices correspond to the irreducible representations of $\\Gamma$, denoted by $\\rho_j$, where $j$ is an index in $\\Z_n$. We have $$(\\alpha,\\beta)=((\\alpha_1,...,\\alpha_n),(\\beta_1,...,\\beta_n)),$$ where $\\alpha_j$, $\\beta_j$ are the edge maps connecting vertices $\\rho_{j-1}$ and $\\rho_j$. We let $\\alpha_j^*$, $\\beta_j^*$ denote the conjugate transposes of $\\alpha_j$ and $\\beta_j$; note that $\\alpha_j^*$, $\\beta_j^*$ reverse the arrows of $\\alpha_j$ and $\\beta_j$.\n\nWhen $\\Gamma$ is an odd cyclic subgroup of $SU(2)$ with $|\\Gamma|=n=2m+1$, $\\widetilde{\\C^2/\\Gamma}$ has $n$ isolated $S^1$-fixed points indexed by $j$, with the property that $\\alpha_j=\\beta_j=0$. It's evident to see that the arrow directions and the edge values are completely determined up to gauge equivalence, once $j$ is fixed. The reason for the existence of some $j$ such that both $\\alpha_j$ and $\\beta_j$ are $0$ is that the restriction of an $S^1$-fixed point. Indeed, in order to be an $S^1$-fixed point, we must have $$f(\\varphi)\\alpha_jf(\\varphi)^{-1}= f_{j-1}(\\varphi)f_{j}^{-1}(\\varphi)\\alpha_j=e^{i\\varphi}\\alpha_j,$$$$f(\\varphi)\\beta_jf(\\varphi)^{-1}=f_{j-1}^{-1}(\\varphi)f_{j}(\\varphi)\\beta_j=e^{i\\varphi}\\beta_j,$$ for all $j$. This condition forces $f(\\varphi)$ to act by weights differing by $1$ at adjacent vertices with the arrow pointing to the lower weight. We see that if the number of vertices is odd, for this condition to hold at all vertices, there must be some j such that $\\alpha_j=\\beta_j=0.$ On the other hand, there can be at most such a $j$ since in order for (5.17) to hold. It's not hard to see that there is a unique global minimal which is given by the critical point with the same number of $\\alpha$-edges and $\\beta$-edges, and hence, it must be the unique index-$0$ critical point. Since $\\Phi$ is perfect, the rest of the critical points must all be of index-$2$.\n\n\\begin{proof}[Proof of Proposition \\ref{prop}]\nWe first prove the proposition for $\\Theta$ an $S^1$-fixed point. Recall that for $\\Theta=(\\alpha,\\beta)$, and $\\gamma=\\begin{pmatrix}\nu & v \\\\\n-v^* & u^* \n\\end{pmatrix},$ we have the following identities: \n\\begin{align}\nR(\\gamma^{-1})\\alpha R(\\gamma) &= u\\alpha + v\\beta, \\\\\nR(\\gamma^{-1})\\beta R(\\gamma) &= -v^*\\alpha + u^*\\beta. \\label{eq:conj-beta}\n\\end{align}\nIt follows from Schur's lemma that $\\alpha$, $\\beta$ will decompose into edge maps along the McKay quiver of $\\Gamma$. Except for when $\\Gamma$ is the trivial group where the proposition holds trivially, the McKay quiver of $\\Gamma$ contains no self loops at any vertices. As a result, $\\alpha$, $\\beta$ both have vanishing diagonal blocks as matrices in $End(R)$, where the blocks are given by the irreducible representations of $\\Gamma$. Now, if $(\\alpha,\\beta)$ is an $S^1$-fixed point, there can be at most a single nonzero edge connecting any two vertices. Recall, the holonomy representation is given by $\\rho_\\Theta=\\rho_{(\\alpha,\\beta)}: \\Gamma \\to U(R)$, $$\\gamma \\mapsto R(\\gamma)\\exp((v_1-(uv_1-\\bar{v}v_2))\\alpha+(v_2-(\\bar{u}v_2+vv_1))\\beta).$$ We now focus on the term $\\exp((v_1-(uv_1-\\bar{v}v_2))\\alpha+(v_2-(\\bar{u}v_2+vv_1))\\beta)$ appearing in the expression. First, we note that a term of the form $\\alpha^k\\beta^\\ell$ can be understood via paths along the directed edges. Since there can be at most a single nonzero edge connecting any two vertices, by following any path defined by edge maps with the prescribed arrows, one can never come back to the same vertex. In particular, $\\alpha$, $\\beta$ must be nilpotent matrices, which is a well-known fact for $S^1$-fixed points. This observation implies that $\\exp((v_1-(uv_1-\\bar{v}v_2))\\alpha+(v_2-(\\bar{u}v_2+vv_1))\\beta)$ also has vanishing diagonal blocks. Hence, $R(\\gamma)\\exp((v_1-(uv_1-\\bar{v}v_2))\\alpha+(v_2-(\\bar{u}v_2+vv_1))\\beta)$ must have the same diagonal blocks as $R(\\gamma)$. This implies that $\\rho_\\Theta$ has the same character as the regular representation $R$ which means they are isomorphic.\n\n\\begin{conj}[c.f. Conjecture \\ref{conj}]\nLet $\\Gamma$ be a finite subgroup of $SU(2)$. Let $X_\\zeta$ be as in Theorem \\ref{thm}. Let $S_1$,...$S_r$ be the components of $S^1$-fixed points of $X_\\zeta$ generating $H^2(X_\\zeta)$, and let $\\rho_j(t)=P_j(t)RP_j(t)^{-1}, 1\\leq j\\leq r$ be the $1$-parameter family of holonomy representations with the property that $$\\rho_j(t)=\\rho_{(\\alpha_j(t),\\beta_j(t))},$$ $$\\lim_{t\\to-\\infty}(\\alpha_j(t),\\beta_j(t))=(\\alpha_j,\\beta_j),$$ with $(\\alpha_j,\\beta_j)\\in S_j$. Then $$P_j^{\\lim}=\\lim_{t\\to\\infty}P_j(t)$$ is equal to the projector onto the nontrivial irreducible representation $\\rho_j$ of $\\Gamma$, and every nontrivial irreducible representation of $\\Gamma$ arises this way. \n\\end{conj}", "post_theorem_intro_text_len": 1868, "post_theorem_intro_text": "In Section 6, we provide detailed verification of Conjecture \\ref{conj} in low rank cyclic cases $\\Z_2$, $\\Z_3$. The method is indicative of the general verification for all $A_n$-type ALE spaces, which will be contained in a subsequent version of this paper. The verification of Conjecture \\ref{conj} for nonabelian $\\Gamma$ requires more conceptual inputs as the computation for $S^1$-fixed points becomes cumbersome in those cases. A tentative approach for future work is to realize a gradient flow line as a connection on the ALE space or on $(S^3/\\Gamma) \\times \\mathbb{R}$ in temporal gauge, in light of the geometric interpretation of the McKay correspondence via APS-index theory \\cite{kronheimer3} and the construction of instanton Floer theory \\cite{floer}.\n\nAnother direction for future work is to compare this framework constructed from gauge theory and Morse theory to the symplectic and contact geometric interpretation given in \\cite{M-R} and \\cite{D-N}. The robust feature of the geometry present in the McKay correspondence might potentially be an orifice to exploring structural parallel between symplectic field theory and gauge theory, echoing a long lasting endeavor to relate the two subfields. \n\nThe organization of the paper is the following. Section 2 reviews Kronoheimer's construction of $4$-dimensional hyperkähler ALE spaces as finite-dimensional hyperkähler quotients and the gauge-theoretic interpretation of Kronoheimer's construction. Section 3 and Section 4 develop the main setup for this paper where we construct a flat connection from any point of an ALE space and calculate its holonomy representation. Section 5 formally introduces the $S^1$-action and the Morse-Bott function, and analyzes the properties of the Morse-Bott function and its critical points in detail. Section 6 proves Conjecture \\ref{conj} for cyclic cases.", "sketch": "In Section 6, the authors \"provide detailed verification of Conjecture \\ref{conj} in low rank cyclic cases $\\Z_2$, $\\Z_3$\"; this method is said to be \"indicative of the general verification for all $A_n$-type ALE spaces\" (to appear later). For nonabelian $\\Gamma$, they note verification \"requires more conceptual inputs\" since \"the computation for $S^1$-fixed points becomes cumbersome\"; a proposed approach is \"to realize a gradient flow line as a connection on the ALE space or on $(S^3/\\Gamma) \\times \\mathbb{R}$ in temporal gauge\", motivated by \"the geometric interpretation of the McKay correspondence via APS-index theory\" and \"the construction of instanton Floer theory.\"", "expanded_sketch": "Next, the authors “provide detailed verification of \\begin{conj}\\label{conj}\nLet $\\Gamma$ be a finite subgroup of $SU(2)$. Let $X$ be the minimal resolution of $\\C^2/\\Gamma$. Let $S_1$,...$S_r$ be the components of $S^1$-fixed points of $X$ generating $H^2(X)$, and let $\\rho_j(t)=P_j(t)RP_j(t)^{-1}, 1\\leq j\\leq r$ be the $1$-parameter family of holonomy representations with the property that $$\\rho_j(t)=\\rho_{(\\alpha_j(t),\\beta_j(t))},$$ $$\\lim_{t\\to-\\infty}(\\alpha_j(t),\\beta_j(t))=(\\alpha_j,\\beta_j),$$ with $(\\alpha_j,\\beta_j)\\in S_j$. Then $$P_j^{\\lim}=\\lim_{t\\to\\infty}P_j(t)$$ is equal to the projector onto the nontrivial irreducible representation $\\rho_j$ of $\\Gamma$, and every nontrivial irreducible representation of $\\Gamma$ arises this way. \n\\end{conj}\nin low rank cyclic cases $\\Z_2$, $\\Z_3$; this method is said to be “indicative of the general verification for all $A_n$-type ALE spaces” (to appear later). For nonabelian $\\Gamma$, they note verification “requires more conceptual inputs” since “the computation for $S^1$-fixed points becomes cumbersome”; a proposed approach is “to realize a gradient flow line as a connection on the ALE space or on $(S^3/\\Gamma) \\times \\mathbb{R}$ in temporal gauge”, motivated by “the geometric interpretation of the McKay correspondence via APS-index theory” and “the construction of instanton Floer theory.”", "expanded_theorem": "\\label{conj}\nLet $\\Gamma$ be a finite subgroup of $SU(2)$. Let $X$ be the minimal resolution of $\\mathbb C^2/\\Gamma$. Let $S_1$,...$S_r$ be the components of $S^1$-fixed points of $X$ generating $H^2(X)$, and let $\\rho_j(t)=P_j(t)RP_j(t)^{-1}, 1\\leq j\\leq r$ be the $1$-parameter family of holonomy representations with the property that $$\\rho_j(t)=\\rho_{(\\alpha_j(t),\\beta_j(t))},$$ $$\\lim_{t\\to-\\infty}(\\alpha_j(t),\\beta_j(t))=(\\alpha_j,\\beta_j),$$ with $(\\alpha_j,\\beta_j)\\in S_j$. Then $$P_j^{\\lim}=\\lim_{t\\to\\infty}P_j(t)$$ is equal to the projector onto the nontrivial irreducible representation $\\rho_j$ of $\\Gamma$, and every nontrivial irreducible representation of $\\Gamma$ arises this way.,", "theorem_type": ["Universal", "Classification or Bijection"], "mcq": {"question": "Let $\\Gamma\\subset SU(2)$ be a finite subgroup, let $X$ be the minimal resolution of $\\mathbb C^2/\\Gamma$, and let $S_1,\\dots,S_r$ be the components of the $S^1$-fixed-point set of $X$ that generate $H^2(X)$. For each $j$, suppose there is a one-parameter family of holonomy representations on the regular representation $R$ of $\\Gamma$ of the form\n$$\\rho_j(t)=\\rho_{(\\alpha_j(t),\\beta_j(t))}=P_j(t)RP_j(t)^{-1},$$\nwith\n$$\\lim_{t\\to-\\infty}(\\alpha_j(t),\\beta_j(t))=(\\alpha_j,\\beta_j)\\in S_j.$$ \nIf the limit\n$$P_j^{\\lim}:=\\lim_{t\\to\\infty}P_j(t)$$\nexists, which representations of $\\Gamma$ are obtained in this way?", "correct_choice": {"label": "A", "text": "Exactly the nontrivial irreducible representations of $\\Gamma$: for each $j$, the limit $P_j^{\\lim}$ is the projector onto a nontrivial irreducible representation of $\\Gamma$, and every nontrivial irreducible representation of $\\Gamma$ arises from some $P_j^{\\lim}$ in this manner."}, "choices": [{"label": "B", "text": "Exactly the irreducible representations of $\\Gamma$, including the trivial representation: for each $j$, the limit $P_j^{\\lim}$ is the projector onto an irreducible representation of $\\Gamma$, and every irreducible representation of $\\Gamma$ arises from some $P_j^{\\lim}$ in this manner."}, {"label": "C", "text": "Only nontrivial irreducible representations of $\\Gamma$ can occur: for each $j$, if $P_j^{\\lim}$ exists, then it is the projector onto a nontrivial irreducible representation of $\\Gamma$."}, {"label": "D", "text": "Exactly the nontrivial irreducible representations of $\\Gamma$ are obtained, and this identification is uniform in the sense that for each nontrivial irreducible representation $\\rho$ of $\\Gamma$ there is a unique index $j$ with $P_j^{\\lim}$ equal to the projector onto $\\rho$."}, {"label": "E", "text": "Exactly the nontrivial isotypic summands of the regular representation $R$ are obtained: for each $j$, the limit $P_j^{\\lim}$ is the projector onto the full nontrivial isotypic component corresponding to some irreducible representation of $\\Gamma$, and every such nontrivial isotypic component arises in this manner."}], "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": "exclusion_of_trivial_representation", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "surjectivity_onto_all_nontrivial_irreducibles", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "computational_check", "tampered_component": "existence_without_uniqueness", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "irreducible_projector_vs_isotypic_projector", "template_used": "wildcard"}]}} +{"id": "2601.08461v1", "paper_link": "http://arxiv.org/abs/2601.08461v1", "theorems_cnt": 1, "theorem": {"env_name": "conjecture", "content": "~\\cite{raayoni2021generating}~\n\\begin{equation}\\label{eq:target_intro}\n-\\frac{\\pi}{4} = \\cfrac{1}{-1 + \\cfrac{1}{-4 + \\cfrac{-2}{-7 + \\cfrac{-9}{-10 + \\cfrac{-20}{-13 + \\dots}}}}}\n\\end{equation}", "start_pos": 2549, "end_pos": 2778, "label": "eq:target_intro"}, "ref_dict": {"eq:target_intro": "\\begin{equation}\\label{eq:target_intro}\n-\\frac{\\pi}{4} = \\cfrac{1}{-1 + \\cfrac{1}{-4 + \\cfrac{-2}{-7 + \\cfrac{-9}{-10 + \\cfrac{-20}{-13 + \\dots}}}}}\n\\end{equation}"}, "pre_theorem_intro_text_len": 705, "pre_theorem_intro_text": "The representation of fundamental transcendental constants through generalized continued fractions $\\K_{n=1}^{\\infty}(a_n/b_n)$ has remained a cornerstone of classical analysis since the foundational work of Euler and Gauss. Traditionally, such identities are derived from the analytic theory of hypergeometric functions or the spectral theory of discrete operators. Recently, however, algorithmic induction frameworks—most notably the Ramanujan Machine~\\cite{raayoni2021generating}—have introduced a class of non-canonical conjectures that lack immediate analytic origins.\n\nIn this paper, we establish a formal proof for a striking identity concerning $-\\pi/4$ discovered through these heuristic methods:", "context": "The representation of fundamental transcendental constants through generalized continued fractions $\\K_{n=1}^{\\infty}(a_n/b_n)$ has remained a cornerstone of classical analysis since the foundational work of Euler and Gauss. Traditionally, such identities are derived from the analytic theory of hypergeometric functions or the spectral theory of discrete operators. Recently, however, algorithmic induction frameworks—most notably the Ramanujan Machine~\\cite{raayoni2021generating}—have introduced a class of non-canonical conjectures that lack immediate analytic origins.\n\nIn this paper, we establish a formal proof for a striking identity concerning $-\\pi/4$ discovered through these heuristic methods:", "full_context": "The representation of fundamental transcendental constants through generalized continued fractions $\\K_{n=1}^{\\infty}(a_n/b_n)$ has remained a cornerstone of classical analysis since the foundational work of Euler and Gauss. Traditionally, such identities are derived from the analytic theory of hypergeometric functions or the spectral theory of discrete operators. Recently, however, algorithmic induction frameworks—most notably the Ramanujan Machine~\\cite{raayoni2021generating}—have introduced a class of non-canonical conjectures that lack immediate analytic origins.\n\nIn this paper, we establish a formal proof for a striking identity concerning $-\\pi/4$ discovered through these heuristic methods:\n\n\\begin{abstract}\nWe provide a formal analytic proof for a class of non-canonical polynomial continued fractions representing $-\\pi/4$, originally conjectured by the Ramanujan Machine using algorithmic induction~\\cite{raayoni2021generating}. \nBy establishing an explicit correspondence with the ratio of contiguous Gaussian hypergeometric functions ${}_2F_1(a, b; c; z)$, we show that these identities can be derived via a discrete sequence of equivalence transformations. We further prove that the conjectured integer coefficients constitute a symbolically minimal realization of the underlying analytic kernel. Stability analysis confirms that the resulting limit-periodic structures reside strictly within the Worpitzky convergence disk, ensuring absolute convergence. This work demonstrates that such algorithmically discovered identities are not isolated numerical artifacts, but are deeply rooted in the classical theory of hypergeometric transformations.\n\\end{abstract}\n\nThe representation of fundamental transcendental constants through generalized continued fractions $\\K_{n=1}^{\\infty}(a_n/b_n)$ has remained a cornerstone of classical analysis since the foundational work of Euler and Gauss. Traditionally, such identities are derived from the analytic theory of hypergeometric functions or the spectral theory of discrete operators. Recently, however, algorithmic induction frameworks—most notably the Ramanujan Machine~\\cite{raayoni2021generating}—have introduced a class of non-canonical conjectures that lack immediate analytic origins.\n\nThe analytic significance of \\eqref{eq:target_intro} lies in its departure from unit-denominator Gaussian forms. The identity involves a transition from rational coefficients typically found in hypergeometric ratios to a symbolically sparse representation with integer-based polynomials. This transition poses a non-trivial challenge in verification: one must prove that the sequence $\\{a_n, b_n\\}$ functions as an asymptotically equivalent attractor to a specific hypergeometric kernel.\n\n\\subsection{Phase 1: Derivation of the Hypergeometric Kernel}\nThe analytic foundation of the conjectured identity lies in the ratio of contiguous Gauss hypergeometric functions, defined as:\n\\begin{equation}\n\\mathcal{R}(a, b, c; z) := \\frac{{}_2F_1(a, b+1; c+1; z)}{{}_2F_1(a, b; c; z)}\n\\end{equation}\nwhere the hypergeometric function ${}_2F_1$ is given by the classic power series representation:\n\\begin{equation}\n{}_2F_1(a, b; c; z) = \\sum_{n=0}^{\\infty} \\frac{(a)_n (b)_n}{(c)_n} \\frac{z^n}{n!}, \\quad |z| < 1\n\\end{equation}\n\nand its analytic continuation elsewhere. According to the foundational theory of Gauss \\cite{gauss1813disquisitiones}, this ratio admits a generalized continued fraction expansion:\n\\begin{equation}\\label{eq:gauss_fundamental}\n\\mathcal{R}(a, b, c; z) = \\frac{1}{1 + \\K_{n=1}^{\\infty} \\left( \\frac{d_n z}{1} \\right)} = \\cfrac{1}{1 + \\cfrac{d_1 z}{1 + \\cfrac{d_2 z}{1 + \\dots}}}\n\\end{equation}\nThe partial coefficients $\\{d_n\\}$ follow the alternating laws:\n\\begin{equation}\nd_{2k} = \\frac{(b+k)(c-a+k)}{(c+2k-1)(c+2k)}, \\quad d_{2k+1} = \\frac{(a+k)(c-b+k)}{(c+2k)(c+2k+1)}\n\\end{equation}\nSubstituting the specific parameter set $(a, b, c) = (1/2, 0, 1/2)$, the coefficients simplify as follows:\n\\begin{itemize}\n \\item For $n=2k$: \n \\begin{equation}\n d_{2k} = \\frac{k(1/2-1/2+k)}{(1/2+2k-1)(1/2+2k)} = \\frac{k^2}{(2k-1/2)(2k+1/2)} = \\frac{(2k)^2}{4(2k)^2-1}\n \\end{equation}\n \\item For $n=2k+1$: \n \\begin{equation}\n d_{2k+1} = \\frac{(1/2+k)(1/2-0+k)}{(1/2+2k)(1/2+2k+1)} = \\frac{(k+1/2)^2}{(2k+1/2)(2k+3/2)} = \\frac{(2k+1)^2}{4(2k+1)^2-1}\n \\end{equation}\n\\end{itemize}\nThus, $d_n = \\frac{n^2}{4n^2-1}$ for all $n \\in \\mathbb{N}$. To evaluate the kernel at the boundary $z=-1$, we observe that the denominator ${}_2F_1(1/2, 0; 1/2; -1)$ reduces to unity. The numerator evaluates to:\n\\begin{equation}\n{}_2F_1\\left(\\frac{1}{2}, 1; \\frac{3}{2}; -1\\right) = \\sum_{n=0}^{\\infty} \\frac{(\\frac{1}{2})_n (1)_n}{(\\frac{3}{2})_n} \\frac{(-1)^n}{n!} = \\sum_{n=0}^{\\infty} \\frac{(-1)^n}{2n+1} = \\frac{\\pi}{4}\n\\end{equation}\nThis establishes the transcendental evaluation $\\mathcal{R}(1/2, 0, 1/2; -1) = \\pi/4$, yielding the unit-denominator Gaussian numerators $a_n^* = -d_{n-1} = \\frac{-(n-1)^2}{4(n-1)^2-1}$ for $n \\ge 2$.\n\nAccording to the fundamental theory of continued fractions \\cite{wall1948analytic}, a scaling sequence $\\{r_n\\}_{n=0}^{\\infty}$ (with $r_n \\neq 0$ and $r_0=1$) maps the standard pair $(a_n^*, b_n^*)$ to a new set $(\\tilde{a}_n, \\tilde{b}_n)$ via the following relations:\n\\begin{equation}\\label{eq:equiv_trans}\n\\tilde{b}_n = r_n b_n^*, \\quad \\tilde{a}_n = r_n r_{n-1} a_n^*, \\quad n \\ge 1\n\\end{equation}\nThe quadratic dependence of the numerator $\\tilde{a}_n$ on the scaling factors $(r_n, r_{n-1})$ originates from the recursive nesting of the fraction. Specifically, a scaling of the $(n-1)$-th denominator by $r_{n-1}$ propagates a factor into the $n$-th level numerator, which must then be balanced by the $n$-th scaling factor $r_n$ to satisfy the required value of $\\tilde{b}_n$.\n\nSubstituting the specific forms $r_n = -(3n-2)$, $r_{n-1} = -(3n-5)$, and the Gaussian coefficients $a_n^* = -d_{n-1} = -(n-1)^2 / (4(n-1)^2 - 1)=-(n-1)^2 / (2n-3)(2n-1)$, we perform the following substitution for $n \\ge 2$:\n\\begin{align}\\label{eq:exact_an_detail}\n\\tilde{a}_n &= [-(3n-2)] \\cdot [-(3n-5)] \\cdot \\left( \\frac{-(n-1)^2}{(2n-3)(2n-1)} \\right) \\nonumber \\\\\n&= - \\frac{(3n-2)(3n-5)(n-1)^2}{(2n-3)(2n-1)}\n\\end{align}\nThe first few terms of this exact sequence are:\n\\begin{itemize}\n \\item \\textbf{Term $n=2$}: $\\tilde{a}_2 = - \\frac{(4)(1)(1^2)}{1 \\cdot 3} = -\\frac{4}{3}$\n \\item \\textbf{Term $n=3$}: $\\tilde{a}_3 = - \\frac{(7)(4)(4)}{3 \\cdot 5} = -\\frac{112}{15}$\n\\end{itemize}\nThe sequence $\\tilde{a}_n$ establishes the rigorous analytic baseline for the continued fraction. Although these terms are rational functions of $n$, the conjecture employs an \\textit{arithmetically sparse} representation $a_n = -(n-1)(2n-5)$. Phase 3 demonstrates that this substitution constitutes a vanishing perturbation in the transformation space; specifically, both sequences share the same asymptotic limit within the Worpitzky convergence disk, thereby ensuring that the transcendental value $-\\pi/4$ remains invariant under this symbolic simplification\n\n\\subsection{The Convergence Factor}\nThe geometric decay factor $\\sigma$, defined as the ratio of consecutive errors $|f - f_n|$, is calculated via the characteristic root of the limit-periodic tail \\cite{lorentzen1992continued}:\n\\begin{equation}\n\\sigma = \\left| \\frac{1 - \\sqrt{1 - 4|L|}}{1 + \\sqrt{1 - 4|L|}} \\right| = \\frac{1 - 1/3}{1 + 1/3} = \\frac{1}{2}\n\\end{equation}\nThis confirms that the identity provides approximately 3.01 decimal digits of precision per 10 iterations.\n\n\\begin{equation}\\label{eq:target_intro}\n-\\frac{\\pi}{4} = \\cfrac{1}{-1 + \\cfrac{1}{-4 + \\cfrac{-2}{-7 + \\cfrac{-9}{-10 + \\cfrac{-20}{-13 + \\dots}}}}}\n\\end{equation}", "post_theorem_intro_text_len": 1604, "post_theorem_intro_text": "More formally, conjecture~\\eqref{eq:target_intro} corresponds to the limit of the convergents of $b_0 + \\K_{n=1}^{\\infty}(a_n/b_n)$, where $b_0 = 0$, $a_1 = 1$, and for $n \\ge 1$, the partial quotients are defined by the linear denominator sequence $b_n = -(3n-2)$ and the piecewise quadratic numerator sequence:\n\\begin{equation}\\label{eq:an_def}\na_n = \n\\begin{cases} \n1, & n = 1, 2 \\\\\n-(n-1)(2n-5), & n \\ge 3\n\\end{cases}\n\\end{equation}\n\nThe analytic significance of \\eqref{eq:target_intro} lies in its departure from unit-denominator Gaussian forms. The identity involves a transition from rational coefficients typically found in hypergeometric ratios to a symbolically sparse representation with integer-based polynomials. This transition poses a non-trivial challenge in verification: one must prove that the sequence $\\{a_n, b_n\\}$ functions as an asymptotically equivalent attractor to a specific hypergeometric kernel.\n\nOur derivation proceeds by identifying \\eqref{eq:target_intro} as the result of a discrete equivalence transformation applied to the Gaussian ratio $\\mathcal{R}(1/2, 0, 1/2; -1)$. We demonstrate that the transformation sequence $\\{r_n\\}$ not only forces the linear growth of the denominators but also facilitates a polynomial regularization of the partial numerators. By analyzing the limit-periodicity of the Worpitzky parameter $\\rho_n$, we prove that the identity converges absolutely to the desired branch of the arctangent function. This work thus reconciles the experimental results of algorithmic induction with the deductive rigor of analytic continued fraction theory.", "sketch": "To verify \\eqref{eq:target_intro}, “one must prove that the sequence $\\{a_n, b_n\\}$ functions as an asymptotically equivalent attractor to a specific hypergeometric kernel.” The derivation “proceeds by identifying \\eqref{eq:target_intro} as the result of a discrete equivalence transformation applied to the Gaussian ratio $\\mathcal{R}(1/2, 0, 1/2; -1)$.” One then “demonstrate[s] that the transformation sequence $\\{r_n\\}$ not only forces the linear growth of the denominators but also facilitates a polynomial regularization of the partial numerators.” Finally, “by analyzing the limit-periodicity of the Worpitzky parameter $\\rho_n$, we prove that the identity converges absolutely to the desired branch of the arctangent function.”", "expanded_sketch": "To verify the main identity, “one must prove that the sequence $\\{a_n, b_n\\}$ functions as an asymptotically equivalent attractor to a specific hypergeometric kernel.” The derivation “proceeds by identifying the main identity as the result of a discrete equivalence transformation applied to the Gaussian ratio $\\mathcal{R}(1/2, 0, 1/2; -1)$.” One then “demonstrate[s] that the transformation sequence $\\{r_n\\}$ not only forces the linear growth of the denominators but also facilitates a polynomial regularization of the partial numerators.” Finally, “by analyzing the limit-periodicity of the Worpitzky parameter $\\rho_n$, we prove that the identity converges absolutely to the desired branch of the arctangent function.”", "expanded_theorem": "~\\cite{raayoni2021generating}~\n\\begin{equation}\\label{eq:target_intro}\n-\\frac{\\pi}{4} = \\cfrac{1}{-1 + \\cfrac{1}{-4 + \\cfrac{-2}{-7 + \\cfrac{-9}{-10 + \\cfrac{-20}{-13 + \\dots}}}}}\n\\end{equation},", "theorem_type": ["Existence", "Equality or Identity"], "mcq": {"question": "Which explicit identity is satisfied by the generalized continued fraction \\[\\cfrac{1}{-1 + \\cfrac{1}{-4 + \\cfrac{-2}{-7 + \\cfrac{-9}{-10 + \\cfrac{-20}{-13 + \\dots}}}}}\\,?\\]", "correct_choice": {"label": "A", "text": "It converges to \\(-\\pi/4\\); equivalently, \\[ -\\frac{\\pi}{4} = \\cfrac{1}{-1 + \\cfrac{1}{-4 + \\cfrac{-2}{-7 + \\cfrac{-9}{-10 + \\cfrac{-20}{-13 + \\dots}}}}}. \\]"}, "choices": [{"label": "B", "text": "It converges to \\(\\pi/4\\); equivalently, \\[ \\frac{\\pi}{4} = \\cfrac{1}{-1 + \\cfrac{1}{-4 + \\cfrac{-2}{-7 + \\cfrac{-9}{-10 + \\cfrac{-20}{-13 + \\dots}}}}}. \\]"}, {"label": "C", "text": "It converges absolutely to a finite real value."}, {"label": "D", "text": "It converges to \\(-\\arctan(1/2)\\); equivalently, \\[ -\\arctan\\!\\left(\\frac12\\right) = \\cfrac{1}{-1 + \\cfrac{1}{-4 + \\cfrac{-2}{-7 + \\cfrac{-9}{-10 + \\cfrac{-20}{-13 + \\dots}}}}}. \\]"}, {"label": "E", "text": "It converges to \\(-\\pi/4\\) only conditionally, since the continued fraction lies on the boundary of the Worpitzky convergence disk; equivalently, \\[ -\\frac{\\pi}{4} = \\cfrac{1}{-1 + \\cfrac{1}{-4 + \\cfrac{-2}{-7 + \\cfrac{-9}{-10 + \\cfrac{-20}{-13 + \\dots}}}}}, \\quad \\text{but the convergence is not absolute}. \\]"}], "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": "desired branch/sign from the hypergeometric kernel evaluation", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "drops the explicit evaluation \\(-\\pi/4\\) and retains only absolute convergence to a real limit", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "misidentifies the arctangent branch/value produced by the kernel", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "strictly within Worpitzky disk versus merely on its boundary; absolute versus conditional convergence", "template_used": "boundary_range"}]}} +{"id": "2601.08461v1", "paper_link": "http://arxiv.org/abs/2601.08461v1", "theorems_cnt": 1, "theorem": {"env_name": "conjecture", "content": "~\\cite{raayoni2021generating}~\n\\begin{equation}\\label{eq:target_intro}\n-\\frac{\\pi}{4} = \\cfrac{1}{-1 + \\cfrac{1}{-4 + \\cfrac{-2}{-7 + \\cfrac{-9}{-10 + \\cfrac{-20}{-13 + \\dots}}}}}\n\\end{equation}", "start_pos": 2549, "end_pos": 2778, "label": "eq:target_intro"}, "ref_dict": {"eq:target_intro": "\\begin{equation}\\label{eq:target_intro}\n-\\frac{\\pi}{4} = \\cfrac{1}{-1 + \\cfrac{1}{-4 + \\cfrac{-2}{-7 + \\cfrac{-9}{-10 + \\cfrac{-20}{-13 + \\dots}}}}}\n\\end{equation}"}, "pre_theorem_intro_text_len": 705, "pre_theorem_intro_text": "The representation of fundamental transcendental constants through generalized continued fractions $\\K_{n=1}^{\\infty}(a_n/b_n)$ has remained a cornerstone of classical analysis since the foundational work of Euler and Gauss. Traditionally, such identities are derived from the analytic theory of hypergeometric functions or the spectral theory of discrete operators. Recently, however, algorithmic induction frameworks—most notably the Ramanujan Machine~\\cite{raayoni2021generating}—have introduced a class of non-canonical conjectures that lack immediate analytic origins.\n\nIn this paper, we establish a formal proof for a striking identity concerning $-\\pi/4$ discovered through these heuristic methods:", "context": "The representation of fundamental transcendental constants through generalized continued fractions $\\K_{n=1}^{\\infty}(a_n/b_n)$ has remained a cornerstone of classical analysis since the foundational work of Euler and Gauss. Traditionally, such identities are derived from the analytic theory of hypergeometric functions or the spectral theory of discrete operators. Recently, however, algorithmic induction frameworks—most notably the Ramanujan Machine~\\cite{raayoni2021generating}—have introduced a class of non-canonical conjectures that lack immediate analytic origins.\n\nIn this paper, we establish a formal proof for a striking identity concerning $-\\pi/4$ discovered through these heuristic methods:", "full_context": "The representation of fundamental transcendental constants through generalized continued fractions $\\K_{n=1}^{\\infty}(a_n/b_n)$ has remained a cornerstone of classical analysis since the foundational work of Euler and Gauss. Traditionally, such identities are derived from the analytic theory of hypergeometric functions or the spectral theory of discrete operators. Recently, however, algorithmic induction frameworks—most notably the Ramanujan Machine~\\cite{raayoni2021generating}—have introduced a class of non-canonical conjectures that lack immediate analytic origins.\n\nIn this paper, we establish a formal proof for a striking identity concerning $-\\pi/4$ discovered through these heuristic methods:\n\n\\begin{abstract}\nWe provide a formal analytic proof for a class of non-canonical polynomial continued fractions representing $-\\pi/4$, originally conjectured by the Ramanujan Machine using algorithmic induction~\\cite{raayoni2021generating}. \nBy establishing an explicit correspondence with the ratio of contiguous Gaussian hypergeometric functions ${}_2F_1(a, b; c; z)$, we show that these identities can be derived via a discrete sequence of equivalence transformations. We further prove that the conjectured integer coefficients constitute a symbolically minimal realization of the underlying analytic kernel. Stability analysis confirms that the resulting limit-periodic structures reside strictly within the Worpitzky convergence disk, ensuring absolute convergence. This work demonstrates that such algorithmically discovered identities are not isolated numerical artifacts, but are deeply rooted in the classical theory of hypergeometric transformations.\n\\end{abstract}\n\nThe representation of fundamental transcendental constants through generalized continued fractions $\\K_{n=1}^{\\infty}(a_n/b_n)$ has remained a cornerstone of classical analysis since the foundational work of Euler and Gauss. Traditionally, such identities are derived from the analytic theory of hypergeometric functions or the spectral theory of discrete operators. Recently, however, algorithmic induction frameworks—most notably the Ramanujan Machine~\\cite{raayoni2021generating}—have introduced a class of non-canonical conjectures that lack immediate analytic origins.\n\nThe analytic significance of \\eqref{eq:target_intro} lies in its departure from unit-denominator Gaussian forms. The identity involves a transition from rational coefficients typically found in hypergeometric ratios to a symbolically sparse representation with integer-based polynomials. This transition poses a non-trivial challenge in verification: one must prove that the sequence $\\{a_n, b_n\\}$ functions as an asymptotically equivalent attractor to a specific hypergeometric kernel.\n\n\\subsection{Phase 1: Derivation of the Hypergeometric Kernel}\nThe analytic foundation of the conjectured identity lies in the ratio of contiguous Gauss hypergeometric functions, defined as:\n\\begin{equation}\n\\mathcal{R}(a, b, c; z) := \\frac{{}_2F_1(a, b+1; c+1; z)}{{}_2F_1(a, b; c; z)}\n\\end{equation}\nwhere the hypergeometric function ${}_2F_1$ is given by the classic power series representation:\n\\begin{equation}\n{}_2F_1(a, b; c; z) = \\sum_{n=0}^{\\infty} \\frac{(a)_n (b)_n}{(c)_n} \\frac{z^n}{n!}, \\quad |z| < 1\n\\end{equation}\n\nand its analytic continuation elsewhere. According to the foundational theory of Gauss \\cite{gauss1813disquisitiones}, this ratio admits a generalized continued fraction expansion:\n\\begin{equation}\\label{eq:gauss_fundamental}\n\\mathcal{R}(a, b, c; z) = \\frac{1}{1 + \\K_{n=1}^{\\infty} \\left( \\frac{d_n z}{1} \\right)} = \\cfrac{1}{1 + \\cfrac{d_1 z}{1 + \\cfrac{d_2 z}{1 + \\dots}}}\n\\end{equation}\nThe partial coefficients $\\{d_n\\}$ follow the alternating laws:\n\\begin{equation}\nd_{2k} = \\frac{(b+k)(c-a+k)}{(c+2k-1)(c+2k)}, \\quad d_{2k+1} = \\frac{(a+k)(c-b+k)}{(c+2k)(c+2k+1)}\n\\end{equation}\nSubstituting the specific parameter set $(a, b, c) = (1/2, 0, 1/2)$, the coefficients simplify as follows:\n\\begin{itemize}\n \\item For $n=2k$: \n \\begin{equation}\n d_{2k} = \\frac{k(1/2-1/2+k)}{(1/2+2k-1)(1/2+2k)} = \\frac{k^2}{(2k-1/2)(2k+1/2)} = \\frac{(2k)^2}{4(2k)^2-1}\n \\end{equation}\n \\item For $n=2k+1$: \n \\begin{equation}\n d_{2k+1} = \\frac{(1/2+k)(1/2-0+k)}{(1/2+2k)(1/2+2k+1)} = \\frac{(k+1/2)^2}{(2k+1/2)(2k+3/2)} = \\frac{(2k+1)^2}{4(2k+1)^2-1}\n \\end{equation}\n\\end{itemize}\nThus, $d_n = \\frac{n^2}{4n^2-1}$ for all $n \\in \\mathbb{N}$. To evaluate the kernel at the boundary $z=-1$, we observe that the denominator ${}_2F_1(1/2, 0; 1/2; -1)$ reduces to unity. The numerator evaluates to:\n\\begin{equation}\n{}_2F_1\\left(\\frac{1}{2}, 1; \\frac{3}{2}; -1\\right) = \\sum_{n=0}^{\\infty} \\frac{(\\frac{1}{2})_n (1)_n}{(\\frac{3}{2})_n} \\frac{(-1)^n}{n!} = \\sum_{n=0}^{\\infty} \\frac{(-1)^n}{2n+1} = \\frac{\\pi}{4}\n\\end{equation}\nThis establishes the transcendental evaluation $\\mathcal{R}(1/2, 0, 1/2; -1) = \\pi/4$, yielding the unit-denominator Gaussian numerators $a_n^* = -d_{n-1} = \\frac{-(n-1)^2}{4(n-1)^2-1}$ for $n \\ge 2$.\n\nAccording to the fundamental theory of continued fractions \\cite{wall1948analytic}, a scaling sequence $\\{r_n\\}_{n=0}^{\\infty}$ (with $r_n \\neq 0$ and $r_0=1$) maps the standard pair $(a_n^*, b_n^*)$ to a new set $(\\tilde{a}_n, \\tilde{b}_n)$ via the following relations:\n\\begin{equation}\\label{eq:equiv_trans}\n\\tilde{b}_n = r_n b_n^*, \\quad \\tilde{a}_n = r_n r_{n-1} a_n^*, \\quad n \\ge 1\n\\end{equation}\nThe quadratic dependence of the numerator $\\tilde{a}_n$ on the scaling factors $(r_n, r_{n-1})$ originates from the recursive nesting of the fraction. Specifically, a scaling of the $(n-1)$-th denominator by $r_{n-1}$ propagates a factor into the $n$-th level numerator, which must then be balanced by the $n$-th scaling factor $r_n$ to satisfy the required value of $\\tilde{b}_n$.\n\nSubstituting the specific forms $r_n = -(3n-2)$, $r_{n-1} = -(3n-5)$, and the Gaussian coefficients $a_n^* = -d_{n-1} = -(n-1)^2 / (4(n-1)^2 - 1)=-(n-1)^2 / (2n-3)(2n-1)$, we perform the following substitution for $n \\ge 2$:\n\\begin{align}\\label{eq:exact_an_detail}\n\\tilde{a}_n &= [-(3n-2)] \\cdot [-(3n-5)] \\cdot \\left( \\frac{-(n-1)^2}{(2n-3)(2n-1)} \\right) \\nonumber \\\\\n&= - \\frac{(3n-2)(3n-5)(n-1)^2}{(2n-3)(2n-1)}\n\\end{align}\nThe first few terms of this exact sequence are:\n\\begin{itemize}\n \\item \\textbf{Term $n=2$}: $\\tilde{a}_2 = - \\frac{(4)(1)(1^2)}{1 \\cdot 3} = -\\frac{4}{3}$\n \\item \\textbf{Term $n=3$}: $\\tilde{a}_3 = - \\frac{(7)(4)(4)}{3 \\cdot 5} = -\\frac{112}{15}$\n\\end{itemize}\nThe sequence $\\tilde{a}_n$ establishes the rigorous analytic baseline for the continued fraction. Although these terms are rational functions of $n$, the conjecture employs an \\textit{arithmetically sparse} representation $a_n = -(n-1)(2n-5)$. Phase 3 demonstrates that this substitution constitutes a vanishing perturbation in the transformation space; specifically, both sequences share the same asymptotic limit within the Worpitzky convergence disk, thereby ensuring that the transcendental value $-\\pi/4$ remains invariant under this symbolic simplification\n\n\\subsection{The Convergence Factor}\nThe geometric decay factor $\\sigma$, defined as the ratio of consecutive errors $|f - f_n|$, is calculated via the characteristic root of the limit-periodic tail \\cite{lorentzen1992continued}:\n\\begin{equation}\n\\sigma = \\left| \\frac{1 - \\sqrt{1 - 4|L|}}{1 + \\sqrt{1 - 4|L|}} \\right| = \\frac{1 - 1/3}{1 + 1/3} = \\frac{1}{2}\n\\end{equation}\nThis confirms that the identity provides approximately 3.01 decimal digits of precision per 10 iterations.\n\n\\begin{equation}\\label{eq:target_intro}\n-\\frac{\\pi}{4} = \\cfrac{1}{-1 + \\cfrac{1}{-4 + \\cfrac{-2}{-7 + \\cfrac{-9}{-10 + \\cfrac{-20}{-13 + \\dots}}}}}\n\\end{equation}", "post_theorem_intro_text_len": 1604, "post_theorem_intro_text": "More formally, conjecture~\\eqref{eq:target_intro} corresponds to the limit of the convergents of $b_0 + \\K_{n=1}^{\\infty}(a_n/b_n)$, where $b_0 = 0$, $a_1 = 1$, and for $n \\ge 1$, the partial quotients are defined by the linear denominator sequence $b_n = -(3n-2)$ and the piecewise quadratic numerator sequence:\n\\begin{equation}\\label{eq:an_def}\na_n = \n\\begin{cases} \n1, & n = 1, 2 \\\\\n-(n-1)(2n-5), & n \\ge 3\n\\end{cases}\n\\end{equation}\n\nThe analytic significance of \\eqref{eq:target_intro} lies in its departure from unit-denominator Gaussian forms. The identity involves a transition from rational coefficients typically found in hypergeometric ratios to a symbolically sparse representation with integer-based polynomials. This transition poses a non-trivial challenge in verification: one must prove that the sequence $\\{a_n, b_n\\}$ functions as an asymptotically equivalent attractor to a specific hypergeometric kernel.\n\nOur derivation proceeds by identifying \\eqref{eq:target_intro} as the result of a discrete equivalence transformation applied to the Gaussian ratio $\\mathcal{R}(1/2, 0, 1/2; -1)$. We demonstrate that the transformation sequence $\\{r_n\\}$ not only forces the linear growth of the denominators but also facilitates a polynomial regularization of the partial numerators. By analyzing the limit-periodicity of the Worpitzky parameter $\\rho_n$, we prove that the identity converges absolutely to the desired branch of the arctangent function. This work thus reconciles the experimental results of algorithmic induction with the deductive rigor of analytic continued fraction theory.", "sketch": "To verify \\eqref{eq:target_intro}, “one must prove that the sequence $\\{a_n, b_n\\}$ functions as an asymptotically equivalent attractor to a specific hypergeometric kernel.” The derivation “proceeds by identifying \\eqref{eq:target_intro} as the result of a discrete equivalence transformation applied to the Gaussian ratio $\\mathcal{R}(1/2, 0, 1/2; -1)$.” One then “demonstrate[s] that the transformation sequence $\\{r_n\\}$ not only forces the linear growth of the denominators but also facilitates a polynomial regularization of the partial numerators.” Finally, “by analyzing the limit-periodicity of the Worpitzky parameter $\\rho_n$, we prove that the identity converges absolutely to the desired branch of the arctangent function.”", "expanded_sketch": "To verify the main identity, “one must prove that the sequence $\\{a_n, b_n\\}$ functions as an asymptotically equivalent attractor to a specific hypergeometric kernel.” The derivation “proceeds by identifying the main identity as the result of a discrete equivalence transformation applied to the Gaussian ratio $\\mathcal{R}(1/2, 0, 1/2; -1)$.” One then “demonstrate[s] that the transformation sequence $\\{r_n\\}$ not only forces the linear growth of the denominators but also facilitates a polynomial regularization of the partial numerators.” Finally, “by analyzing the limit-periodicity of the Worpitzky parameter $\\rho_n$, we prove that the identity converges absolutely to the desired branch of the arctangent function.”", "expanded_theorem": "~\\cite{raayoni2021generating}~\n\\begin{equation}\\label{eq:target_intro}\n-\\frac{\\pi}{4} = \\cfrac{1}{-1 + \\cfrac{1}{-4 + \\cfrac{-2}{-7 + \\cfrac{-9}{-10 + \\cfrac{-20}{-13 + \\dots}}}}}\n\\end{equation},", "theorem_type": ["Existence", "Equality or Identity"], "mcq": {"question": "Consider the generalized continued fraction with successive denominators [1m\\( -1,-4,-7,-10,-13,\\dots \\)[0m (equivalently, \\(b_n=-(3n-2)\\) for \\(n\\ge 1\\)) and successive inner numerators \\(1,-2,-9,-20,\\dots\\) (equivalently, \\(a_n=-(n-1)(2n-5)\\) for \\(n\\ge 2\\)). Which explicit continued-fraction identity holds?", "correct_choice": {"label": "A", "text": "\\[ -\\frac{\\pi}{4} = \\cfrac{1}{-1 + \\cfrac{1}{-4 + \\cfrac{-2}{-7 + \\cfrac{-9}{-10 + \\cfrac{-20}{-13 + \\dots}}}}}. \\]"}, "choices": [{"label": "B", "text": "\\[ \\frac{\\pi}{4} = \\cfrac{1}{-1 + \\cfrac{1}{-4 + \\cfrac{-2}{-7 + \\cfrac{-9}{-10 + \\cfrac{-20}{-13 + \\dots}}}}}. \\]"}, {"label": "C", "text": "\\[ \\cfrac{1}{-1 + \\cfrac{1}{-4 + \\cfrac{-2}{-7 + \\cfrac{-9}{-10 + \\cfrac{-20}{-13 + \\dots}}}}} \\text{ converges absolutely to a finite real value.} \\]"}, {"label": "D", "text": "\\[ -\\frac{\\pi}{4} = \\cfrac{1}{1 + \\cfrac{1}{4 + \\cfrac{-2}{7 + \\cfrac{-9}{10 + \\cfrac{-20}{13 + \\dots}}}}}. \\]"}, {"label": "E", "text": "\\[ -\\frac{\\pi}{4} = \\cfrac{1}{-1 + \\cfrac{1}{-4 + \\cfrac{-\\frac{4}{3}}{-7 + \\cfrac{-\\frac{112}{15}}{-10 + \\cfrac{-\\frac{432}{35}}{-13 + \\dots}}}}}. \\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "desired branch/sign at z=-1", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "explicit evaluation to -\\pi/4 dropped, retaining only absolute convergence to a finite real limit", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "equivalence-transformation sign pattern in the linear denominators", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "polynomial regularization replaced by the exact transformed hypergeometric numerators", "template_used": "wildcard"}]}} +{"id": "2601.08810v1", "paper_link": "http://arxiv.org/abs/2601.08810v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:bndrankinv-intro}\nFor any $k,R\\in \\mathbb{N}$ and $\\delta>0$, there is $\\varepsilon>0$ and a finite collection $\\mathcal{N}_{k,R,\\delta}$ of degree-$k$ filtered nilmanifolds $G/\\Gamma$, each equipped with a smooth Riemannian metric and with connected and simply-connected ambient group $G$, such that the following holds. For any finite abelian group $\\mathrm{Z}$ of rank at most $R$, and any 1-bounded function $f:\\mathrm{Z}\\to\\mathbb{C}$ with $\\|f\\|_{U^{k+1}}\\geq \\delta$, there exists $G/\\Gamma\\in \\mathcal{N}_{k,R,\\delta}$, a Lipschitz 1-bounded function $F:G/\\Gamma\\to\\mathbb{C}$ of Lipschitz norm $O_{k,R,\\delta}(1)$, and a polynomial map $g:\\mathrm{Z}\\to G/\\Gamma$, such that $|\\mathbb{E}_{x\\in \\mathrm{Z}} f(x)\\overline{F(g(x))}| \\geq \\varepsilon$.", "start_pos": 10415, "end_pos": 11158, "label": "thm:bndrankinv-intro"}, "ref_dict": {"ex:nonext": "\\begin{example}[A non-extendable polynomial map]\\label{ex:nonext}\nFix any prime $p$ and let $\\ab=\\mb{Z}_{p^2}\\times \\mb{Z}_p$ and $\\ab_0=(p\\mb{Z}_{p^2})\\times \\mb{Z}_p\\leq \\ab$. Consider the morphism $g\\in \\hom(\\mc{D}_1(\\ab_0),\\mc{D}_2(\\mb{T}))$ defined by $g(px,y)=\\frac{xy}{p}$, where the point $\\frac{xy}{p}\\in \\mb{T}$ is obtained by taking any integer representatives of $x$ (mod $p^2$) and $y$ (mod $p$) and then taking $\\frac{xy}{p}\\mod 1$ (note that this is well-defined and indeed a quadratic map, i.e.\\ a morphism into $\\mc{D}_2(\\mb{T})$ as claimed). We claim that there is no morphism $\\wt{g}:\\mc{D}_1(\\ab)\\to\\mc{D}_2(\\mb{T})$ such that $\\wt{g}|_{\\ab_0}=g$. Indeed, suppose for a contradiction that such a map $\\wt{g}$ exists. Then, composing it with the natural surjective homomorphism $\\mb{Z}^2\\to \\ab$ (i.e.\\ reduction mod $p^2$ in the first coordinate and mod $p$ in the second), we obtain a morphism $h:\\mc{D}_1(\\mb{Z}^2)\\to \\mc{D}_2(\\mb{T})$ which is $p^2$-periodic in the first coordinate, and $p$-periodic in the second coordinate. Moreover, from the assumption that $\\wt{g}(px,y)=g(px,y)=\\frac{xy}{p}\\mod 1$, we deduce that for all integers $x,y$ we have $h(px,y)= \\frac{xy}{p}$. We thus have the following system of constraints:\n\\begin{align}\n \\forall x,y\\in \\mb{Z}, & \\quad h(x+p^2,y)=h(x,y) \\label{eq:circ1} \\\\ \n \\forall x,y\\in \\mb{Z}, & \\quad h(x,y+p)=h(x,y) \\label{eq:circ2} \\\\\n \\forall x,y\\in \\mb{Z}, & \\quad h(px,y)=\\frac{xy}{p}.\\label{eq:circ3}\n\\end{align}\nOn the other hand, since $h$ is a quadratic map into $\\mb{T}$, it has a Taylor expansion of the form $h(x,y)=a_1+a_2x+a_3y+a_4\\binom{x}{2}+a_5\\binom{y}{2}+a_6xy$ for some $a_i\\in \\mb{T}$. Now note that from equation \\eqref{eq:circ3} evaluated successively at $(x,y)=(0,0),(0,1),(0,2)$, we deduce that $a_1=0$, then $a_3=0$, and then $a_5=0$. Hence $h(x,y)=a_2x+a_4\\binom{x}{2}+a_6xy$. Now, applying this in \\eqref{eq:circ2} we deduce that $a_6xp=h(x,y+p)-h(x,y)=0$ for all $x$, whence $a_6p=0$. Then, using this back in \\eqref{eq:circ3} we obtain that $a_2px+a_4\\binom{px}{2}=\\frac{xy}{p}$, which, setting $x=1$, gives us the absurd claim that the constant $a_2p+a_4\\binom{p}{2}$ equals $\\frac{y}{p}$ for all $y$. This proves that the extension $\\wt{g}$ cannot exist.\\footnote{Note that we did not use equation \\eqref{eq:circ1} to derive the contradiction, only \\eqref{eq:circ2} and \\eqref{eq:circ3}. This is not concerning because it can be checked that equation \\eqref{eq:circ1} is actually redundant; more precisely, this equation can be deduced from equation \\eqref{eq:circ3} by evaluating the latter further at $(1,0)$ and $(2,0)$ and deducing from this that $a_2$, $a_4$ and $a_6$ must be such that $h$ is forced to be $p^2$-periodic in $x$.}\n\\end{example}", "def:quasitoral": "\\begin{defn}[Quasitoral nilspaces]\\label{def:quasitoral}\nWe say that a $k$-step \\textsc{cfr} nilspace $\\ns$ is \\emph{quasitoral} if for every $i\\in [2,k]$ the structure group $\\ab_i(\\ns)$ is a torus.\n\\end{defn}", "thm:bndrankinv-intro": "\\begin{theorem}\\label{thm:bndrankinv-intro}\nFor any $k,R\\in \\mb{N}$ and $\\delta>0$, there is $\\varepsilon>0$ and a finite collection $\\mc{N}_{k,R,\\delta}$ of degree-$k$ filtered nilmanifolds $G/\\Gamma$, each equipped with a smooth Riemannian metric and with connected and simply-connected ambient group $G$, such that the following holds. For any finite abelian group $\\ab$ of rank at most $R$, and any 1-bounded function $f:\\ab\\to\\mb{C}$ with $\\|f\\|_{U^{k+1}}\\geq \\delta$, there exists $G/\\Gamma\\in \\mc{N}_{k,R,\\delta}$, a Lipschitz 1-bounded function $F:G/\\Gamma\\to\\mb{C}$ of Lipschitz norm $O_{k,R,\\delta}(1)$, and a polynomial map $g:\\ab\\to G/\\Gamma$, such that $|\\mb{E}_{x\\in \\ab} f(x)\\overline{F(g(x))}| \\geq \\varepsilon$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 1571, "pre_theorem_intro_text": "\\noindent The Jamneshan-Tao conjecture \\cite[Conjecture 1.11]{J&T} has become one of the central objectives in higher-order Fourier analysis in recent years. It posits that nilsequences based on polynomial maps into nilmanifolds suffice (among more general morphisms into compact nilspaces) to obtain inverse theorems for Gowers norms on all finite abelian groups. Confirming this conjecture would generalize, using only nilmanifolds, earlier central results in this direction such as the Green--Tao--Ziegler inverse theorem in the integer setting \\cite{GTZ} (see also \\cite{GT08, GTZ-U4, Manners2}), and the inverse theorem in the finite-field setting by Bergelson--Tao--Ziegler \\cite{BTZ} and Tao--Ziegler \\cite{T&Z-Low}.\n\nRecently, several works have made progress towards the Jamneshan--Tao conjecture, including an inverse theorem for finite abelian groups in terms of projected nilsequences \\cite{CGSS-projnil}, and a proof of the conjecture in the case of groups of bounded exponent \\cite{JST-bndexp}. \n\nWe refer to all the aforementioned works for further background on inverse theorems for the Gowers norms and on higher-order Fourier analysis more generally. We also refer to works in the quantitative direction of improving the bounds for the inverse theorem, for specific classes of finite abelian groups \\cite{GM, LSS, Manners, M2}, or for specific Gowers norms on all such groups \\cite{M}.\n\nThe main result of this paper is the following inverse theorem for Gowers norms, proving the Jamneshan--Tao conjecture in the case of abelian groups of bounded rank.", "context": "\\noindent The Jamneshan-Tao conjecture \\cite[Conjecture 1.11]{J&T} has become one of the central objectives in higher-order Fourier analysis in recent years. It posits that nilsequences based on polynomial maps into nilmanifolds suffice (among more general morphisms into compact nilspaces) to obtain inverse theorems for Gowers norms on all finite abelian groups. Confirming this conjecture would generalize, using only nilmanifolds, earlier central results in this direction such as the Green--Tao--Ziegler inverse theorem in the integer setting \\cite{GTZ} (see also \\cite{GT08, GTZ-U4, Manners2}), and the inverse theorem in the finite-field setting by Bergelson--Tao--Ziegler \\cite{BTZ} and Tao--Ziegler \\cite{T&Z-Low}.\n\nRecently, several works have made progress towards the Jamneshan--Tao conjecture, including an inverse theorem for finite abelian groups in terms of projected nilsequences \\cite{CGSS-projnil}, and a proof of the conjecture in the case of groups of bounded exponent \\cite{JST-bndexp}.\n\nWe refer to all the aforementioned works for further background on inverse theorems for the Gowers norms and on higher-order Fourier analysis more generally. We also refer to works in the quantitative direction of improving the bounds for the inverse theorem, for specific classes of finite abelian groups \\cite{GM, LSS, Manners, M2}, or for specific Gowers norms on all such groups \\cite{M}.\n\nThe main result of this paper is the following inverse theorem for Gowers norms, proving the Jamneshan--Tao conjecture in the case of abelian groups of bounded rank.", "full_context": "\\noindent The Jamneshan-Tao conjecture \\cite[Conjecture 1.11]{J&T} has become one of the central objectives in higher-order Fourier analysis in recent years. It posits that nilsequences based on polynomial maps into nilmanifolds suffice (among more general morphisms into compact nilspaces) to obtain inverse theorems for Gowers norms on all finite abelian groups. Confirming this conjecture would generalize, using only nilmanifolds, earlier central results in this direction such as the Green--Tao--Ziegler inverse theorem in the integer setting \\cite{GTZ} (see also \\cite{GT08, GTZ-U4, Manners2}), and the inverse theorem in the finite-field setting by Bergelson--Tao--Ziegler \\cite{BTZ} and Tao--Ziegler \\cite{T&Z-Low}.\n\nRecently, several works have made progress towards the Jamneshan--Tao conjecture, including an inverse theorem for finite abelian groups in terms of projected nilsequences \\cite{CGSS-projnil}, and a proof of the conjecture in the case of groups of bounded exponent \\cite{JST-bndexp}.\n\nWe refer to all the aforementioned works for further background on inverse theorems for the Gowers norms and on higher-order Fourier analysis more generally. We also refer to works in the quantitative direction of improving the bounds for the inverse theorem, for specific classes of finite abelian groups \\cite{GM, LSS, Manners, M2}, or for specific Gowers norms on all such groups \\cite{M}.\n\nThe main result of this paper is the following inverse theorem for Gowers norms, proving the Jamneshan--Tao conjecture in the case of abelian groups of bounded rank.\n\nThe main result of this paper is the following inverse theorem for Gowers norms, proving the Jamneshan--Tao conjecture in the case of abelian groups of bounded rank.\n\n\\begin{proposition}\\label{prop:mainstep1}\nFor every $k,R\\in \\mb{N}$ and $\\delta>0$, there exists $C_0>0$ and $\\varepsilon_0>0$ such that the following holds. Let $\\ab$ be a finite abelian group of rank at most $R$, and let $f:\\ab\\to\\mb{C}$ be a 1-bounded function satisfying $\\|f\\|_{U^{k+1}}\\geq \\delta$. Then there exists a subgroup $\\ab_0\\leq \\ab$ of index at most $C_0$, a filtered nilmanifold $G_0/\\Gamma_0$ of degree $k$ where the filtration on $G_0$ consists of connected and simply-connected Lie groups, such that the associated toral nilspace $\\nss=G_0/\\Gamma_0$ is of complexity at most $C_0$, a 1-bounded Lipschitz function $F_0:\\nss\\to \\mb{C}$ of Lipschitz norm at most $C_0$, a morphism $\\varphi_0:\\mc{D}_1(\\ab_0)\\to\\nss$ and some element $t_0\\in \\ab$ such that \n\\begin{equation}\\label{eq:mainstep1}\n|\\mb{E}_{y\\in \\ab_0}f(t_0+y) F_0(\\varphi_0(y))|\\geq \\varepsilon_0.\n\\end{equation}\n\\end{proposition}\n\nLet $G/\\Gamma$ be a degree-$k$ filtered nilmanifold of complexity at most $m$ where $G$ is connected and simply-connected, let $\\ab\\cong \\prod_{i=1}^r \\mb{Z}_{n_i}$ be a finite abelian group, let $\\varphi:\\mc{D}_1(\\ab)\\to G$ be a morphism, and let $F:G/\\Gamma\\to \\mb{C}$ be a Lipschitz function with Lipschitz norm at most $M$. Then there exists a degree-$k$ filtered nilmanifold $H/\\Lambda$ of complexity at most $W(m,r)$, with $H$ connected and simply-connected, and pairwise commuting elements $h_1,\\ldots,h_r\\in H$ with $h_i^{n_i}\\in \\Lambda$ for $i\\in[r]$, and a $Q(m,r,M)$-Lipschitz function $F':H/\\Lambda\\to \\mb{C}$, such that\n\\begin{equation}\\label{eq:good-repre}\nF(\\varphi(\\underline{z})\\Gamma)=F'(\\prod_{i\\in[r]}h_i^{z_i}\\Lambda)\\quad \\text{for all }\\underline{z}\\in\\prod_{i\\in[r]}\\mb{Z}_{n_i}\\cong \\ab. \n\\end{equation}\n\\end{proposition}\n\\noindent This is a multivariable extension of \\cite[Proposition C.2]{GTZ} (the latter is the 1-variable case $r=1$).\n\\begin{proof}\nLet $H',\\Lambda'$ be as given by Proposition \\ref{lem:growing-nilmanifold} applied with $G_\\bullet$, and let $H,\\Lambda$ be as given by that proposition applied with $G_\\bullet^{+1}$. Note that $H'/\\Lambda'$ is filtered of degree $k+1$, whereas the subnilmanifold $H/\\Lambda$ is filtered of degree $k$. Note also that, by construction, if $G/\\Gamma$ has complexity $m$ then $H/\\Lambda$ has complexity at most $W(m,r)$ for some function $W$.\n\n\\begin{corollary}[Extending in the non-split case]\\label{cor:non-split-case}\nLet $k,d,p,r\\in \\mb{N}$, let $A$ be a finite abelian group of rank $r-1$, and let $(G/\\Gamma,G_\\bullet)$ be a filtered nilmanifold of degree $k$ where the nilpotent Lie group $G$ is connected and simply-connected. Then for every nilsequence $F(\\varphi(\\cdot)\\Gamma):(p\\mb{Z}_{pd})\\oplus A\\to\\mb{C}$ there exists another nilsequence $F'(\\phi(\\cdot)\\Lambda):\\mb{Z}_{pd}\\oplus A\\to\\mb{C}$ such that\n\\begin{equation}\\label{eq:corext}\n\\forall\\, \\underline{z}\\in (p\\cdot \\mb{Z}_{pd})\\oplus A,\\quad F(\\varphi(\\underline{z})\\Gamma)=F'(\\phi(\\underline{z})\\Lambda).\n\\end{equation}\nMoreover, if for some $m\\ge 0$ we have that the nilsequence $F(\\varphi(\\cdot)\\Gamma)$ has complexity $m$, then $F'(\\phi(\\cdot)\\Lambda)$ has complexity at most $K(m,r)$ for some fixed function $K:\\mb{Z}_{\\ge 0}^2\\to \\mb{Z}_{\\ge 0}$.\n\\end{corollary}\n\\begin{proof}\nWe can assume that the finite abelian group $\\ab:=(p\\mb{Z}_{pd})\\times A$ has an expression $\\ab =\\prod_{i=1}^r\\mb{Z}_{n_i}$ where $n_1$ is the order $d$ of the first component $p\\mb{Z}_{pd}\\cong \\mb{Z}_d$ of $\\ab$.\n\n\\begin{theorem}\\label{thm:bndrankinv}\nFor every $k,R\\in \\mb{N}$ and $\\delta>0$, there exist $C>0$ and $\\varepsilon>0$ such that the following holds. Let $\\ab$ be a finite abelian group of rank at most $R$, and let $f:\\ab\\to\\mb{C}$ be a 1-bounded function satisfying $\\|f\\|_{U^{k+1}}\\geq \\delta$. Then there exists a connected and simply-connected filtered Lie group $(G,G_\\bullet)$ of degree at most $k$, and a discrete cocompact subgroup $\\Gamma\\leq G$, such that the toral nilspace associated with the filtered nilmanifold $(G/\\Gamma,G_\\bullet)$ has complexity at most $C$, and there is a polynomial map $g:\\ab\\to G/\\Gamma$, and a continuous 1-bounded function $F:G/\\Gamma\\to\\mb{C}$ of Lipschitz constant at most $C$, such that\n\\begin{equation}\\label{eq:invlobound}\n|\\mb{E}_{x\\in \\ab} f(x)\\overline{F(g(x))}| \\geq \\varepsilon. \n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nBy Proposition \\ref{prop:mainstep1}, we have a connected filtered nilmanifold $G_0/\\Gamma_0$ of degree $k$ and complexity at most $C_0(\\delta,k,R)$, a subgroup $\\ab_0\\leq \\ab$ with index $|\\ab|/|\\ab_0|=O_{C_0}(1)$, and a nilsequence $F_0(g_0(\\cdot)\\Gamma_0)$ on $\\ab_0$ where $F_0:G_0/\\Gamma_0\\to\\mb{C}$ has Lipschitz constant at most $C_0$ and $g_0(\\cdot)\\Gamma_0\\in \\hom(\\mc{D}_1(\\ab_0),(G_0/\\Gamma_0,(G_{0\\bullet}))$, such that for some $t_0\\in \\ab$ we have\n\\begin{equation}\\label{eq:startingbnd}\n \\mb{E}_{x\\in \\ab_0} f(x+t_0)F_0(g_0(x)\\Gamma_0) \\geq \\varepsilon_0. \n\\end{equation}\nLet $F'(g'(\\cdot)\\Gamma')$ be the nilsequence on $\\ab$ extending $F_0(g_0(\\cdot)\\Gamma_0)$, given by Theorem \\ref{thm:mainextension}.\n\nBy basic Fourier analysis, the indicator function $1_{\\ab_0}$ on $\\ab$ satisfies (noting that $\\frac{1}{|\\ab_0^\\perp|}=\\frac{|\\ab_0|}{|\\ab|}$) $ 1_{\\ab_0}(x)=\\sum_{\\chi\\in \\ab_0^\\perp}\\tfrac{|\\ab_0|}{|\\ab|} \\chi(x)=\\mb{E}_{\\chi\\in \\ab_0^\\perp}\\chi(x)$. Hence, starting from \\eqref{eq:startingbnd}, we have\n\\begin{align*}\n& \\varepsilon_0 \\leq \\mb{E}_{x\\in \\ab_0}f(x+t_0) F_0(g_0(x)\\Gamma_0) = \\tfrac{|\\ab|}{|\\ab_0|} \\mb{E}_{x\\in \\ab} f(x+t_0) F'(g'(x)\\Gamma') 1_{\\ab_0}(x)\\\\\n& = \\tfrac{|\\ab|}{|\\ab_0|} \\mb{E}_{\\chi\\in \\ab_0^\\perp} \\mb{E}_{x\\in \\ab}f(x+t_0) F'(g'(x)\\Gamma') \\chi(x) \\leq \\tfrac{|\\ab|}{|\\ab_0|} \\max_{\\chi\\in \\ab_0^\\perp} | \\mb{E}_{x\\in \\ab}f(x) F'(g'(x-t_0)\\Gamma') \\chi(x-t_0)|.\n\\end{align*}\nNow fix any $\\chi\\in \\ab_0^\\perp$ attaining this last maximum, let us relabel the shifted polynomial map $x\\mapsto g'(x-t_0)$ as a polynomial map $g''(x)$, and let us ignore the multiplicative constant $ \\chi(t_0)$ of modulus 1, thus concluding that\n\\begin{equation}\\label{eq:correl2}\n | \\mb{E}_{x\\in \\ab}f(x) F'(g''(x)\\Gamma') \\chi(x)| \\geq \\varepsilon_0 \\tfrac{|\\ab_0|}{|\\ab|} =\\Omega_{C_0}(\\varepsilon_0)=:\\varepsilon.\n\\end{equation}\nBy the standard representation of characters via non-degenerate symmetric bilinear forms (see \\cite[Ch.\\ 4]{T-V}), there exists such a form $\\cdot:\\ab\\times\\ab\\to \\mb{T}$ and some $\\xi\\in \\ab$ such that $\\chi(x)=e(\\xi\\cdot x)$. We can now define the nilmanifold $G/\\Gamma$ as $(G=G'\\times \\mb{R})/(\\Gamma=\\Gamma' \\times \\mb{Z})$, which is isomorphic as a compact nilspace to $(G'/\\Gamma')\\times \\mb{T}$, and define the map $g(\\cdot)\\Gamma:\\ab\\to G/\\Gamma$ by $g(x)\\Gamma:= (g''(x)\\Gamma',\\xi\\cdot x)$, which is indeed a nilspace morphism $\\mc{D}_1(\\ab)\\to (G'/\\Gamma')\\times \\mc{D}_1(\\mb{T})$. We define also the 1-bounded Lipschitz function $F:G/\\Gamma\\to\\mb{C}$ by $F(x\\Gamma',z)=F'(x\\Gamma')e(z)$ for any $x\\Gamma'\\in G'/\\Gamma'$ and $z\\in\\mb{T}$, and it can be checked easily that $F$ has bounded Lipschitz constant if $F_0$ does (using that both $F_0$ and $e(\\cdot)$ are 1-bounded). Thus, the inequality \\eqref{eq:correl2} is rewritten as $\n|\\mb{E}_{x\\in \\ab}f(x) F(g(x)\\Gamma)| \\geq \\varepsilon$. This completes the proof.\\end{proof}", "post_theorem_intro_text_len": 3967, "post_theorem_intro_text": "\\begin{remark}\nIn the bounded-rank setting, there is a subcase consisting of any infinite family $\\mathcal{F}$ of finite abelian groups of rank at most $R$ with the additional property that every prime $p$ divides the order of only \\emph{finitely} many groups in $\\mathcal{F}$. An inverse theorem confirming the Jamneshan--Tao conjecture for any such family $\\mathcal{F}$ is obtained by straightforward adaptations of previous literature (see \\cite[Remark 1.11]{CSinverse}). The strictly more general Theorem \\ref{thm:bndrankinv-intro} does not follow from such prior work, a central reason being that the type of nilspaces that emerge in this setting (see Definition \\ref{def:quasitoral} below) was not sufficiently analyzed previously.\n\\end{remark}\nOur proof of Theorem \\ref{thm:bndrankinv-intro} can be divided into two main steps. \n\nThe first step consists in applying the inverse theorem \\cite[Theorem 5.2]{CSinverse} and showing that, in the bounded-rank setting, the resuling nilspace is what we call a \\emph{quasitoral} nilspace. (All the following basic terminology from nilspace theory is recalled in Section \\ref{sec:prelims} below.)\n\nRecall from \\cite{CamSzeg,Cand:Notes2} that a $k$-step compact finite-rank (\\textsc{cfr}) nilspace $\\mathrm{X}$ is \\emph{toral} if its structure groups $\\ab_1(\\mathrm{X}),\\ab_2(\\mathrm{X}),\\ldots,\\ab_k(\\mathrm{X})$ are all tori.\n\n\\begin{defn}[Quasitoral nilspaces]\\label{def:quasitoral}\nWe say that a $k$-step \\textsc{cfr} nilspace $\\mathrm{X}$ is \\emph{quasitoral} if for every $i\\in [2,k]$ the structure group $\\ab_i(\\mathrm{X})$ is a torus.\n\\end{defn}\n\\noindent We show that quasitoral nilspaces are disjoint unions of toral nilspaces (which are connected nilmanifolds). In the case of the quasitoral nilspace $\\mathrm{X}$ obtained by applying \\cite[Theorem 5.2]{CSinverse}, the number of its toral-nilspace components depends only on the complexity of $\\mathrm{X}$, and is therefore adequately bounded. As a consequence, we deduce that on some bounded-index subgroup $\\mathrm{Z}'$ of the original abelian group $\\mathrm{Z}$, some shift of the initial function $f$ correlates non-trivially with a bounded-complexity nilsequence defined on $\\mathrm{Z}'$. This completes this first step.\n\nThe second main step consists principally in proving that the nilsequence on $\\mathrm{Z}'$ can always be extended to a nilsequence on the full group $\\mathrm{Z}$, while ensuring that the latter nilsequence still has bounded complexity. To solve this extension problem in general, it is necessary to modify the nilmanifold underlying the nilsequence. We prove this necessity with an example of a degree-2 polynomial map which cannot be extended to the whole group from a subgroup while conserving its target nilspace; see Example \\ref{ex:nonext}.\n\nThe paper has the following outline. In Section \\ref{sec:prelims} we gather some basic tools and concepts from nilspace theory and higher-order Fourier analysis. In Section \\ref{sec:quasitoral} we carry out the first main step described above, and in Section \\ref{sec:extend} we solve the nilsequence-extension problem. The proof of Theorem \\ref{thm:bndrankinv-intro} (including the control on the final correlation bound), is given in Section \\ref{sec:mainproof}.\n\n\\medskip \n\n\\noindent \\textbf{Acknowledgments.} All authors used funding from project PID2024-156180NB-I00 funded by Spain's MICIU/AEI. The second-named author was funded by HORIZON-MSCA-2024-PF-01, AlgHOF 101202161, funded by the European Union.\\footnote{Views and opinions expressed are those of the author(s) only and do not reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them.} The third-named author was also supported partially by the Hungarian Ministry of Innovation and Technology NRDI Office within the framework of the Artificial Intelligence National Laboratory Program (MILAB, RRF-2.3.1-21-2022-00004).", "sketch": "Our proof of Theorem \\ref{thm:bndrankinv-intro} can be divided into two main steps.\n\n(1) \"The first step consists in applying the inverse theorem \\cite[Theorem 5.2]{CSinverse} and showing that, in the bounded-rank setting, the resuling nilspace is what we call a \\emph{quasitoral} nilspace.\" The text then states: \"We show that quasitoral nilspaces are disjoint unions of toral nilspaces (which are connected nilmanifolds).\" For the quasitoral nilspace \\(\\mathrm{X}\\) arising from \\cite[Theorem 5.2]{CSinverse}, \"the number of its toral-nilspace components depends only on the complexity of \\(\\mathrm{X}\\), and is therefore adequately bounded.\" Hence, \"on some bounded-index subgroup \\(\\mathrm{Z}'\\) of the original abelian group \\(\\mathrm{Z}\\), some shift of the initial function \\(f\\) correlates non-trivially with a bounded-complexity nilsequence defined on \\(\\mathrm{Z}'\\).\" \"This completes this first step.\"\n\n(2) \"The second main step consists principally in proving that the nilsequence on \\(\\mathrm{Z}'\\) can always be extended to a nilsequence on the full group \\(\\mathrm{Z}\\), while ensuring that the latter nilsequence still has bounded complexity.\" The introduction notes that \"to solve this extension problem in general, it is necessary to modify the nilmanifold underlying the nilsequence,\" and that this necessity is illustrated by \"an example of a degree-2 polynomial map which cannot be extended to the whole group from a subgroup while conserving its target nilspace\" (Example \\ref{ex:nonext}).\n\nFinally, it is stated that \"The proof of Theorem \\ref{thm:bndrankinv-intro} (including the control on the final correlation bound), is given in Section \\ref{sec:mainproof}.\"", "expanded_sketch": "Our proof of the main theorem can be divided into two main steps.\n\n(1) \"The first step consists in applying the inverse theorem \\cite[Theorem 5.2]{CSinverse} and showing that, in the bounded-rank setting, the resuling nilspace is what we call a \\emph{quasitoral} nilspace.\" The text then states: \"We show that quasitoral nilspaces are disjoint unions of toral nilspaces (which are connected nilmanifolds).\" For the quasitoral nilspace \\(\\mathrm{X}\\) arising from \\cite[Theorem 5.2]{CSinverse}, \"the number of its toral-nilspace components depends only on the complexity of \\(\\mathrm{X}\\), and is therefore adequately bounded.\" Hence, \"on some bounded-index subgroup \\(\\mathrm{Z}'\\) of the original abelian group \\(\\mathrm{Z}\\), some shift of the initial function \\(f\\) correlates non-trivially with a bounded-complexity nilsequence defined on \\(\\mathrm{Z}'\\).\" \"This completes this first step.\"\n\n(2) \"The second main step consists principally in proving that the nilsequence on \\(\\mathrm{Z}'\\) can always be extended to a nilsequence on the full group \\(\\mathrm{Z}\\), while ensuring that the latter nilsequence still has bounded complexity.\" The introduction notes that \"to solve this extension problem in general, it is necessary to modify the nilmanifold underlying the nilsequence,\" and that this necessity is illustrated by the following example.\n\n\\begin{example}[A non-extendable polynomial map]\\label{ex:nonext}\nFix any prime $p$ and let $\\ab=\\mb{Z}_{p^2}\\times \\mb{Z}_p$ and $\\ab_0=(p\\mb{Z}_{p^2})\\times \\mb{Z}_p\\leq \\ab$. Consider the morphism $g\\in \\hom(\\mc{D}_1(\\ab_0),\\mc{D}_2(\\mb{T}))$ defined by $g(px,y)=\\frac{xy}{p}$, where the point $\\frac{xy}{p}\\in \\mb{T}$ is obtained by taking any integer representatives of $x$ (mod $p^2$) and $y$ (mod $p$) and then taking $\\frac{xy}{p}\\mod 1$ (note that this is well-defined and indeed a quadratic map, i.e.\\ a morphism into $\\mc{D}_2(\\mb{T})$ as claimed). We claim that there is no morphism $\\wt{g}:\\mc{D}_1(\\ab)\\to\\mc{D}_2(\\mb{T})$ such that $\\wt{g}|_{\\ab_0}=g$. Indeed, suppose for a contradiction that such a map $\\wt{g}$ exists. Then, composing it with the natural surjective homomorphism $\\mb{Z}^2\\to \\ab$ (i.e.\\ reduction mod $p^2$ in the first coordinate and mod $p$ in the second), we obtain a morphism $h:\\mc{D}_1(\\mb{Z}^2)\\to \\mc{D}_2(\\mb{T})$ which is $p^2$-periodic in the first coordinate, and $p$-periodic in the second coordinate. Moreover, from the assumption that $\\wt{g}(px,y)=g(px,y)=\\frac{xy}{p}\\mod 1$, we deduce that for all integers $x,y$ we have $h(px,y)= \\frac{xy}{p}$. We thus have the following system of constraints:\n\\begin{align}\n \\forall x,y\\in \\mb{Z}, & \\quad h(x+p^2,y)=h(x,y) \\label{eq:circ1} \\\\ \n \\forall x,y\\in \\mb{Z}, & \\quad h(x,y+p)=h(x,y) \\label{eq:circ2} \\\\\n \\forall x,y\\in \\mb{Z}, & \\quad h(px,y)=\\frac{xy}{p}.\\label{eq:circ3}\n\\end{align}\nOn the other hand, since $h$ is a quadratic map into $\\mb{T}$, it has a Taylor expansion of the form $h(x,y)=a_1+a_2x+a_3y+a_4\\binom{x}{2}+a_5\\binom{y}{2}+a_6xy$ for some $a_i\\in \\mb{T}$. Now note that from equation \\eqref{eq:circ3} evaluated successively at $(x,y)=(0,0),(0,1),(0,2)$, we deduce that $a_1=0$, then $a_3=0$, and then $a_5=0$. Hence $h(x,y)=a_2x+a_4\\binom{x}{2}+a_6xy$. Now, applying this in \\eqref{eq:circ2} we deduce that $a_6xp=h(x,y+p)-h(x,y)=0$ for all $x$, whence $a_6p=0$. Then, using this back in \\eqref{eq:circ3} we obtain that $a_2px+a_4\\binom{px}{2}=\\frac{xy}{p}$, which, setting $x=1$, gives us the absurd claim that the constant $a_2p+a_4\\binom{p}{2}$ equals $\\frac{y}{p}$ for all $y$. This proves that the extension $\\wt{g}$ cannot exist.\\footnote{Note that we did not use equation \\eqref{eq:circ1} to derive the contradiction, only \\eqref{eq:circ2} and \\eqref{eq:circ3}. This is not concerning because it can be checked that equation \\eqref{eq:circ1} is actually redundant; more precisely, this equation can be deduced from equation \\eqref{eq:circ3} by evaluating the latter further at $(1,0)$ and $(2,0)$ and deducing from this that $a_2$, $a_4$ and $a_6$ must be such that $h$ is forced to be $p^2$-periodic in $x$.}\n\\end{example}\n\nFinally, it is stated that \"In establishing the main theorem (including the control on the final correlation bound), the proof is given later.\"", "expanded_theorem": "\\label{thm:bndrankinv-intro}\nFor any $k,R\\in \\mathbb{N}$ and $\\delta>0$, there is $\\varepsilon>0$ and a finite collection $\\mathcal{N}_{k,R,\\delta}$ of degree-$k$ filtered nilmanifolds $G/\\Gamma$, each equipped with a smooth Riemannian metric and with connected and simply-connected ambient group $G$, such that the following holds. For any finite abelian group $\\mathrm{Z}$ of rank at most $R$, and any 1-bounded function $f:\\mathrm{Z}\\to\\mathbb{C}$ with $\\|f\\|_{U^{k+1}}\\geq \\delta$, there exists $G/\\Gamma\\in \\mathcal{N}_{k,R,\\delta}$, a Lipschitz 1-bounded function $F:G/\\Gamma\\to\\mathbb{C}$ of Lipschitz norm $O_{k,R,\\delta}(1)$, and a polynomial map $g:\\mathrm{Z}\\to G/\\Gamma$, such that $|\\mathbb{E}_{x\\in \\mathrm{Z}} f(x)\\overline{F(g(x))}| \\geq \\varepsilon$.", "theorem_type": ["Existential–Universal", "Universal–Existential"], "mcq": {"question": "Fix integers $k,R\\in\\mathbb N$ and $\\delta>0$. Let a finite abelian group have rank at most $R$ (that is, it can be generated by at most $R$ elements), and let a function be 1-bounded if $|f(x)|\\le 1$ for all $x$. A degree-$k$ filtered nilmanifold means a nilmanifold $G/\\Gamma$ equipped with a degree-$k$ filtration, and a polynomial map $g:\\mathrm Z\\to G/\\Gamma$ is taken with respect to that filtration. Which statement holds for every finite abelian group $\\mathrm Z$ of rank at most $R$ and every 1-bounded function $f:\\mathrm Z\\to\\mathbb C$ with Gowers norm $\\|f\\|_{U^{k+1}}\\ge \\delta$?", "correct_choice": {"label": "A", "text": "There exist a constant $\\varepsilon>0$ and a finite collection $\\mathcal N_{k,R,\\delta}$ of degree-$k$ filtered nilmanifolds $G/\\Gamma$, each equipped with a smooth Riemannian metric and with connected and simply-connected ambient group $G$, such that for every such $\\mathrm Z$ and $f$ there are some $G/\\Gamma\\in\\mathcal N_{k,R,\\delta}$, a 1-bounded Lipschitz function $F:G/\\Gamma\\to\\mathbb C$ with Lipschitz norm $O_{k,R,\\delta}(1)$, and a polynomial map $g:\\mathrm Z\\to G/\\Gamma$ for which\n\\[\n\\left|\\mathbb E_{x\\in \\mathrm Z} f(x)\\,\\overline{F(g(x))}\\right|\\ge \\varepsilon.\n\\]"}, "choices": [{"label": "B", "text": "There exist a constant $\\varepsilon>0$ and a finite collection $\\mathcal N_{k,R,\\delta}$ of degree-$(k+1)$ filtered nilmanifolds $G/\\Gamma$, each equipped with a smooth Riemannian metric and with connected and simply-connected ambient group $G$, such that for every such $\\mathrm Z$ and $f$ there are some $G/\\Gamma\\in\\mathcal N_{k,R,\\delta}$, a 1-bounded Lipschitz function $F:G/\\Gamma\\to\\mathbb C$ with Lipschitz norm $O_{k,R,\\delta}(1)$, and a polynomial map $g:\\mathrm Z\\to G/\\Gamma$ for which\n\\[\n\\left|\\mathbb E_{x\\in \\mathrm Z} f(x)\\,\\overline{F(g(x))}\\right|\\ge \\varepsilon.\n\\]"}, {"label": "C", "text": "There exist a constant $\\varepsilon>0$ such that for every such $\\mathrm Z$ and $f$ there are a degree-$k$ filtered nilmanifold $G/\\Gamma$ with connected and simply-connected ambient group $G$, a 1-bounded Lipschitz function $F:G/\\Gamma\\to\\mathbb C$, and a polynomial map $g:\\mathrm Z\\to G/\\Gamma$ for which\n\\[\n\\left|\\mathbb E_{x\\in \\mathrm Z} f(x)\\,\\overline{F(g(x))}\\right|\\ge \\varepsilon.\n\\]"}, {"label": "D", "text": "There exist a constant $\\varepsilon>0$ and a finite collection $\\mathcal N_{k,R,\\delta}$ of degree-$k$ filtered nilmanifolds $G/\\Gamma$, each equipped with a smooth Riemannian metric and with connected and simply-connected ambient group $G$, such that for every such $\\mathrm Z$ and $f$ there are some $G/\\Gamma\\in\\mathcal N_{k,R,\\delta}$, a 1-bounded Lipschitz function $F:G/\\Gamma\\to\\mathbb C$ with Lipschitz norm $O_{k,R,\\delta}(1)$, and a group homomorphism $g:\\mathrm Z\\to G/\\Gamma$ for which\n\\[\n\\left|\\mathbb E_{x\\in \\mathrm Z} f(x)\\,\\overline{F(g(x))}\\right|\\ge \\varepsilon.\n\\]"}, {"label": "E", "text": "There exist a constant $\\varepsilon>0$ and a finite collection $\\mathcal N_{k,R,\\delta}$ of degree-$k$ filtered nilmanifolds $G/\\Gamma$, each equipped with a smooth Riemannian metric and with connected and simply-connected ambient group $G$, such that for every such $\\mathrm Z$ and $f$ there are some $G/\\Gamma\\in\\mathcal N_{k,R,\\delta}$, a 1-bounded Lipschitz function $F:G/\\Gamma\\to\\mathbb C$ with Lipschitz norm $O_{k,R,\\delta}(1)$, and a polynomial map $g:\\mathrm Z\\to G/\\Gamma$ for which\n\\[\n\\mathbb E_{x\\in \\mathrm Z} f(x)\\,\\overline{F(g(x))}\\ge \\varepsilon.\n\\]"}], "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": "degree_preservation_in_extension", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "finite_collection_and_uniform_lipschitz_bound", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "polynomial_map_vs_homomorphism", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "absolute_value_in_correlation", "template_used": "wildcard"}]}} +{"id": "2601.09443v1", "paper_link": "http://arxiv.org/abs/2601.09443v1", "theorems_cnt": 3, "theorem": {"env_name": "thmx", "content": "\\label{intro:thma}(\\ref{prop:pgdimineq},\\ref{prop:ringepi}, \\ref{prop:hringepi})\nLet $R$ be a ring, then there is an inequality $\\mathsf{pgdim}(\\mathsf{D}(R))\\geq \\mathsf{pgdim}(R)$. As such, $\\mathsf{D}(R)$ is not a brown category whenever $\\mathsf{pgdim}(R)\\geq 2$. Moreover, if $f\\colon R\\to S$ a ring homomorphism, then\n\\begin{enumerate}\n\\item if $f$ is a ring epimorphism, one has $\\mathsf{pgdim}(S)\\leq \\mathsf{pgdim}(R)$;\n\\item if $f$ is a homological ring epimorphism, one has $\\mathsf{pgdim}(\\mathsf{D}(S))\\leq \\mathsf{pgdim}(\\mathsf{D}(R))$.\n\\end{enumerate}", "start_pos": 9770, "end_pos": 10307, "label": "intro:thma"}, "ref_dict": {"thm:BAfunctoriality": "\\begin{thm}\\label{thm:BAfunctoriality}\nLet $\\T$ and $\\U$ be compactly generated triangulated categories and $F\\colon\\Flat{\\T^{\\c}}\\to\\Flat{\\U^{\\c}}$ a definable functor.\n\\begin{enumerate}\n\\item Suppose $F$ is essentially surjective up to direct summands, then if $\\T$ is a Brown category so is $\\U$.\n\\item Suppose $F$ is fully faithful, then if $\\U$ is a Brown category so is $\\T$.\n\\end{enumerate}\n\\end{thm}", "intro:thma": "\\begin{thmx}\\label{intro:thma}(\\ref{prop:pgdimineq},\\ref{prop:ringepi}, \\ref{prop:hringepi})\nLet $R$ be a ring, then there is an inequality $\\msf{pgdim}(\\D(R))\\geq \\msf{pgdim}(R)$. As such, $\\D(R)$ is not a brown category whenever $\\msf{pgdim}(R)\\geq 2$. Moreover, if $f\\colon R\\to S$ a ring homomorphism, then\n\\begin{enumerate}\n\\item if $f$ is a ring epimorphism, one has $\\msf{pgdim}(S)\\leq \\msf{pgdim}(R)$;\n\\item if $f$ is a homological ring epimorphism, one has $\\msf{pgdim}(\\D(S))\\leq \\msf{pgdim}(\\D(R))$.\n\\end{enumerate}\n\\end{thmx}", "prop:brownschemes": "\\begin{prop}\\label{prop:brownschemes}\nLet $X$ be a qcqs scheme. Then $\\D_{\\mrm{qc}}(X)$ is a Brown category only if $\\msf{pgdim}(\\scr{O}_{X,x})\\leq 1$, or $\\scr{O}_{X,x}^{\\mrm{ab}}$ is a hereditary von Neumann regular ring, for all $x\\in X$.\n\\end{prop}", "prop:telescope": "\\begin{prop}\\label{prop:telescope}\nLet $R$ be a von Neumann regular ring, then every Serre subcategory of $\\mod{\\D(R)^{\\c}}$ is perfect. Consequently, there is a bijection between\n\\begin{enumerate}\n\\item thick subcategories of $\\D(R)^{\\c}$;\n\\item Serre subcategories of $\\mod{R}$;\n\\item $\\Sigma$-invariant Serre subcategories of $\\mod{\\D(R)^{\\c}}$;\n\\item smashing localisations of $\\D(R)$;\n\\end{enumerate}\nand all these are in bijection with two-sided ideals of $R$. In particular, the telescope conjecture holds in $\\D(R)$.\n\\end{prop}", "prop:pgdimineq": "\\begin{prop}\\label{prop:pgdimineq}\nLet $R$ be a ring. Then $\\msf{pgdim}(\\D(R))\\geq \\msf{pgdim}(R)$.\n\\end{prop}", "prop:fghholds": "\\begin{prop}\\label{prop:fghholds}\nThe generating hypothesis holds for $\\D(R)$ if and only if $\\scr{S}\\cap\\mod{\\D(R)^{\\c}}=0$. In particular, if $R$ is right coherent, then $\\D(R)$ satisfies the generating hypothesis if and only if $R$ is von Neumann regular.\n\\end{prop}", "ex:fdalgebras": "\\begin{ex}[Finite dimensional local algebras]\\label{ex:fdalgebras}\nLet $k$ be an uncountable algebraically closed field and $A$ a finite dimensional local $k$-algebra. If $A$ is of finite representation type, then it is pure semisimple, meaning every $A$-module is pure injective, and thus $\\msf{pgdim}(A)=0$. If $A$ is not of finite representation type, then $\\msf{pgdim}(A)\\geq 2$ by \\cite[Proposition 5.3]{BBL}, and thus $\\D(A)$ is not a Brown category by \\cref{prop:pgdimineq}. \n\nIt is not clear whether $A$ being pure semisimple implies that $\\D(A)$ is Brown without an assumption of $A$ being hereditary. See \\cite[\\S 12]{belrhap} for this consideration.\n\\end{ex}", "intro:thmb": "\\begin{thmx}\\label{intro:thmb}(\\ref{prop:pextisos})\nLet $F\\colon\\A\\to\\B$ be a definable functor between finitely accessible categories with products, and suppose that $F$ admits a left adjoint $\\Lambda\\colon\\B\\to\\A$. Then there are isomorphisms\n\\[\n\\mrm{PExt}_{\\msf{B}}^{n}(B,F(A))\\simeq \\mrm{PExt}_{\\A}^{n}(\\Lambda(B),A)\n\\]\nfor all $A\\in\\A, B\\in\\B$ and $n\\geq 0$.\n\\end{thmx}", "prop:vnrdeterminesbrown": "\\begin{prop}\\label{prop:vnrdeterminesbrown}\nLet $R$ be a ring. If $R^{\\mrm{ab}}\\neq 0$, then if $R^{\\mrm{ab}}$ is not hereditary, $\\D(R)$ is not a Brown category.\n\\end{prop}", "prop:hringepi": "\\begin{prop}\\label{prop:hringepi}\nLet $f\\colon R\\to S$ be a homological ring epimorphism, then $\\msf{pgdim}(\\D(S))\\leq \\msf{pgdim}(\\D(R))$.\n\\end{prop}", "prop:cosilting": "\\begin{prop}\\label{prop:cosilting}\nLet $\\T$ be a compactly generated triangulated category and suppose that $\\mbb{t}$ is a compactly generated t-structure that restricts to compact objects. Then $\\msf{pgdim}(\\T)\\geq\\msf{pgdim}(\\msf{fpInj}(\\scr{H}_{\\mbb{t}}))$. If, additionally $\\mbb{t}$ is non-degenerate, then $\\msf{pgdim}(\\T)\\geq\\msf{pgdim}(\\scr{H}_{\\mbb{t}})$.\n\\end{prop}", "prop:ringepi": "\\begin{prop}\\label{prop:ringepi}\nLet $f\\colon R\\to S$ be a ring epimorphism. Then $\\msf{pgdim}(S)\\leq \\msf{pgdim}(R)$.\n\\end{prop}", "prop:pextisos": "\\begin{prop}\\label{prop:pextisos}\nLet $F\\colon\\A\\to\\B$ be a definable functor between finitely accessible categories with products, and suppose that $F$ admits a left adjoint $\\Lambda\\colon\\B\\to\\A$. Then there are isomorphisms\n\\[\n\\mrm{PExt}_{\\msf{B}}^{n}(B,F(A))\\simeq \\mrm{PExt}_{\\A}^{n}(\\Lambda(B),A)\n\\]\nfor all $A\\in\\A, B\\in\\B$ and $n\\geq 0$.\n\\end{prop}", "intro:thmc": "\\begin{thmx}\\label{intro:thmc}(\\ref{thm:BAfunctoriality})\nLet $\\T$ and $\\U$ be compactly generated triangulated categories, and suppose $F\\colon\\Flat{\\T^{\\c}}\\to\\Flat{\\U^{\\c}}$ is a definable functor. \n\\begin{enumerate}\n\\item If $F$ is essentially surjective up to direct summands and $\\T$ is a Brown category, so is $\\U$;\n\\item If $F$ is fully faithful and $\\U$ is a Brown category, then so is $\\T$.\n\\end{enumerate}\n\\end{thmx}"}, "pre_theorem_intro_text_len": 3539, "pre_theorem_intro_text": "A renowned result in homotopy theory, due to Brown \\cite{brown} and Adams \\cite{adams}, is that the stable homotopy category of spectra $\\mathsf{Sp}$ satisfies Brown--Adams representability. This means that any cohomological functor $F\\colon(\\mathsf{Sp}^{\\omega})^{\\mrm{op}}\\to\\Mod{\\mathbb{Z}}$ on finite spectra $\\mathsf{Sp}^{\\omega}$ is naturally isomorphic to $\\Hom_{\\mathsf{Sp}}(-,X)\\vert_{\\mathsf{Sp}^{\\omega}}$ for some $X\\in\\mathsf{Sp}$, and if $\\varphi\\colon F\\to G$ is a natural transformation between cohomological functors, then $\\varphi\\simeq\\Hom_{\\mathsf{Sp}}(-,f)$, where $f\\colon X\\to Y$ is a map in $\\mathsf{Sp}$ between objects representing $F$ and $G$.\n\nThis statement can be translated into any compactly generated triangulated category $\\mathsf{T}$: if $\\mathsf{T}^{\\mrm{c}}$ denotes the compact objects of $\\mathsf{T}$, one says that $\\mathsf{T}$ satisfies Brown--Adams representability (or is a Brown category), provided every cohomological functor $F\\colon(\\mathsf{T}^{\\mrm{c}})^{\\mrm{op}}\\to\\Mod{\\mathbb{Z}}$ is of the form $\\Hom_{\\mathsf{T}}(-,X)\\vert_{\\mathsf{T}^{\\mrm{c}}}$ for some object $X$, and any map $\\varphi\\colon\\Hom_{\\mathsf{T}}(-,X)\\vert_{\\mathsf{T}^{\\mrm{c}}}\\to\\Hom_{\\mathsf{T}}(-,Y)\\vert_{\\mathsf{T}^{\\mrm{c}}}$ is of the form $\\Hom_{\\mathsf{T}}(-,f)$ for some $f\\colon X\\to Y$. In other words, $\\mathsf{T}$ is a Brown category, by definition, if and only if the restricted Yoneda functor\n\\[\n\\msf{y}\\colon\\mathsf{T}\\to\\Flat{\\mathsf{T}^{\\mrm{c}}}, \\, X\\mapsto \\Hom_{\\mathsf{T}}(-,X)\\vert_{\\mathsf{T}^{\\mrm{c}}}\n\\]\nis essentially surjective and full. Here $\\Flat{\\mathsf{T}^{\\mrm{c}}}$ denotes the category of cohomological functors $(\\mathsf{T}^{\\mrm{c}})^{\\mrm{op}}\\to\\msf{Ab}$, which is expressed this way as it is the ind-completion of the category $\\mathsf{T}^{\\mrm{c}}$.\n\nIt is known that not every compactly generated $\\mathsf{T}$ is a Brown category, and determining when a specific one is, or is not, is not elementary. Criteria to approach this problem were proved by Neeman \\cite{neemantba} and later expanded by Beligiannis \\cite{belrhap}: it was shown that $\\mathsf{T}$ satisfies Brown--Adams representability if and only if $\\mathsf{pgdim}(\\mathsf{T})$, the pure global dimension of $\\mathsf{T}$, is at most one. \n\nIt is through considering pure global dimension, or more specifically its ascent and descent, that we approach the question of whether or not a triangulated category is Brown. Our main investigation is in revisiting the question of when $\\mathsf{D}(R)$ is a Brown category for a ring $R$, and before giving more details and stating results, we provide some historical context.\n\nThe question of $\\mathsf{D}(R)$ being Brown was first considered in Neeman \\cite{neemantba}, where an example credited to Keller showed $\\mathsf{D}(\\mathbb{C}[x,y])$ is not Brown; the pertinent point is that this ring has a flat module of projective dimension at least two. In \\cite{belrhap}, Beligiannis gave some further counterexamples, and provided a relationship between the pure global dimension of $\\msf{Mod}(R)$ and $\\mathsf{D}(R)$ in the case that $R$ is a (right) hereditary ring, showing the two are equal. This (among other questions) was furthered by Christensen, Keller and Neeman in \\cite{ckn} to show that if $R$ is a coherent ring such that every finitely presented $R$-module has finite projective dimension, then $\\mathsf{pgdim}(R)\\leq\\mathsf{pgdim}(\\mathsf{D}(R))$. In this paper, the main result in this direction is the following, which removes all assumptions on rings.", "context": "A renowned result in homotopy theory, due to Brown \\cite{brown} and Adams \\cite{adams}, is that the stable homotopy category of spectra $\\mathsf{Sp}$ satisfies Brown--Adams representability. This means that any cohomological functor $F\\colon(\\mathsf{Sp}^{\\omega})^{\\mrm{op}}\\to\\Mod{\\mathbb{Z}}$ on finite spectra $\\mathsf{Sp}^{\\omega}$ is naturally isomorphic to $\\Hom_{\\mathsf{Sp}}(-,X)\\vert_{\\mathsf{Sp}^{\\omega}}$ for some $X\\in\\mathsf{Sp}$, and if $\\varphi\\colon F\\to G$ is a natural transformation between cohomological functors, then $\\varphi\\simeq\\Hom_{\\mathsf{Sp}}(-,f)$, where $f\\colon X\\to Y$ is a map in $\\mathsf{Sp}$ between objects representing $F$ and $G$.\n\nThis statement can be translated into any compactly generated triangulated category $\\mathsf{T}$: if $\\mathsf{T}^{\\mrm{c}}$ denotes the compact objects of $\\mathsf{T}$, one says that $\\mathsf{T}$ satisfies Brown--Adams representability (or is a Brown category), provided every cohomological functor $F\\colon(\\mathsf{T}^{\\mrm{c}})^{\\mrm{op}}\\to\\Mod{\\mathbb{Z}}$ is of the form $\\Hom_{\\mathsf{T}}(-,X)\\vert_{\\mathsf{T}^{\\mrm{c}}}$ for some object $X$, and any map $\\varphi\\colon\\Hom_{\\mathsf{T}}(-,X)\\vert_{\\mathsf{T}^{\\mrm{c}}}\\to\\Hom_{\\mathsf{T}}(-,Y)\\vert_{\\mathsf{T}^{\\mrm{c}}}$ is of the form $\\Hom_{\\mathsf{T}}(-,f)$ for some $f\\colon X\\to Y$. In other words, $\\mathsf{T}$ is a Brown category, by definition, if and only if the restricted Yoneda functor\n\\[\n\\msf{y}\\colon\\mathsf{T}\\to\\Flat{\\mathsf{T}^{\\mrm{c}}}, \\, X\\mapsto \\Hom_{\\mathsf{T}}(-,X)\\vert_{\\mathsf{T}^{\\mrm{c}}}\n\\]\nis essentially surjective and full. Here $\\Flat{\\mathsf{T}^{\\mrm{c}}}$ denotes the category of cohomological functors $(\\mathsf{T}^{\\mrm{c}})^{\\mrm{op}}\\to\\msf{Ab}$, which is expressed this way as it is the ind-completion of the category $\\mathsf{T}^{\\mrm{c}}$.\n\nIt is known that not every compactly generated $\\mathsf{T}$ is a Brown category, and determining when a specific one is, or is not, is not elementary. Criteria to approach this problem were proved by Neeman \\cite{neemantba} and later expanded by Beligiannis \\cite{belrhap}: it was shown that $\\mathsf{T}$ satisfies Brown--Adams representability if and only if $\\mathsf{pgdim}(\\mathsf{T})$, the pure global dimension of $\\mathsf{T}$, is at most one.\n\nIt is through considering pure global dimension, or more specifically its ascent and descent, that we approach the question of whether or not a triangulated category is Brown. Our main investigation is in revisiting the question of when $\\mathsf{D}(R)$ is a Brown category for a ring $R$, and before giving more details and stating results, we provide some historical context.\n\nThe question of $\\mathsf{D}(R)$ being Brown was first considered in Neeman \\cite{neemantba}, where an example credited to Keller showed $\\mathsf{D}(\\mathbb{C}[x,y])$ is not Brown; the pertinent point is that this ring has a flat module of projective dimension at least two. In \\cite{belrhap}, Beligiannis gave some further counterexamples, and provided a relationship between the pure global dimension of $\\msf{Mod}(R)$ and $\\mathsf{D}(R)$ in the case that $R$ is a (right) hereditary ring, showing the two are equal. This (among other questions) was furthered by Christensen, Keller and Neeman in \\cite{ckn} to show that if $R$ is a coherent ring such that every finitely presented $R$-module has finite projective dimension, then $\\mathsf{pgdim}(R)\\leq\\mathsf{pgdim}(\\mathsf{D}(R))$. In this paper, the main result in this direction is the following, which removes all assumptions on rings.\n\n\\begin{prop}\\label{prop:hringepi}\nLet $f\\colon R\\to S$ be a homological ring epimorphism, then $\\msf{pgdim}(\\D(S))\\leq \\msf{pgdim}(\\D(R))$.\n\\end{prop}\n\n\\begin{prop}\\label{prop:pgdimineq}\nLet $R$ be a ring. Then $\\msf{pgdim}(\\D(R))\\geq \\msf{pgdim}(R)$.\n\\end{prop}\n\n\\begin{prop}\\label{prop:ringepi}\nLet $f\\colon R\\to S$ be a ring epimorphism. Then $\\msf{pgdim}(S)\\leq \\msf{pgdim}(R)$.\n\\end{prop}", "full_context": "A renowned result in homotopy theory, due to Brown \\cite{brown} and Adams \\cite{adams}, is that the stable homotopy category of spectra $\\mathsf{Sp}$ satisfies Brown--Adams representability. This means that any cohomological functor $F\\colon(\\mathsf{Sp}^{\\omega})^{\\mrm{op}}\\to\\Mod{\\mathbb{Z}}$ on finite spectra $\\mathsf{Sp}^{\\omega}$ is naturally isomorphic to $\\Hom_{\\mathsf{Sp}}(-,X)\\vert_{\\mathsf{Sp}^{\\omega}}$ for some $X\\in\\mathsf{Sp}$, and if $\\varphi\\colon F\\to G$ is a natural transformation between cohomological functors, then $\\varphi\\simeq\\Hom_{\\mathsf{Sp}}(-,f)$, where $f\\colon X\\to Y$ is a map in $\\mathsf{Sp}$ between objects representing $F$ and $G$.\n\nThis statement can be translated into any compactly generated triangulated category $\\mathsf{T}$: if $\\mathsf{T}^{\\mrm{c}}$ denotes the compact objects of $\\mathsf{T}$, one says that $\\mathsf{T}$ satisfies Brown--Adams representability (or is a Brown category), provided every cohomological functor $F\\colon(\\mathsf{T}^{\\mrm{c}})^{\\mrm{op}}\\to\\Mod{\\mathbb{Z}}$ is of the form $\\Hom_{\\mathsf{T}}(-,X)\\vert_{\\mathsf{T}^{\\mrm{c}}}$ for some object $X$, and any map $\\varphi\\colon\\Hom_{\\mathsf{T}}(-,X)\\vert_{\\mathsf{T}^{\\mrm{c}}}\\to\\Hom_{\\mathsf{T}}(-,Y)\\vert_{\\mathsf{T}^{\\mrm{c}}}$ is of the form $\\Hom_{\\mathsf{T}}(-,f)$ for some $f\\colon X\\to Y$. In other words, $\\mathsf{T}$ is a Brown category, by definition, if and only if the restricted Yoneda functor\n\\[\n\\msf{y}\\colon\\mathsf{T}\\to\\Flat{\\mathsf{T}^{\\mrm{c}}}, \\, X\\mapsto \\Hom_{\\mathsf{T}}(-,X)\\vert_{\\mathsf{T}^{\\mrm{c}}}\n\\]\nis essentially surjective and full. Here $\\Flat{\\mathsf{T}^{\\mrm{c}}}$ denotes the category of cohomological functors $(\\mathsf{T}^{\\mrm{c}})^{\\mrm{op}}\\to\\msf{Ab}$, which is expressed this way as it is the ind-completion of the category $\\mathsf{T}^{\\mrm{c}}$.\n\nIt is known that not every compactly generated $\\mathsf{T}$ is a Brown category, and determining when a specific one is, or is not, is not elementary. Criteria to approach this problem were proved by Neeman \\cite{neemantba} and later expanded by Beligiannis \\cite{belrhap}: it was shown that $\\mathsf{T}$ satisfies Brown--Adams representability if and only if $\\mathsf{pgdim}(\\mathsf{T})$, the pure global dimension of $\\mathsf{T}$, is at most one.\n\nIt is through considering pure global dimension, or more specifically its ascent and descent, that we approach the question of whether or not a triangulated category is Brown. Our main investigation is in revisiting the question of when $\\mathsf{D}(R)$ is a Brown category for a ring $R$, and before giving more details and stating results, we provide some historical context.\n\nThe question of $\\mathsf{D}(R)$ being Brown was first considered in Neeman \\cite{neemantba}, where an example credited to Keller showed $\\mathsf{D}(\\mathbb{C}[x,y])$ is not Brown; the pertinent point is that this ring has a flat module of projective dimension at least two. In \\cite{belrhap}, Beligiannis gave some further counterexamples, and provided a relationship between the pure global dimension of $\\msf{Mod}(R)$ and $\\mathsf{D}(R)$ in the case that $R$ is a (right) hereditary ring, showing the two are equal. This (among other questions) was furthered by Christensen, Keller and Neeman in \\cite{ckn} to show that if $R$ is a coherent ring such that every finitely presented $R$-module has finite projective dimension, then $\\mathsf{pgdim}(R)\\leq\\mathsf{pgdim}(\\mathsf{D}(R))$. In this paper, the main result in this direction is the following, which removes all assumptions on rings.\n\n\\begin{prop}\\label{prop:hringepi}\nLet $f\\colon R\\to S$ be a homological ring epimorphism, then $\\msf{pgdim}(\\D(S))\\leq \\msf{pgdim}(\\D(R))$.\n\\end{prop}\n\n\\begin{prop}\\label{prop:pgdimineq}\nLet $R$ be a ring. Then $\\msf{pgdim}(\\D(R))\\geq \\msf{pgdim}(R)$.\n\\end{prop}\n\n\\begin{prop}\\label{prop:ringepi}\nLet $f\\colon R\\to S$ be a ring epimorphism. Then $\\msf{pgdim}(S)\\leq \\msf{pgdim}(R)$.\n\\end{prop}\n\nAs such, we see that the pure global dimension of $\\D(R)$ is bounded below by that of $\\Mod{R}$ for any ring, and this is an optimal lower bound. As an example of a consequence, see \\ref{ex:fdalgebras}, one obtains that a finite dimensional local algebra over an algebraically closed field has a Brown derived category if and only if every module is pure injective. We also note that the first itemised statement recovers some classic results in \\cite{BL}.\n\n\\begin{thmx}\\label{intro:thmc}(\\ref{thm:BAfunctoriality})\nLet $\\T$ and $\\U$ be compactly generated triangulated categories, and suppose $F\\colon\\Flat{\\T^{\\c}}\\to\\Flat{\\U^{\\c}}$ is a definable functor. \n\\begin{enumerate}\n\\item If $F$ is essentially surjective up to direct summands and $\\T$ is a Brown category, so is $\\U$;\n\\item If $F$ is fully faithful and $\\U$ is a Brown category, then so is $\\T$.\n\\end{enumerate}\n\\end{thmx}\n\n\\begin{prop}\\label{prop:ringepi}\nLet $f\\colon R\\to S$ be a ring epimorphism. Then $\\msf{pgdim}(S)\\leq \\msf{pgdim}(R)$.\n\\end{prop}\n\\begin{proof}\nThe fact that $f$ is a ring epimorphism is equivalent to there being an isomorphism $N\\simeq S\\otimes_{R}(N)_{R}$ for any $N\\in\\Mod{S}$, see \\cite[Proposition XI.1.2]{stenstrom}. As such, the isomorphisms of \\cref{eqn:ringmap} give $\\t{PExt}_{R}^{i}((M)_{R},(N)_{R})\\simeq \\t{PExt}_{S}^{i}(M,N)$ for all $M,N\\in\\Mod{S}$, from which it is immediate that $\\msf{pgdim}(S)\\leq \\msf{pgdim}(R)$.\n\\end{proof}\n\n\\begin{lem}\\label{lem:localisation}\nLet $F\\colon \\A\\to \\B$ be a definable functor with left adjoint $\\Lambda$. \n\\begin{enumerate}\n \\item If $F$ is fully faithful, then $\\msf{pgdim}(\\A)\\leq \\msf{pgdim}(\\B)$.\n \\item If $\\Lambda$ is fully faithful, then $\\msf{pgdim}(\\B)\\leq \\msf{pgdim}(\\A)$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nBy \\cref{prop:pextisos}, there are isomorphisms\n\\[\n\\t{PExt}_{\\B}^{n}(FM,FM')\\simeq \\t{PExt}_{\\A}^{n}(\\Lambda FM,M')\n\\]\nand\n\\[\n\\t{PExt}_{\\A}^{n}(\\Lambda N,\\Lambda N')\\simeq \\t{PExt}_{\\B}^{n}(N,F\\Lambda N')\n\\]\nfor all $M,M'\\in\\A$ and $N,N'\\in \\B$. Now, $F$ (respectively $\\Lambda$) is fully faithful if and only if $\\Lambda F\\simeq \\t{Id}_{\\A}$ (respectively $F\\Lambda\\simeq \\t{Id}_{\\B}$), see \\cite[Theorem IV.3.1]{maclane}. The claims are now immediate.\n\\end{proof}\n\n\\begin{prop}\\label{prop:pgdimineq}\nLet $R$ be a ring. Then $\\msf{pgdim}(\\D(R))\\geq \\msf{pgdim}(R)$.\n\\end{prop}\n\\begin{proof}\nConsider the fully faithful functor $i\\colon R\\to \\D(R)^{\\c}$ which, by \\cref{prel:modules}, induces an adjoint triple\n\\[\n\\begin{tikzcd}\n\\Mod{\\D(R)^{\\c}} \\arrow[r, \"i^{*}\" description] \\arrow[r, leftarrow, shift left = 2ex, \"i^{!}\"] \\arrow[r, leftarrow, shift right = 2ex, swap, \"i_{*}\"]& \\Mod{R}. \n\\end{tikzcd}\n\\]\nSince $i$ is fully faithful, so is $i^{!}$, and thus so is $i_{*}$, in other words the above diagram is the right hand side of a recollement. Now, consider $X\\in\\Mod{R}$, then viewing $X$ as an object of $\\D(R)$, we obtain a functor $\\y X\\in\\Flat{\\T^{\\c}}$. Applying \\cref{prop:pextisos}, we obtain isomorphisms\n\\[\n\\t{PExt}_{R}^{n}(M,i^{*}\\y X)\\simeq \\t{PExt}_{\\D(R)^{\\c}}^{n}(i^{!}M,\\y X)\\simeq \\t{Ext}_{\\D(R)^{\\c}}^{n}(i^{!}M,\\y X),\n\\]\nwhere the last isomorphism holds because $\\y X\\in\\Flat{\\T^{\\c}}=\\msf{fpInj}(\\T^{\\c})$. Yet since $i^{*}(F)=F(R)$ for all $F\\in\\Mod{\\T^{\\c}}$, we have that $i^{*}(\\y X)=\\Hom_{\\D(R)}(R,X)\\simeq X$, and thus $\\t{PExt}_{R}^{n}(M,X)\\simeq\\t{Ext}_{\\D(R)^{\\c}}^{n}(i^{!}M,\\y X)$. Consequently, we see that $\\msf{pid}(X\\in\\Mod{R})\\leq \\msf{id}(\\y X)=\\msf{pid}(X\\in\\D(R))$, and so $\\msf{pgdim}(R)\\leq \\msf{pgdim}(\\D(R))$.\n\\end{proof}\n\n\\begin{prop}\\label{prop:hringepi}\nLet $f\\colon R\\to S$ be a homological ring epimorphism, then $\\msf{pgdim}(\\D(S))\\leq \\msf{pgdim}(\\D(R))$.\n\\end{prop}\n\\begin{proof}\nIf $f$ is homological then $\\Phi(-):=(-)_{R}\\colon\\D(S)\\to\\D(R)$ is a product and coproduct preserving triangulated functor, which is therefore definable by \\cite[Proposition 4.18]{bwdeffun}. Let $F\\colon\\Mod{\\D(S)^{\\c}}\\to\\Mod{\\D(R)^{\\c}}$ be the induced functor from \\cref{ex:tridef} extending $\\Phi$. First, note that $F$ has both a right and left adjoint, by \\cite[Theorem 4.21]{bwdeffun}, and the left adjoint restricts to flat objects as $F$ is exact. Let $M,N\\in\\Mod{\\D(S)^{\\c}}$, and write $M\\simeq \\msf{colim}_{I}\\,\\y A_{i}$ as a colimit of projective objects and using \\cref{prel:bergman}, write $N\\simeq \\llim_{J} \\y X_{j}$ as an inverse limit of injective objects, where $A_{i}\\in\\D(S)^{\\c}$ and $X_{j}$ are pure injective. We claim that $F$ is fully faithful. To prove this, observe there are isomorphisms\n\\begin{align*}\n\\Hom(FM,FN)&\\simeq \\Hom(F(\\msf{colim}_{I}\\,\\y A_{i}), F(\\llim_{J}\\y X_{i})) \n\\\\ &\\simeq \\Hom(\\msf{colim}_{I}\\, F\\y A_{i}, \\llim_{J} F\\y X_{j}) \\\\\n&\\simeq \\lim_{I}\\llim_{J}\\Hom(\\y \\Phi A_{i},\\y \\Phi X_{j}) \\\\\n&\\simeq \\lim_{I}\\llim_{J}\\Hom(A_{i},X_{j}) \\simeq \\Hom(M,N),\n\\end{align*}\nwhere the second row isomorphism follows from the fact $F$ preserves all limits and colimits, the third follows from the fact that $F\\y\\simeq \\y f_{*}$, the fourth row from the fact that both $\\y$ and $f_{*}$ are fully faithful on these objects since $X_{j}$ is pure injective, as discussed in \\cref{prel:puretri}. This proves the fully faithfulness of $F$, as claimed. As such, $F\\colon\\Flat{\\D(S)^{\\c}}\\to\\Flat{\\D(R)^{\\c}}$ is fully faithful and definable with a left adjoint, from which we apply \\cref{lem:localisation} to deduce that $\\msf{pgdim}(\\D(S)):=\\msf{pgdim}(\\Flat{\\D(S)^{\\c}})\\leq \\msf{pgdim}(\\Flat{\\D(R)^{\\c}})=\\msf{pgdim}(\\D(R))$, which finishes the proof.\n\\end{proof}\n\n\\begin{thm}\\label{thm:monogenic}\nLet $\\T$ be a monogenic compactly generated triangulated category generated by $S\\in\\T^{\\c}$. If $A=\\mrm{End}_{\\T}(S)$ has a flat module of projective dimension greater than one, then $\\T$ is not a Brown category.\n\\end{thm}\n\\begin{proof}\nConsider the functor $F\\colon \\T\\to \\Mod{A}$ given by $X\\mapsto \\Hom_{\\T}(S,X)$; as $S\\in\\T^{\\c}$, this functor preserves coproducts, products and pure triangles and is thus a coherent functor, in the sense of \\cite[\\S 3]{bwdeffun}. As such, by \\cite[Theorem 3.2]{bwdeffun} there is a unique definable functor $\\hat{F}\\colon\\Flat{\\T^{\\c}}\\to\\Mod{A}$ such that $\\hat{F}\\circ y\\simeq F$. However, since $F$ is cohomological, there is also an adjoint triple\n\\[\n\\begin{tikzcd}[column sep = 2cm]\n\\Mod{\\T^{\\c}} \\arrow[r, \"\\bar{F}\" description] & \\Mod{A} \\arrow[l, shift left = 2ex, hook', \"\\rho\"] \\arrow[l, shift right = 2ex, swap, hook', \"\\Lambda\"]\n\\end{tikzcd}\n\\]\nwhere $\\bar{F}\\circ \\y\\simeq F$ and $\\bar{F}\\vert_{\\Flat{\\T^{\\c}}}=\\hat{F}$. The existence of the triple can either be induced by \\cite[Theorem 3.26]{bwdeffun} and the comments following, or from the fact that $\\bar{F}$ is nothing other than the restriction along the fully faithful embedding $S\\hookrightarrow \\T^{\\c}$, once one views $S$ as a singleton subcategory with endomorphisms $A$. In particular, this viewpoint illustrates $\\Lambda$ and $\\rho$ as fully faithful.\nFurthermore, $\\Lambda$ itself will preserve flat objects, and thus there is an adjoint pair\n\\[\n\\begin{tikzcd}\n\\Flat{\\T^{\\c}} \\arrow[r, shift right = 1ex,swap, \"\\hat{F}\"]& \\Flat{A} \\arrow[l, shift right = 1ex,swap, \"\\Lambda\"]\n\\end{tikzcd}\n\\]\nwith $\\hat{F}$ a definable functor. Since $\\Lambda$ is fully faithful, we may apply \\cref{lem:localisation}, and thus $\\msf{pgdim}(\\T)\\geq \\msf{pgdim}(\\Flat{A})$. The claim now follows immediately from \\cref{neemancrit}.\n\\end{proof}", "post_theorem_intro_text_len": 6883, "post_theorem_intro_text": "As such, we see that the pure global dimension of $\\mathsf{D}(R)$ is bounded below by that of $\\msf{Mod}(R)$ for any ring, and this is an optimal lower bound. As an example of a consequence, see \\ref{ex:fdalgebras}, one obtains that a finite dimensional local algebra over an algebraically closed field has a Brown derived category if and only if every module is pure injective. We also note that the first itemised statement recovers some classic results in \\cite{BL}.\n\nLet us now discuss the pure structure and pure global dimension. The restricted Yoneda embedding $\\msf{y}$ described above sends triangles to long exact sequences; a triangle is sent to a short exact sequence precisely when the connecting morphism vanishes under $\\msf{y}$, that is, the connecting morphism is a phantom map. These are the pure triangles, and thus the pure structure of $\\mathsf{T}$ is the intersection of the short exact sequences in $\\Mod{\\mathsf{T}^{\\mrm{c}}}$, the category of additive functors $(\\mathsf{T}^{\\mrm{c}})^{\\mrm{op}}\\to\\msf{Ab}$, and the triangles in in $\\mathsf{T}$, or equivalently purity can be thought of as the study of phantom maps. \n\nIn fact, $\\msf{y}$ establishes an equivalence between the pure structure on $\\mathsf{T}$ and the pure structure on $\\Flat{\\mathsf{T}^{\\mrm{c}}}$, and the latter has enough pure injective and pure projective objects. Accordingly, one can construct resolutions in $\\Flat{\\mathsf{T}^{\\mrm{c}}}$ with respect to these objects from which one defines the pure projective and pure injective dimensions, which in turn yield the pure global dimension of $\\Flat{\\mathsf{T}^{\\mrm{c}}}$. The pure global dimension of $\\mathsf{T}$ is defined to be the pure global dimension of $\\Flat{\\mathsf{T}^{\\mrm{c}}}$.\n\nThe main tool used to prove \\cref{intro:thma}, among other things, is how pure homological dimension transfers across functors which preserve the pure structure. These are the definable, or interpretation, functors, which, in the triangulated setting were introduced in \\cite{bwdeffun}. A functor $F\\colon\\mathsf{T}\\to \\mathsf{U}$ between compactly generated triangulated categories is definable if (and only if) it preserves coproducts, products, and pure triangles. Any such functor induces a unique product and direct limit preserving functor $\\widehat{F}\\colon\\Flat{\\mathsf{T}^{\\mrm{c}}}\\to\\Flat{\\mathsf{U}^{\\mrm{c}}}$ that is compatible with $\\msf{y}$, and as such, understanding the transfer of pure homological dimension in this accessible setting is good enough to understand what happens on the triangulated level. This is our first main result, which is the workhorse of the paper.\n\n\\begin{thmx}\\label{intro:thmb}(\\ref{prop:pextisos})\nLet $F\\colon\\mathsf{A}\\to\\mathsf{B}$ be a definable functor between finitely accessible categories with products, and suppose that $F$ admits a left adjoint $\\Lambda\\colon\\mathsf{B}\\to\\mathsf{A}$. Then there are isomorphisms\n\\[\n\\mathrm{PExt}_{\\mathsf{B}}^{n}(B,F(A))\\simeq \\mathrm{PExt}_{\\mathsf{A}}^{n}(\\Lambda(B),A)\n\\]\nfor all $A\\in\\mathsf{A}, B\\in\\mathsf{B}$ and $n\\geq 0$.\n\\end{thmx}\n\nHere, $\\text{PExt}$ denotes the derived functors of Hom with respect to pure projective and pure injective resolutions, which measure the pure homological dimension of objects. We note that any definable functor of triangulated categories, as many other definable functors which appear naturally, translate into the above framework. In relation to \\cref{intro:thma}, it is through the above isomorphisms applied to particular choices of definable functors between $\\msf{Mod}(R)$, $\\mathsf{D}(R)$, and $\\mathsf{D}(S)$ that one obtains the result.\n\nHowever, while the inequalities of \\cref{intro:thma} are more suited to illustrate when Brown--Adams representability fails, we are still able to use them, as well as \\cref{intro:thmb}, to provide certain positive examples of when it holds.\n\n\\begin{thmx}\\label{intro:thmc}(\\ref{thm:BAfunctoriality})\nLet $\\mathsf{T}$ and $\\mathsf{U}$ be compactly generated triangulated categories, and suppose $F\\colon\\Flat{\\mathsf{T}^{\\mrm{c}}}\\to\\Flat{\\mathsf{U}^{\\mrm{c}}}$ is a definable functor. \n\\begin{enumerate}\n\\item If $F$ is essentially surjective up to direct summands and $\\mathsf{T}$ is a Brown category, so is $\\mathsf{U}$;\n\\item If $F$ is fully faithful and $\\mathsf{U}$ is a Brown category, then so is $\\mathsf{T}$.\n\\end{enumerate}\n\\end{thmx}\n\nThe combination of \\cref{intro:thma} and \\cref{intro:thmc} also enables a straightforward test as to whether, for example, the category $\\D_{\\text{qc}}(X)$ fails to be a Brown category, where $X$ is a quasi-compact quasi-separated scheme. This is done in \\cref{prop:brownschemes}, where it is shown that if $\\D_{\\text{qc}}(X)$ is Brown, then $\\mathsf{pgdim}(\\mathscr{O}_{X,x})\\leq 1$ for all $x\\in X$. \n\nReturning to derived categories of rings, as \\cref{intro:thma} illustrates, understanding ring epimorphisms is enough to determine whether Brown representability fails. And in fact, for many rings, such as all commutative rings, it is possible to test this on a single ring epimorphism. A classic result of Olivier \\cite{olivier} demonstrates how every commutative ring $A$ admits a unique non-zero epimorphism $A\\to A^{\\text{ab}}$ where $A^{\\text{ab}}$ is a commutative von Neumann regular ring. These rings are particularly relevant since they are those for which pure global dimension coincides with the usual global dimension. Olivier's result was recently expanded to all rings in \\cite{herzinn}, and thus for any ring $R$ there is a ring epimorphism $R\\to R^{\\text{ab}}$ where $R^{\\text{ab}}$ is an abelian von Neumann regular ring (although it can be the case in this generality that $R^{\\text{ab}}$ vanishes). As such, as shown in \\cref{prop:vnrdeterminesbrown}, von Neumann regular rings determine whether derived categories of many rings are Brown categories.\n\nAlong the way to getting to this point, we investigate some parts of the structure of the derived category of von Neumann regular rings. As a consequence, we obtain, among other things, a new and very short proof that a coherent ring satisfies Freyd's generating hypothesis if and only if it is von Neumann regular (see \\cref{prop:fghholds}), as well as a new proof of the telescope conjecture for such rings in \\cref{prop:telescope}. Lastly, in \\cref{sec:beyondderived}, we show how some of the results concerning bounding the pure global dimension derived categories of rings can be extended to, in some cases, all compactly generated triangulated categories; this includes a consideration of the hearts of cosilting t-structures in \\cref{prop:cosilting}.\n\n\\subsection*{Acknowledgements}\nThis research was supported by project PRIMUS/23/SCI/006 from Charles University, and by Charles University Research Centre program No. UNCE/24/SCI/022. I am grateful to M. Prest, S. Virili, J. Williamson and A. Zvonareva for constructive comments.", "sketch": "The post-theorem discussion indicates that the proof of \\cref{intro:thma} is obtained by transferring pure homological dimension across functors that preserve the pure structure. Concretely:\n\\begin{itemize}\n\\item One interprets purity in a compactly generated triangulated category $\\mathsf{T}$ via the restricted Yoneda embedding $\\msf{y}$: it sends triangles to long exact sequences, and a triangle goes to a short exact sequence precisely when the connecting morphism is a phantom map; these are the \\emph{pure triangles}. Thus “purity can be thought of as the study of phantom maps.”\n\\item The embedding $\\msf{y}$ “establishes an equivalence between the pure structure on $\\mathsf{T}$ and the pure structure on $\\Flat{\\mathsf{T}^{\\mrm{c}}}$,” where there are enough pure injectives/projectives. One therefore defines pure projective/injective dimensions and the pure global dimension on $\\Flat{\\mathsf{T}^{\\mrm{c}}}$, and “the pure global dimension of $\\mathsf{T}$ is defined to be the pure global dimension of $\\Flat{\\mathsf{T}^{\\mrm{c}}}$.”\n\\item The “main tool used to prove \\cref{intro:thma}” is to analyze how pure homological dimension transfers along \\emph{definable} (interpretation) functors, i.e. functors preserving coproducts, products, and pure triangles. Any such $F\\colon\\mathsf{T}\\to\\mathsf{U}$ induces a functor $\\widehat{F}\\colon\\Flat{\\mathsf{T}^{\\mrm{c}}}\\to\\Flat{\\mathsf{U}^{\\mrm{c}}}$ compatible with $\\msf{y}$, so it suffices to work in this accessible setting.\n\\item The transfer is implemented via \\cref{intro:thmb}: for a definable $F\\colon\\mathsf{A}\\to\\mathsf{B}$ with left adjoint $\\Lambda$, there are isomorphisms\n\\[\\mathrm{PExt}_{\\mathsf{B}}^{n}(B,F(A))\\simeq \\mathrm{PExt}_{\\mathsf{A}}^{n}(\\Lambda(B),A)\\]\nfor all $n\\ge 0$, where $\\mathrm{PExt}$ is computed using pure projective/pure injective resolutions and “measure[s] the pure homological dimension of objects.”\n\\item “In relation to \\cref{intro:thma}, it is through the above isomorphisms applied to particular choices of definable functors between $\\msf{Mod}(R)$, $\\mathsf{D}(R)$, and $\\mathsf{D}(S)$ that one obtains the result.”\n\\end{itemize}\nThis is the only proof-outline content given for \\cref{intro:thma} in the provided post-theorem text.", "expanded_sketch": "The post-theorem discussion indicates that the proof of the main theorem is obtained by transferring pure homological dimension across functors that preserve the pure structure. Concretely:\n\\begin{itemize}\n\\item One interprets purity in a compactly generated triangulated category $\\mathsf{T}$ via the restricted Yoneda embedding $\\msf{y}$: it sends triangles to long exact sequences, and a triangle goes to a short exact sequence precisely when the connecting morphism is a phantom map; these are the \\emph{pure triangles}. Thus “purity can be thought of as the study of phantom maps.”\n\\item The embedding $\\msf{y}$ “establishes an equivalence between the pure structure on $\\mathsf{T}$ and the pure structure on $\\Flat{\\mathsf{T}^{\\mrm{c}}}$,” where there are enough pure injectives/projectives. One therefore defines pure projective/injective dimensions and the pure global dimension on $\\Flat{\\mathsf{T}^{\\mrm{c}}}$, and “the pure global dimension of $\\mathsf{T}$ is defined to be the pure global dimension of $\\Flat{\\mathsf{T}^{\\mrm{c}}}$.\"\n\\item The “main tool used to prove the main theorem” is to analyze how pure homological dimension transfers along \\emph{definable} (interpretation) functors, i.e. functors preserving coproducts, products, and pure triangles. Any such $F\\colon\\mathsf{T}\\to\\mathsf{U}$ induces a functor $\\widehat{F}\\colon\\Flat{\\mathsf{T}^{\\mrm{c}}}\\to\\Flat{\\mathsf{U}^{\\mrm{c}}}$ compatible with $\\msf{y}$, so it suffices to work in this accessible setting.\n\\item The transfer is implemented via the following theorem.\n\n\\begin{thmx}\\label{intro:thmb}(\\ref{prop:pextisos})\nLet $F\\colon\\A\\to\\B$ be a definable functor between finitely accessible categories with products, and suppose that $F$ admits a left adjoint $\\Lambda\\colon\\B\\to\\A$. Then there are isomorphisms\n\\[\n\\mrm{PExt}_{\\msf{B}}^{n}(B,F(A))\\simeq \\mrm{PExt}_{\\A}^{n}(\\Lambda(B),A)\n\\]\nfor all $A\\in\\A, B\\in\\B$ and $n\\geq 0$.\n\\end{thmx}\n\nIn particular, for a definable $F\\colon\\mathsf{A}\\to\\mathsf{B}$ with left adjoint $\\Lambda$, there are isomorphisms\n\\[\\mathrm{PExt}_{\\mathsf{B}}^{n}(B,F(A))\\simeq \\mathrm{PExt}_{\\mathsf{A}}^{n}(\\Lambda(B),A)\\]\nfor all $n\\ge 0$, where $\\mathrm{PExt}$ is computed using pure projective/pure injective resolutions and “measure[s] the pure homological dimension of objects.”\n\\item “In relation to the main theorem, it is through the above isomorphisms applied to particular choices of definable functors between $\\msf{Mod}(R)$, $\\mathsf{D}(R)$, and $\\mathsf{D}(S)$ that one obtains the result.”\n\\end{itemize}\nThis is the only proof-outline content given for the main theorem in the provided post-theorem text.", "expanded_theorem": "\\label{intro:thma}(\\begin{prop}\\label{prop:pgdimineq}\nLet $R$ be a ring. Then $\\msf{pgdim}(\\D(R))\\geq \\msf{pgdim}(R)$.\n\\end{prop},\\begin{prop}\\label{prop:ringepi}\nLet $f\\colon R\\to S$ be a ring epimorphism. Then $\\msf{pgdim}(S)\\leq \\msf{pgdim}(R)$.\n\\end{prop}, \\begin{prop}\\label{prop:hringepi}\nLet $f\\colon R\\to S$ be a homological ring epimorphism, then $\\msf{pgdim}(\\D(S))\\leq \\msf{pgdim}(\\D(R))$.\n\\end{prop})\nLet $R$ be a ring. Then $\\msf{pgdim}(\\D(R))\\geq \\msf{pgdim}(R)$. As such, $\\mathsf{D}(R)$ is not a brown category whenever $\\mathsf{pgdim}(R)\\geq 2$. Moreover, if $f\\colon R\\to S$ a ring homomorphism, then\n\\begin{enumerate}\n\\item if $f$ is a ring epimorphism, one has $\\mathsf{pgdim}(S)\\leq \\mathsf{pgdim}(R)$;\n\\item if $f$ is a homological ring epimorphism, one has $\\mathsf{pgdim}(\\mathsf{D}(S))\\leq \\mathsf{pgdim}(\\mathsf{D}(R))$.\n\\end{enumerate}", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let $R$ be a ring, and write $\\mathsf{D}(R)$ for the derived category of $R$-modules. Let $\\mathsf{pgdim}(R)$ denote the pure global dimension of the module category $\\mathrm{Mod}(R)$, and let $\\mathsf{pgdim}(\\mathsf{D}(R))$ denote the pure global dimension of the compactly generated triangulated category $\\mathsf{D}(R)$. Recall that a compactly generated triangulated category is called a Brown category if it satisfies Brown--Adams representability. Now let $f\\colon R\\to S$ be a ring homomorphism; if $f$ is a ring epimorphism, this means it is an epimorphism in the category of rings, and if $f$ is a homological ring epimorphism, this means it is a ring epimorphism such that $\\operatorname{Tor}_i^R(S,S)=0$ for all $i>0$. Under these assumptions, which statement is valid?", "correct_choice": {"label": "A", "text": "One has $\\mathsf{pgdim}(\\mathsf{D}(R))\\geq \\mathsf{pgdim}(R)$. Consequently, if $\\mathsf{pgdim}(R)\\geq 2$, then $\\mathsf{D}(R)$ is not a Brown category. Moreover, for a ring homomorphism $f\\colon R\\to S$: if $f$ is a ring epimorphism, then $\\mathsf{pgdim}(S)\\leq \\mathsf{pgdim}(R)$; and if $f$ is a homological ring epimorphism, then $\\mathsf{pgdim}(\\mathsf{D}(S))\\leq \\mathsf{pgdim}(\\mathsf{D}(R))$."}, "choices": [{"label": "B", "text": "One has $\\mathsf{pgdim}(\\mathsf{D}(R))= \\mathsf{pgdim}(R)$. Consequently, if $\\mathsf{pgdim}(R)\\geq 2$, then $\\mathsf{D}(R)$ is not a Brown category. Moreover, for a ring homomorphism $f\\colon R\\to S$: if $f$ is a ring epimorphism, then $\\mathsf{pgdim}(S)=\\mathsf{pgdim}(R)$; and if $f$ is a homological ring epimorphism, then $\\mathsf{pgdim}(\\mathsf{D}(S))=\\mathsf{pgdim}(\\mathsf{D}(R))$."}, {"label": "C", "text": "One has $\\mathsf{pgdim}(\\mathsf{D}(R))\\geq \\mathsf{pgdim}(R)$. Consequently, if $\\mathsf{pgdim}(R)\\geq 2$, then $\\mathsf{D}(R)$ is not a Brown category."}, {"label": "D", "text": "One has $\\mathsf{pgdim}(\\mathsf{D}(R))\\leq \\mathsf{pgdim}(R)$. Consequently, if $\\mathsf{pgdim}(\\mathsf{D}(R))\\geq 2$, then $R$ has pure global dimension at least $2$. Moreover, for a ring homomorphism $f\\colon R\\to S$: if $f$ is a ring epimorphism, then $\\mathsf{pgdim}(S)\\leq \\mathsf{pgdim}(R)$; and if $f$ is a homological ring epimorphism, then $\\mathsf{pgdim}(\\mathsf{D}(S))\\leq \\mathsf{pgdim}(\\mathsf{D}(R))$."}, {"label": "E", "text": "One has $\\mathsf{pgdim}(\\mathsf{D}(R))\\geq \\mathsf{pgdim}(R)$. Consequently, if $\\mathsf{pgdim}(R)\\geq 2$, then $\\mathsf{D}(R)$ is not a Brown category. Moreover, for a ring homomorphism $f\\colon R\\to S$: if $f$ is a ring epimorphism, then $\\mathsf{pgdim}(\\mathsf{D}(S))\\leq \\mathsf{pgdim}(\\mathsf{D}(R))$; and if $f$ is a homological ring epimorphism, then $\\mathsf{pgdim}(S)\\leq \\mathsf{pgdim}(R)$."}], "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": "inequality_vs_equality_under_definable_transfer", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "ring_epimorphism_and_homological_ring_epimorphism_consequences", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "direction_of_pgdim_comparison_between_ModR_and_DR", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "which_conclusion_requires_homological_ring_epimorphism", "template_used": "wildcard"}]}} +{"id": "2601.10171v1", "paper_link": "http://arxiv.org/abs/2601.10171v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{theorem:main}\n A $3$-connected planar graph $G$ has exactly one orientable circuit double cover if and only if $G$ is an Apollonian dual.", "start_pos": 4718, "end_pos": 4894, "label": "theorem:main"}, "ref_dict": {"def:completeaug": "\\begin{definition}\\label{def:completeaug}\nLet $G$ be a $3$-connected planar graph. The graph obtained by augmenting each face of $G$ is denoted by $G^a$ and referred to as the \\textbf{\\emph{complete augmentation}} of $G.$\n\\end{definition}", "def:completetruncation": "\\begin{definition}\\label{def:completetruncation}\nLet $G$ be a $3$-connected planar graph with $V(G)=\\{v_1,\\ldots,v_n\\}.$\nFirst, we define $G_0:=G$ and recursively construct graphs $G_1,\\ldots,G_n$ as follows:\nFor $0\\leq i\\leq n-1$ let $G_i$ already be constructed. \nThe vertex $v_{i+1}$ can be interpreted as a vertex of the graph $G_i$.\nThus, we define the graph $G_{i+1}$ as the graph that results from $G_i$ by applying a truncation at the vertex $v_{i+1}$.\nWe call $G^t:= G_n$ the \\textbf{\\emph{complete truncation}} of $G.$\n\\end{definition}", "fig:proofidea": "\\begin{tikzpicture}[scale=1.6]\n \\tikzset{knoten/.style={circle,fill=black,inner sep=0.6mm}}\n \\node (a1) at (0.3,0) {};\n \\node (a2) at (1.7,0) {};\n \\node (b1) at (0,0.3) {};\n \\node (b2) at (0,1.7) {};\n \\node (c1) at (2,0.3) {};\n \\node (c2) at (2,1.7) {};\n \\node (d1) at (0.3,2) {};\n \\node (d2) at (1.7,2) {};\n\n \\draw[->,thick] (a1) to (a2);\n \\draw[<-,thick] (b1) to (b2); \n \\draw[<-,thick] (c1) to (c2);\n \\draw[->,thick] (d1) to (d2); \n \\node at (0,0) {\\Large $G^\\ast$};\n \\node at (0,2) {\\Large $G$};\n \\node at (2,2) {\\Large $G^t$};\n \\node at (2,0) {\\Large $G'$};\n \\node at (1,-0.25) {\\Large $(\\cdot)^a$};\n \\node at (1,2.25) {\\Large $(\\cdot)^t$};\n \\node at (-0.25,1) {\\Large $(\\cdot)^\\ast$};\n \\node at (2.25,1) {\\Large $(\\cdot)^\\ast$};\n\n \\end{tikzpicture}\n \\caption{Commuting diagram describing the idea of $G'\\cong (G^\\ast)^a\\cong (G^t)^\\ast $}\n \\label{fig:proofidea}\n\\end{figure}\n\n\\begin{figure}[H]\n \\centering\n \\begin{subfigure}{0.4\\textwidth}\n \\begin{tikzpicture}[vertexBall, edgeDouble, faceStyle, scale=2]\n \\tikzset{knoten/.style={circle,fill=black,inner sep=0.6mm}}\n\n \\draw[-,very thick] (-4,0) arc [start angle=180, end angle=0, x radius=1, y radius=1.5];\n\n \\node [knoten,label=right:$v_i$] (V1_3) at (1-3,0) {};\n \\node [knoten,label=$v_k$] (V2_3) at (0.4999999999999999-3, 0.8660254037844386) {};\n \\node [knoten,label=$v_{s_k}$] (V3_3) at (-0.5-3, 0.8660254037844388) {};\n \\node [knoten,label=left:$v_{s_i}$] (V4_3) at (-1-3, 0.) {};\n \\node [knoten] (V5_3) at (-.5-3, -0.8660254037844386) {};\n \\node [knoten] (V6_3) at (0.5000000000000001-3, -0.8660254037844386) {};\n \\node [knoten] (V7_3) at (-3, 0.) {};\n \\node at (-2.7,0.12) {$v_F$};\n\n \\draw[-, very thick] (V1_3) to (V2_3);\n \\draw[-, very thick] (V2_3) to (V3_3);\n \\draw[-, very thick] (V3_3) to (V4_3);\n \\draw[-, very thick] (V4_3) to (V5_3);\n \\draw[-, very thick] (V5_3) to (V6_3);\n \\draw[-, very thick] (V6_3) to (V1_3);\n\n \\draw[-, very thick] (V7_3) to (V1_3);\n \\draw[-, very thick] (V7_3) to (V2_3);\n \\draw[-, very thick] (V7_3) to (V3_3);\n \\draw[-, very thick] (V7_3) to (V4_3);\n \\draw[-, very thick] (V7_3) to (V5_3);\n \\draw[-, very thick] (V7_3) to (V6_3);\n\n \\end{tikzpicture}"}, "pre_theorem_intro_text_len": 2136, "pre_theorem_intro_text": "In 1985, F.\\ Jaeger posed the famous \\textbf{Orientable Strong Embedding Conjecture}, see~\\cite{CycleDoubleCoverConjecture}.\nThis conjecture asserts that every $2$-connected graph has a strong embedding on some orientable surface. Although the conjecture remains unsolved, various partial results on this and related problems have been established over the years. \nFor some results the reader is referred to~\\cite{ELLINGHAM2011495,huseksamal,HAGGKVIST2006183,NEGAMI1988276,ZHA1995259}. The above conjecture has led to different investigations of embeddings of planar graphs on non-spherical surfaces. For instance, \\cite{richter} establishes\nthat every $3$-connected planar graph has a strong embedding on some non-spherical surface. \nIn a recent paper we have investigated the \\textbf{Apollonian duals}, i.e.\\ the dual graphs of Apollonian networks in the context of the Orientable Strong Embedding Conjecture, see~\\cite{UnserPaper}. Here, an \\textbf{Apollonian network} is a planar graph that can be constructed from recursively subdividing the faces of the complete graph $K_4$ into three new faces, see ~\\cite{apolloniannetwork_Intro,fowler,grünbaum} for some references.\nWe have exploited the results on re-embeddings of $3$-connected cubic planar graphs established in \\cite{EnamiEmbeddings,PaperMeikeStrong} to show that Apollonian duals are exactly the $3$-connected cubic planar graphs with exactly one orientable strong embedding. Given the fact that all 3-connected cubic planar graphs with a unique strong orientable embedding are therefore fully classified, the following question naturally arises:\n\\begin{question*}\n Which $3$-connected planar graphs have exactly one orientable strong embedding?\n\\end{question*}\nA strong embedding of a given bridgeless graph can be described by a circuit double cover. In our work, a circuit denotes a connected subgraph in which every vertex has even degree.\n Hence, if a cubic graph has a circuit double cover, the circuits of the cover have to be cycles, i.e.\\ connected $2$-regular subgraphs.\nThese observations allow us to extend \\cite[Theorem 5.4]{UnserPaper} in the following way:", "context": "In 1985, F.\\ Jaeger posed the famous \\textbf{Orientable Strong Embedding Conjecture}, see~\\cite{CycleDoubleCoverConjecture}.\nThis conjecture asserts that every $2$-connected graph has a strong embedding on some orientable surface. Although the conjecture remains unsolved, various partial results on this and related problems have been established over the years. \nFor some results the reader is referred to~\\cite{ELLINGHAM2011495,huseksamal,HAGGKVIST2006183,NEGAMI1988276,ZHA1995259}. The above conjecture has led to different investigations of embeddings of planar graphs on non-spherical surfaces. For instance, \\cite{richter} establishes\nthat every $3$-connected planar graph has a strong embedding on some non-spherical surface. \nIn a recent paper we have investigated the \\textbf{Apollonian duals}, i.e.\\ the dual graphs of Apollonian networks in the context of the Orientable Strong Embedding Conjecture, see~\\cite{UnserPaper}. Here, an \\textbf{Apollonian network} is a planar graph that can be constructed from recursively subdividing the faces of the complete graph $K_4$ into three new faces, see ~\\cite{apolloniannetwork_Intro,fowler,grünbaum} for some references.\nWe have exploited the results on re-embeddings of $3$-connected cubic planar graphs established in \\cite{EnamiEmbeddings,PaperMeikeStrong} to show that Apollonian duals are exactly the $3$-connected cubic planar graphs with exactly one orientable strong embedding. Given the fact that all 3-connected cubic planar graphs with a unique strong orientable embedding are therefore fully classified, the following question naturally arises:\n\\begin{question*}\n Which $3$-connected planar graphs have exactly one orientable strong embedding?\n\\end{question*}\nA strong embedding of a given bridgeless graph can be described by a circuit double cover. In our work, a circuit denotes a connected subgraph in which every vertex has even degree.\n Hence, if a cubic graph has a circuit double cover, the circuits of the cover have to be cycles, i.e.\\ connected $2$-regular subgraphs.\nThese observations allow us to extend \\cite[Theorem 5.4]{UnserPaper} in the following way:", "full_context": "In 1985, F.\\ Jaeger posed the famous \\textbf{Orientable Strong Embedding Conjecture}, see~\\cite{CycleDoubleCoverConjecture}.\nThis conjecture asserts that every $2$-connected graph has a strong embedding on some orientable surface. Although the conjecture remains unsolved, various partial results on this and related problems have been established over the years. \nFor some results the reader is referred to~\\cite{ELLINGHAM2011495,huseksamal,HAGGKVIST2006183,NEGAMI1988276,ZHA1995259}. The above conjecture has led to different investigations of embeddings of planar graphs on non-spherical surfaces. For instance, \\cite{richter} establishes\nthat every $3$-connected planar graph has a strong embedding on some non-spherical surface. \nIn a recent paper we have investigated the \\textbf{Apollonian duals}, i.e.\\ the dual graphs of Apollonian networks in the context of the Orientable Strong Embedding Conjecture, see~\\cite{UnserPaper}. Here, an \\textbf{Apollonian network} is a planar graph that can be constructed from recursively subdividing the faces of the complete graph $K_4$ into three new faces, see ~\\cite{apolloniannetwork_Intro,fowler,grünbaum} for some references.\nWe have exploited the results on re-embeddings of $3$-connected cubic planar graphs established in \\cite{EnamiEmbeddings,PaperMeikeStrong} to show that Apollonian duals are exactly the $3$-connected cubic planar graphs with exactly one orientable strong embedding. Given the fact that all 3-connected cubic planar graphs with a unique strong orientable embedding are therefore fully classified, the following question naturally arises:\n\\begin{question*}\n Which $3$-connected planar graphs have exactly one orientable strong embedding?\n\\end{question*}\nA strong embedding of a given bridgeless graph can be described by a circuit double cover. In our work, a circuit denotes a connected subgraph in which every vertex has even degree.\n Hence, if a cubic graph has a circuit double cover, the circuits of the cover have to be cycles, i.e.\\ connected $2$-regular subgraphs.\nThese observations allow us to extend \\cite[Theorem 5.4]{UnserPaper} in the following way:\n\n\\begin{abstract}\nA circuit double cover of a bridgeless graph is a collection of even subgraphs such that every edge is contained in exactly two subgraphs of the given collection. Such a circuit double cover describes an embedding of the corresponding graph onto a surface. In this paper, we investigate the well-known Orientable Strong Embedding Conjecture. \nThis conjecture proposes that every bridgeless graph has a circuit double cover describing an embedding on an orientable surface. In a recent paper, we have proved that a $3$-connected cubic planar graph $G$ has exactly one orientable circuit double cover if and only if $G$ is the dual graph of an Apollonian network. In this paper, we extend this result by demonstrating that this characterisation applies to any 3-connected planar graph, regardless of whether it is cubic.\n\\end{abstract}\n\n\\section{Preliminaries}\\label{sec:preliminaries}\nWe start this work by introducing some preliminary notions on graphs.\nFor a more detailed description of the theoretical background needed for this paper, we refer the reader to \\cite[Section~2]{UnserPaper}. Here, we assume that all the graphs in this paper are undirected, connected, simple and finite. Let $G$ be such a graph. We denote the vertex and edge set of $G$ by $V(G)$ and $E(G)$, respectively. Next, we formalise the notions of circuits and cycles in a graph. While their definitions vary across the literature, we adopt the following conventions in this work:\nA \\textbf{circuit} in $G$ is an even subgraph of $G$. In the case that a given circuit in $G$ is a 2-regular graph, it is called a \\textbf{cycle} in $G$. A \\textbf{triangle} in $G$ is a cycle of length three. We refer to a triangle $(v_1,v_2,v_3)$ of $G$ as \\textbf{separating}, if $G\\setminus\\{v_1,v_2,v_3\\}$ is no longer connected.\nMoreover, a \\textbf{circuit double cover} of $G$ is a set of circuits in $G$ such that each edge of $E(G)$ is contained in exactly two of these circuits. \nSuch a circuit double cover is called \\textbf{orientable} if there exists an orientation for each of the circuits in the cover, such that for every edge $e$ the two circuits covering $e$ are oriented in opposite directions through $e$. A \\textbf{cycle double cover} and an \\textbf{orientable cycle double cover} are defined analogously by replacing circuits with cycles.\nNote that, in a cubic graph, all circuits are cycles, since all vertices have a degree of three.\nIn order to give an example of a circuit double cover that does not form a cycle double cover, we consider the graph $K_{2,2,2}.$\n\\begin{figure}[H]\n \\centering\n \\input{Figures/oct}\n \\label{fig:placeholder}\n\\end{figure}\nThis graph can be equipped with the following circuit double cover:\n\\begin{align*}\n \\{(1,2,3,1,4,5),(1,2,5,1,3,4),(6,2,3,6,4,5),(6,2,5,6,3,4)\\}.\n\\end{align*}\nWe observe that the circuits of the above set do not form cycles of $K_{2,2,2}$.\n\n\\section{Modifying planar graphs}\\label{sec:modification}\nIn this section, we discuss the complete truncation and the complete augmentation of $3$-connected planar graphs.\nThese modifications will be used to prove that Apollonian duals are exactly the $3$-connected planar graphs with exactly one orientable circuit double cover, see \\Cref{theorem:main}. In order to conclude this result we show that the mentioned modifications preserve the planarity and the $3$-connectivity of a given graph.\n\nFinally, we investigate the class of Apollonian duals and show that these graphs are exactly the $3$-connected planar graphs with exactly one orientable circuit double cover. We need some preparation to conclude this result. We start by showing that the edges of an Apollonian network fall into two classes.\nTo formalise this, we introduce the inverse of augmenting a face of a given $3$-connected planar graph which is the deletion of a vertex. Given a $3$-connected planar graph $G$ and a vertex $v\\in V(G)$, we denote by $G_v$ the graph obtained by removing $v$ along with all edges incident to $v$ from $G$.\n\n\\begin{proposition}\\label{prop:sep3deg}\n Let $G$ be an Apollonian network and $e\\in E(G)$ an edge. Then $e$ is either contained in a separating triangle of $G$ or incident to a vertex of degree three.\n\\end{proposition}\n\\begin{proof}\n We verify this statement by induction over $n:=\\vert V(G)\\vert\\geq 4$. If $n=4,$ then $G$ is isomorphic to the complete graph $K_4$. Hence, it is easy to see that every edge in $E(G)$ is incident to a vertex of degree three. \n Now, we assume that $n>4$ holds. Thus, $G$ contains a vertex $v\\in V(G)$ with $\\deg(v)=3$ such that $G_v$ is a well-defined Apollonian network. Thus, $v$ has three neighbours in $G$, namely $w_1,w_2$ and $w_3$. \n By our inductive argument, every edge in $E(G_v)$ is either contained in a separating triangle or incident to a vertex of degree three in $G_v$. In $G$, all edges except the edges $\\{w_i,w_j\\}$ for $i,j\\in\\{1,2,3\\}$ and $i\\neq j$ are therefore also contained in a separating triangle or incident to a vertex of degree three in $G$.\n Moreover, the edges $\\{w_i,w_j\\}$ for $i,j\\in\\{1,2,3\\}$ and $i\\neq j$ form a separating triangle in $G$. Finally, the edges incident to $v$ in $G$, namely $\\{v,v_1\\},\\{v,v_2\\},\\{v,v_3\\}$ are incident to a vertex of degree three. Thus, the result follows.\n\\end{proof}\nNote that this fact follows directly from the fact that these objects correspond exactly to stacked $2$-spheres, see \\cite{MR4406230} for example. Nevertheless, for completeness, we include a proof in the framework of the present paper.\nNext, we show that dualising, truncating and augmenting can be composed to obtain isomorphic graphs. To proceed, we require the definition of the dual graph. The \\textbf{dual} of a planar graph $G$, denoted by $G^\\ast$, is constructed by replacing each face with a vertex with two vertices adjacent whenever their corresponding faces share an edge. This means that the vertices of $G$ are translated to faces of $G^\\ast$. We show that if $G$ is a $3$-connected planar graph, then the complete augmentation $(G^\\ast)^a$ of $G^\\ast$ forms a graph that is isomorphic to the dual $(G^t)^\\ast$ of the complete truncation $G^t$. This statement is further illustrated in \\Cref{fig:proofidea}.\nThis statement is classical in the setting of polyhedra (see, for example, \\cite{conway}), and by Steinitz’s theorem \\cite{Steinitz} it extends to 3‑connected planar graphs. However, in this paper, we include a proof of this statement for completeness.\n\\input{Figures/diagram}\n\nThe next lemma turns out to be really helpful as it states that orientable circuit double covers of the complete truncation of a 3-connected planar graph $G$ can be translated into orientable circuit double covers of $G$.\n\\begin{lemma}\\label{lemma:transCDC}\n Let $G$ be a 3-connected planar graph. If $G^t$ has an orientable circuit double cover, then $G$ has an orientable circuit double cover.\n\\end{lemma}\n\nTo conclude the result, let us now assume that the other direction does not hold. This implies the existence of a $3$-connected planar graph $G$ that is not isomorphic to an Apollonian dual and that has exactly one orientable circuit double cover. Note that $G$ cannot be cubic because of~\\cite[Theorem 5.4]{UnserPaper}. Furthermore, we notice that different orientable cycle double covers of the complete truncation $G^t$ correspond to different orientable circuit double covers of $G$. By \\Cref{cor:numberscdc} $G^t$ has to be a $3$-connected cubic planar graph with exactly one orientable cycle double cover, i.e.\\ an Apollonian dual by \\cite{UnserPaper}.\nConsequently, $(G^t)^\\ast$ is an Apollonian network. By \\Cref{theorem:dualtruncatedaug}, the graph $(G^\\ast)^a$ is isomorphic to $(G^t)^\\ast$, and therefore $(G^\\ast)^a$ is an Apollonian network. In the following we show that applying a complete augmentation to $G^\\ast$ cannot result in an Apollonian network.", "post_theorem_intro_text_len": 1293, "post_theorem_intro_text": "Our paper is structured as follows: In \\Cref{sec:preliminaries} we introduce notions on graphs and their embeddings which are essential for this work.\nTwo modifications of $3$-connected planar graphs, namely the complete augmentation (\\Cref{def:completeaug}) and the complete truncation (\\Cref{def:completetruncation}) are presented in \\Cref{sec:modification}. Since a graph resulting from one of the above modifications is still $3$-connected and planar, these modifications play a central role in this work. We make use of them to prove the main result of this work.\nIn \\Cref{sec:orientable}, we employ these modifications and demonstrate that they are well-behaved under the dualisation of a $3$-connected planar graph in the following sense: We show that the graph resulting from the complete augmentation of the dual of a $3$-connected planar graph $G$ is isomorphic to the dual of the graph resulting from the complete truncation of $G$, see \\Cref{fig:proofidea} for an illustration of the above statement. We use this statement to show that the Apollonian duals are exactly the $3$-connected planar graphs with exactly one circuit double cover. Note that we investigated Apollonian networks and their properties using the computer algebra systems GAP \\cite{GAP4} and Magma \\cite{magma}.", "sketch": "To prove Theorem~\\ref{theorem:main}, the paper introduces two modifications of $3$-connected planar graphs, the \\emph{complete augmentation} and the \\emph{complete truncation}, and notes that since the resulting graphs are still $3$-connected and planar, these modifications \"play a central role\" and are used \"to prove the main result\". In \\Cref{sec:orientable} it is shown that these modifications are \"well-behaved under the dualisation\" in the sense that \"the graph resulting from the complete augmentation of the dual of a $3$-connected planar graph $G$ is isomorphic to the dual of the graph resulting from the complete truncation of $G$\" (cf. \\Cref{fig:proofidea}). This statement is then used to show that \"the Apollonian duals are exactly the $3$-connected planar graphs with exactly one circuit double cover,\" yielding Theorem~\\ref{theorem:main}.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{theorem:main}\n A $3$-connected planar graph $G$ has exactly one orientable circuit double cover if and only if $G$ is an Apollonian dual.", "theorem_type": ["Biconditional or Equivalence", "Universal"], "mcq": {"question": "Let $G$ be a $3$-connected planar graph. A circuit double cover of $G$ is a collection of connected even subgraphs such that every edge of $G$ lies in exactly two of them; it is orientable if these subgraphs can be oriented so that, for each edge, the two subgraphs containing that edge traverse it in opposite directions. An Apollonian dual is the dual graph of an Apollonian network, where an Apollonian network is a planar graph obtained from $K_4$ by recursively subdividing faces into three new faces. Which of the following statements is equivalent to $G$ having exactly one orientable circuit double cover?", "correct_choice": {"label": "A", "text": "$G$ is an Apollonian dual."}, "choices": [{"label": "B", "text": "$G$ is the dual graph of a $3$-connected planar graph obtained from $K_4$ by recursively subdividing faces into three new faces."}, {"label": "C", "text": "$G$ has at least one orientable circuit double cover."}, {"label": "D", "text": "$G$ is a planar graph with exactly one orientable circuit double cover."}, {"label": "E", "text": "$G$ is an Apollonian network."}], "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": "dual-vs-original object identification", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "uniqueness of the orientable circuit double cover", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "3-connectedness hypothesis", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "Apollonian dual replaced by Apollonian network", "template_used": "property_confusion"}]}} +{"id": "2601.10298v1", "paper_link": "http://arxiv.org/abs/2601.10298v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "\\label{thm-KF}\nSuppose that $K$ is a number field and $e$ a positive integer. For almost all $\\sigma \\in G_K^e$, any finite extension of $\\overline{K}(\\sigma)$ is both torally Kummer-faithful and AVKF, and thus, Kummer-faithful.", "start_pos": 6541, "end_pos": 6791, "label": "thm-KF"}, "ref_dict": {"cor-thm-KF": "\\begin{corollary}\\label{cor-thm-KF}\nLet $K$ be a number field.\nThere exists an algebraic extension of $K$ that is Kummer-faithful and has abelian absolute Galois group.\n\\end{corollary}", "thm-KF": "\\begin{theorem}\\label{thm-KF}\nSuppose that $K$ is a number field and $e$ a positive integer. For almost all $\\sigma \\in G_K^e$, any finite extension of $\\Kbar(\\sigma)$ is both torally Kummer-faithful and AVKF, and thus, Kummer-faithful.\n\\end{theorem}", "cor-continuummany": "\\begin{corollary}\\label{cor-continuummany}\nLet $K$ be a number field.\nThere are continuum many algebraic extensions of $K$ up to $K$-isomorphism which are Kummer-faithful fields with abelian absolute Galois group.\n\\end{corollary}"}, "pre_theorem_intro_text_len": 4476, "pre_theorem_intro_text": "\\label{sec-intro}\n\nThis paper focuses on algebraic extensions of a number field whose absolute Galois group is finitely generated.\nFix an algebraic closure $\\overline{K}$ of any perfect field $K$ and let $G_K$ be the absolute Galois group $\\Gal(\\overline{K} / K)$ of $K$.\nLet $e$ be a positive integer.\nFor any $\\sigma = (\\sigma_1, \\ldots, \\sigma_e) \\in G_K^e$ (the direct product of $e$ copies of $G_K$), set $\\overline{K}(\\sigma)$ to be the fixed field of $\\sigma$ in $\\overline{K}$, i.e.,\n\\[ \\overline{K}(\\sigma) = \\{x \\in \\overline{K} \\mid \\sigma_i(x) = x \\text{ for all } i\\}. \\]\nEvery field with finite transcendence degree over the field $\\mathbb{Q}$ of rational numbers and with finitely generated absolute Galois group is of this form.\nWe equip the compact group $G_K^e$ with the normalized Haar measure $\\mu = \\mu_{G_K^e}$, which allows $G_K^e$ to be regarded as a probability space~\\cite[Section~21.1]{FriedJ}.\nThe term \\textit{almost all} $\\sigma \\in G_K^e$ is used in the sense of ``all $\\sigma \\in G_K^e$ outside some measure zero set\".\n\nWe investigate in this paper how often the field $\\overline{K}(\\sigma)$ is Kummer-faithful.\nHere we recall the definitions of Kummer-faithfulness and related notions.\n\n\\begin{definition}[{\\cite[Definition~1.5]{Mochizuki15}, \\cite[Definition~6.1~(iii)]{HMT}}]\nA perfect field $K$ is said to be \\textit{Kummer-faithful} (resp.\\ \\textit{torally Kummer-faithful}, \\textit{AVKF} (meaning \\textit{abelian variety Kummer-faithful})) if, for every finite extension $L$ of $K$ and every semiabelian variety (resp.\\ every torus, every abelian variety) $A$ over $L$, the divisible part $A(L)_\\mathrm{div}=\\bigcap_{n \\ge 1} nA(L)$ of the Mordell--Weil group $A(L)$ of $A$ over $L$ is trivial; \n\\[ A(L)_\\mathrm{div} = 0. \\]\n\\end{definition}\n\nNotice that the condition $A(L)_\\mathrm{div} = 0$ in the above definition is equivalent to that the Kummer map associated with $A$ is injective~\\cite[p.~2]{Hoshi}.\nThis property implies that each rational point is uniquely determined by the corresponding Kummer class via the Kummer map, being expected to play an important role in ensuring that the \\'{e}tale fundamental group possesses enough information to allow the reconstruction of various geometric objects in the context of anabelian geometry.\nHoshi~\\cite{Hoshi} proved some versions of the Grothendieck conjecture in anabelian geometry over Kummer-faithful fields satisfying some additional conditions.\n\nA typical example of Kummer-faithful fields is a sub-$p$-adic field, i.e., a field isomorphic to a subfield of a finitely generated field of the field $\\mathbb{Q}_p$ of $p$-adic numbers~\\cite[Remark~1.5.4~(i)]{Mochizuki15}, and it would be interesting to find examples of Kummer-faithful fields that are not sub-$p$-adic.\n\n\\begin{remark}\nIt is obvious by definition that any Kummer-faithful field is both torally Kummer-faithful and AVKF.\nThe converse of each implication is not true; there exist both a torally Kummer-faithful field that is not Kummer-faithful~\\cite[Remark~1.5.3~(ii)]{Mochizuki15} and an AVKF field that is not Kummer-faithful (any finite extension of the maximal abelian extension $\\mathbb{Q}^\\mathrm{ab}$ of $\\mathbb{Q}$ is so~\\cite[p.~11]{HMT}).\nIn the characteristic zero case, Kummer-faithfulness is equivalent to the conjunction of torally Kummer-faithfulness and AVKF-ness~\\cite[Remark~6.1.3]{HMT}.\n\\end{remark}\n\nIf $K$ is finitely generated over $\\mathbb{Q}$, then $\\overline{K}(\\sigma)$ is not sub-$p$-adic for almost all $\\sigma \\in G_K^e$ and any prime number $p$~\\cite[Proposition~5.1]{AT}.\nThis suggests that it would be worth considering $\\overline{K}(\\sigma)$ for finding Kummer-faithful fields that are not sub-$p$-adic.\nJarden and Petersen~\\cite[Theorem~1.3~(ii)]{JP22} proved that any finite extension of $\\overline{K}(\\sigma)$ is AVKF for almost all $\\sigma \\in G_K^e$ if $K$ is finitely generated over $\\mathbb{Q}$ and $e \\ge 2$.\nThe Kummer-faithfulness for a subfield of $\\overline{K}(\\sigma)$ has been studied by Ohtani~\\cite{Ohtani22, Ohtani23} and in joint work of the author and Taguchi~\\cite{AT}.\nBased on these results, we propose the following conjecture.\n\n\\begin{conjecture}\nLet $K$ be a finitely generated field over $\\mathbb{Q}$ and $e$ a positive integer.\nThen any finite extension of $\\overline{K}(\\sigma)$ is Kummer-faithful for almost all $\\sigma \\in G_K^e$.\n\\end{conjecture}\n\nThe main theorem in this paper is that the above conjecture is true if $K$ is a number field.", "context": "This paper focuses on algebraic extensions of a number field whose absolute Galois group is finitely generated.\nFix an algebraic closure $\\overline{K}$ of any perfect field $K$ and let $G_K$ be the absolute Galois group $\\Gal(\\overline{K} / K)$ of $K$.\nLet $e$ be a positive integer.\nFor any $\\sigma = (\\sigma_1, \\ldots, \\sigma_e) \\in G_K^e$ (the direct product of $e$ copies of $G_K$), set $\\overline{K}(\\sigma)$ to be the fixed field of $\\sigma$ in $\\overline{K}$, i.e.,\n\\[ \\overline{K}(\\sigma) = \\{x \\in \\overline{K} \\mid \\sigma_i(x) = x \\text{ for all } i\\}. \\]\nEvery field with finite transcendence degree over the field $\\mathbb{Q}$ of rational numbers and with finitely generated absolute Galois group is of this form.\nWe equip the compact group $G_K^e$ with the normalized Haar measure $\\mu = \\mu_{G_K^e}$, which allows $G_K^e$ to be regarded as a probability space~\\cite[Section~21.1]{FriedJ}.\nThe term \\textit{almost all} $\\sigma \\in G_K^e$ is used in the sense of ``all $\\sigma \\in G_K^e$ outside some measure zero set\".\n\n\\begin{definition}[{\\cite[Definition~1.5]{Mochizuki15}, \\cite[Definition~6.1~(iii)]{HMT}}]\nA perfect field $K$ is said to be \\textit{Kummer-faithful} (resp.\\ \\textit{torally Kummer-faithful}, \\textit{AVKF} (meaning \\textit{abelian variety Kummer-faithful})) if, for every finite extension $L$ of $K$ and every semiabelian variety (resp.\\ every torus, every abelian variety) $A$ over $L$, the divisible part $A(L)_\\mathrm{div}=\\bigcap_{n \\ge 1} nA(L)$ of the Mordell--Weil group $A(L)$ of $A$ over $L$ is trivial; \n\\[ A(L)_\\mathrm{div} = 0. \\]\n\\end{definition}\n\n\\begin{remark}\nIt is obvious by definition that any Kummer-faithful field is both torally Kummer-faithful and AVKF.\nThe converse of each implication is not true; there exist both a torally Kummer-faithful field that is not Kummer-faithful~\\cite[Remark~1.5.3~(ii)]{Mochizuki15} and an AVKF field that is not Kummer-faithful (any finite extension of the maximal abelian extension $\\mathbb{Q}^\\mathrm{ab}$ of $\\mathbb{Q}$ is so~\\cite[p.~11]{HMT}).\nIn the characteristic zero case, Kummer-faithfulness is equivalent to the conjunction of torally Kummer-faithfulness and AVKF-ness~\\cite[Remark~6.1.3]{HMT}.\n\\end{remark}\n\nIf $K$ is finitely generated over $\\mathbb{Q}$, then $\\overline{K}(\\sigma)$ is not sub-$p$-adic for almost all $\\sigma \\in G_K^e$ and any prime number $p$~\\cite[Proposition~5.1]{AT}.\nThis suggests that it would be worth considering $\\overline{K}(\\sigma)$ for finding Kummer-faithful fields that are not sub-$p$-adic.\nJarden and Petersen~\\cite[Theorem~1.3~(ii)]{JP22} proved that any finite extension of $\\overline{K}(\\sigma)$ is AVKF for almost all $\\sigma \\in G_K^e$ if $K$ is finitely generated over $\\mathbb{Q}$ and $e \\ge 2$.\nThe Kummer-faithfulness for a subfield of $\\overline{K}(\\sigma)$ has been studied by Ohtani~\\cite{Ohtani22, Ohtani23} and in joint work of the author and Taguchi~\\cite{AT}.\nBased on these results, we propose the following conjecture.\n\n\\begin{conjecture}\nLet $K$ be a finitely generated field over $\\mathbb{Q}$ and $e$ a positive integer.\nThen any finite extension of $\\overline{K}(\\sigma)$ is Kummer-faithful for almost all $\\sigma \\in G_K^e$.\n\\end{conjecture}\n\nThe main theorem in this paper is that the above conjecture is true if $K$ is a number field.", "full_context": "This paper focuses on algebraic extensions of a number field whose absolute Galois group is finitely generated.\nFix an algebraic closure $\\overline{K}$ of any perfect field $K$ and let $G_K$ be the absolute Galois group $\\Gal(\\overline{K} / K)$ of $K$.\nLet $e$ be a positive integer.\nFor any $\\sigma = (\\sigma_1, \\ldots, \\sigma_e) \\in G_K^e$ (the direct product of $e$ copies of $G_K$), set $\\overline{K}(\\sigma)$ to be the fixed field of $\\sigma$ in $\\overline{K}$, i.e.,\n\\[ \\overline{K}(\\sigma) = \\{x \\in \\overline{K} \\mid \\sigma_i(x) = x \\text{ for all } i\\}. \\]\nEvery field with finite transcendence degree over the field $\\mathbb{Q}$ of rational numbers and with finitely generated absolute Galois group is of this form.\nWe equip the compact group $G_K^e$ with the normalized Haar measure $\\mu = \\mu_{G_K^e}$, which allows $G_K^e$ to be regarded as a probability space~\\cite[Section~21.1]{FriedJ}.\nThe term \\textit{almost all} $\\sigma \\in G_K^e$ is used in the sense of ``all $\\sigma \\in G_K^e$ outside some measure zero set\".\n\n\\begin{definition}[{\\cite[Definition~1.5]{Mochizuki15}, \\cite[Definition~6.1~(iii)]{HMT}}]\nA perfect field $K$ is said to be \\textit{Kummer-faithful} (resp.\\ \\textit{torally Kummer-faithful}, \\textit{AVKF} (meaning \\textit{abelian variety Kummer-faithful})) if, for every finite extension $L$ of $K$ and every semiabelian variety (resp.\\ every torus, every abelian variety) $A$ over $L$, the divisible part $A(L)_\\mathrm{div}=\\bigcap_{n \\ge 1} nA(L)$ of the Mordell--Weil group $A(L)$ of $A$ over $L$ is trivial; \n\\[ A(L)_\\mathrm{div} = 0. \\]\n\\end{definition}\n\n\\begin{remark}\nIt is obvious by definition that any Kummer-faithful field is both torally Kummer-faithful and AVKF.\nThe converse of each implication is not true; there exist both a torally Kummer-faithful field that is not Kummer-faithful~\\cite[Remark~1.5.3~(ii)]{Mochizuki15} and an AVKF field that is not Kummer-faithful (any finite extension of the maximal abelian extension $\\mathbb{Q}^\\mathrm{ab}$ of $\\mathbb{Q}$ is so~\\cite[p.~11]{HMT}).\nIn the characteristic zero case, Kummer-faithfulness is equivalent to the conjunction of torally Kummer-faithfulness and AVKF-ness~\\cite[Remark~6.1.3]{HMT}.\n\\end{remark}\n\nIf $K$ is finitely generated over $\\mathbb{Q}$, then $\\overline{K}(\\sigma)$ is not sub-$p$-adic for almost all $\\sigma \\in G_K^e$ and any prime number $p$~\\cite[Proposition~5.1]{AT}.\nThis suggests that it would be worth considering $\\overline{K}(\\sigma)$ for finding Kummer-faithful fields that are not sub-$p$-adic.\nJarden and Petersen~\\cite[Theorem~1.3~(ii)]{JP22} proved that any finite extension of $\\overline{K}(\\sigma)$ is AVKF for almost all $\\sigma \\in G_K^e$ if $K$ is finitely generated over $\\mathbb{Q}$ and $e \\ge 2$.\nThe Kummer-faithfulness for a subfield of $\\overline{K}(\\sigma)$ has been studied by Ohtani~\\cite{Ohtani22, Ohtani23} and in joint work of the author and Taguchi~\\cite{AT}.\nBased on these results, we propose the following conjecture.\n\n\\begin{conjecture}\nLet $K$ be a finitely generated field over $\\mathbb{Q}$ and $e$ a positive integer.\nThen any finite extension of $\\overline{K}(\\sigma)$ is Kummer-faithful for almost all $\\sigma \\in G_K^e$.\n\\end{conjecture}\n\nThe main theorem in this paper is that the above conjecture is true if $K$ is a number field.\n\nThis paper focuses on algebraic extensions of a number field whose absolute Galois group is finitely generated.\nFix an algebraic closure $\\Kbar$ of any perfect field $K$ and let $G_K$ be the absolute Galois group $\\Gal(\\Kbar / K)$ of $K$.\nLet $e$ be a positive integer.\nFor any $\\sigma = (\\sigma_1, \\ldots, \\sigma_e) \\in G_K^e$ (the direct product of $e$ copies of $G_K$), set $\\Kbar(\\sigma)$ to be the fixed field of $\\sigma$ in $\\Kbar$, i.e.,\n\\[ \\Kbar(\\sigma) = \\{x \\in \\Kbar \\mid \\sigma_i(x) = x \\text{ for all } i\\}. \\]\nEvery field with finite transcendence degree over the field $\\mathbb{Q}$ of rational numbers and with finitely generated absolute Galois group is of this form.\nWe equip the compact group $G_K^e$ with the normalized Haar measure $\\mu = \\mu_{G_K^e}$, which allows $G_K^e$ to be regarded as a probability space~\\cite[Section~21.1]{FriedJ}.\nThe term \\textit{almost all} $\\sigma \\in G_K^e$ is used in the sense of ``all $\\sigma \\in G_K^e$ outside some measure zero set\".\n\nIf $K$ is finitely generated over $\\mathbb{Q}$, then $\\Kbar(\\sigma)$ is not sub-$p$-adic for almost all $\\sigma \\in G_K^e$ and any prime number $p$~\\cite[Proposition~5.1]{AT}.\nThis suggests that it would be worth considering $\\Kbar(\\sigma)$ for finding Kummer-faithful fields that are not sub-$p$-adic.\nJarden and Petersen~\\cite[Theorem~1.3~(ii)]{JP22} proved that any finite extension of $\\Kbar(\\sigma)$ is AVKF for almost all $\\sigma \\in G_K^e$ if $K$ is finitely generated over $\\mathbb{Q}$ and $e \\ge 2$.\nThe Kummer-faithfulness for a subfield of $\\Kbar(\\sigma)$ has been studied by Ohtani~\\cite{Ohtani22, Ohtani23} and in joint work of the author and Taguchi~\\cite{AT}.\nBased on these results, we propose the following conjecture.\n\n\\begin{conjecture}\nLet $K$ be a finitely generated field over $\\mathbb{Q}$ and $e$ a positive integer.\nThen any finite extension of $\\Kbar(\\sigma)$ is Kummer-faithful for almost all $\\sigma \\in G_K^e$.\n\\end{conjecture}\n\nThe main theorem in this paper is that the above conjecture is true if $K$ is a number field.\n\nNotice that the absolute Galois group $G_{\\Kbar(\\sigma)}$ of $\\Kbar(\\sigma)$ is abelian when $e = 1$.\nThis immediately implies that the following corollary.\nActually, more is true: there are continuum many algebraic extensions of $K$ up to $K$-isomorphism which are Kummer-faithful fields with abelian absolute Galois group.\nSee Corollary~\\ref{cor-continuummany} for details.\n\n\\begin{proposition}\\label{prop-ClassificationOfFinExtOfKbarsigma}\nLet $K$ be a perfect field and $\\sigma \\in G_K$.\n\\begin{enumerate}[\\textup{(\\arabic{enumi})}]\n \\item The field $\\Kbar(\\sigma^n)$ is a finite extension over $\\Kbar(\\sigma)$ for any positive integer $n$.\n \\item Conversely, any finite extension $M$ over $\\Kbar(\\sigma)$ is of the form $\\Kbar(\\sigma^n)$ for some positive integer $n$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proposition}\\label{prop-TKF-reduction}\nLet $K$ be a countable perfect field.\nSuppose that, for any finite extension $K' / K$, $a \\in \\overline{K'}^\\times \\smallsetminus \\{1\\}$ with the extension $K'(a) / K'$ cyclic, and a positive integer $n$, the set\n\\[ \\{\\sigma \\in G_{K'} \\mid a \\in \\Gm(\\overline{K'}(\\sigma^n))_\\divisible\\} \\]\nhas measure zero in $G_{K'}$.\nThen any finite extension of $\\Kbar(\\sigma)$ is torally Kummer-faithful for almost all $\\sigma \\in G_K$.\n\\end{proposition}\n\n\\begin{proof}\nBy the definition of torally Kummer-faithfulness, we have\n\\begin{align*}\n S &= \\{\\sigma \\in G_K \\mid \\Gm(M)_\\divisible \\neq 0 \\text{ for some finite extension } M / \\Kbar(\\sigma)\\} \\\\\n &= \\{\\sigma \\in G_K \\mid \\Gm(F \\Kbar(\\sigma))_\\divisible \\neq 0 \\text{ for some finite Galois extension } F / K\\} \\\\\n &= \\bigcup_{\\substack{F / K \\\\ \\text{finite Galois}}} \\{\\sigma \\in G_K \\mid \\Gm(F \\Kbar(\\sigma))_\\divisible \\neq 0\\} \\\\\n &= \\bigcup_{\\substack{F / K \\\\ \\text{finite Galois}}} \\bigcup_{a \\in \\Kbar^\\times \\smallsetminus \\{1\\}} \\{\\sigma \\in G_K \\mid a \\in \\Gm(F \\Kbar(\\sigma))_\\divisible\\}.\n\\end{align*}\nTaking $F$ larger so that it contains $a$, we may assume $a \\in F$, that is, it holds that\n\\[ S = \\bigcup_{\\substack{F / K \\\\ \\text{finite Galois}}} \\bigcup_{a \\in F^\\times \\smallsetminus \\{1\\}} \\{\\sigma \\in G_K \\mid a \\in \\Gm(F \\Kbar(\\sigma))_\\divisible\\}. \\]\nSuppose that $\\sigma \\in G_K$ satisfies $a \\in \\Gm(F \\Kbar(\\sigma))_\\divisible$ for some finite Galois extension $F / K$ and some $a \\in F^\\times \\smallsetminus \\{1\\}$.\nSet $K' = F \\cap \\Kbar(\\sigma)$.\nThen $\\sigma$ fixes $K'$, so we have $\\sigma \\in G_{K'}$ and $\\Kbar(\\sigma) = \\overline{K'}(\\sigma)$.\nBy Proposition~\\ref{prop-FinExtOfKbarsigma-new}, the Galois group\n\\[ \\Gal(F / K') \\cong \\Gal(F \\Kbar(\\sigma) / \\Kbar(\\sigma)) \\]\nis cyclic.\nThis argument implies the equation\n\\[ S = \\bigcup_{\\substack{F / K \\\\ \\text{finite Galois}}} \\bigcup_{a \\in F^\\times \\smallsetminus \\{1\\}} \\bigcup_{\\substack{K \\subset K' \\subset F \\\\ F / K'\\text{: cyclic}}} \\{\\sigma \\in G_{K'} \\mid a \\in \\Gm(F \\overline{K'}(\\sigma))_\\divisible\\}. \\]\nRearranging the order of taking the union and dropping the condition that $F / K$ is Galois yield the inclusion\n\\[ S \\subset \\bigcup_{\\substack{K' / K \\\\ \\text{finite}}} \\bigcup_{\\substack{F / K' \\\\ \\text{cyclic}}} \\bigcup_{a \\in F^\\times \\smallsetminus \\{1\\}} \\{\\sigma \\in G_{K'} \\mid a \\in \\Gm(F \\overline{K'}(\\sigma))_\\divisible\\}. \\]\nSince any subextension of a cyclic extension is again cyclic, we may add the condition that $K'(a) / K'$ is cyclic in the union.\nBy Proposition~\\ref{prop-ClassificationOfFinExtOfKbarsigma}, there is a positive integer $n$ such that $F \\overline{K'}(\\sigma) = \\overline{K'}(\\sigma^n)$.\nRemoving the condition $a \\in F$, we can eliminate $F$ from the union and obtain\n\\[ S \\subset \\bigcup_{\\substack{K' / K \\\\ \\text{finite}}} \\bigcup_{\\substack{a \\in \\overline{K'}^\\times \\smallsetminus \\{1\\} \\\\ K'(a) / K'\\text{: cyclic}}} \\bigcup_{n \\ge 1} \\{\\sigma \\in G_{K'} \\mid a \\in \\Gm(\\overline{K'}(\\sigma^n))_\\divisible\\}. \\]\nSince $K$ is countable, the right hand side is a countable union.\nThe assumption of the proposition says that each component in the union has measure zero.\nHence $S$ has also measure zero.\n\\end{proof}\n\n\\begin{corollary}\\label{cor-continuummany}\nLet $K$ be a number field.\nThere are continuum many algebraic extensions of $K$ up to $K$-isomorphism which are Kummer-faithful fields with abelian absolute Galois group.\n\\end{corollary}", "post_theorem_intro_text_len": 2721, "post_theorem_intro_text": "Notice that the absolute Galois group $G_{\\overline{K}(\\sigma)}$ of $\\overline{K}(\\sigma)$ is abelian when $e = 1$.\nThis immediately implies that the following corollary.\nActually, more is true: there are continuum many algebraic extensions of $K$ up to $K$-isomorphism which are Kummer-faithful fields with abelian absolute Galois group.\nSee Corollary~\\ref{cor-continuummany} for details.\n\n\\begin{corollary}\\label{cor-thm-KF}\nLet $K$ be a number field.\nThere exists an algebraic extension of $K$ that is Kummer-faithful and has abelian absolute Galois group.\n\\end{corollary}\n\n\\begin{remark}\nOne of the assertions in Theorem~1.11 of Mochizuki's paper~\\cite{Mochizuki15} claimed that the absolute Galois group of any Kummer-faithful field of characteristic zero is \\textit{slim}, i.e., every open subgroup has trivial center.\nHe recently informed us that the proof of this assertion has a gap.\nCorollary~\\ref{cor-thm-KF} is incompatible with this assertion and shows the existence of a counterexample in the general case, but the proof is not constructive.\nIt also answers in the negative the questions~\\cite[Remark~2.4.1, Questions~1 and~2]{MT} posed by Minamide and Tsujimura which asked whether the absolute Galois group of any torally Kummer-faithful field is slim.\n\\end{remark}\n\n\\begin{remark}\nThe assertions related to the (torally) Kummer-faithfulness of $\\overline{K}(\\sigma)$ have appeared twice previously, first in~\\cite[Corollary~1]{Ohtani22} and then in the PhD thesis~\\cite[Theorem~4.2.1]{AsayamaThesis} of the author, but their proofs in both works were incorrect.\nThe former has been corrected and the weaker version~\\cite[Corollary~1]{Ohtani23} was proved.\nIn our previous paper~\\cite{AT}, the latter was addressed in p.~5 and the weaker result~\\cite[Theorem~5.3]{AT} was shown.\n\\end{remark}\n\nThe proof of Theorem~\\ref{thm-KF} is carried out by separating it into two parts; the torally Kummer-faithfulness part and the AVKF-ness part.\nWe provide proofs for each part in Sections~\\ref{sec-TKFpart} and~\\ref{sec-AVKFpart}, respectively.\nThe approaches to proving these parts are somewhat similar.\nFirst, we reformulate the problem by reducing it to the calculation of measures.\nFor the AVKF-ness part, this step has been done by Jarden and Petersen~\\cite{JP22}.\nThen the calculation of measures is performed.\nFor the torally Kummer-faithfulness part, this step is based on a refinement of the proof of the weaker version~\\cite[Theorem~5.3]{AT} presented in our previous paper.\nFinally, we conclude the proof by using a combinatorial approach similar to that described by Zywina~\\cite[pp.~495--496]{Zywina16} (see also~\\cite[p.~49]{JP19}).\nSection~\\ref{sec-cor} contains corollaries to Theorem~\\ref{thm-KF}.", "sketch": "The proof of Theorem~\\ref{thm-KF} is split into two parts: the \\emph{torally Kummer-faithfulness} part and the \\emph{AVKF-ness} part, proved in Sections~\\ref{sec-TKFpart} and~\\ref{sec-AVKFpart}, respectively. The approaches are described as similar. First, the problem is reformulated by reducing it to a \\emph{calculation of measures} (for the AVKF-ness part, this reduction was done by Jarden and Petersen~\\cite{JP22}; for the torally Kummer-faithfulness part, it uses a refinement of the proof of the weaker version~\\cite[Theorem~5.3]{AT}). Then, the required measure calculations are carried out. Finally, the proof is concluded via a \\emph{combinatorial approach} similar to Zywina~\\cite[pp.~495--496]{Zywina16} (see also~\\cite[p.~49]{JP19}).", "expanded_sketch": "The proof of the main theorem is split into two parts: the \\emph{torally Kummer-faithfulness} part and the \\emph{AVKF-ness} part, proved next and then later, respectively. The approaches are described as similar. First, the problem is reformulated by reducing it to a \\emph{calculation of measures} (for the AVKF-ness part, this reduction was done by Jarden and Petersen~\\cite{JP22}; for the torally Kummer-faithfulness part, it uses a refinement of the proof of the weaker version~\\cite[Theorem~5.3]{AT}). Then, the required measure calculations are carried out. Finally, the proof is concluded via a \\emph{combinatorial approach} similar to Zywina~\\cite[pp.~495--496]{Zywina16} (see also~\\cite[p.~49]{JP19}).", "expanded_theorem": "\\label{thm-KF}\nSuppose that $K$ is a number field and $e$ a positive integer. For almost all $\\sigma \\in G_K^e$, any finite extension of $\\overline{K}(\\sigma)$ is both torally Kummer-faithful and AVKF, and thus, Kummer-faithful.,", "theorem_type": ["Universal", "Implication"], "mcq": {"question": "Let $K$ be a number field, let $\\overline{K}$ be an algebraic closure of $K$, let $G_K=\\operatorname{Gal}(\\overline{K}/K)$, and let $e\\ge 1$. For $\\sigma=(\\sigma_1,\\dots,\\sigma_e)\\in G_K^e$, define the fixed field\n\\[\n\\overline{K}(\\sigma)=\\{x\\in \\overline{K}\\mid \\sigma_i(x)=x\\text{ for all }i\\}.\n\\]\nEquip $G_K^e$ with its normalized Haar measure, and interpret “almost all $\\sigma\\in G_K^e$” as “all $\\sigma$ outside a measure-zero subset.” A field $F$ is called torally Kummer-faithful if for every finite extension $L/F$ and every torus $T$ over $L$, the divisible subgroup\n\\[\nT(L)_{\\mathrm{div}}=\\bigcap_{n\\ge 1} nT(L)\n\\]\nis zero; $F$ is called AVKF if for every finite extension $L/F$ and every abelian variety $A$ over $L$,\n\\[\nA(L)_{\\mathrm{div}}=\\bigcap_{n\\ge 1} nA(L)=0.\n\\]\nWhich statement holds for almost all $\\sigma\\in G_K^e$?", "correct_choice": {"label": "A", "text": "For almost all $\\sigma\\in G_K^e$, every finite extension $E$ of $\\overline{K}(\\sigma)$ is both torally Kummer-faithful and AVKF; equivalently, for every finite extension $L/E$, every torus $T$ over $L$ and every abelian variety $A$ over $L$ satisfy $T(L)_{\\mathrm{div}}=0$ and $A(L)_{\\mathrm{div}}=0$. In particular, every such finite extension $E$ is Kummer-faithful."}, "choices": [{"label": "B", "text": "For almost all $\\sigma\\in G_K^e$, the fixed field $\\overline{K}(\\sigma)$ itself is both torally Kummer-faithful and AVKF; equivalently, for every finite extension $L/\\overline{K}(\\sigma)$, every torus $T$ over $L$ and every abelian variety $A$ over $L$ satisfy $T(L)_{\\mathrm{div}}=0$ and $A(L)_{\\mathrm{div}}=0$, but no assertion is made for arbitrary finite extensions $E$ of $\\overline{K}(\\sigma)$ as base fields."}, {"label": "C", "text": "For almost all $\\sigma\\in G_K^e$, every finite extension $E$ of $\\overline{K}(\\sigma)$ is AVKF; equivalently, for every finite extension $L/E$ and every abelian variety $A$ over $L$, one has $A(L)_{\\mathrm{div}}=0$."}, {"label": "D", "text": "For every finite extension $E$ of $\\overline{K}(\\sigma)$, there exists a measure-zero subset $N_E\\subset G_K^e$ such that for all $\\sigma\\notin N_E$, the field $E$ is both torally Kummer-faithful and AVKF; equivalently, outside a null set depending on $E$, for every finite extension $L/E$, every torus $T$ over $L$ and every abelian variety $A$ over $L$ satisfy $T(L)_{\\mathrm{div}}=0$ and $A(L)_{\\mathrm{div}}=0$."}, {"label": "E", "text": "For almost all $\\sigma\\in G_K^e$, every algebraic extension $E$ of $\\overline{K}(\\sigma)$ is both torally Kummer-faithful and AVKF; equivalently, for every algebraic extension $L/E$, every torus $T$ over $L$ and every abelian variety $A$ over $L$ satisfy $T(L)_{\\mathrm{div}}=0$ and $A(L)_{\\mathrm{div}}=0$. In particular, every algebraic extension of $\\overline{K}(\\sigma)$ is Kummer-faithful."}], "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": "scope_from_all_finite_extensions_to_base_field_only", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped_torally_Kummer_faithful_clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "uniform_null_set_independent_of_E", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "finite_extension_replaced_by_arbitrary_algebraic_extension", "template_used": "stronger_trap"}]}} +{"id": "2601.10613v1", "paper_link": "http://arxiv.org/abs/2601.10613v1", "theorems_cnt": 1, "theorem": {"env_name": "proposition", "content": "\\label{polarization}\nThe polarization of $\\mathcal{L}\\mathcal{S}\\$ is\n\\begin{gather}\\label{pol1}\n[[a,b],c]+[[b,c],a]+[[c,a],b]=0 \n\\end{gather}\nand\n\\begin{gather}\\label{pol2}\n\\{\\{a,b\\},c\\}=-\\{[a,b],c\\}-2\\{[a,c],b\\}+[\\{a,b\\},c]-[[a,c],b]+\\{a,\\{b,c\\}\\}-\\{a,[b,c]\\}+[a,\\{b,c\\}].\n\\end{gather}", "start_pos": 8236, "end_pos": 8567, "label": "polarization"}, "ref_dict": {}, "pre_theorem_intro_text_len": 5425, "pre_theorem_intro_text": "The class of left-symmetric algebras (also called pre-Lie algebras) arose in differential geometry and deformation theory \\cite{Gerst,Kosz,Vin}. Recall that a left-symmetric algebra is a vector space \\(A\\) with a bilinear product \\((a,b) \\mapsto ab\\) such that its associator\n\\[\n(a,b,c) = (ab)c - a(bc)\n\\]\nis symmetric in the first two arguments, that is,\n\\[\n(a,b,c) = (b,a,c)\n\\]\nfor all \\(a,b,c \\in A\\).\n\nIn this paper, we study three types of left-symmetric algebras based on Malcev's classification. As shown in \\cite{Mal'cev}, the space of invariant relations of degree $3$ decomposes as a sum\n\\[\n\\mathfrak{L}=\\mathfrak{L}_1+\\mathfrak{L}_2,\n\\]\nwhere $\\mathfrak{L}_1:=\\mathfrak{A}_1+\\mathfrak{B}_1+\\mathfrak{C}_1+\\mathfrak{D}_1$, and the subspaces are defined by the following polynomial relations:\n\\begin{gather}\n\\mathfrak{A}_1:\\;(ab)c+(ba)c+(ac)b+(ca)b+(bc)a+(cb)a=0,\n\\label{as1} \\\\\n\\mathfrak{B}_1:\\;(ab)c-(ba)c-(ac)b+(ca)b+(bc)a-(cb)a=0,\n\\label{as2}\n\\end{gather}\n\\begin{gather}\\label{as3}\n\\mathfrak{C}_1:\\;\n\\begin{cases}\n(ab)c+(ba)c-(bc)a-(cb)a=0, \\\\\n(ac)b+(ca)b-(cb)a-(bc)a=0,\n\\end{cases}\n\\end{gather}\n\\begin{gather}\\label{as4}\n\\mathfrak{D}_1:\\;\n\\begin{cases}\n(ab)c+(ac)b-(cb)a-(ca)b=0, \\\\\n(ab)c+(ac)b-(bc)a-(ba)c=0.\n\\end{cases}\n\\end{gather}\nAnalogously, set $\\mathfrak{L}_2:=\\mathfrak{A}_2+\\mathfrak{B}_2+\\mathfrak{C}_2+\\mathfrak{D}_2$, defined by the same relations written with right-normed bracketing. We denote by $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}$ the variety of left-symmetric algebras with identity $\\mathfrak{Z}_i$, where $\\mathfrak{Z}=\\mathfrak{A},\\mathfrak{B},\\mathfrak{C},\\mathfrak{D}$ and $i=1,2$. The Malcev classification was studied for the varieties of associative and alternative algebras, see \\cite{KazMat,MalcAs1,MalcAs2}. In \\cite{MalcAs1} it was shown that, in the associative case, the operads obtained from the Malcev classification via the Koszul dual computation are exactly the alternative, assosymmetric and $(-1,1)$ operads. For further results on the importance of these operads, see \\cite{assos1,Dzhuma,As1,As2,binaryperm,-1-1}.\n\nWe restrict ourselves to the varieties\n\\[\n\\mathcal{L}\\mathcal{S}_{\\mathcal{A}_1},\\mathcal{L}\\mathcal{S}_{\\mathcal{B}_1} \\;\\textrm{and}\\; \\mathcal{L}\\mathcal{S}_{\\mathcal{A}_2}.\n\\]\nThere is no need to consider $\\mathcal{L}\\mathcal{S}_{\\mathcal{B}_2}$ separately, since it coincides with $\\mathcal{L}\\mathcal{S}_{\\mathcal{B}_1}$. Moreover, we shall see that the varieties $\\mathcal{L}\\mathcal{S}_{\\mathcal{A}_1}$, $\\mathcal{L}\\mathcal{S}_{\\mathcal{B}_1}$ and $\\mathcal{L}\\mathcal{S}_{\\mathcal{A}_2}$\nare pairwise distinct.\n\nFor each variety $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}$ under consideration, we determine its Koszul dual operad in the sense of \\cite{GK94}. We then describe all polynomial identities of degrees up to~$4$ satisfied by the commutator and anti-commutator operations. Moreover, we construct an explicit basis for each of the obtained Koszul dual operads. Since the Koszul dual of the left-symmetric operad is a perm operad (associative operad with a left-commutative identity), for $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}$ and $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}^{\\;(!)}$, we describe the following diagram:\n\\begin{gather}\\label{diagram}\n\\begin{picture}(400,90)\n \\put(120,75){$\\mathcal{L}\\mathcal{S}$}\n \\put(255,75){$\\mathcal{\\mathcal{P}}erm$} \n \\put(125,57){$\\cup$}\n \\put(256,57){$\\cap$}\n \\put(120,40){$\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}$}\n \\put(250,40){$\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}^{\\;(!)}$}\n \\put(256,20){$\\cap$}\n \\put(256,5){$\\mathcal{A}$}\n \\put(175,43){\\vector(1,0){55}}\n \\put(210,43){\\vector(-1,0){55}}\n \\put(175,79){\\vector(1,0){55}}\n \\put(210,79){\\vector(-1,0){55}}\n\\end{picture}\n\\end{gather}\nArtin's theorem states that every alternative algebra generated by two elements is associative, i.e., the variety of binary associative algebras coincides with the variety of alternative algebras \\cite{Artin}. Also, it was shown in \\cite{binaryperm} that the variety of binary perm algebras (which we denote by $\\mathcal{P}erm_2$) coincides with the variety of alternative algebras satisfying an additional identity of degree $3$, which is\n\\[\na(bc)+a(cb)=b(ac)+c(ab).\n\\]\nIn the cases $\\mathfrak{Z}_i = \\mathfrak{A}_1$ and $\\mathfrak{Z}_i = \\mathfrak{A}_2$, it turns out that the operad $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}^{\\;(!)}$ is an alternative satisfying an additional identity of degree $3$.\nMoreover, in the case $\\mathfrak{Z}_i = \\mathfrak{A}_1$, the additional identity is again left-commutative. From the general observations, we obtain\n\\[\n\\mathcal{P}erm\\subset \\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_1}^{\\;(!)}\\subset\\mathcal{P}erm_2.\n\\]\nAlthough this is not immediate, direct computations show that\n\\[\n\\mathcal{P}erm\\subset \\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_2}^{\\;(!)}\\subset\\mathcal{P}erm_2.\n\\]\nAlso, we shall see that\n\\[\n\\mathcal{P}erm\\subset \\mathcal{L}\\mathcal{S}_{\\mathfrak{B}_1}^{\\;(!)}\\not\\subset\\mathcal{P}erm_2.\n\\]\nTo compute the Koszul dual operad of $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}$, we need the multilinear basis of degree $3$ of the algebra $\\mathcal{L}\\mathcal{S}\\$. We use the following rewriting rules:\n\\[\na(bc)=(ab)c-(ba)c+b(ac),\n\\]\n\\[\na(cb)=(ac)b-(ca)b+c(ab)\n\\]\nand\n\\[\nb(ca)=(bc)a-(cb)a+c(ba).\n\\]\nLet us compute the polarization of $\\mathcal{L}\\mathcal{S}$ following \\cite{MarklRemm}; this construction will be used throughout the paper.", "context": "The class of left-symmetric algebras (also called pre-Lie algebras) arose in differential geometry and deformation theory \\cite{Gerst,Kosz,Vin}. Recall that a left-symmetric algebra is a vector space \\(A\\) with a bilinear product \\((a,b) \\mapsto ab\\) such that its associator\n\\[\n(a,b,c) = (ab)c - a(bc)\n\\]\nis symmetric in the first two arguments, that is,\n\\[\n(a,b,c) = (b,a,c)\n\\]\nfor all \\(a,b,c \\in A\\).\n\nIn this paper, we study three types of left-symmetric algebras based on Malcev's classification. As shown in \\cite{Mal'cev}, the space of invariant relations of degree $3$ decomposes as a sum\n\\[\n\\mathfrak{L}=\\mathfrak{L}_1+\\mathfrak{L}_2,\n\\]\nwhere $\\mathfrak{L}_1:=\\mathfrak{A}_1+\\mathfrak{B}_1+\\mathfrak{C}_1+\\mathfrak{D}_1$, and the subspaces are defined by the following polynomial relations:\n\\begin{gather}\n\\mathfrak{A}_1:\\;(ab)c+(ba)c+(ac)b+(ca)b+(bc)a+(cb)a=0,\n\\label{as1} \\\\\n\\mathfrak{B}_1:\\;(ab)c-(ba)c-(ac)b+(ca)b+(bc)a-(cb)a=0,\n\\label{as2}\n\\end{gather}\n\\begin{gather}\\label{as3}\n\\mathfrak{C}_1:\\;\n\\begin{cases}\n(ab)c+(ba)c-(bc)a-(cb)a=0, \\\\\n(ac)b+(ca)b-(cb)a-(bc)a=0,\n\\end{cases}\n\\end{gather}\n\\begin{gather}\\label{as4}\n\\mathfrak{D}_1:\\;\n\\begin{cases}\n(ab)c+(ac)b-(cb)a-(ca)b=0, \\\\\n(ab)c+(ac)b-(bc)a-(ba)c=0.\n\\end{cases}\n\\end{gather}\nAnalogously, set $\\mathfrak{L}_2:=\\mathfrak{A}_2+\\mathfrak{B}_2+\\mathfrak{C}_2+\\mathfrak{D}_2$, defined by the same relations written with right-normed bracketing. We denote by $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}$ the variety of left-symmetric algebras with identity $\\mathfrak{Z}_i$, where $\\mathfrak{Z}=\\mathfrak{A},\\mathfrak{B},\\mathfrak{C},\\mathfrak{D}$ and $i=1,2$. The Malcev classification was studied for the varieties of associative and alternative algebras, see \\cite{KazMat,MalcAs1,MalcAs2}. In \\cite{MalcAs1} it was shown that, in the associative case, the operads obtained from the Malcev classification via the Koszul dual computation are exactly the alternative, assosymmetric and $(-1,1)$ operads. For further results on the importance of these operads, see \\cite{assos1,Dzhuma,As1,As2,binaryperm,-1-1}.\n\nWe restrict ourselves to the varieties\n\\[\n\\mathcal{L}\\mathcal{S}_{\\mathcal{A}_1},\\mathcal{L}\\mathcal{S}_{\\mathcal{B}_1} \\;\\textrm{and}\\; \\mathcal{L}\\mathcal{S}_{\\mathcal{A}_2}.\n\\]\nThere is no need to consider $\\mathcal{L}\\mathcal{S}_{\\mathcal{B}_2}$ separately, since it coincides with $\\mathcal{L}\\mathcal{S}_{\\mathcal{B}_1}$. Moreover, we shall see that the varieties $\\mathcal{L}\\mathcal{S}_{\\mathcal{A}_1}$, $\\mathcal{L}\\mathcal{S}_{\\mathcal{B}_1}$ and $\\mathcal{L}\\mathcal{S}_{\\mathcal{A}_2}$\nare pairwise distinct.\n\nFor each variety $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}$ under consideration, we determine its Koszul dual operad in the sense of \\cite{GK94}. We then describe all polynomial identities of degrees up to~$4$ satisfied by the commutator and anti-commutator operations. Moreover, we construct an explicit basis for each of the obtained Koszul dual operads. Since the Koszul dual of the left-symmetric operad is a perm operad (associative operad with a left-commutative identity), for $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}$ and $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}^{\\;(!)}$, we describe the following diagram:\n\\begin{gather}\\label{diagram}\n\\begin{picture}(400,90)\n \\put(120,75){$\\mathcal{L}\\mathcal{S}$}\n \\put(255,75){$\\mathcal{\\mathcal{P}}erm$} \n \\put(125,57){$\\cup$}\n \\put(256,57){$\\cap$}\n \\put(120,40){$\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}$}\n \\put(250,40){$\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}^{\\;(!)}$}\n \\put(256,20){$\\cap$}\n \\put(256,5){$\\mathcal{A}$}\n \\put(175,43){\\vector(1,0){55}}\n \\put(210,43){\\vector(-1,0){55}}\n \\put(175,79){\\vector(1,0){55}}\n \\put(210,79){\\vector(-1,0){55}}\n\\end{picture}\n\\end{gather}\nArtin's theorem states that every alternative algebra generated by two elements is associative, i.e., the variety of binary associative algebras coincides with the variety of alternative algebras \\cite{Artin}. Also, it was shown in \\cite{binaryperm} that the variety of binary perm algebras (which we denote by $\\mathcal{P}erm_2$) coincides with the variety of alternative algebras satisfying an additional identity of degree $3$, which is\n\\[\na(bc)+a(cb)=b(ac)+c(ab).\n\\]\nIn the cases $\\mathfrak{Z}_i = \\mathfrak{A}_1$ and $\\mathfrak{Z}_i = \\mathfrak{A}_2$, it turns out that the operad $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}^{\\;(!)}$ is an alternative satisfying an additional identity of degree $3$.\nMoreover, in the case $\\mathfrak{Z}_i = \\mathfrak{A}_1$, the additional identity is again left-commutative. From the general observations, we obtain\n\\[\n\\mathcal{P}erm\\subset \\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_1}^{\\;(!)}\\subset\\mathcal{P}erm_2.\n\\]\nAlthough this is not immediate, direct computations show that\n\\[\n\\mathcal{P}erm\\subset \\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_2}^{\\;(!)}\\subset\\mathcal{P}erm_2.\n\\]\nAlso, we shall see that\n\\[\n\\mathcal{P}erm\\subset \\mathcal{L}\\mathcal{S}_{\\mathfrak{B}_1}^{\\;(!)}\\not\\subset\\mathcal{P}erm_2.\n\\]\nTo compute the Koszul dual operad of $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}$, we need the multilinear basis of degree $3$ of the algebra $\\mathcal{L}\\mathcal{S}\\$. We use the following rewriting rules:\n\\[\na(bc)=(ab)c-(ba)c+b(ac),\n\\]\n\\[\na(cb)=(ac)b-(ca)b+c(ab)\n\\]\nand\n\\[\nb(ca)=(bc)a-(cb)a+c(ba).\n\\]\nLet us compute the polarization of $\\mathcal{L}\\mathcal{S}$ following \\cite{MarklRemm}; this construction will be used throughout the paper.", "full_context": "The class of left-symmetric algebras (also called pre-Lie algebras) arose in differential geometry and deformation theory \\cite{Gerst,Kosz,Vin}. Recall that a left-symmetric algebra is a vector space \\(A\\) with a bilinear product \\((a,b) \\mapsto ab\\) such that its associator\n\\[\n(a,b,c) = (ab)c - a(bc)\n\\]\nis symmetric in the first two arguments, that is,\n\\[\n(a,b,c) = (b,a,c)\n\\]\nfor all \\(a,b,c \\in A\\).\n\nIn this paper, we study three types of left-symmetric algebras based on Malcev's classification. As shown in \\cite{Mal'cev}, the space of invariant relations of degree $3$ decomposes as a sum\n\\[\n\\mathfrak{L}=\\mathfrak{L}_1+\\mathfrak{L}_2,\n\\]\nwhere $\\mathfrak{L}_1:=\\mathfrak{A}_1+\\mathfrak{B}_1+\\mathfrak{C}_1+\\mathfrak{D}_1$, and the subspaces are defined by the following polynomial relations:\n\\begin{gather}\n\\mathfrak{A}_1:\\;(ab)c+(ba)c+(ac)b+(ca)b+(bc)a+(cb)a=0,\n\\label{as1} \\\\\n\\mathfrak{B}_1:\\;(ab)c-(ba)c-(ac)b+(ca)b+(bc)a-(cb)a=0,\n\\label{as2}\n\\end{gather}\n\\begin{gather}\\label{as3}\n\\mathfrak{C}_1:\\;\n\\begin{cases}\n(ab)c+(ba)c-(bc)a-(cb)a=0, \\\\\n(ac)b+(ca)b-(cb)a-(bc)a=0,\n\\end{cases}\n\\end{gather}\n\\begin{gather}\\label{as4}\n\\mathfrak{D}_1:\\;\n\\begin{cases}\n(ab)c+(ac)b-(cb)a-(ca)b=0, \\\\\n(ab)c+(ac)b-(bc)a-(ba)c=0.\n\\end{cases}\n\\end{gather}\nAnalogously, set $\\mathfrak{L}_2:=\\mathfrak{A}_2+\\mathfrak{B}_2+\\mathfrak{C}_2+\\mathfrak{D}_2$, defined by the same relations written with right-normed bracketing. We denote by $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}$ the variety of left-symmetric algebras with identity $\\mathfrak{Z}_i$, where $\\mathfrak{Z}=\\mathfrak{A},\\mathfrak{B},\\mathfrak{C},\\mathfrak{D}$ and $i=1,2$. The Malcev classification was studied for the varieties of associative and alternative algebras, see \\cite{KazMat,MalcAs1,MalcAs2}. In \\cite{MalcAs1} it was shown that, in the associative case, the operads obtained from the Malcev classification via the Koszul dual computation are exactly the alternative, assosymmetric and $(-1,1)$ operads. For further results on the importance of these operads, see \\cite{assos1,Dzhuma,As1,As2,binaryperm,-1-1}.\n\nWe restrict ourselves to the varieties\n\\[\n\\mathcal{L}\\mathcal{S}_{\\mathcal{A}_1},\\mathcal{L}\\mathcal{S}_{\\mathcal{B}_1} \\;\\textrm{and}\\; \\mathcal{L}\\mathcal{S}_{\\mathcal{A}_2}.\n\\]\nThere is no need to consider $\\mathcal{L}\\mathcal{S}_{\\mathcal{B}_2}$ separately, since it coincides with $\\mathcal{L}\\mathcal{S}_{\\mathcal{B}_1}$. Moreover, we shall see that the varieties $\\mathcal{L}\\mathcal{S}_{\\mathcal{A}_1}$, $\\mathcal{L}\\mathcal{S}_{\\mathcal{B}_1}$ and $\\mathcal{L}\\mathcal{S}_{\\mathcal{A}_2}$\nare pairwise distinct.\n\nFor each variety $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}$ under consideration, we determine its Koszul dual operad in the sense of \\cite{GK94}. We then describe all polynomial identities of degrees up to~$4$ satisfied by the commutator and anti-commutator operations. Moreover, we construct an explicit basis for each of the obtained Koszul dual operads. Since the Koszul dual of the left-symmetric operad is a perm operad (associative operad with a left-commutative identity), for $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}$ and $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}^{\\;(!)}$, we describe the following diagram:\n\\begin{gather}\\label{diagram}\n\\begin{picture}(400,90)\n \\put(120,75){$\\mathcal{L}\\mathcal{S}$}\n \\put(255,75){$\\mathcal{\\mathcal{P}}erm$} \n \\put(125,57){$\\cup$}\n \\put(256,57){$\\cap$}\n \\put(120,40){$\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}$}\n \\put(250,40){$\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}^{\\;(!)}$}\n \\put(256,20){$\\cap$}\n \\put(256,5){$\\mathcal{A}$}\n \\put(175,43){\\vector(1,0){55}}\n \\put(210,43){\\vector(-1,0){55}}\n \\put(175,79){\\vector(1,0){55}}\n \\put(210,79){\\vector(-1,0){55}}\n\\end{picture}\n\\end{gather}\nArtin's theorem states that every alternative algebra generated by two elements is associative, i.e., the variety of binary associative algebras coincides with the variety of alternative algebras \\cite{Artin}. Also, it was shown in \\cite{binaryperm} that the variety of binary perm algebras (which we denote by $\\mathcal{P}erm_2$) coincides with the variety of alternative algebras satisfying an additional identity of degree $3$, which is\n\\[\na(bc)+a(cb)=b(ac)+c(ab).\n\\]\nIn the cases $\\mathfrak{Z}_i = \\mathfrak{A}_1$ and $\\mathfrak{Z}_i = \\mathfrak{A}_2$, it turns out that the operad $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}^{\\;(!)}$ is an alternative satisfying an additional identity of degree $3$.\nMoreover, in the case $\\mathfrak{Z}_i = \\mathfrak{A}_1$, the additional identity is again left-commutative. From the general observations, we obtain\n\\[\n\\mathcal{P}erm\\subset \\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_1}^{\\;(!)}\\subset\\mathcal{P}erm_2.\n\\]\nAlthough this is not immediate, direct computations show that\n\\[\n\\mathcal{P}erm\\subset \\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_2}^{\\;(!)}\\subset\\mathcal{P}erm_2.\n\\]\nAlso, we shall see that\n\\[\n\\mathcal{P}erm\\subset \\mathcal{L}\\mathcal{S}_{\\mathfrak{B}_1}^{\\;(!)}\\not\\subset\\mathcal{P}erm_2.\n\\]\nTo compute the Koszul dual operad of $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}$, we need the multilinear basis of degree $3$ of the algebra $\\mathcal{L}\\mathcal{S}\\$. We use the following rewriting rules:\n\\[\na(bc)=(ab)c-(ba)c+b(ac),\n\\]\n\\[\na(cb)=(ac)b-(ca)b+c(ab)\n\\]\nand\n\\[\nb(ca)=(bc)a-(cb)a+c(ba).\n\\]\nLet us compute the polarization of $\\mathcal{L}\\mathcal{S}$ following \\cite{MarklRemm}; this construction will be used throughout the paper.\n\nFor each variety $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}$ under consideration, we determine its Koszul dual operad in the sense of \\cite{GK94}. We then describe all polynomial identities of degrees up to~$4$ satisfied by the commutator and anti-commutator operations. Moreover, we construct an explicit basis for each of the obtained Koszul dual operads. Since the Koszul dual of the left-symmetric operad is a perm operad (associative operad with a left-commutative identity), for $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}$ and $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}^{\\;(!)}$, we describe the following diagram:\n\\begin{gather}\\label{diagram}\n\\begin{picture}(400,90)\n \\put(120,75){$\\mathcal{L}\\mathcal{S}$}\n \\put(255,75){$\\mathcal{\\mathcal{P}}erm$} \n \\put(125,57){$\\cup$}\n \\put(256,57){$\\cap$}\n \\put(120,40){$\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}$}\n \\put(250,40){$\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}^{\\;(!)}$}\n \\put(256,20){$\\cap$}\n \\put(256,5){$\\mathcal{A}$}\n \\put(175,43){\\vector(1,0){55}}\n \\put(210,43){\\vector(-1,0){55}}\n \\put(175,79){\\vector(1,0){55}}\n \\put(210,79){\\vector(-1,0){55}}\n\\end{picture}\n\\end{gather}\nArtin's theorem states that every alternative algebra generated by two elements is associative, i.e., the variety of binary associative algebras coincides with the variety of alternative algebras \\cite{Artin}. Also, it was shown in \\cite{binaryperm} that the variety of binary perm algebras (which we denote by $\\mathcal{P}erm_2$) coincides with the variety of alternative algebras satisfying an additional identity of degree $3$, which is\n\\[\na(bc)+a(cb)=b(ac)+c(ab).\n\\]\nIn the cases $\\mathfrak{Z}_i = \\mathfrak{A}_1$ and $\\mathfrak{Z}_i = \\mathfrak{A}_2$, it turns out that the operad $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}^{\\;(!)}$ is an alternative satisfying an additional identity of degree $3$.\nMoreover, in the case $\\mathfrak{Z}_i = \\mathfrak{A}_1$, the additional identity is again left-commutative. From the general observations, we obtain\n\\[\n\\mathcal{P}erm\\subset \\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_1}^{\\;(!)}\\subset\\mathcal{P}erm_2.\n\\]\nAlthough this is not immediate, direct computations show that\n\\[\n\\mathcal{P}erm\\subset \\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_2}^{\\;(!)}\\subset\\mathcal{P}erm_2.\n\\]\nAlso, we shall see that\n\\[\n\\mathcal{P}erm\\subset \\mathcal{L}\\mathcal{S}_{\\mathfrak{B}_1}^{\\;(!)}\\not\\subset\\mathcal{P}erm_2.\n\\]\nTo compute the Koszul dual operad of $\\mathcal{L}\\mathcal{S}_{\\mathfrak{Z}_i}$, we need the multilinear basis of degree $3$ of the algebra $\\mathcal{L}\\mathcal{S}\\$. We use the following rewriting rules:\n\\[\na(bc)=(ab)c-(ba)c+b(ac),\n\\]\n\\[\na(cb)=(ac)b-(ca)b+c(ab)\n\\]\nand\n\\[\nb(ca)=(bc)a-(cb)a+c(ba).\n\\]\nLet us compute the polarization of $\\mathcal{L}\\mathcal{S}$ following \\cite{MarklRemm}; this construction will be used throughout the paper.\n\nSince there is a one-to-one correspondence between a variety of algebras\ndefined by identities of degrees two and three $\\Var$ and the associated quadratic operad,\nwe will use the same terminology for both objects throughout the paper. We consider all algebras over a field $K$ of characteristic $0$.\n\n\\begin{proposition}\nThe polarization of $\\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_1}\\$ is defined by (\\ref{pol1}), (\\ref{pol2}) and\n\\begin{multline*}\n\\{a,\\{b,c\\}\\}=\\{[a,b],c\\}+\\{[a,c],b\\}-3/2[\\{a,b\\},c]-3/2[\\{a,c\\},b]\\\\\n+3/2[[a,c],b]-1/3[a,\\{b,c\\}]+1/3[a,[b,c]].\n\\end{multline*}\n\\end{proposition}\n\\begin{proof}\nThe proof is analogous to Proposition \\ref{polarization}.\n\\end{proof}\n\nLet $\\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_1}^{\\;\\;(!)}\\$ be a free algebra of the variety $\\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_1}^{\\;\\;(!)}$.\n\\begin{lemma}\\label{lemmaA1}\nIn algebra $\\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_1}^{\\;\\;(!)}\\$ the following identities hold:\n\\[\n((ba)c)d=((ac)b)d,\n\\]\n\\[\n(((ab)c)d)e=(((ac)b)d)e\n\\]\nand\n\\[\n(((ab)c)d)e=(((ab)d)c)e.\n\\]\n\\end{lemma}\n\\begin{proof}\nIt can be proved using computer algebra as software programs Albert \\cite{Albert}. \\end{proof}\n\nTo prove the linear independence of the set $\\overline{\\mathfrak{A}^{((!))}}_1$, we construct the multiplication table for the algebra $A_1\\$ with the basis $\\overline{\\mathfrak{A}^{((!))}}_1$ and show that such an algebra $A_1\\$ belongs to the variety $\\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_1}^{\\;\\;(!)}\\$. Up to degree 4, we define multiplication in $A_1\\$ that is consistent with the defining identities of $\\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_1}^{\\;\\;(!)}\\$. Starting from degree $5$, the multiplication is defined as follows:\n\\begin{multline}\\label{mult1}\n(((\\cdots((x_{i_1}x_{i_2})x_{i_3})\\cdots )x_{i_{n-1}})x_{i_n})(((\\cdots((x_{j_1}x_{j_2})x_{j_3})\\cdots )x_{j_{m-1}})x_{j_m})=\\\\\n((\\cdots((x_{k_1}x_{k_2})x_{k_3})\\cdots )x_{k_{n+m-1}})x_{j_{m}},\n\\end{multline}\nwhere $\\{k_1,k_2,\\ldots,k_{n+m-1}\\}=\\{i_1,i_2,\\ldots,i_{n},j_1,j_2,\\ldots,j_{m-1}\\}$ and $k_1\\leq k_2\\leq\\ldots\\leq k_{n+m-1}$. It remains to check that the algebra $A_1\\$ satisfies the left-alternative, right-alternative, and left-commutative identities.\n\n\\begin{proposition}\nThe polarization of $\\mathcal{L}\\mathcal{S}_{\\mathfrak{B}_1}\\$ is defined by (\\ref{pol1}), (\\ref{pol2}) and\n\\[\n\\{[a,b],c\\}+\\{[b,c],a\\}+\\{[c,a],b\\}=0.\n\\]\n\\end{proposition}\n\\begin{proof}\nThe proof is analogous to Proposition \\ref{polarization}.\n\\end{proof}\n\n\\section{$\\mathfrak{A}_2$-left-symmetric algebras}\nIn this section, we consider algebras from the variety $\\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_2}$. Let us compute the dual operad for $\\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_2}$. The Lie-admissibility condition for $S\\otimes U$ gives the defining identities of the operad $\\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_2}^{\\;\\;(!)}$, where $S$ is an algebra from $\\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_2}$. So, we have\n\\begin{multline*}\n[[a\\otimes u,b\\otimes v],c\\otimes w]=(ab)c\\otimes (uv)w-(ba)c\\otimes (vu)w-\nc(ab)\\otimes w(uv)+c(ba)\\otimes w(vu)=\\\\\n((ba)c-(ac)b+(ca)b-(bc)a+(cb)a-2b(ac)-2c(ab)-2c(ba))\\otimes (uv)w\\\\\n-(ba)c\\otimes (vu)w-c(ab)\\otimes w(uv)+c(ba)\\otimes w(vu),\n\\end{multline*}\n\\begin{multline*}\n[[b\\otimes v,c\\otimes w],a\\otimes u]=(bc)a\\otimes (vw)u-(cb)a\\otimes (wv)u-\na(bc)\\otimes u(vw)+a(cb)\\otimes u(wv)=\\\\\n(bc)a\\otimes (vw)u-(cb)a\\otimes (wv)u\n-(-(ac)b+(ca)b-(bc)a+(cb)a-b(ac)-2c(ab)-2c(ba))\\otimes u(vw)\\\\\n+((ac)b-(ca)b+c(ab))\\otimes u(wv)\n\\end{multline*}\nand\n\\begin{multline*}\n[[c\\otimes w,a\\otimes u],b\\otimes v]=(ca)b\\otimes (wu)v-(ac)b\\otimes (uw)v-\nb(ca)\\otimes v(wu)+b(ac)\\otimes v(uw)=\\\\\n (ca)b\\otimes (wu)v-(ac)b\\otimes (uw)v-\n((bc)a-(cb)a+c(ba))\\otimes v(wu)+b(ac)\\otimes v(uw).\n\\end{multline*}\nThe sum of the same elements on the left side of the tensors, we obtain the following result: \n\\begin{proposition}\nThe operad $\\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_2}^{\\;\\;(!)}$ is the alternative operad with an identity\n\\[\n(ab)c=(ba)c\n\\]\n\\end{proposition}\n\\begin{proposition}\nThe polarization of $\\mathcal{L}\\mathcal{S}_{\\mathfrak{A}_2}\\$ is defined by (\\ref{pol1}), (\\ref{pol2}) and\n\\[\n\\{a,\\{b,c\\}\\}=\\{[a,b],c\\}+\\{[a,c],b\\}+[\\{b,c\\},a]+2/3[[a,c],b]+1/3[a,[b,c]].\n\\]\n\\end{proposition}\n\\begin{proof}\nThe proof is analogous to Proposition \\ref{polarization}.\n\\end{proof}", "post_theorem_intro_text_len": 2253, "post_theorem_intro_text": "\\begin{proof}\nFirstly, we rewrite each identity of $\\mathcal{LS}$ using commutators and anti-commutators as follows:\n\\[\nab=1/2([a,b]+\\{a,b\\}).\n\\]\nWe define an order on monomials with operations $[\\cdot,\\cdot]$ and $\\{\\cdot,\\cdot\\}$ by\n\\begin{itemize}\n \\item $[[\\cdot,\\cdot],\\cdot]>[\\{\\cdot,\\cdot\\},\\cdot]>\\{[\\cdot,\\cdot],\\cdot\\}>\\{\\{\\cdot,\\cdot\\},\\cdot\\}$;\n \\item the remaining monomials are ordered by lexicographical order;\n\\end{itemize}\nLet $\\mathcal{R}$ be a set of relations in $\\mathcal{LS}$ in terms of $[\\cdot,\\cdot]$ and $\\{\\cdot,\\cdot\\}$, i.e., we have\n\\begin{multline*}\n(ab)c-a(bc)+b(ac)-(ba)c=1/4([[a,b],c]+[\\{a,b\\},c]+\\{[a,b],c\\}+\\{\\{a,b\\},c\\}\\\\\n+[[b,c],a]-\\{[b,c],a\\}+[\\{b,c\\},a]-\\{\\{b,c\\},a\\}\\\\\n+[[a,b],c]-[\\{a,b\\},c]+\\{[a,b],c\\}-\\{\\{a,b\\},c\\}\\\\\n-[[a,c],b]+\\{[a,c],b\\}-[\\{a,c\\},b]+\\{\\{a,c\\},b\\}),\n\\end{multline*}\n\\begin{multline*}\n(ac)b-a(cb)-(ca)b+c(ab)=1/4([[a,c],b]+[\\{a,c\\},b]+\\{[a,c],b\\}+\\{\\{a,c\\},b\\}\\\\\n-[[b,c],a]+\\{[b,c],a\\}+[\\{b,c\\},a]-\\{\\{b,c\\},a\\}\\\\\n+[[a,c],b]-[\\{a,c\\},b]+\\{[a,c],b\\}-\\{\\{a,c\\},b\\}\\\\\n-[[a,b],c]+\\{[a,b],c\\}-[\\{a,b\\},c]+\\{\\{a,b\\},c\\})),\\\\ \n\\end{multline*}\nand\n\\begin{multline*}\n(bc)a-b(ca)-(cb)a+c(ba)=1/4([[b,c],a]+[\\{b,c\\},a]+\\{[b,c],a\\}+\\{\\{b,c\\},a\\}\\\\\n-[[a,c],b]+\\{[a,c],b\\}+[\\{a,c\\},b]-\\{\\{a,c\\},b\\}\\\\\n+[[b,c],a]-[\\{b,c\\},a]+\\{[b,c],a\\}-\\{\\{b,c\\},a\\}\\\\\n+[[a,b],c]-\\{[a,b],c\\}-[\\{a,b\\},c]+\\{\\{a,b\\},c\\}).\\\\\n\\end{multline*}\nFor the set $\\mathcal{R}$, we construct a matrix $[\\mathcal{R}]$ representing the identities as row vectors relative to the given order.\n\\[ \\left(\n\\begin{array}{cccccccccccc}\n2/4 & -1/4 & 1/4 & 0 & -1/4 & 1/4 & 2/4 & 1/4 & -1/4 & 0 & 1/4 & -1/4\\\\\n-1/4 & 2/4 & -1/4 & -1/4 & 0 & 1/4 & 1/4 & 2/4 & 1/4 & 1/4 & 0 & -1/4\\\\\n 1/4 & -1/4 & 2/4 & -1/4 & 1/4 & 0 & -1/4 & 1/4 & 2/4 & 1/4 & -1/4 & 0\n\\end{array} \\right)\n\\]\nBy applying Gröbner basis theory, the matrix $[\\mathcal{R}]$ can be brought to an echelon form, and we obtain the result.\n\\end{proof}\n\nSince there is a one-to-one correspondence between a variety of algebras\ndefined by identities of degrees two and three $\\mathrm{Var}$ and the associated quadratic operad,\nwe will use the same terminology for both objects throughout the paper. We consider all algebras over a field $K$ of characteristic $0$.", "sketch": "Firstly, rewrite each identity of $\\mathcal{LS}$ using commutators and anti-commutators via\n\\[ab=1/2([a,b]+\\{a,b\\}).\\]\nDefine an order on monomials in the operations $[\\cdot,\\cdot]$ and $\\{\\cdot,\\cdot\\}$ by\n\\begin{itemize}\n\\item $[[\\cdot,\\cdot],\\cdot]>[\\{\\cdot,\\cdot\\},\\cdot]>\\{[\\cdot,\\cdot],\\cdot\\}>\\{\\{\\cdot,\\cdot\\},\\cdot\\}$;\n\\item order the remaining monomials lexicographically.\n\\end{itemize}\nLet $\\mathcal{R}$ be the resulting set of relations in terms of $[\\cdot,\\cdot]$ and $\\{\\cdot,\\cdot\\}$ (obtained by expanding the degree-3 identities into linear combinations of the ordered monomials). Construct a matrix $[\\mathcal{R}]$ representing these identities as row vectors relative to the chosen order. Then, “by applying Gröbner basis theory,” bring $[\\mathcal{R}]$ to echelon form; from this reduced form “we obtain the result,” i.e. the polarized identities \\eqref{pol1} and \\eqref{pol2}.", "expanded_sketch": "Firstly, rewrite each identity of $\\mathcal{LS}$ using commutators and anti-commutators via\n\\[ab=1/2([a,b]+\\{a,b\\}).\\]\nDefine an order on monomials in the operations $[\\cdot,\\cdot]$ and $\\{\\cdot,\\cdot\\}$ by\n\\begin{itemize}\n\\item $[[\\cdot,\\cdot],\\cdot]>[\\{\\cdot,\\cdot\\},\\cdot]>\\{[\\cdot,\\cdot],\\cdot\\}>\\{\\{\\cdot,\\cdot\\},\\cdot\\}$;\n\\item order the remaining monomials lexicographically.\n\\end{itemize}\nLet $\\mathcal{R}$ be the resulting set of relations in terms of $[\\cdot,\\cdot]$ and $\\{\\cdot,\\cdot\\}$ (obtained by expanding the degree-3 identities into linear combinations of the ordered monomials). Construct a matrix $[\\mathcal{R}]$ representing these identities as row vectors relative to the chosen order. Then, by applying Gröbner basis theory, bring $[\\mathcal{R}]$ to echelon form; from this reduced form we obtain the polarized identities \\eqref{pol1} and \\eqref{pol2}.", "expanded_theorem": "\\label{polarization}\nThe polarization of $\\mathcal{L}\\mathcal{S}\\$ is\n\\begin{gather}\\label{pol1}\n[[a,b],c]+[[b,c],a]+[[c,a],b]=0 \n\\end{gather}\nand\n\\begin{gather}\\label{pol2}\n\\{\\{a,b\\},c\\}=-\\{[a,b],c\\}-2\\{[a,c],b\\}+[\\{a,b\\},c]-[[a,c],b]+\\{a,\\{b,c\\}\\}-\\{a,[b,c]\\}+[a,\\{b,c\\}].\n\\end{gather}", "theorem_type": ["Universal", "Equivalence"], "mcq": {"question": "Let \\(\\mathcal{LS}\\langle X\\rangle\\) be the free left-symmetric (pre-Lie) algebra on a set \\(X\\). For elements \\(a,b,c\\), define the commutator and anti-commutator by \\([a,b]=ab-ba\\) and \\(\\{a,b\\}=ab+ba\\). Which identities describe the polarization of \\(\\mathcal{LS}\\langle X\\rangle\\)?", "correct_choice": {"label": "A", "text": "The polarization is given by\n\\[\n[[a,b],c]+[[b,c],a]+[[c,a],b]=0\n\\]\nand\n\\[\n\\{\\{a,b\\},c\\}=-\\{[a,b],c\\}-2\\{[a,c],b\\}+[\\{a,b\\},c]-[[a,c],b]+\\{a,\\{b,c\\}\\}-\\{a,[b,c]\\}+[a,\\{b,c\\}].\n\\]"}, "choices": [{"label": "B", "text": "The polarization is given by\n\\[\n[[a,b],c]+[[b,c],a]+[[c,a],b]=0\n\\]\nand\n\\[\n\\{\\{a,b\\},c\\}=-\\{[a,b],c\\}-2\\{[a,c],b\\}+[\\{a,b\\},c]-[[a,b],c]+\\{a,\\{b,c\\}\\}-\\{a,[b,c]\\}+[a,\\{b,c\\}].\n\\]"}, {"label": "C", "text": "The polarization implies\n\\[\n[[a,b],c]+[[b,c],a]+[[c,a],b]=0.\n\\]"}, {"label": "D", "text": "The polarization is given by\n\\[\n[[a,b],c]+[[b,c],a]+[[c,a],b]=0\n\\]\nand\n\\[\n\\{\\{a,b\\},c\\}=-\\{[a,b],c\\}-\\{[a,c],b\\}+[\\{a,b\\},c]-[[a,c],b]+\\{a,\\{b,c\\}\\}-\\{a,[b,c]\\}+[a,\\{b,c\\}].\n\\]"}, {"label": "E", "text": "The polarization is given by\n\\[\n[[a,b],c]+[[b,c],a]+[[c,a],b]=0\n\\]\nand\n\\[\n\\{\\{a,b\\},c\\}=-\\{[a,b],c\\}-2\\{[a,c],b\\}+[\\{a,b\\},c]-[[a,c],b]+\\{a,\\{b,c\\}\\}+\\{a,[b,c]\\}+[a,\\{b,c\\}].\n\\]"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "B"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "computational_check", "tampered_component": "ordered-monomial coefficient alignment in the bracket term", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "computational_check", "tampered_component": "drops the second polarized identity", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "computational_check", "tampered_component": "coefficient of \\{[a,c],b\\}", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "computational_check", "tampered_component": "sign of \\{a,[b,c]\\}", "template_used": "property_confusion"}]}} +{"id": "2601.10624v1", "paper_link": "http://arxiv.org/abs/2601.10624v1", "theorems_cnt": 7, "theorem": {"env_name": "theorem", "content": "\\label{thm:SRW}\n For any infinite vertex-transitive strongly transient graph $\\mathbb{H}$ of finite degree,\n \\begin{equation*}\n \\liminf_{n \\to \\infty} \\PP_v\\big(\\Psi(G_n^\\mathbb{H}) = v\\big) > 0.\n \\end{equation*}\n In particular, source localisation is possible if $\\mathbb{H} = \\ZZ^d$ for $d \\ge 5$.", "start_pos": 5348, "end_pos": 5675, "label": "thm:SRW"}, "ref_dict": {"thm:SRWoptimal": "\\begin{theorem}\n\\label{thm:SRWoptimal}\n We have\n \\begin{equation*}\n \\liminf_{n\\to\\infty} \\inf_{v \\in \\ZZ^d} \\PP_v\\big(\\Psi\\big(G_n^{\\ZZ^d}\\big) = v\\big) = 1/2 - O(1/d) \\ \\text{ as } \\ d \\to \\infty.\n \\end{equation*}\n Moreover, for every estimator $\\Phi$, we have \n \\begin{equation*}\n \\limsup_{n\\to\\infty} \\inf_{v \\in \\ZZ^d} \\PP_v(\\Phi(G_n^{\\ZZ^d}) = v) = 1/2 - \\Omega(1/d) \\ \\text{ as } \\ d \\to \\infty.\n \\end{equation*}\n\\end{theorem}", "thm:SRW2": "\\begin{theorem}\n\\label{thm:SRW2}\n Let $d \\ge 5$. Then $\\liminf_{n \\to \\infty} \\PP_v\\big(\\Psi(\\widetilde{G}_n^d) = v\\big) > 0$.\n\\end{theorem}", "app:strongtrans": "\\begin{proof}\n By \\Cref{lem:pd}, it suffices to show that $c(d) \\ge s(d)$. For each $d \\in \\NN$, we have\n \\begin{align*}\n 1 - c(d) &= \\PP(\\overline{X}(-\\infty,0] \\cap \\overline{X}[1,\\infty) \\ne \\emptyset)\\\\\n &= \\PP(\\overline{X}_i = \\overline{X}_j \\text{ for some integers } i \\le 0, j \\ge 1)\\\\\n &\\le \\PP(\\overline{X}_i = \\overline{X}_j \\text{ for some integers } i \\le 0, j \\ge 0 \\text{ such that } (i,j) \\ne (0, 0))\\\\\n &= 1 - s(d). \\qedhere\n \\end{align*}\n\\end{proof}\n\n\\section{Strongly transient walks}\n\\label{app:strongtrans}\n\nLet $\\gH$ be an infinite vertex-transitive graph of finite degree. Suppose the simple random walk $(X_n)_{n \\ge 0}$ on $\\gH$ is transient. Then it is said to be \\emph{strongly transient} if\n\\begin{equation}\n\\label{eqn:st}\n \\EE[\\tau_v^+ \\mid X_0 = v, \\tau_v^+ < \\infty] < \\infty,\n\\end{equation}\nwhere $\\tau_v^+ \\coloneqq \\inf \\{t > 0: X_t = v\\}$ is the first return time to the starting vertex $v$. Let\n\\begin{equation*}\n p_n = \\PP(\\tau_v^+ = n \\mid X_0 = v) \\text{ and } q_n = \\PP(X_n = v \\mid X_0 = v),\n\\end{equation*}\nfor all $n \\in \\ZZ_{\\ge 0}$. The expectation in \\eqref{eqn:st} can be rewritten as\n\\begin{equation*}\n \\EE[\\tau_v^+ \\mid X_0 = v, \\tau_v^+ < \\infty] = \\sum_{n = 0}^\\infty n \\PP(\\tau_v^+ = n \\mid X_0 = v, \\tau_v^+ < \\infty) = \\frac{\\sum_{n = 0}^\\infty n p_n}{\\PP(\\tau_v^+ < \\infty \\mid X_0 = v)}.\n\\end{equation*}\nNote that the denominator of the last expression is positive. Thus the walk is strongly transient if and only if $\\sum_{n=0}^\\infty n p_n < \\infty$.\n\n\\begin{lemma}\n\\label{lem:st}\n The random walk $(X_n)_{n \\ge 0}$ is strongly transient if and only if $\\sum_{n = 0}^\\infty n q_n < \\infty$.\n\\end{lemma}\n\n\\begin{proof}\n Suppose that $\\sum_{n = 0}^\\infty n q_n < \\infty$. Then we have $\\sum_{n = 0}^\\infty n p_n < \\infty$ since $0 \\le p_n \\le q_n$ for all $n \\in \\ZZ_{\\ge 0}$. This implies that the walk is strongly transient.\n\n Conversely, suppose the walk is strongly transient. Consider the two power series\n \\begin{equation*}\n f(x) = \\sum_{n = 0}^\\infty p_n x^n \\quad\\text{ and }\\quad g(x) = \\sum_{n = 0}^\\infty q_n x^n.\n \\end{equation*}\n Their radii of convergence are both at least $1$, as their coefficients are probabilities, so in $[0,1]$. By transience, we also know that the first power series is convergent at $x = 1$. Furthermore, the definitions of $(p_n)_{n \\ge 0}$ and $(q_n)_{n \\ge 0}$ yield the relation\n \\begin{equation*}\n q_n = \\sum_{k = 0}^n p_k q_{n - k}\n \\end{equation*}\n for all positive integers $n$. By the above and the fact that $q_0 = 1$, for $|x| < 1$, we have\n \\begin{equation*}\n g(x) = \\frac{1}{1 - f(x)}.\n \\end{equation*}\n Now consider the power series\n \\begin{equation*}\n f_1(x) = \\sum_{n = 0}^\\infty n p_n x^n \\quad \\text{ and } \\quad g_1(x) = \\sum_{n = 0}^\\infty n q_n x^n.\n \\end{equation*}\n Then $f_1(x) = f^\\prime(x)$ and $g_1(x) = g^\\prime(x) = f'(x)/(1-f(x))^2$ for $|x| < 1$. As the walk is strongly transient, the power series for $f_1(x)$ is convergent at $x = 1$. Then by Abel's theorem, we have\n \\begin{equation}\n \\label{eqn:f1conv}\n \\lim_{x \\rightarrow 1^-} f^\\prime(x) = f_1(1).\n \\end{equation}\n For any positive integer $N$, we have\n \\begin{equation*}\n \\sum_{n = 0}^N n q_n = \\lim_{x \\rightarrow 1^-} \\sum_{n = 0}^N n q_n x^n \\le \\lim_{x \\rightarrow 1^-} \\sum_{n = 0}^\\infty n q_n x^n \n = \\lim_{x \\to 1^-} \\frac{f^\\prime(x)}{(1 - f(x))^2} = \\frac{f_1(1)}{(1 - f(1))^2},\n \\end{equation*}\n where the last equality follows from \\eqref{eqn:f1conv} and Abel's theorem applied to the power series defining $f(x)$. Thus the power series for $g_1(x)$ converges at $x = 1$, giving $\\sum_{n = 0}^\\infty n q_n < \\infty$.\n\\end{proof}", "thm:SRW": "\\begin{theorem}\n\\label{thm:SRW}\n For any infinite vertex-transitive strongly transient graph $\\gH$ of finite degree,\n \\begin{equation*}\n \\liminf_{n \\to \\infty} \\PP_v\\big(\\Psi(G_n^\\gH) = v\\big) > 0.\n \\end{equation*}\n In particular, source localisation is possible if $\\gH = \\ZZ^d$ for $d \\ge 5$.\n\\end{theorem}", "thm:SRWrecurrent": "\\begin{theorem}\n\\label{thm:SRWrecurrent}\n Suppose $\\gH$ is a recurrent, infinite, connected, vertex-transitive graph of finite degree. Then the simple random walk on $\\gH$ is amnesiac.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 3329, "pre_theorem_intro_text": "\\label{sec:intro}\n\nGrowth processes are ubiquitous in nature, with examples ranging from the growth of cancer tumours in a lab to polymer formation~\\cite{Tur}. These phenomena have inspired many mathematical models of random growth processes, including Diffusion Limited Aggregation (DLA) \\cite{WS}, First Passage Percolation (FPP) \\cite{HWe}, Dielectric Breakdown Model (DBM) \\cite{NPW}, Eden growth process \\cite{Eden}, Internal DLA (IDLA) \\cite{LBG}, and the Simple Random Walk (SRW) viewed at first passage times. Depending on the underlying graph $\\mathbb{H}$ and the stochastic growth rule, the growth cluster may exhibit very different kinds of typical structure. In particular, on lattices $\\mathbb{H} = \\ZZ^d$, the growth clusters in DLA, IDLA, and the high-dimensional simple random walk viewed at first passage times have fractal, Euclidean ball, and quasi-linear structures, respectively.\n\nOur focus in this paper is the question of locating the source (starting vertex) of a growth process given a snapshot of the cluster at some large (or infinite) time. For a random process, this is naturally a statistical question, in which one aims for a prediction that is successful with constant or high probability for cluster size $n \\to \\infty$. There is an extensive empirical literature (see the survey \\cite{SaCh}) in the context of rumor source detection on various real-world networks and also theoretical analysis of some models, such as the Eden growth process on the infinite $d$-regular tree~\\cite{ShZ,ShZ2} and an extensive literature on finding the root of randomly growing trees; see~\\cite{BDL17,BH23,BB22,CCL+24}. We consider this question for simple random walks.\n\n\\subsection{Localisation for the SRW}\nThroughout the paper, we consider the simple random walk $(X_n)_{n \\ge 0}$ on an infinite vertex-transitive graph $\\mathbb{H}$ of finite degree. This walk is called \\emph{recurrent} if it returns to the source almost surely, or \\emph{transient} otherwise. It is called \\emph{strongly transient} if it is transient and the expected return time, given that it returns, is finite; for example, lattices $\\ZZ^d$ with $d \\ge 5$ are strongly transient (see \\Cref{app:strongtrans} for details). Let $G_n = G^{\\mathbb{H}}_n$ be the $n$-step \\emph{trace} of the walk, namely, the graph that consists of vertices and edges visited by the walk in the first $n$ steps, embedded inside the host graph $\\mathbb{H}$. We write $\\PP_v$ for the law of the walk started at a vertex $X_0 = v \\in \\mathbb{H}$, which we call the \\emph{source}. An \\emph{estimator} for the source is a function $\\Phi(G)$ that takes an embedded graph $G$ as input and outputs a vertex (or a set of vertices) in $\\mathbb{H}$, which is possibly random conditional on the trace. We evaluate the performance of $\\Phi$ by $ \\inf_{v \\in V(\\mathbb{H})} \\PP_v(\\Phi(G_n) = v)$. If $\\Phi$ is translation invariant, we may simply write $\\PP_v(\\Phi(G_n) = v)$.\n\nOur main result shows that if $\\mathbb{H}$ is strongly transient, then the source can be located with constant probability, using the following natural estimator $\\Psi$: given an input graph $G$, select a uniformly random diametrical (maximum distance) path and output one of its endpoints uniformly at random. We note that this estimator only depends on $G$, not its embedding in $\\mathbb{H}$.", "context": "\\label{sec:intro}\n\nGrowth processes are ubiquitous in nature, with examples ranging from the growth of cancer tumours in a lab to polymer formation~\\cite{Tur}. These phenomena have inspired many mathematical models of random growth processes, including Diffusion Limited Aggregation (DLA) \\cite{WS}, First Passage Percolation (FPP) \\cite{HWe}, Dielectric Breakdown Model (DBM) \\cite{NPW}, Eden growth process \\cite{Eden}, Internal DLA (IDLA) \\cite{LBG}, and the Simple Random Walk (SRW) viewed at first passage times. Depending on the underlying graph $\\mathbb{H}$ and the stochastic growth rule, the growth cluster may exhibit very different kinds of typical structure. In particular, on lattices $\\mathbb{H} = \\ZZ^d$, the growth clusters in DLA, IDLA, and the high-dimensional simple random walk viewed at first passage times have fractal, Euclidean ball, and quasi-linear structures, respectively.\n\nOur focus in this paper is the question of locating the source (starting vertex) of a growth process given a snapshot of the cluster at some large (or infinite) time. For a random process, this is naturally a statistical question, in which one aims for a prediction that is successful with constant or high probability for cluster size $n \\to \\infty$. There is an extensive empirical literature (see the survey \\cite{SaCh}) in the context of rumor source detection on various real-world networks and also theoretical analysis of some models, such as the Eden growth process on the infinite $d$-regular tree~\\cite{ShZ,ShZ2} and an extensive literature on finding the root of randomly growing trees; see~\\cite{BDL17,BH23,BB22,CCL+24}. We consider this question for simple random walks.\n\n\\subsection{Localisation for the SRW}\nThroughout the paper, we consider the simple random walk $(X_n)_{n \\ge 0}$ on an infinite vertex-transitive graph $\\mathbb{H}$ of finite degree. This walk is called \\emph{recurrent} if it returns to the source almost surely, or \\emph{transient} otherwise. It is called \\emph{strongly transient} if it is transient and the expected return time, given that it returns, is finite; for example, lattices $\\ZZ^d$ with $d \\ge 5$ are strongly transient (see \\Cref{app:strongtrans} for details). Let $G_n = G^{\\mathbb{H}}_n$ be the $n$-step \\emph{trace} of the walk, namely, the graph that consists of vertices and edges visited by the walk in the first $n$ steps, embedded inside the host graph $\\mathbb{H}$. We write $\\PP_v$ for the law of the walk started at a vertex $X_0 = v \\in \\mathbb{H}$, which we call the \\emph{source}. An \\emph{estimator} for the source is a function $\\Phi(G)$ that takes an embedded graph $G$ as input and outputs a vertex (or a set of vertices) in $\\mathbb{H}$, which is possibly random conditional on the trace. We evaluate the performance of $\\Phi$ by $ \\inf_{v \\in V(\\mathbb{H})} \\PP_v(\\Phi(G_n) = v)$. If $\\Phi$ is translation invariant, we may simply write $\\PP_v(\\Phi(G_n) = v)$.\n\nOur main result shows that if $\\mathbb{H}$ is strongly transient, then the source can be located with constant probability, using the following natural estimator $\\Psi$: given an input graph $G$, select a uniformly random diametrical (maximum distance) path and output one of its endpoints uniformly at random. We note that this estimator only depends on $G$, not its embedding in $\\mathbb{H}$.", "full_context": "\\label{sec:intro}\n\nGrowth processes are ubiquitous in nature, with examples ranging from the growth of cancer tumours in a lab to polymer formation~\\cite{Tur}. These phenomena have inspired many mathematical models of random growth processes, including Diffusion Limited Aggregation (DLA) \\cite{WS}, First Passage Percolation (FPP) \\cite{HWe}, Dielectric Breakdown Model (DBM) \\cite{NPW}, Eden growth process \\cite{Eden}, Internal DLA (IDLA) \\cite{LBG}, and the Simple Random Walk (SRW) viewed at first passage times. Depending on the underlying graph $\\mathbb{H}$ and the stochastic growth rule, the growth cluster may exhibit very different kinds of typical structure. In particular, on lattices $\\mathbb{H} = \\ZZ^d$, the growth clusters in DLA, IDLA, and the high-dimensional simple random walk viewed at first passage times have fractal, Euclidean ball, and quasi-linear structures, respectively.\n\nOur focus in this paper is the question of locating the source (starting vertex) of a growth process given a snapshot of the cluster at some large (or infinite) time. For a random process, this is naturally a statistical question, in which one aims for a prediction that is successful with constant or high probability for cluster size $n \\to \\infty$. There is an extensive empirical literature (see the survey \\cite{SaCh}) in the context of rumor source detection on various real-world networks and also theoretical analysis of some models, such as the Eden growth process on the infinite $d$-regular tree~\\cite{ShZ,ShZ2} and an extensive literature on finding the root of randomly growing trees; see~\\cite{BDL17,BH23,BB22,CCL+24}. We consider this question for simple random walks.\n\n\\subsection{Localisation for the SRW}\nThroughout the paper, we consider the simple random walk $(X_n)_{n \\ge 0}$ on an infinite vertex-transitive graph $\\mathbb{H}$ of finite degree. This walk is called \\emph{recurrent} if it returns to the source almost surely, or \\emph{transient} otherwise. It is called \\emph{strongly transient} if it is transient and the expected return time, given that it returns, is finite; for example, lattices $\\ZZ^d$ with $d \\ge 5$ are strongly transient (see \\Cref{app:strongtrans} for details). Let $G_n = G^{\\mathbb{H}}_n$ be the $n$-step \\emph{trace} of the walk, namely, the graph that consists of vertices and edges visited by the walk in the first $n$ steps, embedded inside the host graph $\\mathbb{H}$. We write $\\PP_v$ for the law of the walk started at a vertex $X_0 = v \\in \\mathbb{H}$, which we call the \\emph{source}. An \\emph{estimator} for the source is a function $\\Phi(G)$ that takes an embedded graph $G$ as input and outputs a vertex (or a set of vertices) in $\\mathbb{H}$, which is possibly random conditional on the trace. We evaluate the performance of $\\Phi$ by $ \\inf_{v \\in V(\\mathbb{H})} \\PP_v(\\Phi(G_n) = v)$. If $\\Phi$ is translation invariant, we may simply write $\\PP_v(\\Phi(G_n) = v)$.\n\nOur main result shows that if $\\mathbb{H}$ is strongly transient, then the source can be located with constant probability, using the following natural estimator $\\Psi$: given an input graph $G$, select a uniformly random diametrical (maximum distance) path and output one of its endpoints uniformly at random. We note that this estimator only depends on $G$, not its embedding in $\\mathbb{H}$.\n\nOur main result shows that if $\\gH$ is strongly transient, then the source can be located with constant probability, using the following natural estimator $\\Psi$: given an input graph $G$, select a uniformly random diametrical (maximum distance) path and output one of its endpoints uniformly at random. We note that this estimator only depends on $G$, not its embedding in $\\gH$.\n\nFor $\\gH = \\ZZ^d$, the estimator $\\Psi$ yields a localisation probability of $1/2 - O(1/d)$ as $d \\rightarrow \\infty$; we show that this is sharp.\n\n\\begin{theorem}\n\\label{thm:SRWoptimal}\n We have\n \\begin{equation*}\n \\liminf_{n\\to\\infty} \\inf_{v \\in \\ZZ^d} \\PP_v\\big(\\Psi\\big(G_n^{\\ZZ^d}\\big) = v\\big) = 1/2 - O(1/d) \\ \\text{ as } \\ d \\to \\infty.\n \\end{equation*}\n Moreover, for every estimator $\\Phi$, we have \n \\begin{equation*}\n \\limsup_{n\\to\\infty} \\inf_{v \\in \\ZZ^d} \\PP_v(\\Phi(G_n^{\\ZZ^d}) = v) = 1/2 - \\Omega(1/d) \\ \\text{ as } \\ d \\to \\infty.\n \\end{equation*}\n\\end{theorem}\n\n\\begin{theorem}\n\\label{thm:SRWconst}\n Let $d \\ge 5$. For each $\\varepsilon \\in (0, 1)$, there is a constant $K \\coloneqq K(\\varepsilon) \\in \\NN$ such that\n \\begin{equation*}\n \\liminf_{n \\to \\infty} \\PP_v \\big(v \\in \\Lambda_{K}(G_n^d)\\big) \\ge 1 - \\varepsilon.\n \\end{equation*}\n\\end{theorem}\n\n\\begin{theorem}\n\\label{thm:infinite}\n Let $d \\ge 5$. Then there is an estimator $\\Gamma$ such that\n \\begin{equation*}\n I_d \\coloneqq \\inf_{v \\in \\ZZ^d} \\PP_v(\\Gamma(G_\\infty^d) = v) \\text{ satisfies } I_d > 0 \\text{ and } I_d = 1-O(1/d) \\text{ as } d \\to \\infty.\n \\end{equation*}\n Furthermore, $\\inf_{v \\in \\ZZ^d} \\PP_v(\\Phi(G_\\infty^d) = v) = 1 - \\Omega(1/d)$ as $d \\rightarrow \\infty$, for any estimator $\\Phi$.\n\\end{theorem}\n\n\\begin{theorem}\n\\label{thm:SRW2}\n Let $d \\ge 5$. Then $\\liminf_{n \\to \\infty} \\PP_v\\big(\\Psi(\\widetilde{G}_n^d) = v\\big) > 0$.\n\\end{theorem}\n\n\\begin{proof}[Proof of \\Cref{thm:SRW}]\n\\label{prf:thmSRW1}\n Let $n \\in \\ZZ_+$. Suppose that the event $\\cC_{n-1}$ occurs. \n Then each pair $(\\bfu, \\bfv)$ of vertices in $G_n$ with $d_{G_n}(\\bfu, \\bfv) = \\diam(G_n)$ contains $X_n$.\n As $\\Psi(G)$ selects uniformly at random one endpoint of some diameter of $G$, \n by reversibility we have\n \\begin{equation*}\n \\PP(\\Psi(G_n) = X_0) = \\PP(\\Psi(G_n) = X_n) \\ge \\frac{1}{2} \\PP(\\cC_{n-1}).\n \\end{equation*}\n Taking $\\liminf_{n \\to \\infty}$, the result follows using \\Cref{prop:PC} and $\\cgH > 0$ by \\Cref{lem:intprob}.\n\\end{proof}\n\nLet $\\gH$ be an infinite vertex-transitive graph of finite degree. Suppose the simple random walk $(X_n)_{n \\ge 0}$ on $\\gH$ is transient. Then it is said to be \\emph{strongly transient} if\n\\begin{equation}\n\\label{eqn:st}\n \\EE[\\tau_v^+ \\mid X_0 = v, \\tau_v^+ < \\infty] < \\infty,\n\\end{equation}\nwhere $\\tau_v^+ \\coloneqq \\inf \\{t > 0: X_t = v\\}$ is the first return time to the starting vertex $v$. Let\n\\begin{equation*}\n p_n = \\PP(\\tau_v^+ = n \\mid X_0 = v) \\text{ and } q_n = \\PP(X_n = v \\mid X_0 = v),\n\\end{equation*}\nfor all $n \\in \\ZZ_{\\ge 0}$. The expectation in \\eqref{eqn:st} can be rewritten as\n\\begin{equation*}\n \\EE[\\tau_v^+ \\mid X_0 = v, \\tau_v^+ < \\infty] = \\sum_{n = 0}^\\infty n \\PP(\\tau_v^+ = n \\mid X_0 = v, \\tau_v^+ < \\infty) = \\frac{\\sum_{n = 0}^\\infty n p_n}{\\PP(\\tau_v^+ < \\infty \\mid X_0 = v)}.\n\\end{equation*}\nNote that the denominator of the last expression is positive. Thus the walk is strongly transient if and only if $\\sum_{n=0}^\\infty n p_n < \\infty$.\n\n\\begin{theorem}\n\\label{thm:SRW}\n For any infinite vertex-transitive strongly transient graph $\\gH$ of finite degree,\n \\begin{equation*}\n \\liminf_{n \\to \\infty} \\PP_v\\big(\\Psi(G_n^\\gH) = v\\big) > 0.\n \\end{equation*}\n In particular, source localisation is possible if $\\gH = \\ZZ^d$ for $d \\ge 5$.\n\\end{theorem}", "post_theorem_intro_text_len": 7264, "post_theorem_intro_text": "For $\\mathbb{H} = \\ZZ^d$, the estimator $\\Psi$ yields a localisation probability of $1/2 - O(1/d)$ as $d \\rightarrow \\infty$; we show that this is sharp.\n\n\\begin{theorem}\n\\label{thm:SRWoptimal}\n We have\n \\begin{equation*}\n \\liminf_{n\\to\\infty} \\inf_{v \\in \\ZZ^d} \\PP_v\\big(\\Psi\\big(G_n^{\\ZZ^d}\\big) = v\\big) = 1/2 - O(1/d) \\ \\text{ as } \\ d \\to \\infty.\n \\end{equation*}\n Moreover, for every estimator $\\Phi$, we have \n \\begin{equation*}\n \\limsup_{n\\to\\infty} \\inf_{v \\in \\ZZ^d} \\PP_v(\\Phi(G_n^{\\ZZ^d}) = v) = 1/2 - \\Omega(1/d) \\ \\text{ as } \\ d \\to \\infty.\n \\end{equation*}\n\\end{theorem}\n\nOur next result concerns the impossibility of localising the source. We say that the SRW on $\\mathbb{H}$ is \\emph{amnesiac} if $\\limsup_{n \\to \\infty} \\inf_{v \\in V(\\mathbb{H})} \\PP_v(\\Phi(G^\\gH_n) = v) = 0$ for every estimator~$\\Phi$.\n\n\\begin{theorem}\n\\label{thm:SRWrecurrent}\n Suppose $\\mathbb{H}$ is a recurrent, infinite, connected, vertex-transitive graph of finite degree. Then the simple random walk on $\\mathbb{H}$ is amnesiac.\n\\end{theorem}\n\nIn particular, the above theorem rules out localisation in $\\ZZ^1$ and $\\ZZ^2$, but leaves $\\ZZ^3$ and $\\ZZ^4$ unresolved (see \\Cref{sec:conclude}).\n\n\\subsection{Overview of ideas and methods}\n\\label{sec:overview}\n\nThe starting point for constructing an estimator for locating the source in \\Cref{thm:SRW} is the observation that by reversibility the source vertex $X_0$ and the final vertex $X_n$ are indistinguishable, so it is equivalent (and more convenient) to locate $X_n$ rather than $X_0$. The success of the estimator $\\Psi$ based on diametrical paths may be intuitively understood from the phenomenon that in a strongly transient graph, the number of cut-edges of the trace $G_n$ grows linearly in $n$. This suggests that there should be a constant probability of the event $\\cC_{n-1}$ that step $n$ increases the diameter of the trace, in which case $X_n$ is an endpoint of all diametrical paths.\n\nHowever, there are significant challenges in making this heuristic rigorous. The most significant is that the only straightforward conclusion (via the ergodic theorem) is a lower bound for the Ces\\`aro averages $\\frac1n \\sum_{i=0}^{n-1} \\PP(\\cC_i)$. The key technical ingredients of the proof concern the distribution of the set $L$ of cut edges of the trace, in particular, showing that with high probability, it has a large intersection with every diametrical path. This allows us to control the variation of $\\PP(\\cC_i)$ on linear scales, thus boosting Ces\\`aro control to a lower bound on $\\liminf_n \\PP(\\cC_n)$.\n\nOur upper bounds on the localisation probability are based on coupling arguments. The simplest such approach is a global coupling based on reversibility, which implies that no estimator $\\Phi$ for the SRW in $\\ZZ^d$ can succeed with probability greater than $1/2$. We do not assume $\\Phi$ to be translation invariant, which makes the argument slightly cumbersome. In our proof, we consider a large box $B \\subseteq \\ZZ^d$ and the random variable $W = \\sum_{v \\in B} [\\mathbf{1}(\\Phi(G_n + v) = v) + \\mathbf{1}(\\Phi(G'_n + v + X_n) = v + X_n)]$, where $G_n$ is the trace of $(X_0,\\dots,X_n)$ and $G'_n$ is the trace of the reversed walk. Then $G_n + v$ and $G'_n + v + X_n$ are identical, so whenever $X_n \\ne X_0$, we have $|W| \\le |B|$. We then obtain the required bound from $\\mathbb{E}W \\ge (2+o(1)) |B| \\inf_{v \\in \\ZZ^d} \\PP(\\Phi(G_n+v)=v)$ as $|B| \\to \\infty$; here the $o(1)$ term is justified by amenability of $\\ZZ^d$. A more sophisticated coupling based on re-routing the ends of a walk leads to the improved bound required for \\Cref{thm:SRWoptimal}.\n\nFor recurrent graphs, we obtain the bound in \\Cref{thm:SRWrecurrent} by a similar calculation as above in combination with the idea that several walks starting at distinct vertices can, with high probability, be coupled so that they have identical traces.\n\n\\subsection{Further results}\n\nHere we consider several natural variants of the source localisation problem for the simple random walk. For simplicity, we only consider $\\mathbb{H} = \\ZZ^d$ for $d \\ge 5$, but our arguments work for any strongly transient~$\\mathbb{H}$. For brevity, we shall write $G_n^d$ for $G_n^{\\ZZ^d}$.\n\nFirst we consider the \\emph{high accuracy} problem, where given a target accuracy $\\varepsilon > 0$, we construct an estimator $\\Lambda_K(G_n^d)$ which is a set of $K = K(\\varepsilon)$ vertices that contains the source with probability at least $1 - \\varepsilon$. Given an input graph $G$ and $k \\ge 2$ even, we define $\\Lambda_k(G)$ by selecting a uniformly random pair $(u,v)$ of diametric vertices in $G$ and outputting the $k/2$ closest vertices to $u$ and the $k/2$ closest vertices to $v$.\n\n\\begin{theorem}\n\\label{thm:SRWconst}\n Let $d \\ge 5$. For each $\\varepsilon \\in (0, 1)$, there is a constant $K \\coloneqq K(\\varepsilon) \\in \\NN$ such that\n \\begin{equation*}\n \\liminf_{n \\to \\infty} \\PP_v \\big(v \\in \\Lambda_{K}(G_n^d)\\big) \\ge 1 - \\varepsilon.\n \\end{equation*}\n\\end{theorem}\n\nNext we consider the problem of source localisation given the \\emph{total} (infinite) trace $G_\\infty^d$.\n\n\\begin{theorem}\n\\label{thm:infinite}\n Let $d \\ge 5$. Then there is an estimator $\\Gamma$ such that\n \\begin{equation*}\n I_d \\coloneqq \\inf_{v \\in \\ZZ^d} \\PP_v(\\Gamma(G_\\infty^d) = v) \\text{ satisfies } I_d > 0 \\text{ and } I_d = 1-O(1/d) \\text{ as } d \\to \\infty.\n \\end{equation*}\n Furthermore, $\\inf_{v \\in \\ZZ^d} \\PP_v(\\Phi(G_\\infty^d) = v) = 1 - \\Omega(1/d)$ as $d \\rightarrow \\infty$, for any estimator $\\Phi$.\n\\end{theorem}\n\nWe also show that the estimator $\\Psi$ in \\Cref{thm:SRW} is effective when given a trace with exactly $n$ vertices: formally $R_n^d \\coloneqq G^d_{\\tau_n}$ where $\\tau_n = \\inf \\{t \\ge 0 \\colon |V(G_t^d)| = n\\}$.\n\n\\begin{theorem}\n\\label{thm:range}\n Let $d \\ge 5$. Then $\\liminf_{n\\to\\infty} \\PP_v(\\Psi(R_n^d)=v) > 0$.\n\\end{theorem}\n\nOur final variant concerns the harder source localisation problem where one cannot see the edges of the trace, only its vertex set $V_n$. It turns out that the same estimator $\\Psi$ in \\Cref{thm:SRW} applied to the induced subgraph $\\widetilde{G}_n^d = \\ZZ^d[V_n]$ is still effective!\n\n\\begin{theorem}\n\\label{thm:SRW2}\n Let $d \\ge 5$. Then $\\liminf_{n \\to \\infty} \\PP_v\\big(\\Psi(\\widetilde{G}_n^d) = v\\big) > 0$.\n\\end{theorem}\n\nIn fact, \\Cref{thm:infinite,thm:SRWconst,thm:range} also admit stronger versions with the vertex trace, which can be proved by the same proof technique as in \\Cref{thm:SRW2}, but we omit this for simplicity.\n\n\\subsection{Organisation}\n\nThe rest of this paper is organised as follows. The proofs appear in the same order as the statements of the corresponding theorems in the introduction. The main results regarding the simple random walk are proved in \\Cref{sec:SRW}. In \\Cref{sec:variants}, we establish the variants of the SRW source localisation. \\Cref{sec:conclude} contains some concluding remarks and open problems. We also include \\Cref{app:cd,app:strongtrans}, concerning (A) estimates for the probability of a cut-edge in the two-sided simple random walk on $\\ZZ^d$, and (B) strongly transient walks, including the equivalence of two different definitions used in the literature.", "sketch": "To prove Theorem~\\ref{thm:SRW}, the starting point is that “by reversibility the source vertex $X_0$ and the final vertex $X_n$ are indistinguishable, so it is equivalent (and more convenient) to locate $X_n$ rather than $X_0$.” The estimator $\\Psi$ “based on diametrical paths may be intuitively understood” using that “in a strongly transient graph, the number of cut-edges of the trace $G_n$ grows linearly in $n$,” which “suggests that there should be a constant probability of the event $\\cC_{n-1}$ that step $n$ increases the diameter of the trace, in which case $X_n$ is an endpoint of all diametrical paths.”\n\nThe main difficulty is that “the only straightforward conclusion (via the ergodic theorem) is a lower bound for the Ces\\`aro averages $\\frac1n \\sum_{i=0}^{n-1} \\PP(\\cC_i)$.” The “key technical ingredients” are to analyze “the distribution of the set $L$ of cut edges of the trace,” specifically “showing that with high probability, it has a large intersection with every diametrical path.” This “allows us to control the variation of $\\PP(\\cC_i)$ on linear scales,” thereby “boosting Ces\\`aro control to a lower bound on $\\liminf_n \\PP(\\cC_n)$,” which yields a positive $\\liminf$ success probability for $\\Psi$.", "expanded_sketch": "To prove the main theorem, the starting point is that “by reversibility the source vertex $X_0$ and the final vertex $X_n$ are indistinguishable, so it is equivalent (and more convenient) to locate $X_n$ rather than $X_0$.” The estimator $\\Psi$ “based on diametrical paths may be intuitively understood” using that “in a strongly transient graph, the number of cut-edges of the trace $G_n$ grows linearly in $n$,” which “suggests that there should be a constant probability of the event $\\cC_{n-1}$ that step $n$ increases the diameter of the trace, in which case $X_n$ is an endpoint of all diametrical paths.”\n\nThe main difficulty is that “the only straightforward conclusion (via the ergodic theorem) is a lower bound for the Ces\\`aro averages $\\frac1n \\sum_{i=0}^{n-1} \\PP(\\cC_i)$.” The “key technical ingredients” are to analyze “the distribution of the set $L$ of cut edges of the trace,” specifically “showing that with high probability, it has a large intersection with every diametrical path.” This “allows us to control the variation of $\\PP(\\cC_i)$ on linear scales,” thereby “boosting Ces\\`aro control to a lower bound on $\\liminf_n \\PP(\\cC_n)$,” which yields a positive $\\liminf$ success probability for $\\Psi$.", "expanded_theorem": "\\label{thm:SRW}\n For any infinite vertex-transitive strongly transient graph $\\mathbb{H}$ of finite degree,\n \\begin{equation*}\n \\liminf_{n \\to \\infty} \\PP_v\\big(\\Psi(G_n^\\mathbb{H}) = v\\big) > 0.\n \\end{equation*}\n In particular, source localisation is possible if $\\mathbb{H} = \\ZZ^d$ for $d \\ge 5$.", "theorem_type": ["Asymptotic or Limit", "Inequality or Bound"], "mcq": {"question": "Let \\(\\mathbb H\\) be an infinite vertex-transitive graph of finite degree, and let \\((X_n)_{n\\ge 0}\\) be the simple random walk on \\(\\mathbb H\\) started from a source vertex \\(v\\), with law \\(\\mathbb P_v\\). Assume the walk is strongly transient, meaning it is transient and\n\\[\n\\mathbb E[\\tau_v^+\\mid X_0=v,\\ \\tau_v^+<\\infty]<\\infty,\n\\]\nwhere \\(\\tau_v^+=\\inf\\{t>0:X_t=v\\}\\) is the first return time to \\(v\\). Let \\(G_n^{\\mathbb H}\\) be the trace graph formed by the vertices and edges visited in the first \\(n\\) steps. Define the estimator \\(\\Psi\\) as follows: given a finite graph \\(G\\), choose uniformly at random a diametrical path of \\(G\\) (that is, a path whose endpoints are at graph distance equal to \\(\\operatorname{diam}(G)\\)), and then output one of its two endpoints uniformly at random. Under these assumptions, which quantitative estimate holds for the probability that \\(\\Psi\\) correctly identifies the source from the trace \\(G_n^{\\mathbb H}\\)?", "correct_choice": {"label": "A", "text": "For any infinite vertex-transitive strongly transient graph \\(\\mathbb H\\) of finite degree,\n\\[\n\\liminf_{n\\to\\infty}\\mathbb P_v\\big(\\Psi(G_n^{\\mathbb H})=v\\big)>0.\n\\]\nIn particular, source localisation is possible when \\(\\mathbb H=\\mathbb Z^d\\) for \\(d\\ge 5\\)."}, "choices": [{"label": "B", "text": "For any infinite vertex-transitive transient graph \\(\\mathbb H\\) of finite degree,\n\\[\n\\liminf_{n\\to\\infty}\\mathbb P_v\\big(\\Psi(G_n^{\\mathbb H})=v\\big)>0.\n\\]\nIn particular, source localisation is possible when \\(\\mathbb H=\\mathbb Z^d\\) for every \\(d\\ge 3\\)."}, {"label": "C", "text": "For any infinite vertex-transitive strongly transient graph \\(\\mathbb H\\) of finite degree, there exists an estimator for the source based on the trace \\(G_n^{\\mathbb H}\\) such that\n\\[\n\\liminf_{n\\to\\infty}\\mathbb P_v\\big(\\Phi(G_n^{\\mathbb H})=v\\big)>0.\n\\]\nIn particular, source localisation is possible when \\(\\mathbb H=\\mathbb Z^d\\) for \\(d\\ge 5\\)."}, {"label": "D", "text": "For any infinite vertex-transitive strongly transient graph \\(\\mathbb H\\) of finite degree,\n\\[\n\\inf_{n\\ge 1}\\mathbb P_v\\big(\\Psi(G_n^{\\mathbb H})=v\\big)>0.\n\\]\nIn particular, source localisation is possible when \\(\\mathbb H=\\mathbb Z^d\\) for \\(d\\ge 5\\)."}, {"label": "E", "text": "For any infinite vertex-transitive strongly transient graph \\(\\mathbb H\\) of finite degree,\n\\[\n\\lim_{n\\to\\infty}\\mathbb P_v\\big(\\Psi(G_n^{\\mathbb H})=v\\big)>0.\n\\]\nIn particular, source localisation is possible when \\(\\mathbb H=\\mathbb Z^d\\) for \\(d\\ge 5\\)."}], "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": "strong_transience_hypothesis", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "specific_estimator_Psi_replaced_by_existence_of_some_estimator", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "liminf_in_n_replaced_by_uniform_bound_over_all_n", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "Cesaro_to_liminf_boost_misread_as_full_limit_existence", "template_used": "wildcard"}]}} +{"id": "2601.10624v1", "paper_link": "http://arxiv.org/abs/2601.10624v1", "theorems_cnt": 7, "theorem": {"env_name": "theorem", "content": "\\label{thm:SRW}\n For any infinite vertex-transitive strongly transient graph $\\mathbb{H}$ of finite degree,\n \\begin{equation*}\n \\liminf_{n \\to \\infty} \\PP_v\\big(\\Psi(G_n^\\mathbb{H}) = v\\big) > 0.\n \\end{equation*}\n In particular, source localisation is possible if $\\mathbb{H} = \\ZZ^d$ for $d \\ge 5$.", "start_pos": 5348, "end_pos": 5675, "label": "thm:SRW"}, "ref_dict": {"thm:SRWoptimal": "\\begin{theorem}\n\\label{thm:SRWoptimal}\n We have\n \\begin{equation*}\n \\liminf_{n\\to\\infty} \\inf_{v \\in \\ZZ^d} \\PP_v\\big(\\Psi\\big(G_n^{\\ZZ^d}\\big) = v\\big) = 1/2 - O(1/d) \\ \\text{ as } \\ d \\to \\infty.\n \\end{equation*}\n Moreover, for every estimator $\\Phi$, we have \n \\begin{equation*}\n \\limsup_{n\\to\\infty} \\inf_{v \\in \\ZZ^d} \\PP_v(\\Phi(G_n^{\\ZZ^d}) = v) = 1/2 - \\Omega(1/d) \\ \\text{ as } \\ d \\to \\infty.\n \\end{equation*}\n\\end{theorem}", "thm:SRW2": "\\begin{theorem}\n\\label{thm:SRW2}\n Let $d \\ge 5$. Then $\\liminf_{n \\to \\infty} \\PP_v\\big(\\Psi(\\widetilde{G}_n^d) = v\\big) > 0$.\n\\end{theorem}", "app:strongtrans": "\\begin{proof}\n By \\Cref{lem:pd}, it suffices to show that $c(d) \\ge s(d)$. For each $d \\in \\NN$, we have\n \\begin{align*}\n 1 - c(d) &= \\PP(\\overline{X}(-\\infty,0] \\cap \\overline{X}[1,\\infty) \\ne \\emptyset)\\\\\n &= \\PP(\\overline{X}_i = \\overline{X}_j \\text{ for some integers } i \\le 0, j \\ge 1)\\\\\n &\\le \\PP(\\overline{X}_i = \\overline{X}_j \\text{ for some integers } i \\le 0, j \\ge 0 \\text{ such that } (i,j) \\ne (0, 0))\\\\\n &= 1 - s(d). \\qedhere\n \\end{align*}\n\\end{proof}\n\n\\section{Strongly transient walks}\n\\label{app:strongtrans}\n\nLet $\\gH$ be an infinite vertex-transitive graph of finite degree. Suppose the simple random walk $(X_n)_{n \\ge 0}$ on $\\gH$ is transient. Then it is said to be \\emph{strongly transient} if\n\\begin{equation}\n\\label{eqn:st}\n \\EE[\\tau_v^+ \\mid X_0 = v, \\tau_v^+ < \\infty] < \\infty,\n\\end{equation}\nwhere $\\tau_v^+ \\coloneqq \\inf \\{t > 0: X_t = v\\}$ is the first return time to the starting vertex $v$. Let\n\\begin{equation*}\n p_n = \\PP(\\tau_v^+ = n \\mid X_0 = v) \\text{ and } q_n = \\PP(X_n = v \\mid X_0 = v),\n\\end{equation*}\nfor all $n \\in \\ZZ_{\\ge 0}$. The expectation in \\eqref{eqn:st} can be rewritten as\n\\begin{equation*}\n \\EE[\\tau_v^+ \\mid X_0 = v, \\tau_v^+ < \\infty] = \\sum_{n = 0}^\\infty n \\PP(\\tau_v^+ = n \\mid X_0 = v, \\tau_v^+ < \\infty) = \\frac{\\sum_{n = 0}^\\infty n p_n}{\\PP(\\tau_v^+ < \\infty \\mid X_0 = v)}.\n\\end{equation*}\nNote that the denominator of the last expression is positive. Thus the walk is strongly transient if and only if $\\sum_{n=0}^\\infty n p_n < \\infty$.\n\n\\begin{lemma}\n\\label{lem:st}\n The random walk $(X_n)_{n \\ge 0}$ is strongly transient if and only if $\\sum_{n = 0}^\\infty n q_n < \\infty$.\n\\end{lemma}\n\n\\begin{proof}\n Suppose that $\\sum_{n = 0}^\\infty n q_n < \\infty$. Then we have $\\sum_{n = 0}^\\infty n p_n < \\infty$ since $0 \\le p_n \\le q_n$ for all $n \\in \\ZZ_{\\ge 0}$. This implies that the walk is strongly transient.\n\n Conversely, suppose the walk is strongly transient. Consider the two power series\n \\begin{equation*}\n f(x) = \\sum_{n = 0}^\\infty p_n x^n \\quad\\text{ and }\\quad g(x) = \\sum_{n = 0}^\\infty q_n x^n.\n \\end{equation*}\n Their radii of convergence are both at least $1$, as their coefficients are probabilities, so in $[0,1]$. By transience, we also know that the first power series is convergent at $x = 1$. Furthermore, the definitions of $(p_n)_{n \\ge 0}$ and $(q_n)_{n \\ge 0}$ yield the relation\n \\begin{equation*}\n q_n = \\sum_{k = 0}^n p_k q_{n - k}\n \\end{equation*}\n for all positive integers $n$. By the above and the fact that $q_0 = 1$, for $|x| < 1$, we have\n \\begin{equation*}\n g(x) = \\frac{1}{1 - f(x)}.\n \\end{equation*}\n Now consider the power series\n \\begin{equation*}\n f_1(x) = \\sum_{n = 0}^\\infty n p_n x^n \\quad \\text{ and } \\quad g_1(x) = \\sum_{n = 0}^\\infty n q_n x^n.\n \\end{equation*}\n Then $f_1(x) = f^\\prime(x)$ and $g_1(x) = g^\\prime(x) = f'(x)/(1-f(x))^2$ for $|x| < 1$. As the walk is strongly transient, the power series for $f_1(x)$ is convergent at $x = 1$. Then by Abel's theorem, we have\n \\begin{equation}\n \\label{eqn:f1conv}\n \\lim_{x \\rightarrow 1^-} f^\\prime(x) = f_1(1).\n \\end{equation}\n For any positive integer $N$, we have\n \\begin{equation*}\n \\sum_{n = 0}^N n q_n = \\lim_{x \\rightarrow 1^-} \\sum_{n = 0}^N n q_n x^n \\le \\lim_{x \\rightarrow 1^-} \\sum_{n = 0}^\\infty n q_n x^n \n = \\lim_{x \\to 1^-} \\frac{f^\\prime(x)}{(1 - f(x))^2} = \\frac{f_1(1)}{(1 - f(1))^2},\n \\end{equation*}\n where the last equality follows from \\eqref{eqn:f1conv} and Abel's theorem applied to the power series defining $f(x)$. Thus the power series for $g_1(x)$ converges at $x = 1$, giving $\\sum_{n = 0}^\\infty n q_n < \\infty$.\n\\end{proof}", "thm:SRW": "\\begin{theorem}\n\\label{thm:SRW}\n For any infinite vertex-transitive strongly transient graph $\\gH$ of finite degree,\n \\begin{equation*}\n \\liminf_{n \\to \\infty} \\PP_v\\big(\\Psi(G_n^\\gH) = v\\big) > 0.\n \\end{equation*}\n In particular, source localisation is possible if $\\gH = \\ZZ^d$ for $d \\ge 5$.\n\\end{theorem}", "thm:SRWrecurrent": "\\begin{theorem}\n\\label{thm:SRWrecurrent}\n Suppose $\\gH$ is a recurrent, infinite, connected, vertex-transitive graph of finite degree. Then the simple random walk on $\\gH$ is amnesiac.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 3329, "pre_theorem_intro_text": "\\label{sec:intro}\n\nGrowth processes are ubiquitous in nature, with examples ranging from the growth of cancer tumours in a lab to polymer formation~\\cite{Tur}. These phenomena have inspired many mathematical models of random growth processes, including Diffusion Limited Aggregation (DLA) \\cite{WS}, First Passage Percolation (FPP) \\cite{HWe}, Dielectric Breakdown Model (DBM) \\cite{NPW}, Eden growth process \\cite{Eden}, Internal DLA (IDLA) \\cite{LBG}, and the Simple Random Walk (SRW) viewed at first passage times. Depending on the underlying graph $\\mathbb{H}$ and the stochastic growth rule, the growth cluster may exhibit very different kinds of typical structure. In particular, on lattices $\\mathbb{H} = \\ZZ^d$, the growth clusters in DLA, IDLA, and the high-dimensional simple random walk viewed at first passage times have fractal, Euclidean ball, and quasi-linear structures, respectively.\n\nOur focus in this paper is the question of locating the source (starting vertex) of a growth process given a snapshot of the cluster at some large (or infinite) time. For a random process, this is naturally a statistical question, in which one aims for a prediction that is successful with constant or high probability for cluster size $n \\to \\infty$. There is an extensive empirical literature (see the survey \\cite{SaCh}) in the context of rumor source detection on various real-world networks and also theoretical analysis of some models, such as the Eden growth process on the infinite $d$-regular tree~\\cite{ShZ,ShZ2} and an extensive literature on finding the root of randomly growing trees; see~\\cite{BDL17,BH23,BB22,CCL+24}. We consider this question for simple random walks.\n\n\\subsection{Localisation for the SRW}\nThroughout the paper, we consider the simple random walk $(X_n)_{n \\ge 0}$ on an infinite vertex-transitive graph $\\mathbb{H}$ of finite degree. This walk is called \\emph{recurrent} if it returns to the source almost surely, or \\emph{transient} otherwise. It is called \\emph{strongly transient} if it is transient and the expected return time, given that it returns, is finite; for example, lattices $\\ZZ^d$ with $d \\ge 5$ are strongly transient (see \\Cref{app:strongtrans} for details). Let $G_n = G^{\\mathbb{H}}_n$ be the $n$-step \\emph{trace} of the walk, namely, the graph that consists of vertices and edges visited by the walk in the first $n$ steps, embedded inside the host graph $\\mathbb{H}$. We write $\\PP_v$ for the law of the walk started at a vertex $X_0 = v \\in \\mathbb{H}$, which we call the \\emph{source}. An \\emph{estimator} for the source is a function $\\Phi(G)$ that takes an embedded graph $G$ as input and outputs a vertex (or a set of vertices) in $\\mathbb{H}$, which is possibly random conditional on the trace. We evaluate the performance of $\\Phi$ by $ \\inf_{v \\in V(\\mathbb{H})} \\PP_v(\\Phi(G_n) = v)$. If $\\Phi$ is translation invariant, we may simply write $\\PP_v(\\Phi(G_n) = v)$.\n\nOur main result shows that if $\\mathbb{H}$ is strongly transient, then the source can be located with constant probability, using the following natural estimator $\\Psi$: given an input graph $G$, select a uniformly random diametrical (maximum distance) path and output one of its endpoints uniformly at random. We note that this estimator only depends on $G$, not its embedding in $\\mathbb{H}$.", "context": "\\label{sec:intro}\n\nGrowth processes are ubiquitous in nature, with examples ranging from the growth of cancer tumours in a lab to polymer formation~\\cite{Tur}. These phenomena have inspired many mathematical models of random growth processes, including Diffusion Limited Aggregation (DLA) \\cite{WS}, First Passage Percolation (FPP) \\cite{HWe}, Dielectric Breakdown Model (DBM) \\cite{NPW}, Eden growth process \\cite{Eden}, Internal DLA (IDLA) \\cite{LBG}, and the Simple Random Walk (SRW) viewed at first passage times. Depending on the underlying graph $\\mathbb{H}$ and the stochastic growth rule, the growth cluster may exhibit very different kinds of typical structure. In particular, on lattices $\\mathbb{H} = \\ZZ^d$, the growth clusters in DLA, IDLA, and the high-dimensional simple random walk viewed at first passage times have fractal, Euclidean ball, and quasi-linear structures, respectively.\n\nOur focus in this paper is the question of locating the source (starting vertex) of a growth process given a snapshot of the cluster at some large (or infinite) time. For a random process, this is naturally a statistical question, in which one aims for a prediction that is successful with constant or high probability for cluster size $n \\to \\infty$. There is an extensive empirical literature (see the survey \\cite{SaCh}) in the context of rumor source detection on various real-world networks and also theoretical analysis of some models, such as the Eden growth process on the infinite $d$-regular tree~\\cite{ShZ,ShZ2} and an extensive literature on finding the root of randomly growing trees; see~\\cite{BDL17,BH23,BB22,CCL+24}. We consider this question for simple random walks.\n\n\\subsection{Localisation for the SRW}\nThroughout the paper, we consider the simple random walk $(X_n)_{n \\ge 0}$ on an infinite vertex-transitive graph $\\mathbb{H}$ of finite degree. This walk is called \\emph{recurrent} if it returns to the source almost surely, or \\emph{transient} otherwise. It is called \\emph{strongly transient} if it is transient and the expected return time, given that it returns, is finite; for example, lattices $\\ZZ^d$ with $d \\ge 5$ are strongly transient (see \\Cref{app:strongtrans} for details). Let $G_n = G^{\\mathbb{H}}_n$ be the $n$-step \\emph{trace} of the walk, namely, the graph that consists of vertices and edges visited by the walk in the first $n$ steps, embedded inside the host graph $\\mathbb{H}$. We write $\\PP_v$ for the law of the walk started at a vertex $X_0 = v \\in \\mathbb{H}$, which we call the \\emph{source}. An \\emph{estimator} for the source is a function $\\Phi(G)$ that takes an embedded graph $G$ as input and outputs a vertex (or a set of vertices) in $\\mathbb{H}$, which is possibly random conditional on the trace. We evaluate the performance of $\\Phi$ by $ \\inf_{v \\in V(\\mathbb{H})} \\PP_v(\\Phi(G_n) = v)$. If $\\Phi$ is translation invariant, we may simply write $\\PP_v(\\Phi(G_n) = v)$.\n\nOur main result shows that if $\\mathbb{H}$ is strongly transient, then the source can be located with constant probability, using the following natural estimator $\\Psi$: given an input graph $G$, select a uniformly random diametrical (maximum distance) path and output one of its endpoints uniformly at random. We note that this estimator only depends on $G$, not its embedding in $\\mathbb{H}$.", "full_context": "\\label{sec:intro}\n\nGrowth processes are ubiquitous in nature, with examples ranging from the growth of cancer tumours in a lab to polymer formation~\\cite{Tur}. These phenomena have inspired many mathematical models of random growth processes, including Diffusion Limited Aggregation (DLA) \\cite{WS}, First Passage Percolation (FPP) \\cite{HWe}, Dielectric Breakdown Model (DBM) \\cite{NPW}, Eden growth process \\cite{Eden}, Internal DLA (IDLA) \\cite{LBG}, and the Simple Random Walk (SRW) viewed at first passage times. Depending on the underlying graph $\\mathbb{H}$ and the stochastic growth rule, the growth cluster may exhibit very different kinds of typical structure. In particular, on lattices $\\mathbb{H} = \\ZZ^d$, the growth clusters in DLA, IDLA, and the high-dimensional simple random walk viewed at first passage times have fractal, Euclidean ball, and quasi-linear structures, respectively.\n\nOur focus in this paper is the question of locating the source (starting vertex) of a growth process given a snapshot of the cluster at some large (or infinite) time. For a random process, this is naturally a statistical question, in which one aims for a prediction that is successful with constant or high probability for cluster size $n \\to \\infty$. There is an extensive empirical literature (see the survey \\cite{SaCh}) in the context of rumor source detection on various real-world networks and also theoretical analysis of some models, such as the Eden growth process on the infinite $d$-regular tree~\\cite{ShZ,ShZ2} and an extensive literature on finding the root of randomly growing trees; see~\\cite{BDL17,BH23,BB22,CCL+24}. We consider this question for simple random walks.\n\n\\subsection{Localisation for the SRW}\nThroughout the paper, we consider the simple random walk $(X_n)_{n \\ge 0}$ on an infinite vertex-transitive graph $\\mathbb{H}$ of finite degree. This walk is called \\emph{recurrent} if it returns to the source almost surely, or \\emph{transient} otherwise. It is called \\emph{strongly transient} if it is transient and the expected return time, given that it returns, is finite; for example, lattices $\\ZZ^d$ with $d \\ge 5$ are strongly transient (see \\Cref{app:strongtrans} for details). Let $G_n = G^{\\mathbb{H}}_n$ be the $n$-step \\emph{trace} of the walk, namely, the graph that consists of vertices and edges visited by the walk in the first $n$ steps, embedded inside the host graph $\\mathbb{H}$. We write $\\PP_v$ for the law of the walk started at a vertex $X_0 = v \\in \\mathbb{H}$, which we call the \\emph{source}. An \\emph{estimator} for the source is a function $\\Phi(G)$ that takes an embedded graph $G$ as input and outputs a vertex (or a set of vertices) in $\\mathbb{H}$, which is possibly random conditional on the trace. We evaluate the performance of $\\Phi$ by $ \\inf_{v \\in V(\\mathbb{H})} \\PP_v(\\Phi(G_n) = v)$. If $\\Phi$ is translation invariant, we may simply write $\\PP_v(\\Phi(G_n) = v)$.\n\nOur main result shows that if $\\mathbb{H}$ is strongly transient, then the source can be located with constant probability, using the following natural estimator $\\Psi$: given an input graph $G$, select a uniformly random diametrical (maximum distance) path and output one of its endpoints uniformly at random. We note that this estimator only depends on $G$, not its embedding in $\\mathbb{H}$.\n\nOur main result shows that if $\\gH$ is strongly transient, then the source can be located with constant probability, using the following natural estimator $\\Psi$: given an input graph $G$, select a uniformly random diametrical (maximum distance) path and output one of its endpoints uniformly at random. We note that this estimator only depends on $G$, not its embedding in $\\gH$.\n\nFor $\\gH = \\ZZ^d$, the estimator $\\Psi$ yields a localisation probability of $1/2 - O(1/d)$ as $d \\rightarrow \\infty$; we show that this is sharp.\n\n\\begin{theorem}\n\\label{thm:SRWoptimal}\n We have\n \\begin{equation*}\n \\liminf_{n\\to\\infty} \\inf_{v \\in \\ZZ^d} \\PP_v\\big(\\Psi\\big(G_n^{\\ZZ^d}\\big) = v\\big) = 1/2 - O(1/d) \\ \\text{ as } \\ d \\to \\infty.\n \\end{equation*}\n Moreover, for every estimator $\\Phi$, we have \n \\begin{equation*}\n \\limsup_{n\\to\\infty} \\inf_{v \\in \\ZZ^d} \\PP_v(\\Phi(G_n^{\\ZZ^d}) = v) = 1/2 - \\Omega(1/d) \\ \\text{ as } \\ d \\to \\infty.\n \\end{equation*}\n\\end{theorem}\n\n\\begin{theorem}\n\\label{thm:SRWconst}\n Let $d \\ge 5$. For each $\\varepsilon \\in (0, 1)$, there is a constant $K \\coloneqq K(\\varepsilon) \\in \\NN$ such that\n \\begin{equation*}\n \\liminf_{n \\to \\infty} \\PP_v \\big(v \\in \\Lambda_{K}(G_n^d)\\big) \\ge 1 - \\varepsilon.\n \\end{equation*}\n\\end{theorem}\n\n\\begin{theorem}\n\\label{thm:infinite}\n Let $d \\ge 5$. Then there is an estimator $\\Gamma$ such that\n \\begin{equation*}\n I_d \\coloneqq \\inf_{v \\in \\ZZ^d} \\PP_v(\\Gamma(G_\\infty^d) = v) \\text{ satisfies } I_d > 0 \\text{ and } I_d = 1-O(1/d) \\text{ as } d \\to \\infty.\n \\end{equation*}\n Furthermore, $\\inf_{v \\in \\ZZ^d} \\PP_v(\\Phi(G_\\infty^d) = v) = 1 - \\Omega(1/d)$ as $d \\rightarrow \\infty$, for any estimator $\\Phi$.\n\\end{theorem}\n\n\\begin{theorem}\n\\label{thm:SRW2}\n Let $d \\ge 5$. Then $\\liminf_{n \\to \\infty} \\PP_v\\big(\\Psi(\\widetilde{G}_n^d) = v\\big) > 0$.\n\\end{theorem}\n\n\\begin{proof}[Proof of \\Cref{thm:SRW}]\n\\label{prf:thmSRW1}\n Let $n \\in \\ZZ_+$. Suppose that the event $\\cC_{n-1}$ occurs. \n Then each pair $(\\bfu, \\bfv)$ of vertices in $G_n$ with $d_{G_n}(\\bfu, \\bfv) = \\diam(G_n)$ contains $X_n$.\n As $\\Psi(G)$ selects uniformly at random one endpoint of some diameter of $G$, \n by reversibility we have\n \\begin{equation*}\n \\PP(\\Psi(G_n) = X_0) = \\PP(\\Psi(G_n) = X_n) \\ge \\frac{1}{2} \\PP(\\cC_{n-1}).\n \\end{equation*}\n Taking $\\liminf_{n \\to \\infty}$, the result follows using \\Cref{prop:PC} and $\\cgH > 0$ by \\Cref{lem:intprob}.\n\\end{proof}\n\nLet $\\gH$ be an infinite vertex-transitive graph of finite degree. Suppose the simple random walk $(X_n)_{n \\ge 0}$ on $\\gH$ is transient. Then it is said to be \\emph{strongly transient} if\n\\begin{equation}\n\\label{eqn:st}\n \\EE[\\tau_v^+ \\mid X_0 = v, \\tau_v^+ < \\infty] < \\infty,\n\\end{equation}\nwhere $\\tau_v^+ \\coloneqq \\inf \\{t > 0: X_t = v\\}$ is the first return time to the starting vertex $v$. Let\n\\begin{equation*}\n p_n = \\PP(\\tau_v^+ = n \\mid X_0 = v) \\text{ and } q_n = \\PP(X_n = v \\mid X_0 = v),\n\\end{equation*}\nfor all $n \\in \\ZZ_{\\ge 0}$. The expectation in \\eqref{eqn:st} can be rewritten as\n\\begin{equation*}\n \\EE[\\tau_v^+ \\mid X_0 = v, \\tau_v^+ < \\infty] = \\sum_{n = 0}^\\infty n \\PP(\\tau_v^+ = n \\mid X_0 = v, \\tau_v^+ < \\infty) = \\frac{\\sum_{n = 0}^\\infty n p_n}{\\PP(\\tau_v^+ < \\infty \\mid X_0 = v)}.\n\\end{equation*}\nNote that the denominator of the last expression is positive. Thus the walk is strongly transient if and only if $\\sum_{n=0}^\\infty n p_n < \\infty$.\n\n\\begin{theorem}\n\\label{thm:SRW}\n For any infinite vertex-transitive strongly transient graph $\\gH$ of finite degree,\n \\begin{equation*}\n \\liminf_{n \\to \\infty} \\PP_v\\big(\\Psi(G_n^\\gH) = v\\big) > 0.\n \\end{equation*}\n In particular, source localisation is possible if $\\gH = \\ZZ^d$ for $d \\ge 5$.\n\\end{theorem}", "post_theorem_intro_text_len": 7264, "post_theorem_intro_text": "For $\\mathbb{H} = \\ZZ^d$, the estimator $\\Psi$ yields a localisation probability of $1/2 - O(1/d)$ as $d \\rightarrow \\infty$; we show that this is sharp.\n\n\\begin{theorem}\n\\label{thm:SRWoptimal}\n We have\n \\begin{equation*}\n \\liminf_{n\\to\\infty} \\inf_{v \\in \\ZZ^d} \\PP_v\\big(\\Psi\\big(G_n^{\\ZZ^d}\\big) = v\\big) = 1/2 - O(1/d) \\ \\text{ as } \\ d \\to \\infty.\n \\end{equation*}\n Moreover, for every estimator $\\Phi$, we have \n \\begin{equation*}\n \\limsup_{n\\to\\infty} \\inf_{v \\in \\ZZ^d} \\PP_v(\\Phi(G_n^{\\ZZ^d}) = v) = 1/2 - \\Omega(1/d) \\ \\text{ as } \\ d \\to \\infty.\n \\end{equation*}\n\\end{theorem}\n\nOur next result concerns the impossibility of localising the source. We say that the SRW on $\\mathbb{H}$ is \\emph{amnesiac} if $\\limsup_{n \\to \\infty} \\inf_{v \\in V(\\mathbb{H})} \\PP_v(\\Phi(G^\\gH_n) = v) = 0$ for every estimator~$\\Phi$.\n\n\\begin{theorem}\n\\label{thm:SRWrecurrent}\n Suppose $\\mathbb{H}$ is a recurrent, infinite, connected, vertex-transitive graph of finite degree. Then the simple random walk on $\\mathbb{H}$ is amnesiac.\n\\end{theorem}\n\nIn particular, the above theorem rules out localisation in $\\ZZ^1$ and $\\ZZ^2$, but leaves $\\ZZ^3$ and $\\ZZ^4$ unresolved (see \\Cref{sec:conclude}).\n\n\\subsection{Overview of ideas and methods}\n\\label{sec:overview}\n\nThe starting point for constructing an estimator for locating the source in \\Cref{thm:SRW} is the observation that by reversibility the source vertex $X_0$ and the final vertex $X_n$ are indistinguishable, so it is equivalent (and more convenient) to locate $X_n$ rather than $X_0$. The success of the estimator $\\Psi$ based on diametrical paths may be intuitively understood from the phenomenon that in a strongly transient graph, the number of cut-edges of the trace $G_n$ grows linearly in $n$. This suggests that there should be a constant probability of the event $\\cC_{n-1}$ that step $n$ increases the diameter of the trace, in which case $X_n$ is an endpoint of all diametrical paths.\n\nHowever, there are significant challenges in making this heuristic rigorous. The most significant is that the only straightforward conclusion (via the ergodic theorem) is a lower bound for the Ces\\`aro averages $\\frac1n \\sum_{i=0}^{n-1} \\PP(\\cC_i)$. The key technical ingredients of the proof concern the distribution of the set $L$ of cut edges of the trace, in particular, showing that with high probability, it has a large intersection with every diametrical path. This allows us to control the variation of $\\PP(\\cC_i)$ on linear scales, thus boosting Ces\\`aro control to a lower bound on $\\liminf_n \\PP(\\cC_n)$.\n\nOur upper bounds on the localisation probability are based on coupling arguments. The simplest such approach is a global coupling based on reversibility, which implies that no estimator $\\Phi$ for the SRW in $\\ZZ^d$ can succeed with probability greater than $1/2$. We do not assume $\\Phi$ to be translation invariant, which makes the argument slightly cumbersome. In our proof, we consider a large box $B \\subseteq \\ZZ^d$ and the random variable $W = \\sum_{v \\in B} [\\mathbf{1}(\\Phi(G_n + v) = v) + \\mathbf{1}(\\Phi(G'_n + v + X_n) = v + X_n)]$, where $G_n$ is the trace of $(X_0,\\dots,X_n)$ and $G'_n$ is the trace of the reversed walk. Then $G_n + v$ and $G'_n + v + X_n$ are identical, so whenever $X_n \\ne X_0$, we have $|W| \\le |B|$. We then obtain the required bound from $\\mathbb{E}W \\ge (2+o(1)) |B| \\inf_{v \\in \\ZZ^d} \\PP(\\Phi(G_n+v)=v)$ as $|B| \\to \\infty$; here the $o(1)$ term is justified by amenability of $\\ZZ^d$. A more sophisticated coupling based on re-routing the ends of a walk leads to the improved bound required for \\Cref{thm:SRWoptimal}.\n\nFor recurrent graphs, we obtain the bound in \\Cref{thm:SRWrecurrent} by a similar calculation as above in combination with the idea that several walks starting at distinct vertices can, with high probability, be coupled so that they have identical traces.\n\n\\subsection{Further results}\n\nHere we consider several natural variants of the source localisation problem for the simple random walk. For simplicity, we only consider $\\mathbb{H} = \\ZZ^d$ for $d \\ge 5$, but our arguments work for any strongly transient~$\\mathbb{H}$. For brevity, we shall write $G_n^d$ for $G_n^{\\ZZ^d}$.\n\nFirst we consider the \\emph{high accuracy} problem, where given a target accuracy $\\varepsilon > 0$, we construct an estimator $\\Lambda_K(G_n^d)$ which is a set of $K = K(\\varepsilon)$ vertices that contains the source with probability at least $1 - \\varepsilon$. Given an input graph $G$ and $k \\ge 2$ even, we define $\\Lambda_k(G)$ by selecting a uniformly random pair $(u,v)$ of diametric vertices in $G$ and outputting the $k/2$ closest vertices to $u$ and the $k/2$ closest vertices to $v$.\n\n\\begin{theorem}\n\\label{thm:SRWconst}\n Let $d \\ge 5$. For each $\\varepsilon \\in (0, 1)$, there is a constant $K \\coloneqq K(\\varepsilon) \\in \\NN$ such that\n \\begin{equation*}\n \\liminf_{n \\to \\infty} \\PP_v \\big(v \\in \\Lambda_{K}(G_n^d)\\big) \\ge 1 - \\varepsilon.\n \\end{equation*}\n\\end{theorem}\n\nNext we consider the problem of source localisation given the \\emph{total} (infinite) trace $G_\\infty^d$.\n\n\\begin{theorem}\n\\label{thm:infinite}\n Let $d \\ge 5$. Then there is an estimator $\\Gamma$ such that\n \\begin{equation*}\n I_d \\coloneqq \\inf_{v \\in \\ZZ^d} \\PP_v(\\Gamma(G_\\infty^d) = v) \\text{ satisfies } I_d > 0 \\text{ and } I_d = 1-O(1/d) \\text{ as } d \\to \\infty.\n \\end{equation*}\n Furthermore, $\\inf_{v \\in \\ZZ^d} \\PP_v(\\Phi(G_\\infty^d) = v) = 1 - \\Omega(1/d)$ as $d \\rightarrow \\infty$, for any estimator $\\Phi$.\n\\end{theorem}\n\nWe also show that the estimator $\\Psi$ in \\Cref{thm:SRW} is effective when given a trace with exactly $n$ vertices: formally $R_n^d \\coloneqq G^d_{\\tau_n}$ where $\\tau_n = \\inf \\{t \\ge 0 \\colon |V(G_t^d)| = n\\}$.\n\n\\begin{theorem}\n\\label{thm:range}\n Let $d \\ge 5$. Then $\\liminf_{n\\to\\infty} \\PP_v(\\Psi(R_n^d)=v) > 0$.\n\\end{theorem}\n\nOur final variant concerns the harder source localisation problem where one cannot see the edges of the trace, only its vertex set $V_n$. It turns out that the same estimator $\\Psi$ in \\Cref{thm:SRW} applied to the induced subgraph $\\widetilde{G}_n^d = \\ZZ^d[V_n]$ is still effective!\n\n\\begin{theorem}\n\\label{thm:SRW2}\n Let $d \\ge 5$. Then $\\liminf_{n \\to \\infty} \\PP_v\\big(\\Psi(\\widetilde{G}_n^d) = v\\big) > 0$.\n\\end{theorem}\n\nIn fact, \\Cref{thm:infinite,thm:SRWconst,thm:range} also admit stronger versions with the vertex trace, which can be proved by the same proof technique as in \\Cref{thm:SRW2}, but we omit this for simplicity.\n\n\\subsection{Organisation}\n\nThe rest of this paper is organised as follows. The proofs appear in the same order as the statements of the corresponding theorems in the introduction. The main results regarding the simple random walk are proved in \\Cref{sec:SRW}. In \\Cref{sec:variants}, we establish the variants of the SRW source localisation. \\Cref{sec:conclude} contains some concluding remarks and open problems. We also include \\Cref{app:cd,app:strongtrans}, concerning (A) estimates for the probability of a cut-edge in the two-sided simple random walk on $\\ZZ^d$, and (B) strongly transient walks, including the equivalence of two different definitions used in the literature.", "sketch": "To prove Theorem~\\ref{thm:SRW}, the starting point is that “by reversibility the source vertex $X_0$ and the final vertex $X_n$ are indistinguishable, so it is equivalent (and more convenient) to locate $X_n$ rather than $X_0$.” The estimator $\\Psi$ “based on diametrical paths may be intuitively understood” using that “in a strongly transient graph, the number of cut-edges of the trace $G_n$ grows linearly in $n$,” which “suggests that there should be a constant probability of the event $\\cC_{n-1}$ that step $n$ increases the diameter of the trace, in which case $X_n$ is an endpoint of all diametrical paths.”\n\nThe main difficulty is that “the only straightforward conclusion (via the ergodic theorem) is a lower bound for the Ces\\`aro averages $\\frac1n \\sum_{i=0}^{n-1} \\PP(\\cC_i)$.” The “key technical ingredients” are to analyze “the distribution of the set $L$ of cut edges of the trace,” specifically “showing that with high probability, it has a large intersection with every diametrical path.” This “allows us to control the variation of $\\PP(\\cC_i)$ on linear scales,” thereby “boosting Ces\\`aro control to a lower bound on $\\liminf_n \\PP(\\cC_n)$,” which yields a positive $\\liminf$ success probability for $\\Psi$.", "expanded_sketch": "To prove the main theorem, the starting point is that “by reversibility the source vertex $X_0$ and the final vertex $X_n$ are indistinguishable, so it is equivalent (and more convenient) to locate $X_n$ rather than $X_0$.” The estimator $\\Psi$ “based on diametrical paths may be intuitively understood” using that “in a strongly transient graph, the number of cut-edges of the trace $G_n$ grows linearly in $n$,” which “suggests that there should be a constant probability of the event $\\cC_{n-1}$ that step $n$ increases the diameter of the trace, in which case $X_n$ is an endpoint of all diametrical paths.”\n\nThe main difficulty is that “the only straightforward conclusion (via the ergodic theorem) is a lower bound for the Ces\\`aro averages $\\frac1n \\sum_{i=0}^{n-1} \\PP(\\cC_i)$.” The “key technical ingredients” are to analyze “the distribution of the set $L$ of cut edges of the trace,” specifically “showing that with high probability, it has a large intersection with every diametrical path.” This “allows us to control the variation of $\\PP(\\cC_i)$ on linear scales,” thereby “boosting Ces\\`aro control to a lower bound on $\\liminf_n \\PP(\\cC_n)$,” which yields a positive $\\liminf$ success probability for $\\Psi$.", "expanded_theorem": "\\label{thm:SRW}\n For any infinite vertex-transitive strongly transient graph $\\mathbb{H}$ of finite degree,\n \\begin{equation*}\n \\liminf_{n \\to \\infty} \\PP_v\\big(\\Psi(G_n^\\mathbb{H}) = v\\big) > 0.\n \\end{equation*}\n In particular, source localisation is possible if $\\mathbb{H} = \\ZZ^d$ for $d \\ge 5$.", "theorem_type": ["Asymptotic or Limit", "Inequality or Bound"], "mcq": {"question": "Let \\((X_n)_{n\\ge 0}\\) be the simple random walk on an infinite vertex-transitive graph \\(\\mathbb H\\) of finite degree, started from a source vertex \\(v\\). Let \\(G_n^{\\mathbb H}\\) be the \\(n\\)-step trace of the walk, i.e. the subgraph of \\(\\mathbb H\\) consisting of all vertices and edges visited during the first \\(n\\) steps. Write \\(\\mathbb P_v\\) for the law of the walk with \\(X_0=v\\). Assume the walk is strongly transient, meaning it is transient and, with \\(\\tau_v^+=\\inf\\{t>0:X_t=v\\}\\), one has \\(\\mathbb E[\\tau_v^+\\mid X_0=v,\\,\\tau_v^+<\\infty]<\\infty\\). Define the estimator \\(\\Psi\\) as follows: given a graph \\(G\\), choose uniformly at random a diametrical path of \\(G\\) (a path whose endpoints are at graph distance equal to \\(\\operatorname{diam}(G)\\)), and then output one of the two endpoints uniformly at random. As \\(n\\to\\infty\\), which limiting statement holds for the probability that \\(\\Psi(G_n^{\\mathbb H})\\) correctly recovers the source?", "correct_choice": {"label": "A", "text": "For every infinite vertex-transitive strongly transient graph \\(\\mathbb H\\) of finite degree, one has \\[\\liminf_{n\\to\\infty}\\mathbb P_v\\big(\\Psi(G_n^{\\mathbb H})=v\\big)>0.\\] In particular, taking \\(\\mathbb H=\\mathbb Z^d\\) with \\(d\\ge 5\\), the same positive lower-limit conclusion holds, so source localisation is possible in that case."}, "choices": [{"label": "B", "text": "For every infinite vertex-transitive transient graph \\(\\mathbb H\\) of finite degree, one has \\[\\liminf_{n\\to\\infty}\\mathbb P_v\\big(\\Psi(G_n^{\\mathbb H})=v\\big)>0.\\] In particular, taking \\(\\mathbb H=\\mathbb Z^d\\) with \\(d\\ge 3\\), the same positive lower-limit conclusion holds, so source localisation is possible in that case."}, {"label": "C", "text": "For every infinite vertex-transitive strongly transient graph \\(\\mathbb H\\) of finite degree, source localisation is possible in the sense that there exists an estimator \\(\\Phi\\) such that \\[\\liminf_{n\\to\\infty}\\mathbb P_v\\big(\\Phi(G_n^{\\mathbb H})=v\\big)>0.\\] In particular, taking \\(\\mathbb H=\\mathbb Z^d\\) with \\(d\\ge 5\\), there is some estimator with the same positive lower-limit conclusion."}, {"label": "D", "text": "For every infinite vertex-transitive strongly transient graph \\(\\mathbb H\\) of finite degree, there is a constant \\(c>0\\), independent of \\(\\mathbb H\\), such that \\[\\liminf_{n\\to\\infty}\\mathbb P_v\\big(\\Psi(G_n^{\\mathbb H})=v\\big)\\ge c.\\] In particular, taking \\(\\mathbb H=\\mathbb Z^d\\) with \\(d\\ge 5\\), the same uniform positive lower bound holds."}, {"label": "E", "text": "For every infinite vertex-transitive strongly transient graph \\(\\mathbb H\\) of finite degree, one has \\[\\lim_{n\\to\\infty}\\mathbb P_v\\big(\\Psi(G_n^{\\mathbb H})=v\\big)>0.\\] In particular, taking \\(\\mathbb H=\\mathbb Z^d\\) with \\(d\\ge 5\\), the probability converges to a positive limit, so source localisation is possible in that case."}], "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": "strong_transience_hypothesis", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "specific_estimator_Psi", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "graph_dependent_positive_constant", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "liminf_vs_limit", "template_used": "wildcard"}]}} +{"id": "2601.10656v1", "paper_link": "http://arxiv.org/abs/2601.10656v1", "theorems_cnt": 1, "theorem": {"env_name": "namedtheorem", "content": "[(c.f. Theorem \\ref{thm: HK degeneration})]\n Fix generic $\\vec\\beta\\in(0,\\infty)^n$. Let $\\alpha_i(R)=\\frac12-R\\beta_i$, \nlet $\\mathcal{X}(\\vec \\beta)$ be the $n$-hyperpolygon space, let $\\mathcal M_R(\\vec\\alpha(R))$ be the moduli space of solutions to the $R$-rescaled Hitchin's equations on the $n$-punctured sphere with parabolic weights $\\vec{\\alpha}(R)$, and let \n\\[ \\mathcal{T}_R: \\mathcal{X}(\\vec \\beta) \\to \\mathcal M_R(\\vec\\alpha(R))\\]\n be the natural embedding in \\cite{GM11}.\n\nAs $R \\to 0$,\n the pullback of the family of metrics $\\mathcal{T}_R^*(g_{R,\\vec\\alpha(R)})$ on the Hitchin moduli moduli space $\\mathcal M_R(\\vec\\alpha(R))$ converges pointwise to the metric $2\\pi\\cdot g_{\\mathcal X(\\vec\\beta)}$ on hyperpolygon space $\\mathcal X(\\vec\\beta)$.", "start_pos": 13431, "end_pos": 14238, "label": null}, "ref_dict": {"fig: metric degeneration": "\\label{fig: metric degeneration}\n\\end{figure}\n\n\\section{Local Model}\\label{sec: Local Model}\n\nFix $\\beta \\in \\R^+$. In this section, we construct a family of solutions of Hitchin's equations over the"}, "pre_theorem_intro_text_len": 7750, "pre_theorem_intro_text": "A gravitational instanton is a hyperkähler 4-manifold $(X,g,J_1,J_2,J_3,\\omega_1,\\omega_2,\\omega_3)$ with finite energy\n $$E(g)=\\int_X|\\text{Rm}_g|^2\\text{dvol}_g<\\infty.$$\nLet $r$ be a coordinate given by geodesic distance from a fixed point $p_0$ in $X$. If we impose the slightly stronger condition $|\\text{Rm}_g|^2\\in O(r^{-2-\\varepsilon})$ as $r\\to\\infty$, G. Chen and X. Chen prove that $X$ must have a single noncompact end of type ALE, ALF, ALG, ALH according to the volume growth \\cite{CC21}. Namely, we have $\\text{Vol}(B_r(p_0))\\sim r^m$ for some $m\\in\\{1,2,3,4\\}$, and ALE, ALF, ALG, ALH respectively correspond to $m=4, 3, 2, 1$. When we relax the stronger curvature decay condition, the classification expands to include two additional possibilities ALG$^*$ and ALH$^*$.\n\nThe so-called Modularity Conjecture, attributed to Boalch \\cite{aim},\nposits that all of these gravitational instantons can be realized as gauge-theoretic moduli spaces. In particular, the ALG$^{(*)}$ gravitational instantons should be realized as Hitchin moduli spaces on certain punctured Riemann surfaces $C$ with divisor $D$ with special fixed data at $D$. The ALE gravitational instantons can be realized as certain quiver varieties. There are also certain maps between gauge theoretic spaces. A follow-up question to the Modularity Conjecture is:\nCan these gauge-theoretic maps be used to understand degenerations of gravitational instantons from one type to another? \n\nWe consider this question in the particular case of well-tuned families of ALG-$D_4$ gravitational instantons\\footnote{For any fixed modular parameter $\\tau \\in \\mathbb{H}/\\mathrm{PSL}(2,\\mathbb{Z})$, the family of relevant ALG gravitational instantons are all asymptotic to $(\\mathbb{C} \\times T^2_\\tau)/\\Z_2$, where $\\Z_2$ acts by $(z,w) \\mapsto (-z, -w)$. } degenerating to ALE-$D_4$ gravitational instantons\\footnote{\n The relevant ALE gravitational instantons are asymptotic to $\\mathbb{C}^2/Q8$. Under the McKay correspondence, the affine Dynkin diagram arises as the root system of a finite subgroup $\\Gamma$ of $SU(2)$. The relevant subgroup $\\Gamma$ here is the quaternion group $Q8\\simeq \\{\\pm 1, \\pm i, \\pm j, \\pm k\\}$ which is generated inside $SU(2)$ by the elements $\\begin{pmatrix} i & 0 \\\\ 0 & i \\end{pmatrix}$ and $\\begin{pmatrix} 0 & -1 \\\\ 1 & 0 \\end{pmatrix}$.\n}. The ALG-$D_4$ moduli spaces are strongly parabolic $\\SU(2)$-Hitchin moduli spaces on the four-punctured sphere \\cite{FMSW21} and the ALE-$D_4$ are certain Nakajima quiver varieties known as $n=4$ hyperpolygon spaces \\cite{KN90}. Note that we are only considering the subfamily of gravitational instantons that admit a triholomorphic $\\U(1)$-action. The relevant gauge theoretic map from $n=4$ hyperpolygon spaces to strongly parabolic $\\SL(2,\\mathbb{C})$-Higgs bundle moduli spaces (also called $\\SU(2)$-Hitchin moduli spaces) on the four-punctured sphere is described in \\cite{GM11}. It preserves the holomorphic symplectic structures \\cite{BFGM15}. \n\nMoreover, a similar result holds for arbitrary $n$: the hyperk\\\"ahler metrics on well-tuned families of parabolic $\\SU(2)$-Hitchin moduli spaces on the $n$-punctured sphere converge to the $n$-hyperpolygon space. In this case, the hyperk\\\"ahler metric on hyperpolygon space is known to be quasi-asymptotically conical \\cite{DimakisRochon}, a higher-dimensional generalization of ALE. We note that at this time there is no parallel theorem about $\\SU(2)$-Hitchin moduli spaces on the $n$-punctured sphere having a hyperk\\\"ahler type that generalizes ALG.\n\n\\bigskip\n\nThese hyperk\\\"ahler metrics are difficult to access. In particular, the hyperk\\\"ahler metric on the Hitchin moduli space is written in terms of solutions of gauge-theoretic systems of coupled nonlinear elliptic (modulo gauge) partial differential equations. (In \\Cref{sec:preliminaries}, we will introduce many of the relevant notions including parabolic Higgs bundles and describe the hyperk\\\"ahler metric. But for the sake of brevity, we will here assume that the reader has some familiarity with at least ordinary Higgs bundles, in order to give a succinct description of this work and its relation to other works.) The hyperk\\\"ahler metric on parabolic $\\SU(2)$-Hitchin moduli space is understood as one approaches the noncompact end of the Hitchin moduli space in \\cite{MSWW16, Mochizuki, FredricksonSLn, Fre19, FMSW21,mochizuki2024asymptoticbehaviourhitchinmetric,mochizuki2024comparisonhitchinmetricsemiflat} (``$R \\to \\infty$''), as the PDE decouples in the limit. However, in this paper we are taking a very different limit ($``R \\to 0$'', while simultaneously degenerating the boundary conditions near the punctures). \n\nAn ``$R \\to 0$'' type limit appears in \\cite{DFKMMN16, CW18, CFW24}, where the conformal limit of a parabolic Higgs bundle is computed in \\cite{CFW24}. To review, fix a stable parabolic Higgs bundle $(\\mathcal{E}, \\varphi)$ and consider the family of harmonic metrics $h_R$ solving the $R$-rescaled Hitchin equation\n\\begin{equation}\\label{eq:Hitchinintro}\n R^{-1}F_{\\nabla(\\mathcal{E}, h_R)}^\\perp + R [\\varphi, \\varphi^{\\dagger_{h_R}}]=0,\n\\end{equation}\nwhere $\\nabla(\\mathcal{E}, h_R)$ is the Chern connection.\nLet $\\zeta \\in \\mathbb{C}^\\times$ be the twistor parameter where $\\zeta=0$ corresponds to the Higgs bundle moduli space, and fix $\\frac{\\zeta}{R}=\\hbar $.\nCollier--Fredrickson--Wentworth prove that the conformal limit\n\\[ \\lim_{R,\\zeta \\to 0} \\frac{R}{\\zeta} \\varphi + \\nabla(\\mathcal{E}, h_R) + \\zeta R \\varphi^{\\dagger_{h_R}} \\]\nexists in cases that include (1) strongly parabolic Higgs bundles or (2) weakly parabolic Higgs bundles with full flags. Then, they discuss how the conformal limit interacts with the natural stratification of the space of parabolic logarithmic $\\lambda$-connections by $\\mathbb{C}^\\times$-limits. One common feature shared by conformal limit and our limit is that both are joint limits with special tuning. In the conformal limit, $R \\to 0$ and $\\zeta \\to 0$; in our limit, $R \\to 0$ while the parabolic weights $\\alpha_i \\to \\frac{1}{2}$. \n\nThe degeneration of the boundary conditions appears in \\cite{Jud98}, and for parabolic Higgs bundles in \\cite{KW18}. For parabolic Higgs bundles, the harmonic metric $h_R$ is singular at the divisor $D \\subset C$, and the boundary condition for the harmonic metric $h_R$ is determined by weighted flags at those fixed points. In the case of $\\SL(2,\\mathbb{C})$ Higgs bundles, the weighted flag at $p_i \\in D$ is\n$$\\begin{array}[column sep = -5pt]{ccccccc}\n\\hspace{2.9cm} &0 &\\subset &F_i &\\subset &E_{p_i} &\\\\\n\\hspace{2.9cm} & & &1-\\alpha_i &< &\\alpha_i, &\\qquad\\alpha_i \\in (0, \\frac{1}{2}).\n\\end{array}$$\nWe consider a degeneration of the boundary condition $\\alpha_i \\to \\frac{1}{2}$ from the full flag case to the not-full-flag case. In the special cases where a point of the Hitchin moduli space can be interpreted as a uniformization metric on the punctured surface $\\hat C=C\\smallsetminus D$, this is a degeneration from conical singularities to cuspidal singularities at $D$.\n\nThe joint limit has the effect of sending the volume of fibers of the Hitchin fibration to $\\infty$ (``$R \\to 0$'') while preserving the holomorphic symplectic form. We tune the $R \\to 0$ and $\\alpha_i \\to \\frac{1}{2}$ limits so that \n\\[ \\alpha_i(R) = \\frac{1}{2} - R \\beta_i\\]\nfor some fixed $\\beta_i$.\nThis tuning of the degenerating boundary condition $\\alpha_i(R)$ is carefully chosen; when $n=4$, one can interpret this choice as the choice to hold part of the cohomology data of the real symplectic structure constant as $R$ varies (see \\Cref{fig: metric degeneration}).\n\n\\bigskip\n\nWe prove the following result:", "context": "These hyperk\\\"ahler metrics are difficult to access. In particular, the hyperk\\\"ahler metric on the Hitchin moduli space is written in terms of solutions of gauge-theoretic systems of coupled nonlinear elliptic (modulo gauge) partial differential equations. (In \\Cref{sec:preliminaries}, we will introduce many of the relevant notions including parabolic Higgs bundles and describe the hyperk\\\"ahler metric. But for the sake of brevity, we will here assume that the reader has some familiarity with at least ordinary Higgs bundles, in order to give a succinct description of this work and its relation to other works.) The hyperk\\\"ahler metric on parabolic $\\SU(2)$-Hitchin moduli space is understood as one approaches the noncompact end of the Hitchin moduli space in \\cite{MSWW16, Mochizuki, FredricksonSLn, Fre19, FMSW21,mochizuki2024asymptoticbehaviourhitchinmetric,mochizuki2024comparisonhitchinmetricsemiflat} (``$R \\to \\infty$''), as the PDE decouples in the limit. However, in this paper we are taking a very different limit ($``R \\to 0$'', while simultaneously degenerating the boundary conditions near the punctures).\n\nAn ``$R \\to 0$'' type limit appears in \\cite{DFKMMN16, CW18, CFW24}, where the conformal limit of a parabolic Higgs bundle is computed in \\cite{CFW24}. To review, fix a stable parabolic Higgs bundle $(\\mathcal{E}, \\varphi)$ and consider the family of harmonic metrics $h_R$ solving the $R$-rescaled Hitchin equation\n\\begin{equation}\\label{eq:Hitchinintro}\n R^{-1}F_{\\nabla(\\mathcal{E}, h_R)}^\\perp + R [\\varphi, \\varphi^{\\dagger_{h_R}}]=0,\n\\end{equation}\nwhere $\\nabla(\\mathcal{E}, h_R)$ is the Chern connection.\nLet $\\zeta \\in \\mathbb{C}^\\times$ be the twistor parameter where $\\zeta=0$ corresponds to the Higgs bundle moduli space, and fix $\\frac{\\zeta}{R}=\\hbar $.\nCollier--Fredrickson--Wentworth prove that the conformal limit\n\\[ \\lim_{R,\\zeta \\to 0} \\frac{R}{\\zeta} \\varphi + \\nabla(\\mathcal{E}, h_R) + \\zeta R \\varphi^{\\dagger_{h_R}} \\]\nexists in cases that include (1) strongly parabolic Higgs bundles or (2) weakly parabolic Higgs bundles with full flags. Then, they discuss how the conformal limit interacts with the natural stratification of the space of parabolic logarithmic $\\lambda$-connections by $\\mathbb{C}^\\times$-limits. One common feature shared by conformal limit and our limit is that both are joint limits with special tuning. In the conformal limit, $R \\to 0$ and $\\zeta \\to 0$; in our limit, $R \\to 0$ while the parabolic weights $\\alpha_i \\to \\frac{1}{2}$.\n\nThe degeneration of the boundary conditions appears in \\cite{Jud98}, and for parabolic Higgs bundles in \\cite{KW18}. For parabolic Higgs bundles, the harmonic metric $h_R$ is singular at the divisor $D \\subset C$, and the boundary condition for the harmonic metric $h_R$ is determined by weighted flags at those fixed points. In the case of $\\SL(2,\\mathbb{C})$ Higgs bundles, the weighted flag at $p_i \\in D$ is\n$$\\begin{array}[column sep = -5pt]{ccccccc}\n\\hspace{2.9cm} &0 &\\subset &F_i &\\subset &E_{p_i} &\\\\\n\\hspace{2.9cm} & & &1-\\alpha_i &< &\\alpha_i, &\\qquad\\alpha_i \\in (0, \\frac{1}{2}).\n\\end{array}$$\nWe consider a degeneration of the boundary condition $\\alpha_i \\to \\frac{1}{2}$ from the full flag case to the not-full-flag case. In the special cases where a point of the Hitchin moduli space can be interpreted as a uniformization metric on the punctured surface $\\hat C=C\\smallsetminus D$, this is a degeneration from conical singularities to cuspidal singularities at $D$.\n\nThe joint limit has the effect of sending the volume of fibers of the Hitchin fibration to $\\infty$ (``$R \\to 0$'') while preserving the holomorphic symplectic form. We tune the $R \\to 0$ and $\\alpha_i \\to \\frac{1}{2}$ limits so that \n\\[ \\alpha_i(R) = \\frac{1}{2} - R \\beta_i\\]\nfor some fixed $\\beta_i$.\nThis tuning of the degenerating boundary condition $\\alpha_i(R)$ is carefully chosen; when $n=4$, one can interpret this choice as the choice to hold part of the cohomology data of the real symplectic structure constant as $R$ varies (see \\Cref{fig: metric degeneration}).\n\n\\bigskip\n\nWe prove the following result:\n\n\\label{fig: metric degeneration}\n\\end{figure}\n\n\\section{Local Model}\\label{sec: Local Model}\n\nFix $\\beta \\in \\R^+$. In this section, we construct a family of solutions of Hitchin's equations over the", "full_context": "These hyperk\\\"ahler metrics are difficult to access. In particular, the hyperk\\\"ahler metric on the Hitchin moduli space is written in terms of solutions of gauge-theoretic systems of coupled nonlinear elliptic (modulo gauge) partial differential equations. (In \\Cref{sec:preliminaries}, we will introduce many of the relevant notions including parabolic Higgs bundles and describe the hyperk\\\"ahler metric. But for the sake of brevity, we will here assume that the reader has some familiarity with at least ordinary Higgs bundles, in order to give a succinct description of this work and its relation to other works.) The hyperk\\\"ahler metric on parabolic $\\SU(2)$-Hitchin moduli space is understood as one approaches the noncompact end of the Hitchin moduli space in \\cite{MSWW16, Mochizuki, FredricksonSLn, Fre19, FMSW21,mochizuki2024asymptoticbehaviourhitchinmetric,mochizuki2024comparisonhitchinmetricsemiflat} (``$R \\to \\infty$''), as the PDE decouples in the limit. However, in this paper we are taking a very different limit ($``R \\to 0$'', while simultaneously degenerating the boundary conditions near the punctures).\n\nAn ``$R \\to 0$'' type limit appears in \\cite{DFKMMN16, CW18, CFW24}, where the conformal limit of a parabolic Higgs bundle is computed in \\cite{CFW24}. To review, fix a stable parabolic Higgs bundle $(\\mathcal{E}, \\varphi)$ and consider the family of harmonic metrics $h_R$ solving the $R$-rescaled Hitchin equation\n\\begin{equation}\\label{eq:Hitchinintro}\n R^{-1}F_{\\nabla(\\mathcal{E}, h_R)}^\\perp + R [\\varphi, \\varphi^{\\dagger_{h_R}}]=0,\n\\end{equation}\nwhere $\\nabla(\\mathcal{E}, h_R)$ is the Chern connection.\nLet $\\zeta \\in \\mathbb{C}^\\times$ be the twistor parameter where $\\zeta=0$ corresponds to the Higgs bundle moduli space, and fix $\\frac{\\zeta}{R}=\\hbar $.\nCollier--Fredrickson--Wentworth prove that the conformal limit\n\\[ \\lim_{R,\\zeta \\to 0} \\frac{R}{\\zeta} \\varphi + \\nabla(\\mathcal{E}, h_R) + \\zeta R \\varphi^{\\dagger_{h_R}} \\]\nexists in cases that include (1) strongly parabolic Higgs bundles or (2) weakly parabolic Higgs bundles with full flags. Then, they discuss how the conformal limit interacts with the natural stratification of the space of parabolic logarithmic $\\lambda$-connections by $\\mathbb{C}^\\times$-limits. One common feature shared by conformal limit and our limit is that both are joint limits with special tuning. In the conformal limit, $R \\to 0$ and $\\zeta \\to 0$; in our limit, $R \\to 0$ while the parabolic weights $\\alpha_i \\to \\frac{1}{2}$.\n\nThe degeneration of the boundary conditions appears in \\cite{Jud98}, and for parabolic Higgs bundles in \\cite{KW18}. For parabolic Higgs bundles, the harmonic metric $h_R$ is singular at the divisor $D \\subset C$, and the boundary condition for the harmonic metric $h_R$ is determined by weighted flags at those fixed points. In the case of $\\SL(2,\\mathbb{C})$ Higgs bundles, the weighted flag at $p_i \\in D$ is\n$$\\begin{array}[column sep = -5pt]{ccccccc}\n\\hspace{2.9cm} &0 &\\subset &F_i &\\subset &E_{p_i} &\\\\\n\\hspace{2.9cm} & & &1-\\alpha_i &< &\\alpha_i, &\\qquad\\alpha_i \\in (0, \\frac{1}{2}).\n\\end{array}$$\nWe consider a degeneration of the boundary condition $\\alpha_i \\to \\frac{1}{2}$ from the full flag case to the not-full-flag case. In the special cases where a point of the Hitchin moduli space can be interpreted as a uniformization metric on the punctured surface $\\hat C=C\\smallsetminus D$, this is a degeneration from conical singularities to cuspidal singularities at $D$.\n\nThe joint limit has the effect of sending the volume of fibers of the Hitchin fibration to $\\infty$ (``$R \\to 0$'') while preserving the holomorphic symplectic form. We tune the $R \\to 0$ and $\\alpha_i \\to \\frac{1}{2}$ limits so that \n\\[ \\alpha_i(R) = \\frac{1}{2} - R \\beta_i\\]\nfor some fixed $\\beta_i$.\nThis tuning of the degenerating boundary condition $\\alpha_i(R)$ is carefully chosen; when $n=4$, one can interpret this choice as the choice to hold part of the cohomology data of the real symplectic structure constant as $R$ varies (see \\Cref{fig: metric degeneration}).\n\n\\bigskip\n\nWe prove the following result:\n\n\\label{fig: metric degeneration}\n\\end{figure}\n\n\\section{Local Model}\\label{sec: Local Model}\n\nFix $\\beta \\in \\R^+$. In this section, we construct a family of solutions of Hitchin's equations over the\n\n\\bigskip\n\n\\begin{theorem}\\label[theorem]{thm: Morse functions agree at fixed points} \n Let $\\vec\\beta \\in (0, \\infty)^n$ be generic, let $\\alpha_i(R)= \\frac{1}{2} - R \\beta_i$ for $R>0$, and assume $R_\\text{max}$ solves $W_{[n]}(\\vec\\alpha(R_\\text{max}))=(n-2)/2$ as in \\Cref{thm: map from hyperpolygon to Higgs}. For $R\\in(0,R_\\text{max})$, let $M_\\text{HP}$ and $M_R=M_{R,\\vec\\alpha(R)}$ be the Morse-Bott functions on $\\mathcal{X}(\\beta)$ and $\\mathcal{M}_R(\\vec \\alpha(R))$ described in \\eqref{eqn: Morse-Bott functions}.\n For each $\\U(1)$-fixed point $(\\bx,\\by)$ and corresponding $\\U(1)$-fixed harmonic bundle $\\mathcal T_R(\\bx,\\by)=(\\mathcal E,\\varphi,h_R)$, and for all $00$ small enough that $W_{[n]}(\\vec\\alpha(R))>(n-2)/2$, let $\\mathcal T_R\\from\\mathcal X(\\vec\\beta)\\into\\mathcal M_R(\\vec\\alpha)$ be the family of embeddings described above. Then the family of hyperkähler metrics $\\mathcal{T}_R^*(g_R)$ converges pointwise to $2\\pi\\cdot g_\\text{HP}$ as $R \\to 0$.\n\\end{restatable}\n\\begin{proof} Let $\\delta>0$.\n Let $(\\bx,\\by)$ be a unitary hyperpolygon and $(\\dot\\bx,\\dot\\by)$ a unitary deformation of $(\\bx,\\by)$. The image $(\\mathcal E,\\varphi)=\\mathcal T_R(\\bx,\\by)$ is a Higgs bundle with holomorphic structure $\\delbar_{\\mathcal E}=\\delbar$ independent of $(\\bx, \\by)$, while the parabolic flag structure $\\mathcal F$ depends on $(\\bx, \\by)$. \n It suffices to prove that for every unitary deformations $(\\dot \\bx, \\dot \\by) \\in T_{(\\bx, \\by)} \\mathcal{X}(\\vec \\beta)$ and associated harmonic deformation \n $\\mathtt{H}_R \\in T_{\\mathcal{T}_R(\\bx, \\by)} \\mathcal{M}_R(\\vec(\\alpha))$,\n \\begin{equation}\\label{eqn: main goal}\n \\lim_{R\\to0}\\| \\mathtt{H}_R\\|_{g_R}^2=2\\pi\\sum_i\\left(|\\dot x_i|^2+|\\dot y_i|^2\\right).\n \\end{equation}\n Recall that the deformation $(\\dot\\bx,\\dot\\by)$ corresponds to the Higgs bundle deformation $(\\dot{\\eta}, \\dot{\\mathcal F},\\dot\\varphi)$. \n It will be convenient to introduce the notation $\\dot{\\Phi} = \\dot \\Psi^{1,0}$, \n so that \n the associated deformation in the $h_R$-unitary formulation is $(\\dot \\nabla_R^{0,1},\\dot\\Phi_R)=(-\\delbar\\dot\\nu_R,\\dot\\varphi+[\\dot\\nu_R,\\varphi])$ as discussed in \\Cref{sec:hitchinhk}.\nUsing \\eqref{eq:hk expression} and the regularity of $\\gamma_R$,\n \\begin{align}\n \\|(\\dot \\nabla^{0,1}_R,\\dot\\Phi_R)\\|^2\n &=\\sum_{i=1}^n \\left( \\lim_{\\delta'\\to0}\\oint_{\\partial B_{\\delta'}(p_i)}R^{-1}\\langle\\dot\\nu_R,\\delbar\\dot\\nu_R\\rangle_{h_R}\\right) +\\int_{\\P^1}R\\langle\\dot\\varphi,\\dot\\Phi_R\\rangle_{h_R}\n \\notag\\\\\n &=\\sum_{i=1}^n \\left( \\lim_{\\delta'\\to0}\\oint_{\\partial B_{\\delta'}(p_i)}R^{-1}\\langle\\dot\\nu_R,\\delbar\\dot\\nu_R\\rangle_{h_{\\app,R}} \\right)+\\int_{\\P^1}R\\langle\\dot\\varphi,\\dot\\Phi_{\\app,R}\\rangle_{h_{\\app,R}}+O(R^2)\n \\notag\\\\\n &=\\sum_{i=1}^n \\left( \\lim_{\\delta'\\to0}\\oint_{\\partial B_{\\delta'}(p_i)}R^{-1}\\langle\\dot\\nu_{\\text{app},R},\\delbar\\dot\\nu_{\\text{app},R}\\rangle_{h_{\\text{app},R}} \\right)\\\\\n &\\quad+\\int_{\\P^1}R\\langle\\dot\\varphi,\\dot\\Phi_{\\text{app},R}\\rangle_{h_{\\text{app},R}}+O(R^2)\n \\notag\\\\\n &\\quad+\\sum_{i=1}^n \\left( \\lim_{\\delta'\\to0}\\oint_{\\partial B_{\\delta'}(p_i)}\\underbrace{R^{-1}\\langle\\rho_R,\\delbar\\dot\\nu_{\\text{app},R}\\rangle_{h_{\\text{app},R}}+R^{-1}\\langle\\dot\\nu_{\\text{app},R},\\delbar\\rho_R\\rangle_{h_{\\text{app},R}}}_{f_\\text{cor}(\\bar{z-p_i})^{-1}\\de\\bar z} \\right).\n \\end{align}\n The last integrand can be written as $f_\\text{cor}(\\bar{z-p_i})^{-1}\\de\\bar z$. Since $\\rho_R\\in L_{{2-\\eps}}^{2,2}(\\End(\\mathcal E))$, we have $\\rho_R\\in o(r^{1-\\eps})$ and $\\delbar\\rho_R\\in o(r^{-\\eps})$. Thus, $|f_\\text{cor}|$ has leading order terms of the form $r^{1-\\eps-4R\\beta_i}$, so for sufficiently small $R$ we can bound it above by $Cr^{1/2}$ for some constant $C$;\n this constant is uniform as $R\\to0$ by analyticity of $\\rho_R$. Now $Cr^{1/2}(\\bar{z-p_i})^{-1}\\de\\bar z$ vanishes under integration along $\\partial B_{\\delta'}(p_i)$ in the limit $\\delta'\\to0$, so the last integral is zero. Finally, the uniform boundedness of $\\langle\\dot\\varphi,\\dot\\Phi_{\\text{app},R}\\rangle_{h_R}$ on $[0,R_{\\max})\\times K_\\delta$ and \\Cref{prop: local metric pairing} implies \\eqref{eqn: main goal}.\n\\end{proof}\n\nRecall the Morse-Bott functions in \\eqref{eqn: Morse-Bott functions} arising as moment maps with respect to the real symplectic form for the $\\U(1)$-actions on the hyperk\\\"ahler spaces $\\mathcal X(\\vec\\beta)=X\\fourslash_{(0, \\vec \\beta)}$ and $\\mathcal M_R(\\vec\\alpha(R))$:\n $$M_\\text{HP}(\\bx,\\by)=\\frac i2\\sum_i|y_i|^2,\n \\quad M_R(\\mathcal E,\\varphi,h_R)=\\frac i2\\int_{\\C\\P^1}R|\\varphi|_{h_R}^2$$\n In \\Cref{thm: Morse functions agree at fixed points}, we proved that $M_R(\\mathcal{T}_R(\\bx, \\by))=2 \\pi M_\\text{HP}(\\bx, \\by)$ at corresponding $U(1)$-fixed points. Now, we prove that the family of Morse functions converges: \n\\begin{theorem}\n \\label[theorem]{thm: moment maps agree}\n In the setting of \\Cref{thm: HK degeneration}, for any unitary hyperpolygon $(\\bx,\\by)$ and corresponding $(0,R_{\\max})$-family of harmonic bundles $\\mathcal{T}_R(\\bx, \\by)$, we have \\begin{equation} \\lim_{R\\to0}M_R(\\mathcal{T}_R(\\bx,\\by))=2\\pi M_\\text{HP}(\\bx,\\by).\\end{equation}\n\\end{theorem}\n\\begin{proof}\n Let $(\\bx,\\by)$ be a unitary hyperpolygon, and let $\\gamma_R$, $R_\\text{max}$, and $\\eps\\in(0,\\frac12)$ be as in \\Cref{prop: approximate h accuracy}. Our approximation $h_{\\text{app},R}$ is uniformly bounded on $K_\\delta=\\CP^1 \\sm \\bigcup_{i} B_\\delta(p_i)$ as $R\\to0$, and therefore, so is $h_R=e^{\\gamma_R}\\cdot h_{\\text{app},R}$ (recall $\\lim_{R\\to}\\gamma_R=0$ and $\\gamma_R$ is real analytic in $R$). \n Thus, $R|\\varphi|_{h_R}^2\\to0$ on $K_\\delta$ as $R\\to0$.", "post_theorem_intro_text_len": 5808, "post_theorem_intro_text": "We briefly explain the proof.\nHyperpolygon space can be constructed as a hyperk\\\"ahler manifold via a hyperk\\\"ahler quotient construction or as a holomorphic symplectic manifold via a holomorphic symplectic quotient construction. In the first, one must choose representatives which solve both real and complex moment map equations---a ``unitary hyperpolygon'' in our terminology---while in the second, one must only choose a representative which solves the complex moment map equation---a ``hyperpolygon'' in our terminology. The Hitchin moduli space $\\mathcal{M}_R(\\vec \\alpha(R))$ and the Higgs bundle moduli space $\\mathcal{M}^{\\mathrm{Higgs}}(\\vec \\alpha(R))$ are similarly related. Godinho--Mandini \\cite{GM11} construct a map $\\mathcal{T}: \\mathcal{X}(\\vec \\beta) \\to \\mathcal{M}^{\\mathrm{Higgs}}(\\vec \\alpha(R))$ preserving the holomorphic symplectic structures coming from the complex moment maps \\cite{BFGM15}. \n\nThe map $\\mathcal{T}_R$ above factors through $\\mathcal{T}$.\nNamely, a point in $\\mathcal{M}_R(\\vec \\alpha(R))$ is a Higgs bundle in $\\mathcal{M}^{\\mathrm{Higgs}}(\\vec \\alpha(R))$ together with the hermitian metric $h_R$ on the underlying complex bundle that solves $R$-rescaled Hitchin's equations (the real moment map). \nThe key idea in our proof of \\Cref{thm: HK degeneration} involves a delicate construction of approximate solutions to Hitchin's equations (the real moment map), which uses all the hyperpolygon moment map data to ensure the metric is adapted to the parabolic structure and globally defined. \n\nThis result was independently proved by different methods by Lynn Heller, Sebastian Heller, and Claudio Meneses in \\cite{HHM}. Their work builds on \\cite{HHT2025}. A similar observation is made in \\cite[Appendix A]{BTX}, and we would like to better understand the connection with Coulomb branches and Higgs branches, at the level of hyperk\\\"ahler (rather than holomorphic symplectic) geometry. \n\\bigskip\n\nThis more complicated theorem is of a similar type to the simple degeneration of ALF-$A_n$ gravitational to ALE-$A_n$ gravitational instantons. All ALF-$A_n$ and ALE-$A_n$ gravitational instantons can be obtained using the Gibbons-Hawking ansatz, as \nthe total space of a principal $U(1)$ bundle over $\\mathbb{R}^3$. In the case of ALF-$A_n$, the hyperk\\\"ahler multi-Taub-NUT metrics are specified by a harmonic function\n $$V_R(x)=R+\\sum_{i=1}^n\\frac1{4\\pi|x-p_i|}$$\nfor $R>0$ and distinct points $p_1,\\dots,p_{n+1}\\in\\mathbb{R}^3$, up to translation. \nThe hyperk\\\"ahler metric is \n\\begin{equation}\n g_R=V_R \\norm{\\text{d} \\vec{x}}^2 + V_R^{-1} \\left(\\frac{1}{2 \\pi} \\Theta\\right)^2,\n\\end{equation}\nwhere $\\vec{x} \\in \\mathbb{R}^3$ and $\\Theta$ is a connection on the $U(1)$ bundle whose curvature $F_R$ is related to the potential $V_R$ by $F_R = -2 \\pi \\star \\text{d} V_R$. Note that $\\Theta$ doesn't depend on the constant $R$, so the potential determines the entire hyperk\\\"ahler structure. As $R\\to0$, the volume of the $U(1)$-fibers tends toward infinity and the metric degenerates to an ALE-$A_n$ instanton called the multi-Eguchi-Hanson metric.\n\n\\bigskip\n\n\\subsection*{Organization}\nThe paper is organized as follows:\n\nIn \\Cref{sec:preliminaries}, we introduce the $n$-sided hyperpolygon spaces and the strongly parabolic Hitchin moduli spaces on $\\mathbb{CP}^1$ with $n$ punctures, we describe their hyperkähler metrics, and the map from hyperpolygons to strongly parabolic Higgs bundles. \n\nThe $U(1)$-action on each hyperk\\\"ahler space gives rise to a Morse--Bott function. In \\Cref{sec: Torelli Numbers}, we compute the value of this function at the corresponding fixed points and show that they match. Then, we restrict to $n=4$ for concreteness (the real dimension four case), and explore the topology and geometry of the two families of moduli spaces.\n\nIn \\Cref{sec: Local Model}, we introduce local model solutions of the $\\SU(2)$-Hitchin equations on the disk. We conduct a detailed local analysis of these model solutions and their first variation. \n\nIn \\Cref{sec: The Hyperkahler Metrics}, we use the local models to construct approximate harmonic metrics for Higgs bundles on $\\CP^1$ with $n$-punctures. We then use an implicit function theorem argument perturb our approximate metrics to actual solutions, and finally prove our main theorem.\n\n\\subsection*{Notational Conventions} We take hermitian forms $h$ to be conjugate linear in the first slot and $\\mathbb{C}$-linear in the second slot. We identify such forms with their Gram matrices, e.g.\\ $h(u,v)=u^\\dagger hv$. If $h$ is a hermitian form on $V$ and $B\\in\\End(V)$, then the $h$-adjoint is $B^{\\dagger_h}=h^{-1}B^\\dagger h$, where $\\dagger$ (without the $h$) denotes the standard complex conjugation. For the standard 1-forms $\\text{d}\\bar z$ and $\\text{d} z$ in a coordinate chart on a Riemann surface $C$, the Hodge star is given by $\\star\\text{d}\\bar z=i\\text{d} z$ and $\\star\\text{d} z=-i\\text{d}\\bar z$, extended $\\mathcal C_C^\\infty$-antilinearly.\n\nGiven a holomorphic structure $\\delbar_E$ and a metric $h$ on a vector bundle $E$, we denote the Chern connection by $\\nabla(\\delbar_E,h)$.\n\nFor a matrix $A\\in\\mathfrak{gl}(n,\\mathbb{C})$ we use $A^\\perp$ to denote the orthogonal projection onto $\\mathfrak{sl}(n,\\mathbb{C})$. In other words, $A^\\perp:=A-\\frac1n\\tr(A)$ is the trace-free part of $A$.\n\n\\subsection*{Acknowledgements}\n\nThe authors would like to thank Nick Addington, Sergey Cherkis, Sze Hong Kwong, Rafe Mazzeo, Claudio Meneses, Nick Proudfoot, Steve Rayan, Laura Schaposnik, Hartmut Weiss, Richard Wentworth, and Graeme Wilkin for useful discussions. AY particularly thanks Nick Proudfoot for his help with hyperpolygons at the start of this project. LF is partially supported by NSF grant DMS-2005258. AY is supported by NSF grant RTG-2039316.", "sketch": "Hyperpolygon space and the relevant Hitchin/Higgs moduli spaces are compared via their moment-map descriptions: hyperpolygons can be formed either by a hyperk\"ahler quotient (requiring representatives solving both the real and complex moment map equations, i.e. a \"unitary hyperpolygon\") or by a holomorphic symplectic quotient (requiring only the complex moment map equation, i.e. a \"hyperpolygon\"). Likewise, a point of $\\mathcal{M}_R(\\vec \\alpha(R))$ is described as a Higgs bundle in $\\mathcal{M}^{\\mathrm{Higgs}}(\\vec \\alpha(R))$ together with a hermitian metric $h_R$ solving the $R$-rescaled Hitchin equations (the real moment map). Godinho--Mandini construct a holomorphic symplectic map $\\mathcal{T}:\\mathcal{X}(\\vec\\beta)\\to \\mathcal{M}^{\\mathrm{Higgs}}(\\vec\\alpha(R))$ preserving holomorphic symplectic structures, and the embedding $\\mathcal{T}_R$ \"factors through $\\mathcal{T}$.\" The \"key idea\" for proving the theorem is a \"delicate construction of approximate solutions to Hitchin's equations (the real moment map),\" using \"all the hyperpolygon moment map data\" so that the resulting metric is \"adapted to the parabolic structure and globally defined.\" In the later parts of the paper this is implemented by: introducing local model solutions of the $\\mathrm{SU}(2)$ Hitchin equations on a disk and analyzing them and their first variation; then using these local models to build approximate harmonic metrics for Higgs bundles on $\\mathbb{CP}^1$ with $n$ punctures; then applying an implicit function theorem to perturb the approximate metrics to actual solutions; and finally proving the metric degeneration statement.", "expanded_sketch": "Hyperpolygon space and the relevant Hitchin/Higgs moduli spaces are compared via their moment-map descriptions: hyperpolygons can be formed either by a hyperk\"ahler quotient (requiring representatives solving both the real and complex moment map equations, i.e. a \"unitary hyperpolygon\") or by a holomorphic symplectic quotient (requiring only the complex moment map equation, i.e. a \"hyperpolygon\"). Likewise, a point of $\\mathcal{M}_R(\\vec \\alpha(R))$ is described as a Higgs bundle in $\\mathcal{M}^{\\mathrm{Higgs}}(\\vec \\alpha(R))$ together with a hermitian metric $h_R$ solving the $R$-rescaled Hitchin equations (the real moment map). Godinho--Mandini construct a holomorphic symplectic map $\\mathcal{T}:\\mathcal{X}(\\vec\\beta)\\to \\mathcal{M}^{\\mathrm{Higgs}}(\\vec\\alpha(R))$ preserving holomorphic symplectic structures, and the embedding $\\mathcal{T}_R$ \"factors through $\\mathcal{T}$.\" The key idea for proving the theorem is a delicate construction of approximate solutions to Hitchin's equations (the real moment map), using all the hyperpolygon moment map data so that the resulting metric is adapted to the parabolic structure and globally defined. In the later parts of the paper this is implemented by: introducing local model solutions of the $\\mathrm{SU}(2)$ Hitchin equations on a disk and analyzing them and their first variation; then using these local models to build approximate harmonic metrics for Higgs bundles on $\\mathbb{CP}^1$ with $n$ punctures; then applying an implicit function theorem to perturb the approximate metrics to actual solutions; and finally proving the metric degeneration statement.", "expanded_theorem": "[(c.f. Theorem \\ref{thm: HK degeneration})]\n Fix generic $\\vec\\beta\\in(0,\\infty)^n$. Let $\\alpha_i(R)=\\frac12-R\\beta_i$, \nlet $\\mathcal{X}(\\vec \\beta)$ be the $n$-hyperpolygon space, let $\\mathcal M_R(\\vec\\alpha(R))$ be the moduli space of solutions to the $R$-rescaled Hitchin's equations on the $n$-punctured sphere with parabolic weights $\\vec{\\alpha}(R)$, and let \n\\[ \\mathcal{T}_R: \\mathcal{X}(\\vec \\beta) \\to \\mathcal M_R(\\vec\\alpha(R))\\]\n be the natural embedding in \\cite{GM11}.\n\nAs $R \\to 0$,\n the pullback of the family of metrics $\\mathcal{T}_R^*(g_{R,\\vec\\alpha(R)})$ on the Hitchin moduli moduli space $\\mathcal M_R(\\vec\\alpha(R))$ converges pointwise to the metric $2\\pi\\cdot g_{\\mathcal X(\\vec\\beta)}$ on hyperpolygon space $\\mathcal X(\\vec\\beta)$.,", "theorem_type": ["Asymptotic or Limit", "Universal"], "mcq": {"question": "Fix a generic vector \\(\\vec\\beta=(\\beta_1,\\dots,\\beta_n)\\in(0,\\infty)^n\\), and for each \\(R>0\\) define parabolic weights by \\(\\alpha_i(R)=\\tfrac12-R\\beta_i\\). Let \\(\\mathcal X(\\vec\\beta)\\) be the \\(n\\)-hyperpolygon space, let \\(\\mathcal M_R(\\vec\\alpha(R))\\) be the moduli space of solutions of the \\(R\\)-rescaled Hitchin equations on the \\(n\\)-punctured sphere with parabolic weights \\(\\vec\\alpha(R)\\), and let\n\\[\n\\mathcal T_R:\\mathcal X(\\vec\\beta)\\to \\mathcal M_R(\\vec\\alpha(R))\n\\]\nbe the natural embedding. If \\(g_{R,\\vec\\alpha(R)}\\) denotes the hyperk\\\"ahler metric on \\(\\mathcal M_R(\\vec\\alpha(R))\\) and \\(g_{\\mathcal X(\\vec\\beta)}\\) denotes the hyperk\\\"ahler metric on \\(\\mathcal X(\\vec\\beta)\\), which statement holds as \\(R\\to 0\\)?", "correct_choice": {"label": "A", "text": "The pulled-back metrics \\(\\mathcal T_R^*(g_{R,\\vec\\alpha(R)})\\) converge pointwise on \\(\\mathcal X(\\vec\\beta)\\) to \\(2\\pi\\, g_{\\mathcal X(\\vec\\beta)}\\); equivalently, for every point \\(x\\in \\mathcal X(\\vec\\beta)\\) and every tangent vectors \\(u,v\\in T_x\\mathcal X(\\vec\\beta)\\),\n\\[\n\\lim_{R\\to 0} \\bigl(\\mathcal T_R^* g_{R,\\vec\\alpha(R)}\\bigr)_x(u,v)=2\\pi\\, \\bigl(g_{\\mathcal X(\\vec\\beta)}\\bigr)_x(u,v).\n\\]"}, "choices": [{"label": "B", "text": "The pulled-back metrics \\(\\mathcal T_R^*(g_{R,\\vec\\alpha(R)})\\) converge pointwise on \\(\\mathcal X(\\vec\\beta)\\) to \\(g_{\\mathcal X(\\vec\\beta)}\\); equivalently, for every point \\(x\\in \\mathcal X(\\vec\\beta)\\) and every tangent vectors \\(u,v\\in T_x\\mathcal X(\\vec\\beta)\\),\n\\[\n\\lim_{R\\to 0} \\bigl(\\mathcal T_R^* g_{R,\\vec\\alpha(R)}\\bigr)_x(u,v)=\\bigl(g_{\\mathcal X(\\vec\\beta)}\\bigr)_x(u,v).\n\\]"}, {"label": "C", "text": "For every point \\(x\\in \\mathcal X(\\vec\\beta)\\) and every tangent vectors \\(u,v\\in T_x\\mathcal X(\\vec\\beta)\\), the limit\n\\[\n\\lim_{R\\to 0} \\bigl(\\mathcal T_R^* g_{R,\\vec\\alpha(R)}\\bigr)_x(u,v)\n\\]\nexists."}, {"label": "D", "text": "The pulled-back metrics \\(\\mathcal T_R^*(g_{R,\\vec\\alpha(R)})\\) converge uniformly on \\(\\mathcal X(\\vec\\beta)\\) to \\(2\\pi\\, g_{\\mathcal X(\\vec\\beta)}\\); equivalently,\n\\[\n\\lim_{R\\to 0}\\sup_{x\\in \\mathcal X(\\vec\\beta)}\\sup_{u,v\\neq 0}\n\\frac{\\left|\\bigl(\\mathcal T_R^* g_{R,\\vec\\alpha(R)}\\bigr)_x(u,v)-2\\pi\\,\\bigl(g_{\\mathcal X(\\vec\\beta)}\\bigr)_x(u,v)\\right|}{\\|u\\|\\,\\|v\\|}=0.\n\\]"}, {"label": "E", "text": "As \\(R\\to 0\\), the pulled-back metrics \\(\\mathcal T_R^*(g_{R,\\vec\\alpha(R)})\\) converge pointwise on \\(\\mathcal X(\\vec\\beta)\\) to \\(2\\pi\\, g_{\\mathcal X(\\vec\\beta)}\\) for every hyperpolygon in \\(\\mathcal X(\\vec\\beta)\\), without needing the genericity assumption on \\(\\vec\\beta\\) or the specific linear tuning \\(\\alpha_i(R)=\\tfrac12-R\\beta_i\\).\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": "normalization constant 2\\pi from symplectic/moment-map comparison", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "identification of the limiting bilinear form as exactly 2\\pi g_{\\mathcal X(\\vec\\beta)}", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "pointwise convergence strengthened to uniform operator-norm convergence", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "dependence on genericity of \\vec\\beta and on the tuned degeneration \\alpha_i(R)=\\tfrac12-R\\beta_i", "template_used": "wildcard"}]}} +{"id": "2601.10656v1", "paper_link": "http://arxiv.org/abs/2601.10656v1", "theorems_cnt": 1, "theorem": {"env_name": "namedtheorem", "content": "[(c.f. Theorem \\ref{thm: HK degeneration})]\n Fix generic $\\vec\\beta\\in(0,\\infty)^n$. Let $\\alpha_i(R)=\\frac12-R\\beta_i$, \nlet $\\mathcal{X}(\\vec \\beta)$ be the $n$-hyperpolygon space, let $\\mathcal M_R(\\vec\\alpha(R))$ be the moduli space of solutions to the $R$-rescaled Hitchin's equations on the $n$-punctured sphere with parabolic weights $\\vec{\\alpha}(R)$, and let \n\\[ \\mathcal{T}_R: \\mathcal{X}(\\vec \\beta) \\to \\mathcal M_R(\\vec\\alpha(R))\\]\n be the natural embedding in \\cite{GM11}.\n\nAs $R \\to 0$,\n the pullback of the family of metrics $\\mathcal{T}_R^*(g_{R,\\vec\\alpha(R)})$ on the Hitchin moduli moduli space $\\mathcal M_R(\\vec\\alpha(R))$ converges pointwise to the metric $2\\pi\\cdot g_{\\mathcal X(\\vec\\beta)}$ on hyperpolygon space $\\mathcal X(\\vec\\beta)$.", "start_pos": 13431, "end_pos": 14238, "label": null}, "ref_dict": {"fig: metric degeneration": "\\label{fig: metric degeneration}\n\\end{figure}\n\n\\section{Local Model}\\label{sec: Local Model}\n\nFix $\\beta \\in \\R^+$. In this section, we construct a family of solutions of Hitchin's equations over the"}, "pre_theorem_intro_text_len": 7750, "pre_theorem_intro_text": "A gravitational instanton is a hyperkähler 4-manifold $(X,g,J_1,J_2,J_3,\\omega_1,\\omega_2,\\omega_3)$ with finite energy\n $$E(g)=\\int_X|\\text{Rm}_g|^2\\text{dvol}_g<\\infty.$$\nLet $r$ be a coordinate given by geodesic distance from a fixed point $p_0$ in $X$. If we impose the slightly stronger condition $|\\text{Rm}_g|^2\\in O(r^{-2-\\varepsilon})$ as $r\\to\\infty$, G. Chen and X. Chen prove that $X$ must have a single noncompact end of type ALE, ALF, ALG, ALH according to the volume growth \\cite{CC21}. Namely, we have $\\text{Vol}(B_r(p_0))\\sim r^m$ for some $m\\in\\{1,2,3,4\\}$, and ALE, ALF, ALG, ALH respectively correspond to $m=4, 3, 2, 1$. When we relax the stronger curvature decay condition, the classification expands to include two additional possibilities ALG$^*$ and ALH$^*$.\n\nThe so-called Modularity Conjecture, attributed to Boalch \\cite{aim},\nposits that all of these gravitational instantons can be realized as gauge-theoretic moduli spaces. In particular, the ALG$^{(*)}$ gravitational instantons should be realized as Hitchin moduli spaces on certain punctured Riemann surfaces $C$ with divisor $D$ with special fixed data at $D$. The ALE gravitational instantons can be realized as certain quiver varieties. There are also certain maps between gauge theoretic spaces. A follow-up question to the Modularity Conjecture is:\nCan these gauge-theoretic maps be used to understand degenerations of gravitational instantons from one type to another? \n\nWe consider this question in the particular case of well-tuned families of ALG-$D_4$ gravitational instantons\\footnote{For any fixed modular parameter $\\tau \\in \\mathbb{H}/\\mathrm{PSL}(2,\\mathbb{Z})$, the family of relevant ALG gravitational instantons are all asymptotic to $(\\mathbb{C} \\times T^2_\\tau)/\\Z_2$, where $\\Z_2$ acts by $(z,w) \\mapsto (-z, -w)$. } degenerating to ALE-$D_4$ gravitational instantons\\footnote{\n The relevant ALE gravitational instantons are asymptotic to $\\mathbb{C}^2/Q8$. Under the McKay correspondence, the affine Dynkin diagram arises as the root system of a finite subgroup $\\Gamma$ of $SU(2)$. The relevant subgroup $\\Gamma$ here is the quaternion group $Q8\\simeq \\{\\pm 1, \\pm i, \\pm j, \\pm k\\}$ which is generated inside $SU(2)$ by the elements $\\begin{pmatrix} i & 0 \\\\ 0 & i \\end{pmatrix}$ and $\\begin{pmatrix} 0 & -1 \\\\ 1 & 0 \\end{pmatrix}$.\n}. The ALG-$D_4$ moduli spaces are strongly parabolic $\\SU(2)$-Hitchin moduli spaces on the four-punctured sphere \\cite{FMSW21} and the ALE-$D_4$ are certain Nakajima quiver varieties known as $n=4$ hyperpolygon spaces \\cite{KN90}. Note that we are only considering the subfamily of gravitational instantons that admit a triholomorphic $\\U(1)$-action. The relevant gauge theoretic map from $n=4$ hyperpolygon spaces to strongly parabolic $\\SL(2,\\mathbb{C})$-Higgs bundle moduli spaces (also called $\\SU(2)$-Hitchin moduli spaces) on the four-punctured sphere is described in \\cite{GM11}. It preserves the holomorphic symplectic structures \\cite{BFGM15}. \n\nMoreover, a similar result holds for arbitrary $n$: the hyperk\\\"ahler metrics on well-tuned families of parabolic $\\SU(2)$-Hitchin moduli spaces on the $n$-punctured sphere converge to the $n$-hyperpolygon space. In this case, the hyperk\\\"ahler metric on hyperpolygon space is known to be quasi-asymptotically conical \\cite{DimakisRochon}, a higher-dimensional generalization of ALE. We note that at this time there is no parallel theorem about $\\SU(2)$-Hitchin moduli spaces on the $n$-punctured sphere having a hyperk\\\"ahler type that generalizes ALG.\n\n\\bigskip\n\nThese hyperk\\\"ahler metrics are difficult to access. In particular, the hyperk\\\"ahler metric on the Hitchin moduli space is written in terms of solutions of gauge-theoretic systems of coupled nonlinear elliptic (modulo gauge) partial differential equations. (In \\Cref{sec:preliminaries}, we will introduce many of the relevant notions including parabolic Higgs bundles and describe the hyperk\\\"ahler metric. But for the sake of brevity, we will here assume that the reader has some familiarity with at least ordinary Higgs bundles, in order to give a succinct description of this work and its relation to other works.) The hyperk\\\"ahler metric on parabolic $\\SU(2)$-Hitchin moduli space is understood as one approaches the noncompact end of the Hitchin moduli space in \\cite{MSWW16, Mochizuki, FredricksonSLn, Fre19, FMSW21,mochizuki2024asymptoticbehaviourhitchinmetric,mochizuki2024comparisonhitchinmetricsemiflat} (``$R \\to \\infty$''), as the PDE decouples in the limit. However, in this paper we are taking a very different limit ($``R \\to 0$'', while simultaneously degenerating the boundary conditions near the punctures). \n\nAn ``$R \\to 0$'' type limit appears in \\cite{DFKMMN16, CW18, CFW24}, where the conformal limit of a parabolic Higgs bundle is computed in \\cite{CFW24}. To review, fix a stable parabolic Higgs bundle $(\\mathcal{E}, \\varphi)$ and consider the family of harmonic metrics $h_R$ solving the $R$-rescaled Hitchin equation\n\\begin{equation}\\label{eq:Hitchinintro}\n R^{-1}F_{\\nabla(\\mathcal{E}, h_R)}^\\perp + R [\\varphi, \\varphi^{\\dagger_{h_R}}]=0,\n\\end{equation}\nwhere $\\nabla(\\mathcal{E}, h_R)$ is the Chern connection.\nLet $\\zeta \\in \\mathbb{C}^\\times$ be the twistor parameter where $\\zeta=0$ corresponds to the Higgs bundle moduli space, and fix $\\frac{\\zeta}{R}=\\hbar $.\nCollier--Fredrickson--Wentworth prove that the conformal limit\n\\[ \\lim_{R,\\zeta \\to 0} \\frac{R}{\\zeta} \\varphi + \\nabla(\\mathcal{E}, h_R) + \\zeta R \\varphi^{\\dagger_{h_R}} \\]\nexists in cases that include (1) strongly parabolic Higgs bundles or (2) weakly parabolic Higgs bundles with full flags. Then, they discuss how the conformal limit interacts with the natural stratification of the space of parabolic logarithmic $\\lambda$-connections by $\\mathbb{C}^\\times$-limits. One common feature shared by conformal limit and our limit is that both are joint limits with special tuning. In the conformal limit, $R \\to 0$ and $\\zeta \\to 0$; in our limit, $R \\to 0$ while the parabolic weights $\\alpha_i \\to \\frac{1}{2}$. \n\nThe degeneration of the boundary conditions appears in \\cite{Jud98}, and for parabolic Higgs bundles in \\cite{KW18}. For parabolic Higgs bundles, the harmonic metric $h_R$ is singular at the divisor $D \\subset C$, and the boundary condition for the harmonic metric $h_R$ is determined by weighted flags at those fixed points. In the case of $\\SL(2,\\mathbb{C})$ Higgs bundles, the weighted flag at $p_i \\in D$ is\n$$\\begin{array}[column sep = -5pt]{ccccccc}\n\\hspace{2.9cm} &0 &\\subset &F_i &\\subset &E_{p_i} &\\\\\n\\hspace{2.9cm} & & &1-\\alpha_i &< &\\alpha_i, &\\qquad\\alpha_i \\in (0, \\frac{1}{2}).\n\\end{array}$$\nWe consider a degeneration of the boundary condition $\\alpha_i \\to \\frac{1}{2}$ from the full flag case to the not-full-flag case. In the special cases where a point of the Hitchin moduli space can be interpreted as a uniformization metric on the punctured surface $\\hat C=C\\smallsetminus D$, this is a degeneration from conical singularities to cuspidal singularities at $D$.\n\nThe joint limit has the effect of sending the volume of fibers of the Hitchin fibration to $\\infty$ (``$R \\to 0$'') while preserving the holomorphic symplectic form. We tune the $R \\to 0$ and $\\alpha_i \\to \\frac{1}{2}$ limits so that \n\\[ \\alpha_i(R) = \\frac{1}{2} - R \\beta_i\\]\nfor some fixed $\\beta_i$.\nThis tuning of the degenerating boundary condition $\\alpha_i(R)$ is carefully chosen; when $n=4$, one can interpret this choice as the choice to hold part of the cohomology data of the real symplectic structure constant as $R$ varies (see \\Cref{fig: metric degeneration}).\n\n\\bigskip\n\nWe prove the following result:", "context": "These hyperk\\\"ahler metrics are difficult to access. In particular, the hyperk\\\"ahler metric on the Hitchin moduli space is written in terms of solutions of gauge-theoretic systems of coupled nonlinear elliptic (modulo gauge) partial differential equations. (In \\Cref{sec:preliminaries}, we will introduce many of the relevant notions including parabolic Higgs bundles and describe the hyperk\\\"ahler metric. But for the sake of brevity, we will here assume that the reader has some familiarity with at least ordinary Higgs bundles, in order to give a succinct description of this work and its relation to other works.) The hyperk\\\"ahler metric on parabolic $\\SU(2)$-Hitchin moduli space is understood as one approaches the noncompact end of the Hitchin moduli space in \\cite{MSWW16, Mochizuki, FredricksonSLn, Fre19, FMSW21,mochizuki2024asymptoticbehaviourhitchinmetric,mochizuki2024comparisonhitchinmetricsemiflat} (``$R \\to \\infty$''), as the PDE decouples in the limit. However, in this paper we are taking a very different limit ($``R \\to 0$'', while simultaneously degenerating the boundary conditions near the punctures).\n\nAn ``$R \\to 0$'' type limit appears in \\cite{DFKMMN16, CW18, CFW24}, where the conformal limit of a parabolic Higgs bundle is computed in \\cite{CFW24}. To review, fix a stable parabolic Higgs bundle $(\\mathcal{E}, \\varphi)$ and consider the family of harmonic metrics $h_R$ solving the $R$-rescaled Hitchin equation\n\\begin{equation}\\label{eq:Hitchinintro}\n R^{-1}F_{\\nabla(\\mathcal{E}, h_R)}^\\perp + R [\\varphi, \\varphi^{\\dagger_{h_R}}]=0,\n\\end{equation}\nwhere $\\nabla(\\mathcal{E}, h_R)$ is the Chern connection.\nLet $\\zeta \\in \\mathbb{C}^\\times$ be the twistor parameter where $\\zeta=0$ corresponds to the Higgs bundle moduli space, and fix $\\frac{\\zeta}{R}=\\hbar $.\nCollier--Fredrickson--Wentworth prove that the conformal limit\n\\[ \\lim_{R,\\zeta \\to 0} \\frac{R}{\\zeta} \\varphi + \\nabla(\\mathcal{E}, h_R) + \\zeta R \\varphi^{\\dagger_{h_R}} \\]\nexists in cases that include (1) strongly parabolic Higgs bundles or (2) weakly parabolic Higgs bundles with full flags. Then, they discuss how the conformal limit interacts with the natural stratification of the space of parabolic logarithmic $\\lambda$-connections by $\\mathbb{C}^\\times$-limits. One common feature shared by conformal limit and our limit is that both are joint limits with special tuning. In the conformal limit, $R \\to 0$ and $\\zeta \\to 0$; in our limit, $R \\to 0$ while the parabolic weights $\\alpha_i \\to \\frac{1}{2}$.\n\nThe degeneration of the boundary conditions appears in \\cite{Jud98}, and for parabolic Higgs bundles in \\cite{KW18}. For parabolic Higgs bundles, the harmonic metric $h_R$ is singular at the divisor $D \\subset C$, and the boundary condition for the harmonic metric $h_R$ is determined by weighted flags at those fixed points. In the case of $\\SL(2,\\mathbb{C})$ Higgs bundles, the weighted flag at $p_i \\in D$ is\n$$\\begin{array}[column sep = -5pt]{ccccccc}\n\\hspace{2.9cm} &0 &\\subset &F_i &\\subset &E_{p_i} &\\\\\n\\hspace{2.9cm} & & &1-\\alpha_i &< &\\alpha_i, &\\qquad\\alpha_i \\in (0, \\frac{1}{2}).\n\\end{array}$$\nWe consider a degeneration of the boundary condition $\\alpha_i \\to \\frac{1}{2}$ from the full flag case to the not-full-flag case. In the special cases where a point of the Hitchin moduli space can be interpreted as a uniformization metric on the punctured surface $\\hat C=C\\smallsetminus D$, this is a degeneration from conical singularities to cuspidal singularities at $D$.\n\nThe joint limit has the effect of sending the volume of fibers of the Hitchin fibration to $\\infty$ (``$R \\to 0$'') while preserving the holomorphic symplectic form. We tune the $R \\to 0$ and $\\alpha_i \\to \\frac{1}{2}$ limits so that \n\\[ \\alpha_i(R) = \\frac{1}{2} - R \\beta_i\\]\nfor some fixed $\\beta_i$.\nThis tuning of the degenerating boundary condition $\\alpha_i(R)$ is carefully chosen; when $n=4$, one can interpret this choice as the choice to hold part of the cohomology data of the real symplectic structure constant as $R$ varies (see \\Cref{fig: metric degeneration}).\n\n\\bigskip\n\nWe prove the following result:\n\n\\label{fig: metric degeneration}\n\\end{figure}\n\n\\section{Local Model}\\label{sec: Local Model}\n\nFix $\\beta \\in \\R^+$. In this section, we construct a family of solutions of Hitchin's equations over the", "full_context": "These hyperk\\\"ahler metrics are difficult to access. In particular, the hyperk\\\"ahler metric on the Hitchin moduli space is written in terms of solutions of gauge-theoretic systems of coupled nonlinear elliptic (modulo gauge) partial differential equations. (In \\Cref{sec:preliminaries}, we will introduce many of the relevant notions including parabolic Higgs bundles and describe the hyperk\\\"ahler metric. But for the sake of brevity, we will here assume that the reader has some familiarity with at least ordinary Higgs bundles, in order to give a succinct description of this work and its relation to other works.) The hyperk\\\"ahler metric on parabolic $\\SU(2)$-Hitchin moduli space is understood as one approaches the noncompact end of the Hitchin moduli space in \\cite{MSWW16, Mochizuki, FredricksonSLn, Fre19, FMSW21,mochizuki2024asymptoticbehaviourhitchinmetric,mochizuki2024comparisonhitchinmetricsemiflat} (``$R \\to \\infty$''), as the PDE decouples in the limit. However, in this paper we are taking a very different limit ($``R \\to 0$'', while simultaneously degenerating the boundary conditions near the punctures).\n\nAn ``$R \\to 0$'' type limit appears in \\cite{DFKMMN16, CW18, CFW24}, where the conformal limit of a parabolic Higgs bundle is computed in \\cite{CFW24}. To review, fix a stable parabolic Higgs bundle $(\\mathcal{E}, \\varphi)$ and consider the family of harmonic metrics $h_R$ solving the $R$-rescaled Hitchin equation\n\\begin{equation}\\label{eq:Hitchinintro}\n R^{-1}F_{\\nabla(\\mathcal{E}, h_R)}^\\perp + R [\\varphi, \\varphi^{\\dagger_{h_R}}]=0,\n\\end{equation}\nwhere $\\nabla(\\mathcal{E}, h_R)$ is the Chern connection.\nLet $\\zeta \\in \\mathbb{C}^\\times$ be the twistor parameter where $\\zeta=0$ corresponds to the Higgs bundle moduli space, and fix $\\frac{\\zeta}{R}=\\hbar $.\nCollier--Fredrickson--Wentworth prove that the conformal limit\n\\[ \\lim_{R,\\zeta \\to 0} \\frac{R}{\\zeta} \\varphi + \\nabla(\\mathcal{E}, h_R) + \\zeta R \\varphi^{\\dagger_{h_R}} \\]\nexists in cases that include (1) strongly parabolic Higgs bundles or (2) weakly parabolic Higgs bundles with full flags. Then, they discuss how the conformal limit interacts with the natural stratification of the space of parabolic logarithmic $\\lambda$-connections by $\\mathbb{C}^\\times$-limits. One common feature shared by conformal limit and our limit is that both are joint limits with special tuning. In the conformal limit, $R \\to 0$ and $\\zeta \\to 0$; in our limit, $R \\to 0$ while the parabolic weights $\\alpha_i \\to \\frac{1}{2}$.\n\nThe degeneration of the boundary conditions appears in \\cite{Jud98}, and for parabolic Higgs bundles in \\cite{KW18}. For parabolic Higgs bundles, the harmonic metric $h_R$ is singular at the divisor $D \\subset C$, and the boundary condition for the harmonic metric $h_R$ is determined by weighted flags at those fixed points. In the case of $\\SL(2,\\mathbb{C})$ Higgs bundles, the weighted flag at $p_i \\in D$ is\n$$\\begin{array}[column sep = -5pt]{ccccccc}\n\\hspace{2.9cm} &0 &\\subset &F_i &\\subset &E_{p_i} &\\\\\n\\hspace{2.9cm} & & &1-\\alpha_i &< &\\alpha_i, &\\qquad\\alpha_i \\in (0, \\frac{1}{2}).\n\\end{array}$$\nWe consider a degeneration of the boundary condition $\\alpha_i \\to \\frac{1}{2}$ from the full flag case to the not-full-flag case. In the special cases where a point of the Hitchin moduli space can be interpreted as a uniformization metric on the punctured surface $\\hat C=C\\smallsetminus D$, this is a degeneration from conical singularities to cuspidal singularities at $D$.\n\nThe joint limit has the effect of sending the volume of fibers of the Hitchin fibration to $\\infty$ (``$R \\to 0$'') while preserving the holomorphic symplectic form. We tune the $R \\to 0$ and $\\alpha_i \\to \\frac{1}{2}$ limits so that \n\\[ \\alpha_i(R) = \\frac{1}{2} - R \\beta_i\\]\nfor some fixed $\\beta_i$.\nThis tuning of the degenerating boundary condition $\\alpha_i(R)$ is carefully chosen; when $n=4$, one can interpret this choice as the choice to hold part of the cohomology data of the real symplectic structure constant as $R$ varies (see \\Cref{fig: metric degeneration}).\n\n\\bigskip\n\nWe prove the following result:\n\n\\label{fig: metric degeneration}\n\\end{figure}\n\n\\section{Local Model}\\label{sec: Local Model}\n\nFix $\\beta \\in \\R^+$. In this section, we construct a family of solutions of Hitchin's equations over the\n\n\\bigskip\n\n\\begin{theorem}\\label[theorem]{thm: Morse functions agree at fixed points} \n Let $\\vec\\beta \\in (0, \\infty)^n$ be generic, let $\\alpha_i(R)= \\frac{1}{2} - R \\beta_i$ for $R>0$, and assume $R_\\text{max}$ solves $W_{[n]}(\\vec\\alpha(R_\\text{max}))=(n-2)/2$ as in \\Cref{thm: map from hyperpolygon to Higgs}. For $R\\in(0,R_\\text{max})$, let $M_\\text{HP}$ and $M_R=M_{R,\\vec\\alpha(R)}$ be the Morse-Bott functions on $\\mathcal{X}(\\beta)$ and $\\mathcal{M}_R(\\vec \\alpha(R))$ described in \\eqref{eqn: Morse-Bott functions}.\n For each $\\U(1)$-fixed point $(\\bx,\\by)$ and corresponding $\\U(1)$-fixed harmonic bundle $\\mathcal T_R(\\bx,\\by)=(\\mathcal E,\\varphi,h_R)$, and for all $00$ small enough that $W_{[n]}(\\vec\\alpha(R))>(n-2)/2$, let $\\mathcal T_R\\from\\mathcal X(\\vec\\beta)\\into\\mathcal M_R(\\vec\\alpha)$ be the family of embeddings described above. Then the family of hyperkähler metrics $\\mathcal{T}_R^*(g_R)$ converges pointwise to $2\\pi\\cdot g_\\text{HP}$ as $R \\to 0$.\n\\end{restatable}\n\\begin{proof} Let $\\delta>0$.\n Let $(\\bx,\\by)$ be a unitary hyperpolygon and $(\\dot\\bx,\\dot\\by)$ a unitary deformation of $(\\bx,\\by)$. The image $(\\mathcal E,\\varphi)=\\mathcal T_R(\\bx,\\by)$ is a Higgs bundle with holomorphic structure $\\delbar_{\\mathcal E}=\\delbar$ independent of $(\\bx, \\by)$, while the parabolic flag structure $\\mathcal F$ depends on $(\\bx, \\by)$. \n It suffices to prove that for every unitary deformations $(\\dot \\bx, \\dot \\by) \\in T_{(\\bx, \\by)} \\mathcal{X}(\\vec \\beta)$ and associated harmonic deformation \n $\\mathtt{H}_R \\in T_{\\mathcal{T}_R(\\bx, \\by)} \\mathcal{M}_R(\\vec(\\alpha))$,\n \\begin{equation}\\label{eqn: main goal}\n \\lim_{R\\to0}\\| \\mathtt{H}_R\\|_{g_R}^2=2\\pi\\sum_i\\left(|\\dot x_i|^2+|\\dot y_i|^2\\right).\n \\end{equation}\n Recall that the deformation $(\\dot\\bx,\\dot\\by)$ corresponds to the Higgs bundle deformation $(\\dot{\\eta}, \\dot{\\mathcal F},\\dot\\varphi)$. \n It will be convenient to introduce the notation $\\dot{\\Phi} = \\dot \\Psi^{1,0}$, \n so that \n the associated deformation in the $h_R$-unitary formulation is $(\\dot \\nabla_R^{0,1},\\dot\\Phi_R)=(-\\delbar\\dot\\nu_R,\\dot\\varphi+[\\dot\\nu_R,\\varphi])$ as discussed in \\Cref{sec:hitchinhk}.\nUsing \\eqref{eq:hk expression} and the regularity of $\\gamma_R$,\n \\begin{align}\n \\|(\\dot \\nabla^{0,1}_R,\\dot\\Phi_R)\\|^2\n &=\\sum_{i=1}^n \\left( \\lim_{\\delta'\\to0}\\oint_{\\partial B_{\\delta'}(p_i)}R^{-1}\\langle\\dot\\nu_R,\\delbar\\dot\\nu_R\\rangle_{h_R}\\right) +\\int_{\\P^1}R\\langle\\dot\\varphi,\\dot\\Phi_R\\rangle_{h_R}\n \\notag\\\\\n &=\\sum_{i=1}^n \\left( \\lim_{\\delta'\\to0}\\oint_{\\partial B_{\\delta'}(p_i)}R^{-1}\\langle\\dot\\nu_R,\\delbar\\dot\\nu_R\\rangle_{h_{\\app,R}} \\right)+\\int_{\\P^1}R\\langle\\dot\\varphi,\\dot\\Phi_{\\app,R}\\rangle_{h_{\\app,R}}+O(R^2)\n \\notag\\\\\n &=\\sum_{i=1}^n \\left( \\lim_{\\delta'\\to0}\\oint_{\\partial B_{\\delta'}(p_i)}R^{-1}\\langle\\dot\\nu_{\\text{app},R},\\delbar\\dot\\nu_{\\text{app},R}\\rangle_{h_{\\text{app},R}} \\right)\\\\\n &\\quad+\\int_{\\P^1}R\\langle\\dot\\varphi,\\dot\\Phi_{\\text{app},R}\\rangle_{h_{\\text{app},R}}+O(R^2)\n \\notag\\\\\n &\\quad+\\sum_{i=1}^n \\left( \\lim_{\\delta'\\to0}\\oint_{\\partial B_{\\delta'}(p_i)}\\underbrace{R^{-1}\\langle\\rho_R,\\delbar\\dot\\nu_{\\text{app},R}\\rangle_{h_{\\text{app},R}}+R^{-1}\\langle\\dot\\nu_{\\text{app},R},\\delbar\\rho_R\\rangle_{h_{\\text{app},R}}}_{f_\\text{cor}(\\bar{z-p_i})^{-1}\\de\\bar z} \\right).\n \\end{align}\n The last integrand can be written as $f_\\text{cor}(\\bar{z-p_i})^{-1}\\de\\bar z$. Since $\\rho_R\\in L_{{2-\\eps}}^{2,2}(\\End(\\mathcal E))$, we have $\\rho_R\\in o(r^{1-\\eps})$ and $\\delbar\\rho_R\\in o(r^{-\\eps})$. Thus, $|f_\\text{cor}|$ has leading order terms of the form $r^{1-\\eps-4R\\beta_i}$, so for sufficiently small $R$ we can bound it above by $Cr^{1/2}$ for some constant $C$;\n this constant is uniform as $R\\to0$ by analyticity of $\\rho_R$. Now $Cr^{1/2}(\\bar{z-p_i})^{-1}\\de\\bar z$ vanishes under integration along $\\partial B_{\\delta'}(p_i)$ in the limit $\\delta'\\to0$, so the last integral is zero. Finally, the uniform boundedness of $\\langle\\dot\\varphi,\\dot\\Phi_{\\text{app},R}\\rangle_{h_R}$ on $[0,R_{\\max})\\times K_\\delta$ and \\Cref{prop: local metric pairing} implies \\eqref{eqn: main goal}.\n\\end{proof}\n\nRecall the Morse-Bott functions in \\eqref{eqn: Morse-Bott functions} arising as moment maps with respect to the real symplectic form for the $\\U(1)$-actions on the hyperk\\\"ahler spaces $\\mathcal X(\\vec\\beta)=X\\fourslash_{(0, \\vec \\beta)}$ and $\\mathcal M_R(\\vec\\alpha(R))$:\n $$M_\\text{HP}(\\bx,\\by)=\\frac i2\\sum_i|y_i|^2,\n \\quad M_R(\\mathcal E,\\varphi,h_R)=\\frac i2\\int_{\\C\\P^1}R|\\varphi|_{h_R}^2$$\n In \\Cref{thm: Morse functions agree at fixed points}, we proved that $M_R(\\mathcal{T}_R(\\bx, \\by))=2 \\pi M_\\text{HP}(\\bx, \\by)$ at corresponding $U(1)$-fixed points. Now, we prove that the family of Morse functions converges: \n\\begin{theorem}\n \\label[theorem]{thm: moment maps agree}\n In the setting of \\Cref{thm: HK degeneration}, for any unitary hyperpolygon $(\\bx,\\by)$ and corresponding $(0,R_{\\max})$-family of harmonic bundles $\\mathcal{T}_R(\\bx, \\by)$, we have \\begin{equation} \\lim_{R\\to0}M_R(\\mathcal{T}_R(\\bx,\\by))=2\\pi M_\\text{HP}(\\bx,\\by).\\end{equation}\n\\end{theorem}\n\\begin{proof}\n Let $(\\bx,\\by)$ be a unitary hyperpolygon, and let $\\gamma_R$, $R_\\text{max}$, and $\\eps\\in(0,\\frac12)$ be as in \\Cref{prop: approximate h accuracy}. Our approximation $h_{\\text{app},R}$ is uniformly bounded on $K_\\delta=\\CP^1 \\sm \\bigcup_{i} B_\\delta(p_i)$ as $R\\to0$, and therefore, so is $h_R=e^{\\gamma_R}\\cdot h_{\\text{app},R}$ (recall $\\lim_{R\\to}\\gamma_R=0$ and $\\gamma_R$ is real analytic in $R$). \n Thus, $R|\\varphi|_{h_R}^2\\to0$ on $K_\\delta$ as $R\\to0$.", "post_theorem_intro_text_len": 5808, "post_theorem_intro_text": "We briefly explain the proof.\nHyperpolygon space can be constructed as a hyperk\\\"ahler manifold via a hyperk\\\"ahler quotient construction or as a holomorphic symplectic manifold via a holomorphic symplectic quotient construction. In the first, one must choose representatives which solve both real and complex moment map equations---a ``unitary hyperpolygon'' in our terminology---while in the second, one must only choose a representative which solves the complex moment map equation---a ``hyperpolygon'' in our terminology. The Hitchin moduli space $\\mathcal{M}_R(\\vec \\alpha(R))$ and the Higgs bundle moduli space $\\mathcal{M}^{\\mathrm{Higgs}}(\\vec \\alpha(R))$ are similarly related. Godinho--Mandini \\cite{GM11} construct a map $\\mathcal{T}: \\mathcal{X}(\\vec \\beta) \\to \\mathcal{M}^{\\mathrm{Higgs}}(\\vec \\alpha(R))$ preserving the holomorphic symplectic structures coming from the complex moment maps \\cite{BFGM15}. \n\nThe map $\\mathcal{T}_R$ above factors through $\\mathcal{T}$.\nNamely, a point in $\\mathcal{M}_R(\\vec \\alpha(R))$ is a Higgs bundle in $\\mathcal{M}^{\\mathrm{Higgs}}(\\vec \\alpha(R))$ together with the hermitian metric $h_R$ on the underlying complex bundle that solves $R$-rescaled Hitchin's equations (the real moment map). \nThe key idea in our proof of \\Cref{thm: HK degeneration} involves a delicate construction of approximate solutions to Hitchin's equations (the real moment map), which uses all the hyperpolygon moment map data to ensure the metric is adapted to the parabolic structure and globally defined. \n\nThis result was independently proved by different methods by Lynn Heller, Sebastian Heller, and Claudio Meneses in \\cite{HHM}. Their work builds on \\cite{HHT2025}. A similar observation is made in \\cite[Appendix A]{BTX}, and we would like to better understand the connection with Coulomb branches and Higgs branches, at the level of hyperk\\\"ahler (rather than holomorphic symplectic) geometry. \n\\bigskip\n\nThis more complicated theorem is of a similar type to the simple degeneration of ALF-$A_n$ gravitational to ALE-$A_n$ gravitational instantons. All ALF-$A_n$ and ALE-$A_n$ gravitational instantons can be obtained using the Gibbons-Hawking ansatz, as \nthe total space of a principal $U(1)$ bundle over $\\mathbb{R}^3$. In the case of ALF-$A_n$, the hyperk\\\"ahler multi-Taub-NUT metrics are specified by a harmonic function\n $$V_R(x)=R+\\sum_{i=1}^n\\frac1{4\\pi|x-p_i|}$$\nfor $R>0$ and distinct points $p_1,\\dots,p_{n+1}\\in\\mathbb{R}^3$, up to translation. \nThe hyperk\\\"ahler metric is \n\\begin{equation}\n g_R=V_R \\norm{\\text{d} \\vec{x}}^2 + V_R^{-1} \\left(\\frac{1}{2 \\pi} \\Theta\\right)^2,\n\\end{equation}\nwhere $\\vec{x} \\in \\mathbb{R}^3$ and $\\Theta$ is a connection on the $U(1)$ bundle whose curvature $F_R$ is related to the potential $V_R$ by $F_R = -2 \\pi \\star \\text{d} V_R$. Note that $\\Theta$ doesn't depend on the constant $R$, so the potential determines the entire hyperk\\\"ahler structure. As $R\\to0$, the volume of the $U(1)$-fibers tends toward infinity and the metric degenerates to an ALE-$A_n$ instanton called the multi-Eguchi-Hanson metric.\n\n\\bigskip\n\n\\subsection*{Organization}\nThe paper is organized as follows:\n\nIn \\Cref{sec:preliminaries}, we introduce the $n$-sided hyperpolygon spaces and the strongly parabolic Hitchin moduli spaces on $\\mathbb{CP}^1$ with $n$ punctures, we describe their hyperkähler metrics, and the map from hyperpolygons to strongly parabolic Higgs bundles. \n\nThe $U(1)$-action on each hyperk\\\"ahler space gives rise to a Morse--Bott function. In \\Cref{sec: Torelli Numbers}, we compute the value of this function at the corresponding fixed points and show that they match. Then, we restrict to $n=4$ for concreteness (the real dimension four case), and explore the topology and geometry of the two families of moduli spaces.\n\nIn \\Cref{sec: Local Model}, we introduce local model solutions of the $\\SU(2)$-Hitchin equations on the disk. We conduct a detailed local analysis of these model solutions and their first variation. \n\nIn \\Cref{sec: The Hyperkahler Metrics}, we use the local models to construct approximate harmonic metrics for Higgs bundles on $\\CP^1$ with $n$-punctures. We then use an implicit function theorem argument perturb our approximate metrics to actual solutions, and finally prove our main theorem.\n\n\\subsection*{Notational Conventions} We take hermitian forms $h$ to be conjugate linear in the first slot and $\\mathbb{C}$-linear in the second slot. We identify such forms with their Gram matrices, e.g.\\ $h(u,v)=u^\\dagger hv$. If $h$ is a hermitian form on $V$ and $B\\in\\End(V)$, then the $h$-adjoint is $B^{\\dagger_h}=h^{-1}B^\\dagger h$, where $\\dagger$ (without the $h$) denotes the standard complex conjugation. For the standard 1-forms $\\text{d}\\bar z$ and $\\text{d} z$ in a coordinate chart on a Riemann surface $C$, the Hodge star is given by $\\star\\text{d}\\bar z=i\\text{d} z$ and $\\star\\text{d} z=-i\\text{d}\\bar z$, extended $\\mathcal C_C^\\infty$-antilinearly.\n\nGiven a holomorphic structure $\\delbar_E$ and a metric $h$ on a vector bundle $E$, we denote the Chern connection by $\\nabla(\\delbar_E,h)$.\n\nFor a matrix $A\\in\\mathfrak{gl}(n,\\mathbb{C})$ we use $A^\\perp$ to denote the orthogonal projection onto $\\mathfrak{sl}(n,\\mathbb{C})$. In other words, $A^\\perp:=A-\\frac1n\\tr(A)$ is the trace-free part of $A$.\n\n\\subsection*{Acknowledgements}\n\nThe authors would like to thank Nick Addington, Sergey Cherkis, Sze Hong Kwong, Rafe Mazzeo, Claudio Meneses, Nick Proudfoot, Steve Rayan, Laura Schaposnik, Hartmut Weiss, Richard Wentworth, and Graeme Wilkin for useful discussions. AY particularly thanks Nick Proudfoot for his help with hyperpolygons at the start of this project. LF is partially supported by NSF grant DMS-2005258. AY is supported by NSF grant RTG-2039316.", "sketch": "Hyperpolygon space and the relevant Hitchin/Higgs moduli spaces are compared via their moment-map descriptions: hyperpolygons can be formed either by a hyperk\"ahler quotient (requiring representatives solving both the real and complex moment map equations, i.e. a \"unitary hyperpolygon\") or by a holomorphic symplectic quotient (requiring only the complex moment map equation, i.e. a \"hyperpolygon\"). Likewise, a point of $\\mathcal{M}_R(\\vec \\alpha(R))$ is described as a Higgs bundle in $\\mathcal{M}^{\\mathrm{Higgs}}(\\vec \\alpha(R))$ together with a hermitian metric $h_R$ solving the $R$-rescaled Hitchin equations (the real moment map). Godinho--Mandini construct a holomorphic symplectic map $\\mathcal{T}:\\mathcal{X}(\\vec\\beta)\\to \\mathcal{M}^{\\mathrm{Higgs}}(\\vec\\alpha(R))$ preserving holomorphic symplectic structures, and the embedding $\\mathcal{T}_R$ \"factors through $\\mathcal{T}$.\" The \"key idea\" for proving the theorem is a \"delicate construction of approximate solutions to Hitchin's equations (the real moment map),\" using \"all the hyperpolygon moment map data\" so that the resulting metric is \"adapted to the parabolic structure and globally defined.\" In the later parts of the paper this is implemented by: introducing local model solutions of the $\\mathrm{SU}(2)$ Hitchin equations on a disk and analyzing them and their first variation; then using these local models to build approximate harmonic metrics for Higgs bundles on $\\mathbb{CP}^1$ with $n$ punctures; then applying an implicit function theorem to perturb the approximate metrics to actual solutions; and finally proving the metric degeneration statement.", "expanded_sketch": "Hyperpolygon space and the relevant Hitchin/Higgs moduli spaces are compared via their moment-map descriptions: hyperpolygons can be formed either by a hyperk\"ahler quotient (requiring representatives solving both the real and complex moment map equations, i.e. a \"unitary hyperpolygon\") or by a holomorphic symplectic quotient (requiring only the complex moment map equation, i.e. a \"hyperpolygon\"). Likewise, a point of $\\mathcal{M}_R(\\vec \\alpha(R))$ is described as a Higgs bundle in $\\mathcal{M}^{\\mathrm{Higgs}}(\\vec \\alpha(R))$ together with a hermitian metric $h_R$ solving the $R$-rescaled Hitchin equations (the real moment map). Godinho--Mandini construct a holomorphic symplectic map $\\mathcal{T}:\\mathcal{X}(\\vec\\beta)\\to \\mathcal{M}^{\\mathrm{Higgs}}(\\vec\\alpha(R))$ preserving holomorphic symplectic structures, and the embedding $\\mathcal{T}_R$ \"factors through $\\mathcal{T}$.\" The key idea for proving the theorem is a delicate construction of approximate solutions to Hitchin's equations (the real moment map), using all the hyperpolygon moment map data so that the resulting metric is adapted to the parabolic structure and globally defined. In the later parts of the paper this is implemented by: introducing local model solutions of the $\\mathrm{SU}(2)$ Hitchin equations on a disk and analyzing them and their first variation; then using these local models to build approximate harmonic metrics for Higgs bundles on $\\mathbb{CP}^1$ with $n$ punctures; then applying an implicit function theorem to perturb the approximate metrics to actual solutions; and finally proving the metric degeneration statement.", "expanded_theorem": "[(c.f. Theorem \\ref{thm: HK degeneration})]\n Fix generic $\\vec\\beta\\in(0,\\infty)^n$. Let $\\alpha_i(R)=\\frac12-R\\beta_i$, \nlet $\\mathcal{X}(\\vec \\beta)$ be the $n$-hyperpolygon space, let $\\mathcal M_R(\\vec\\alpha(R))$ be the moduli space of solutions to the $R$-rescaled Hitchin's equations on the $n$-punctured sphere with parabolic weights $\\vec{\\alpha}(R)$, and let \n\\[ \\mathcal{T}_R: \\mathcal{X}(\\vec \\beta) \\to \\mathcal M_R(\\vec\\alpha(R))\\]\n be the natural embedding in \\cite{GM11}.\n\nAs $R \\to 0$,\n the pullback of the family of metrics $\\mathcal{T}_R^*(g_{R,\\vec\\alpha(R)})$ on the Hitchin moduli moduli space $\\mathcal M_R(\\vec\\alpha(R))$ converges pointwise to the metric $2\\pi\\cdot g_{\\mathcal X(\\vec\\beta)}$ on hyperpolygon space $\\mathcal X(\\vec\\beta)$.,", "theorem_type": ["Asymptotic or Limit", "Universal"], "mcq": {"question": "Fix a generic parameter vector $\\vec\\beta=(\\beta_1,\\dots,\\beta_n)\\in(0,\\infty)^n$, and for $R>0$ set the parabolic weights $\\alpha_i(R)=\\tfrac12-R\\beta_i$. Let $\\mathcal X(\\vec\\beta)$ denote the $n$-hyperpolygon space with metric $g_{\\mathcal X(\\vec\\beta)}$. Let $\\mathcal M_R(\\vec\\alpha(R))$ denote the moduli space of solutions to the $R$-rescaled Hitchin equations on the $n$-punctured sphere with parabolic weights $\\vec\\alpha(R)$, equipped with its metric $g_{R,\\vec\\alpha(R)}$. Let\n\\[\n\\mathcal T_R:\\mathcal X(\\vec\\beta)\\to \\mathcal M_R(\\vec\\alpha(R))\n\\]\nbe the natural embedding. As $R\\to0$, which limiting statement holds for the family of pullback metrics $\\mathcal T_R^*(g_{R,\\vec\\alpha(R)})$ on $\\mathcal X(\\vec\\beta)$?", "correct_choice": {"label": "A", "text": "The pullback metrics converge pointwise to $2\\pi$ times the hyperpolygon metric: for every point $p\\in\\mathcal X(\\vec\\beta)$ and all tangent vectors $u,v\\in T_p\\mathcal X(\\vec\\beta)$,\n\\[\n\\bigl(\\mathcal T_R^* g_{R,\\vec\\alpha(R)}\\bigr)_p(u,v)\\longrightarrow 2\\pi\\,\\bigl(g_{\\mathcal X(\\vec\\beta)}\\bigr)_p(u,v)\n\\qquad\\text{as }R\\to0.\n\\]"}, "choices": [{"label": "B", "text": "The pullback metrics converge pointwise to the hyperpolygon metric without any scaling factor: for every point $p\\in\\mathcal X(\\vec\\beta)$ and all tangent vectors $u,v\\in T_p\\mathcal X(\\vec\\beta)$,\n\\[\n\\bigl(\\mathcal T_R^* g_{R,\\vec\\alpha(R)}\\bigr)_p(u,v)\\longrightarrow \\bigl(g_{\\mathcal X(\\vec\\beta)}\\bigr)_p(u,v)\n\\qquad\\text{as }R\\to0.\n\\]"}, {"label": "C", "text": "For every point $p\\in\\mathcal X(\\vec\\beta)$, the bilinear forms $\\bigl(\\mathcal T_R^* g_{R,\\vec\\alpha(R)}\\bigr)_p$ converge pointwise as $R\\to0$ to a positive constant multiple of the hyperpolygon metric $\\bigl(g_{\\mathcal X(\\vec\\beta)}\\bigr)_p$."}, {"label": "D", "text": "The pullback metrics converge uniformly on $\\mathcal X(\\vec\\beta)$ to $2\\pi$ times the hyperpolygon metric: there exists a constant $C>0$ such that for all sufficiently small $R>0$,\n\\[\n\\sup_{p\\in\\mathcal X(\\vec\\beta)}\\sup_{u,v\\neq 0}\n\\frac{\\left|\\bigl(\\mathcal T_R^* g_{R,\\vec\\alpha(R)}\\bigr)_p(u,v)-2\\pi\\,\\bigl(g_{\\mathcal X(\\vec\\beta)}\\bigr)_p(u,v)\\right|}{\\|u\\|_{g_{\\mathcal X(\\vec\\beta)}}\\,\\|v\\|_{g_{\\mathcal X(\\vec\\beta)}}}\n\\longrightarrow 0\n\\qquad\\text{as }R\\to0.\n\\]"}, {"label": "E", "text": "The pullback metrics converge pointwise to $2\\pi$ times the hyperpolygon metric for every hyperpolygon representative, without requiring the unitary hyperpolygon condition: for every point $p\\in\\mathcal X(\\vec\\beta)$ and all tangent vectors $u,v\\in T_p\\mathcal X(\\vec\\beta)$,\n\\[\n\\bigl(\\mathcal T_R^* g_{R,\\vec\\alpha(R)}\\bigr)_p(u,v)\\longrightarrow 2\\pi\\,\\bigl(g_{\\mathcal X(\\vec\\beta)}\\bigr)_p(u,v)\n\\qquad\\text{as }R\\to0,\n\\]\nwhere the embedding is obtained from the holomorphic symplectic map defined using only the complex moment map data."}], "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": "normalization_constant_2pi", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "exact_identification_of_the_limit_constant_as_2\\pi", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "pointwise_vs_uniform_convergence", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "need_for_real_moment_map_and_actual_harmonic_metric_in_the_embedding", "template_used": "wildcard"}]}} +{"id": "2601.10996v3", "paper_link": "http://arxiv.org/abs/2601.10996v3", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{maintheorem}\n\t\tLet $(M, g)$ be a complete noncompact Riemannian manifold whose Ricci curvature is lower bounded, i.e. $\\textit{Ric} \\geq \\lambda g$ for some constant $\\lambda \\in \\mathbb{R}$. Moreover, we assume that its injectivity radius has a lower bound, i.e. $\\textit{inj} (M,g) \\geq i > 0$ for some constant $i > 0$. Then there exists a positive constant $\\epsilon = \\epsilon (\\lambda , i , n) =O(\\frac{1}{r_H})$ in the reverse order of the harmonic radius $r_H$ such that for every $\\tau > \\epsilon$, there exists a constant $C = C(n, \\tau , M)$ such that \n\n\t\t$$\n\t\t\\sup_{u \\in W^{1,n} (M) , ||u||_{1, \\tau} \\leq 1} \\int_M \\phi_n (\\alpha_n |u|^{\\frac{n}{n-1}}) dV_g \\leq C. \n\t\t$$ \n\t\tMoreover, $\\alpha_n$ is sharp.", "start_pos": 7756, "end_pos": 8504, "label": "maintheorem"}, "ref_dict": {"maintheorem": "\\begin{thm}\\label{maintheorem}\n\t\tLet $(M, g)$ be a complete noncompact Riemannian manifold whose Ricci curvature is lower bounded, i.e. $\\textit{Ric} \\geq \\lambda g$ for some constant $\\lambda \\in \\mathbb{R}$. Moreover, we assume that its injectivity radius has a lower bound, i.e. $\\textit{inj} (M,g) \\geq i > 0$ for some constant $i > 0$. Then there exists a positive constant $\\epsilon = \\epsilon (\\lambda , i , n) =O(\\frac{1}{r_H})$ in the reverse order of the harmonic radius $r_H$ such that for every $\\tau > \\epsilon$, there exists a constant $C = C(n, \\tau , M)$ such that \n\n\t\t$$\n\t\t\\sup_{u \\in W^{1,n} (M) , ||u||_{1, \\tau} \\leq 1} \\int_M \\phi_n (\\alpha_n |u|^{\\frac{n}{n-1}}) dV_g \\leq C. \n\t\t$$ \n\t\tMoreover, $\\alpha_n$ is sharp.\n\t\\end{thm}"}, "pre_theorem_intro_text_len": 1643, "pre_theorem_intro_text": "The classical P\\'olya-Szeg\\\"{o} symmetrization inequality plays a fundamental role in establishing geometric and functional inequalities in the Euclidean space. \nHowever, this symmetrization principle in general fails in many other non-Euclidean settings such as the Heisenberg group or Riemannian manifolds as well as on higher order Sobolev spaces even in the Euclidean space. \n\nTo circumvent this obstacle, \nLam and Lu \n\\cite{LamLu2, LamLu1} developed a symmetrization-free method to prove critical global Trudinger-Moser inequalities on the entire Heisenberg group, and Adams inequalities on the higher order Sobolev spaces on the entire Euclidean spaces. \nLam, Lu and Tang \\cite{LamLuTang} establish the subcritical Trudinger-Moser inequality on the Heisenberg group.\nThe essence of these techniques is to establish the local inequalities on finite domains and then pass them to global inequalities using an argument of level set of functions under consideration. This kind of argument avoids using the P\\'oly-Szeg\\\"{o} inequality and thus is also effective in proving the concentration-compactness principle in settings where such symmtrization principle is absent (\\cite{LiLuZhu-CVPDE, LiLuZhu-ANS}). This argument has been extensively used by a number of authors and we only refer the reader to \nsee also e.g. \\cite{YangQ}, \\cite{ChenLuZhu-PLMS}, \\cite{LuTang-ANS1, LuTang-ANS2} for adapations of this circle of ideas. \n\nThis sort of ideas has also been used in our proofs of the critical Trudinger-Moser inequality (Theorem 1.3) on complete and noncompact Riemannian manifolds \\cite{LiLu-AIM}. To be precise, we established", "context": "The classical P\\'olya-Szeg\\\"{o} symmetrization inequality plays a fundamental role in establishing geometric and functional inequalities in the Euclidean space. \nHowever, this symmetrization principle in general fails in many other non-Euclidean settings such as the Heisenberg group or Riemannian manifolds as well as on higher order Sobolev spaces even in the Euclidean space.\n\nTo circumvent this obstacle, \nLam and Lu \n\\cite{LamLu2, LamLu1} developed a symmetrization-free method to prove critical global Trudinger-Moser inequalities on the entire Heisenberg group, and Adams inequalities on the higher order Sobolev spaces on the entire Euclidean spaces. \nLam, Lu and Tang \\cite{LamLuTang} establish the subcritical Trudinger-Moser inequality on the Heisenberg group.\nThe essence of these techniques is to establish the local inequalities on finite domains and then pass them to global inequalities using an argument of level set of functions under consideration. This kind of argument avoids using the P\\'oly-Szeg\\\"{o} inequality and thus is also effective in proving the concentration-compactness principle in settings where such symmtrization principle is absent (\\cite{LiLuZhu-CVPDE, LiLuZhu-ANS}). This argument has been extensively used by a number of authors and we only refer the reader to \nsee also e.g. \\cite{YangQ}, \\cite{ChenLuZhu-PLMS}, \\cite{LuTang-ANS1, LuTang-ANS2} for adapations of this circle of ideas.\n\nThis sort of ideas has also been used in our proofs of the critical Trudinger-Moser inequality (Theorem 1.3) on complete and noncompact Riemannian manifolds \\cite{LiLu-AIM}. To be precise, we established", "full_context": "The classical P\\'olya-Szeg\\\"{o} symmetrization inequality plays a fundamental role in establishing geometric and functional inequalities in the Euclidean space. \nHowever, this symmetrization principle in general fails in many other non-Euclidean settings such as the Heisenberg group or Riemannian manifolds as well as on higher order Sobolev spaces even in the Euclidean space.\n\nTo circumvent this obstacle, \nLam and Lu \n\\cite{LamLu2, LamLu1} developed a symmetrization-free method to prove critical global Trudinger-Moser inequalities on the entire Heisenberg group, and Adams inequalities on the higher order Sobolev spaces on the entire Euclidean spaces. \nLam, Lu and Tang \\cite{LamLuTang} establish the subcritical Trudinger-Moser inequality on the Heisenberg group.\nThe essence of these techniques is to establish the local inequalities on finite domains and then pass them to global inequalities using an argument of level set of functions under consideration. This kind of argument avoids using the P\\'oly-Szeg\\\"{o} inequality and thus is also effective in proving the concentration-compactness principle in settings where such symmtrization principle is absent (\\cite{LiLuZhu-CVPDE, LiLuZhu-ANS}). This argument has been extensively used by a number of authors and we only refer the reader to \nsee also e.g. \\cite{YangQ}, \\cite{ChenLuZhu-PLMS}, \\cite{LuTang-ANS1, LuTang-ANS2} for adapations of this circle of ideas.\n\nThis sort of ideas has also been used in our proofs of the critical Trudinger-Moser inequality (Theorem 1.3) on complete and noncompact Riemannian manifolds \\cite{LiLu-AIM}. To be precise, we established\n\nThis sort of ideas has also been used in our proofs of the critical Trudinger-Moser inequality (Theorem 1.3) on complete and noncompact Riemannian manifolds \\cite{LiLu-AIM}. To be precise, we established\n\nThe second main result of this note is to establish that Green functions for the Dirichlet problem on any small geodesic balls on Riemannian manifolds with positive injective radius, Ricci curvature lower bounded and sectional curvature upper bounded have uniform estimates independent of the locations of the balls. We believe that this result is of its independent interest. More precisely,\n\n\\begin{thm}\\label{Green}\n Let $(M, g)$ be a complete noncompact Riemannian manifold whose Ricci curvature is lower bounded and sectional curvature is upper bounded, i.e. $\\textit{Ric} \\geq \\lambda g$ and $K \\leq a^2$ for some constant $\\lambda, a \\in \\mathbb{R}$. Moreover, we assume that its injectivity radius has a lower bound, i.e. $\\textit{inj} (M,g) \\geq i > 0$ for some constant $i > 0$. For any $Q >1$ and $0 < \\alpha < 1$, denote the geodesic ball as $B_\\delta (x)$ with $\\delta \\leq r_H(n , i , \\lambda , \\alpha)$, where $r_H(n , i , \\lambda , \\alpha)$ is the harmonic radius. Then the Dirichlet Green's function over $B_{\\delta} (x)$ satisfies the following estimate:\n\n\\begin{lem}\\label{harmonic coordinates}\n Let $\\alpha \\in (0,1)$, $Q > 1$, $\\delta > 0$ and $(M,g)$ satisfy $\\textit{Ric} \\geq \\lambda g$ and $\\textit{inj} (M , g) \\geq i > 0$. Then there exists a positive constant $C = C(n , Q , \\alpha , \\delta , i , \\lambda)$ such that for any $x \\in M$, the $C^{0,\\alpha}$ harmonic radius $r_H(Q , 0 , \\alpha) (x) \\geq C$. \n \\end{lem}\n\n\\begin{thm}\n \\label{improvedAT}\\textit{Let }$n\\geq2$ and $0\\leq\\alpha<\\alpha_{n}.$ Denote $$\n AT\\left( \\alpha,\\beta\\right) =\\sup_{\\left\\Vert \\nabla u\\right\\Vert _{n} \\leq1}\\frac{1}{\\left\\Vert u\\right\\Vert _{n}^{n-\\beta}}\\int_{ \\mathbb{R}\n ^{n}}\\phi_{n}\\left( \\alpha\\left( 1-\\frac{\\beta}{n}\\right) \\left\\vert\n u\\right\\vert ^{\\frac{n}{n-1}}\\right) \\frac{dx}{\\left\\vert x\\right\\vert\n ^{\\beta}}.\n $$\n Then there exist positive constants $c=c\\left( n,\\beta\\right) $ and\n $C=C\\left( n,\\beta\\right) $ such that when $\\alpha$ is close enough to\n $\\alpha_{N}:$\n \\begin{equation}\n \\frac{c\\left( n,\\beta\\right) }{\\left( 1-\\left( \\frac{\\alpha}{\\alpha_{n} }\\right) ^{n-1}\\right) ^{\\left( n-\\beta\\right) /}}\\leq AT\\left(\n \\alpha,\\beta\\right) \\leq\\frac{C\\left( n,\\beta\\right) }{\\left( 1-\\left(\n \\frac{\\alpha}{\\alpha_{n}}\\right) ^{n-1}\\right) ^{\\left( n-\\beta\\right)\n /n}}. \\label{1.3.1} \\end{equation}\n Moreover, the constant $\\alpha_{N}$ is sharp in the sense that $AT\\left(\n \\alpha_{N},\\beta\\right) =\\infty.$\n \\end{thm}\n\n\\begin{thm}[Uniform local estimate]\\label{thm:uniform}\n Given a complete and noncompact Riemannian manifold $M$. Assume there exist $\\delta_0>0$ and $\\Lambda\\ge 1$ such that for every $p\\in M$\n there is a harmonic coordinate chart\n $$\n \\varphi_p: B_{\\delta_0}(p)\\longrightarrow \\R^n\n $$\n whose image contains a Euclidean ball $B_r(0)$ with $r\\in[c_1\\delta_0,c_2\\delta_0]$,\n and such that on $B_{\\delta_0}(p)$\n $$\n \\Lambda^{-1}\\abs{\\xi}^2\\leq g^{ij}(x)\\xi_i\\xi_j\\leq \\Lambda\\abs{\\xi}^2,\n \\qquad\n \\Lambda^{-1}\\leq \\sqrt{\\det(g_{ij}(x))}\\leq \\Lambda.\n $$\n Then there exists $\\delta\\in(0,\\delta_0]$ and $C<\\infty$, depending only on $(n,\\tau,\\Lambda)$,\n such that for every $u\\in W^{1,n}(M)$ with $\\norm{u}_{1,\\tau}\\leq 1$,\n $$\n \\sup_{p\\in M}\\int_{B_{\\delta/2}(p)}\\phi_n\\!\\left(\\alpha_n\\abs{u}^{\\frac{n}{n-1}}\\right)\\,dV_g\n \\leq C.\n $$\n \\end{thm}\n\n\\begin{lem}[Cutoff bounds]\\label{lem:cutoff}\n Let $\\{x_j\\}$ be as in Lemma~\\ref{harmonic coordinates}. For each $j$ choose $\\psi_j\\in C_c^\\infty(B_\\delta(x_j))$\n such that $0\\leq \\psi_j\\leq 1$, $\\psi_j\\equiv 1$ on $B_{\\delta/2}(x_j)$, and $\\abs{\\grad\\psi_j}\\leq 4/\\delta$.\n Then $\\abs{\\grad(\\psi_j^2)}\\leq 8\\psi_j/\\delta$ and for every $u\\in W^{1,n}(M)$ and every $\\tilde\\tau\\geq 0$,\n \\begin{equation}\\label{eq:local-norm}\n \\norm{\\psi_j^2 u}_{1,\\tilde\\tau}\n \\leq\n \\norm{\\grad u}_{L^n(M)}+\\Bigl(\\tilde\\tau+\\frac{8}{\\delta}\\Bigr)\\norm{u}_{L^n(M)}.\n \\end{equation}\n Moreover,\n \\begin{equation}\\label{eq:sum-overlap}\n \\sum_j \\int_M \\psi_j^{2n}\\abs{\\grad u}^n\\,dV_g \\leq N\\int_M \\abs{\\grad u}^n\\,dV_g,\n \\qquad\n \\sum_j \\int_M \\abs{\\grad(\\psi_j^2)}^n\\abs{u}^n\\,dV_g \\leq \\frac{8^n}{\\delta^n}\\,N\\int_M \\abs{u}^n\\,dV_g.\n \\end{equation}\n \\end{lem}\n\n\\smallskip\n Then for every $\\tau\\ge \\tilde\\tau+\\frac{8}{\\delta}$ there exists $C_{\\mathrm{glob}}<\\infty$\n depending only on $(n,\\tau)$ and the geometric data such that for all $u\\in W^{1,n}(M)$ with\n $\\norm{u}_{1,\\tau}\\leq 1$,\n \\begin{equation}\\label{eq:global}\n \\int_M \\phi_n\\!\\left(\\alpha_n\\abs{u}^{\\frac{n}{n-1}}\\right)\\,dV_g \\leq C_{\\mathrm{glob}}.\n \\end{equation}\n \\end{thm}", "post_theorem_intro_text_len": 3427, "post_theorem_intro_text": "In the proof of Theorem \\ref{maintheorem}\n needs to be rigorized about the uniformity of the upper bound of the critical Trudinger-Moser inequality in terms of the bounded domains of the level sets involved. \nIn this note, we will then simply give another simple proof by only establishing the uniformity of the bounds of the critical Trudinger-Moser inequality on geodesic balls within the manifolds by modifying the ideas developed in the earlier work \nby Lam, Lu and Zhang \\cite{LamLuZhang}. In fact, we will achieve this by using a scaling argument and the asymptotic estimates for the supremum of the subcritical Trudinger-Moser functional established in \\cite{LamLuZhang}. As a consequence of this, we can prove the uniform boundedness of the critical Trudinger-Moser inequality on the level set of the functions under consideration in \\cite{LiLu-AIM}. This thus rigorizes the proof of Theorem 1.1 \n(namely, Theorem 1.3 in \\cite{LiLu-AIM}) by covering the level set by the geodesic balls.\n Therefore, the same level argument from local inequalities \n to the global ones used in \\cite{LiLu-AIM} are justified. \n We make use of the equivalence between the subcritical and critical Moser-Trudinger inequalties on geodesic balls. This idea was originally in \\cite{LamLuZhang} and already used in \\cite{LiLu-AIM} to derive subcritical inequalities from critical inequalities. Meanwhile in this note, we will prove in the opposite direction, i.e. from subcritical inequalities to critical inequalities. \n\n The second main result of this note is to establish that Green functions for the Dirichlet problem on any small geodesic balls on Riemannian manifolds with positive injective radius, Ricci curvature lower bounded and sectional curvature upper bounded have uniform estimates independent of the locations of the balls. We believe that this result is of its independent interest. More precisely,\n\n \\begin{thm}\\label{Green}\n Let $(M, g)$ be a complete noncompact Riemannian manifold whose Ricci curvature is lower bounded and sectional curvature is upper bounded, i.e. $\\textit{Ric} \\geq \\lambda g$ and $K \\leq a^2$ for some constant $\\lambda, a \\in \\mathbb{R}$. Moreover, we assume that its injectivity radius has a lower bound, i.e. $\\textit{inj} (M,g) \\geq i > 0$ for some constant $i > 0$. For any $Q >1$ and $0 < \\alpha < 1$, denote the geodesic ball as $B_\\delta (x)$ with $\\delta \\leq r_H(n , i , \\lambda , \\alpha)$, where $r_H(n , i , \\lambda , \\alpha)$ is the harmonic radius. Then the Dirichlet Green's function over $B_{\\delta} (x)$ satisfies the following estimate:\n\n $$\n |\\nabla_y G(x,y)| \\leq \\frac{1}{\\omega_{n-1}} d(x , y)^{1-n} (1 + C d(x , y)),\n $$\n where $C = C(n , \\lambda , Q , \\delta)$.\n \\end{thm}\n\n As a byproduct, this will allow us to conclude that the Trudinger-Moser inequality on any level set is uniformly bounded independent of the location of the level set, within an extra sectional curvature assumption by using the uniform estimates of the Green functions. \n\nSince the main purpose of this short note is to rigorize the proof of Theorem 1.3 in \\cite{LiLu-AIM} and in the meantime to give an alternative proof from the level set argument passing from the local inequalities to global one, we have chosen not to give a comprehensive account of the subject (see e.g. \\cite{LiLu-AIM} for relevant references in the literature).", "sketch": "In the proof of Theorem~\\ref{maintheorem} one “needs to be rigorized about the uniformity of the upper bound of the critical Trudinger--Moser inequality in terms of the bounded domains of the level sets involved.” The note “give[s] another simple proof” by “only establishing the uniformity of the bounds of the critical Trudinger--Moser inequality on geodesic balls within the manifolds by modifying the ideas developed” by Lam--Lu--Zhang~\\cite{LamLuZhang}. This is achieved “by using a scaling argument and the asymptotic estimates for the supremum of the subcritical Trudinger--Moser functional established in~\\cite{LamLuZhang},” yielding “the equivalence between the subcritical and critical Moser--Trudinger inequalties on geodesic balls,” but proved here “in the opposite direction, i.e. from subcritical inequalities to critical inequalities.” As a consequence, one “prove[s] the uniform boundedness of the critical Trudinger--Moser inequality on the level set of the functions under consideration in~\\cite{LiLu-AIM},” thus “rigorize[s] the proof” by “covering the level set by the geodesic balls,” so that “the same level argument from local inequalities to the global ones used in~\\cite{LiLu-AIM} are justified.”", "expanded_sketch": "In the proof of the main theorem one “needs to be rigorized about the uniformity of the upper bound of the critical Trudinger--Moser inequality in terms of the bounded domains of the level sets involved.” The note “give[s] another simple proof” by “only establishing the uniformity of the bounds of the critical Trudinger--Moser inequality on geodesic balls within the manifolds by modifying the ideas developed” by Lam--Lu--Zhang~\\cite{LamLuZhang}. This is achieved “by using a scaling argument and the asymptotic estimates for the supremum of the subcritical Trudinger--Moser functional established in~\\cite{LamLuZhang},” yielding “the equivalence between the subcritical and critical Moser--Trudinger inequalties on geodesic balls,” but proved here “in the opposite direction, i.e. from subcritical inequalities to critical inequalities.” As a consequence, one “prove[s] the uniform boundedness of the critical Trudinger--Moser inequality on the level set of the functions under consideration in~\\cite{LiLu-AIM},” thus “rigorize[s] the proof” by “covering the level set by the geodesic balls,” so that “the same level argument from local inequalities to the global ones used in~\\cite{LiLu-AIM} are justified.”", "expanded_theorem": "\\label{maintheorem}\n\t\tLet $(M, g)$ be a complete noncompact Riemannian manifold whose Ricci curvature is lower bounded, i.e. $\\textit{Ric} \\geq \\lambda g$ for some constant $\\lambda \\in \\mathbb{R}$. Moreover, we assume that its injectivity radius has a lower bound, i.e. $\\textit{inj} (M,g) \\geq i > 0$ for some constant $i > 0$. Then there exists a positive constant $\\epsilon = \\epsilon (\\lambda , i , n) =O(\\frac{1}{r_H})$ in the reverse order of the harmonic radius $r_H$ such that for every $\\tau > \\epsilon$, there exists a constant $C = C(n, \\tau , M)$ such that \n\n\t\t$$\n\t\t\\sup_{u \\in W^{1,n} (M) , ||u||_{1, \\tau} \\leq 1} \\int_M \\phi_n (\\alpha_n |u|^{\\frac{n}{n-1}}) dV_g \\leq C. \n\t\t$$ \n\t\tMoreover, $\\alpha_n$ is sharp.,", "theorem_type": "unknown", "mcq": {"question": "Let $(M,g)$ be a complete noncompact Riemannian manifold with Ricci curvature bounded below by $\\operatorname{Ric}\\ge \\lambda g$ for some $\\lambda\\in\\mathbb R$, and injectivity radius bounded below by $\\operatorname{inj}(M,g)\\ge i>0$. Write\n\\[\\|u\\|_{1,\\tau}:=\\|\\nabla_g u\\|_{L^n(M)}+\\tau\\|u\\|_{L^n(M)},\\]\nand let\n\\[\\phi_n(t):=e^t-\\sum_{k=0}^{n-2}\\frac{t^k}{k!},\\qquad \\alpha_n=n\\,\\omega_{n-1}^{1/(n-1)}.\\]\nIf $r_H$ denotes the harmonic radius, which statement holds under these assumptions?", "correct_choice": {"label": "A", "text": "There exists a positive constant $\\epsilon=\\epsilon(\\lambda,i,n)=O(r_H^{-1})$ such that for every $\\tau>\\epsilon$ there is a constant $C=C(n,\\tau,M)$ satisfying\n\\[\n\\sup_{u\\in W^{1,n}(M),\\,\\|u\\|_{1,\\tau}\\le 1}\\int_M \\phi_n\\!\\left(\\alpha_n |u|^{\\frac{n}{n-1}}\\right)\\,dV_g\\le C.\n\\]\nMoreover, $\\alpha_n$ is sharp, i.e. it is the optimal constant in this inequality."}, "choices": [{"label": "B", "text": "There exists a positive constant $\\epsilon=\\epsilon(\\lambda,i,n)=O(r_H^{-1})$ such that for every $\\tau\\ge \\epsilon$ there is a constant $C=C(n,\\tau,\\lambda,i)$ satisfying\n\\[\n\\sup_{u\\in W^{1,n}(M),\\,\\|u\\|_{1,\\tau}\\le 1}\\int_M \\phi_n\\!\\left(\\alpha_n |u|^{\\frac{n}{n-1}}\\right)\\,dV_g\\le C.\n\\]\nMoreover, $\\alpha_n$ is sharp."}, {"label": "C", "text": "There exists a positive constant $\\epsilon=\\epsilon(\\lambda,i,n)=O(r_H^{-1})$ such that for every $\\tau>\\epsilon$ there is a constant $C=C(n,\\tau,M)$ satisfying\n\\[\n\\sup_{u\\in W^{1,n}(M),\\,\\|u\\|_{1,\\tau}\\le 1}\\int_M \\phi_n\\!\\left(\\alpha |u|^{\\frac{n}{n-1}}\\right)\\,dV_g\\le C\n\\]\nfor every $0\\le \\alpha<\\alpha_n$."}, {"label": "D", "text": "For every $\\tau>0$ there exists a constant $C=C(n,\\tau,M)$ such that\n\\[\n\\sup_{u\\in W^{1,n}(M),\\,\\|u\\|_{1,\\tau}\\le 1}\\int_M \\phi_n\\!\\left(\\alpha_n |u|^{\\frac{n}{n-1}}\\right)\\,dV_g\\le C.\n\\]\nMoreover, $\\alpha_n$ is sharp."}, {"label": "E", "text": "There exists a positive constant $\\epsilon=\\epsilon(\\lambda,i,n)=O(r_H^{-1})$ such that for every $\\tau>\\epsilon$ there is a constant $C=C(n,\\tau,M)$ satisfying\n\\[\n\\sup_{u\\in W^{1,n}(M),\\,\\|u\\|_{1,\\tau}\\le 1}\\int_M \\phi_n\\!\\left(\\alpha_n |u|^{\\frac{n}{n-1}}\\right)\\,dV_g\\le C.\n\\]\nIn addition, the same conclusion remains valid if $\\alpha_n$ is replaced by any $\\alpha>\\alpha_n$."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "uniformity", "tampered_component": "dependence_of_global_constant_on_full_manifold_geometry_and_strict_tau_threshold", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "sharp_critical_constant_replaced_by_subcritical_range_only", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "positive_threshold_epsilon_for_tau", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "sharpness_of_alpha_n_violated_by_supercritical_extension", "template_used": "wildcard"}]}} +{"id": "2601.11131v1", "paper_link": "http://arxiv.org/abs/2601.11131v1", "theorems_cnt": 1, "theorem": {"env_name": "thm", "content": "\\label{thm:main}\nThere exists an algorithm with the following properties. \nIt takes as input integers $N \\geqslant 3$ and $D \\geqslant 1$ with $D < N-1$.\nIt outputs either some $\\alpha\\in\\Z_N^*$ with $\\ord_N(\\alpha) > D$\nor a nontrivial divisor of $N$.\nIts time complexity is\n\\begin{equation}\n\\label{eq:main-complexity}\nO\\left(\\frac{D^{1/2}}{(\\log \\log D)^{1/2}} \\log^2 N\\right).\n\\end{equation}", "start_pos": 9745, "end_pos": 10151, "label": "thm:main"}, "ref_dict": {"thm:main": "\\begin{thm}\n\\label{thm:main}\nThere exists an algorithm with the following properties. \nIt takes as input integers $N \\geq 3$ and $D \\geq 1$ with $D < N-1$.\nIt outputs either some $\\alpha\\in\\Z_N^*$ with $\\ord_N(\\alpha) > D$\nor a nontrivial divisor of $N$.\nIts time complexity is\n\\begin{equation}\n\\label{eq:main-complexity}\nO\\left(\\frac{D^{1/2}}{(\\log \\log D)^{1/2}} \\log^2 N\\right).\n\\end{equation}\n\\end{thm}"}, "pre_theorem_intro_text_len": 5670, "pre_theorem_intro_text": "\\label{sec:intro}\n\nLet $N \\geqslant 2$ be an integer and consider the multiplicative group $\\Z_N^*$ of invertible residues modulo $N$. The order of an element $\\alpha \\in \\Z_N^*$, denoted $\\ord_N(\\alpha)$, is defined to be the smallest positive integer $k$ such that $\\alpha^k = 1$. The problem of finding elements of large or maximal order in $\\Z_N^*$ occurs in various theoretical and applied contexts. Examples include the generation of pseudorandom numbers \\cite{BM-prng} and the discrete logarithm problem modulo a prime $N$, which is the basis of widely used cryptographic protocols \\cite{DH-dlp, E-elgamal}. Given $g, h \\in \\Z_N^*$, the task is to find an integer $x$ such that $g^x = h$, if such a solution exists. To maximise the difficulty of this problem, so as to increase the security of the resulting cryptographic scheme, we should choose $g$ to be a primitive root, i.e., an element of maximal order $\\ord_N(g)=N-1$. If the factorisation of $N-1$ is known, such an element can be found with high probability by testing a few randomly chosen candidates. However, the best known rigorous and unconditional upper bound \\cite{B-leastprimitiveroot} for the smallest primitive root modulo $N$ is still exponential in $\\log N$. So while it is easy to find an element of large order in $\\Z_N^*$ in randomised polynomial time (still assuming that the factorisation of $N-1$ is known), the same problem is much more difficult to solve deterministically and rigorously. \n\nAnother problem in which finding elements of large multiplicative order plays an important role is deterministic integer factorisation. Until recently, the best complexity bounds for rigorously factoring an integer $N$ were of the form $N^{1/4+o(1)}$, based on methods going back to Pollard \\cite{Pol-factoring}, Strassen \\cite{Str-factoring} and Coppersmith \\cite{Cop-lowexponent}. (Here, and for the rest of the paper, time complexity always refers to the number of steps performed by a multitape Turing machine. For a detailed description of this computational model, see \\cite[Ch.\\,2]{Pap-complexity} or \\cite[\\S1.6]{AHU-algorithms}.) The first improvement on these bounds by a superpolynomial factor (i.e., larger than any fixed power of $\\log N$) appeared in \\cite{Hit-BSGS}, and was based on the idea of solving the discrete logarithm problem $\\alpha^x = \\alpha^{N+1}$ in $\\Z_N^*$ for a given element $\\alpha$. If $N$ is a product of two distinct primes, say $N=pq$, then the solution $x=p+q$ immediately reveals the two factors of $N$. However, we can only find this solution if $\\ord_N(\\alpha)$ is sufficiently large. Therefore, the paper also included an algorithm that, for $D\\geqslant N^{2/5}$, either finds an element of order at least $D$, or a nontrivial divisor of $N$, or proves $N$ prime \\cite[Alg.\\,6.2]{Hit-BSGS}.\n\nThe algorithm has two stages. In the first stage, we search for the order of small elements $\\beta=2,3,\\ldots$ via Sutherland's optimisation of the classical babystep-giantstep method \\cite{Sha-classnumber,SutherlandPHD}, assuming that the order is at most $D$. If for some element we fail to compute the order, then the order exceeds $D$ and we are done. If the order $m \\coloneqq \\ord_N(\\beta)$ is actually found, we try to use it to compute a nontrivial divisor of $N$, by examining $\\gcd(\\beta^{m/r} - 1, N)$ for prime divisors $r$ of $m$. If that fails, we deduce that $m$ divides $p-1$ for every prime factor $p$ of $N$. As we compute the exact orders of these small elements, and continue to fail to factorise $N$, we accumulate considerable information about the prime factors of $N$. Eventually, we recover sufficient information to proceed to the second stage of the algorithm, in which we attempt to factorise $N$ directly. If that fails too, we obtain a proof that $N$ is prime. It is noteworthy that the time complexity of this procedure is asymptotically the same as testing whether a single element has order at most $D$, namely\n\\[\nO\\left(\\frac{D^{1/2}}{(\\log \\log D)^{1/2}} \\log^2 N\\right).\n\\]\n\nThe 2018 algorithm for finding elements of large order was later used as a subroutine in a series of works \\cite{Hit-timespace,Har-onefifth,HH-onefifthloglog} that improved the runtime of deterministic factorisation to $N^{1/5+o(1)}$. The original assumption $D \\geqslant N^{2/5}$ was good enough to directly apply the original algorithm in these factorisation methods, but only just. The hypothesis on $D$ was subsequently improved on two occasions, first to $D \\geqslant N^{1/4+o(1)}$ by Gao, Feng, Hu and Pan \\cite[Lem.\\,3.5]{GFHP-factoringlattices} and then to $D \\geqslant N^{1/6}$ by Oznovich and Volk \\cite[Thm.\\,1.1]{OV-highorder}. These improvements are based on a better bound for the number of consecutive elements with the same order $m$ (namely, $O(\\sqrt{m})$ instead of $m$), leading to a faster progression in the loop over the elements $\\beta = 2,3,\\ldots$ in the first stage of the algorithm. Both of these papers (and additionally \\cite{HH-lehmangeneralization}) also considered the case of $N$ having an $r$-power divisor, which allows the assumption to be relaxed further to $D \\geqslant N^{1/6r}$ \\cite[Thm.\\,4.2]{OV-highorder}. Furthermore, both \\cite{GFHP-factoringlattices} and \\cite{OV-highorder} present improved techniques for the second stage, in which $N$ is factorised based on a large known factor of $p-1$ for all primes $p \\mathrel{|} N$. \n\nAll of the results mentioned so far impose a fairly restrictive (in fact, fully exponential) lower bound hypothesis on $D$. The main result of this paper is an algorithm that solves the same problem within the same amount of time, but without any restriction on $D$ at all.", "context": "\\label{sec:intro}\n\nLet $N \\geqslant 2$ be an integer and consider the multiplicative group $\\Z_N^*$ of invertible residues modulo $N$. The order of an element $\\alpha \\in \\Z_N^*$, denoted $\\ord_N(\\alpha)$, is defined to be the smallest positive integer $k$ such that $\\alpha^k = 1$. The problem of finding elements of large or maximal order in $\\Z_N^*$ occurs in various theoretical and applied contexts. Examples include the generation of pseudorandom numbers \\cite{BM-prng} and the discrete logarithm problem modulo a prime $N$, which is the basis of widely used cryptographic protocols \\cite{DH-dlp, E-elgamal}. Given $g, h \\in \\Z_N^*$, the task is to find an integer $x$ such that $g^x = h$, if such a solution exists. To maximise the difficulty of this problem, so as to increase the security of the resulting cryptographic scheme, we should choose $g$ to be a primitive root, i.e., an element of maximal order $\\ord_N(g)=N-1$. If the factorisation of $N-1$ is known, such an element can be found with high probability by testing a few randomly chosen candidates. However, the best known rigorous and unconditional upper bound \\cite{B-leastprimitiveroot} for the smallest primitive root modulo $N$ is still exponential in $\\log N$. So while it is easy to find an element of large order in $\\Z_N^*$ in randomised polynomial time (still assuming that the factorisation of $N-1$ is known), the same problem is much more difficult to solve deterministically and rigorously.\n\nAnother problem in which finding elements of large multiplicative order plays an important role is deterministic integer factorisation. Until recently, the best complexity bounds for rigorously factoring an integer $N$ were of the form $N^{1/4+o(1)}$, based on methods going back to Pollard \\cite{Pol-factoring}, Strassen \\cite{Str-factoring} and Coppersmith \\cite{Cop-lowexponent}. (Here, and for the rest of the paper, time complexity always refers to the number of steps performed by a multitape Turing machine. For a detailed description of this computational model, see \\cite[Ch.\\,2]{Pap-complexity} or \\cite[\\S1.6]{AHU-algorithms}.) The first improvement on these bounds by a superpolynomial factor (i.e., larger than any fixed power of $\\log N$) appeared in \\cite{Hit-BSGS}, and was based on the idea of solving the discrete logarithm problem $\\alpha^x = \\alpha^{N+1}$ in $\\Z_N^*$ for a given element $\\alpha$. If $N$ is a product of two distinct primes, say $N=pq$, then the solution $x=p+q$ immediately reveals the two factors of $N$. However, we can only find this solution if $\\ord_N(\\alpha)$ is sufficiently large. Therefore, the paper also included an algorithm that, for $D\\geqslant N^{2/5}$, either finds an element of order at least $D$, or a nontrivial divisor of $N$, or proves $N$ prime \\cite[Alg.\\,6.2]{Hit-BSGS}.\n\nThe algorithm has two stages. In the first stage, we search for the order of small elements $\\beta=2,3,\\ldots$ via Sutherland's optimisation of the classical babystep-giantstep method \\cite{Sha-classnumber,SutherlandPHD}, assuming that the order is at most $D$. If for some element we fail to compute the order, then the order exceeds $D$ and we are done. If the order $m \\coloneqq \\ord_N(\\beta)$ is actually found, we try to use it to compute a nontrivial divisor of $N$, by examining $\\gcd(\\beta^{m/r} - 1, N)$ for prime divisors $r$ of $m$. If that fails, we deduce that $m$ divides $p-1$ for every prime factor $p$ of $N$. As we compute the exact orders of these small elements, and continue to fail to factorise $N$, we accumulate considerable information about the prime factors of $N$. Eventually, we recover sufficient information to proceed to the second stage of the algorithm, in which we attempt to factorise $N$ directly. If that fails too, we obtain a proof that $N$ is prime. It is noteworthy that the time complexity of this procedure is asymptotically the same as testing whether a single element has order at most $D$, namely\n\\[\nO\\left(\\frac{D^{1/2}}{(\\log \\log D)^{1/2}} \\log^2 N\\right).\n\\]\n\nThe 2018 algorithm for finding elements of large order was later used as a subroutine in a series of works \\cite{Hit-timespace,Har-onefifth,HH-onefifthloglog} that improved the runtime of deterministic factorisation to $N^{1/5+o(1)}$. The original assumption $D \\geqslant N^{2/5}$ was good enough to directly apply the original algorithm in these factorisation methods, but only just. The hypothesis on $D$ was subsequently improved on two occasions, first to $D \\geqslant N^{1/4+o(1)}$ by Gao, Feng, Hu and Pan \\cite[Lem.\\,3.5]{GFHP-factoringlattices} and then to $D \\geqslant N^{1/6}$ by Oznovich and Volk \\cite[Thm.\\,1.1]{OV-highorder}. These improvements are based on a better bound for the number of consecutive elements with the same order $m$ (namely, $O(\\sqrt{m})$ instead of $m$), leading to a faster progression in the loop over the elements $\\beta = 2,3,\\ldots$ in the first stage of the algorithm. Both of these papers (and additionally \\cite{HH-lehmangeneralization}) also considered the case of $N$ having an $r$-power divisor, which allows the assumption to be relaxed further to $D \\geqslant N^{1/6r}$ \\cite[Thm.\\,4.2]{OV-highorder}. Furthermore, both \\cite{GFHP-factoringlattices} and \\cite{OV-highorder} present improved techniques for the second stage, in which $N$ is factorised based on a large known factor of $p-1$ for all primes $p \\mathrel{|} N$.\n\nAll of the results mentioned so far impose a fairly restrictive (in fact, fully exponential) lower bound hypothesis on $D$. The main result of this paper is an algorithm that solves the same problem within the same amount of time, but without any restriction on $D$ at all.", "full_context": "\\label{sec:intro}\n\nLet $N \\geqslant 2$ be an integer and consider the multiplicative group $\\Z_N^*$ of invertible residues modulo $N$. The order of an element $\\alpha \\in \\Z_N^*$, denoted $\\ord_N(\\alpha)$, is defined to be the smallest positive integer $k$ such that $\\alpha^k = 1$. The problem of finding elements of large or maximal order in $\\Z_N^*$ occurs in various theoretical and applied contexts. Examples include the generation of pseudorandom numbers \\cite{BM-prng} and the discrete logarithm problem modulo a prime $N$, which is the basis of widely used cryptographic protocols \\cite{DH-dlp, E-elgamal}. Given $g, h \\in \\Z_N^*$, the task is to find an integer $x$ such that $g^x = h$, if such a solution exists. To maximise the difficulty of this problem, so as to increase the security of the resulting cryptographic scheme, we should choose $g$ to be a primitive root, i.e., an element of maximal order $\\ord_N(g)=N-1$. If the factorisation of $N-1$ is known, such an element can be found with high probability by testing a few randomly chosen candidates. However, the best known rigorous and unconditional upper bound \\cite{B-leastprimitiveroot} for the smallest primitive root modulo $N$ is still exponential in $\\log N$. So while it is easy to find an element of large order in $\\Z_N^*$ in randomised polynomial time (still assuming that the factorisation of $N-1$ is known), the same problem is much more difficult to solve deterministically and rigorously.\n\nAnother problem in which finding elements of large multiplicative order plays an important role is deterministic integer factorisation. Until recently, the best complexity bounds for rigorously factoring an integer $N$ were of the form $N^{1/4+o(1)}$, based on methods going back to Pollard \\cite{Pol-factoring}, Strassen \\cite{Str-factoring} and Coppersmith \\cite{Cop-lowexponent}. (Here, and for the rest of the paper, time complexity always refers to the number of steps performed by a multitape Turing machine. For a detailed description of this computational model, see \\cite[Ch.\\,2]{Pap-complexity} or \\cite[\\S1.6]{AHU-algorithms}.) The first improvement on these bounds by a superpolynomial factor (i.e., larger than any fixed power of $\\log N$) appeared in \\cite{Hit-BSGS}, and was based on the idea of solving the discrete logarithm problem $\\alpha^x = \\alpha^{N+1}$ in $\\Z_N^*$ for a given element $\\alpha$. If $N$ is a product of two distinct primes, say $N=pq$, then the solution $x=p+q$ immediately reveals the two factors of $N$. However, we can only find this solution if $\\ord_N(\\alpha)$ is sufficiently large. Therefore, the paper also included an algorithm that, for $D\\geqslant N^{2/5}$, either finds an element of order at least $D$, or a nontrivial divisor of $N$, or proves $N$ prime \\cite[Alg.\\,6.2]{Hit-BSGS}.\n\nThe algorithm has two stages. In the first stage, we search for the order of small elements $\\beta=2,3,\\ldots$ via Sutherland's optimisation of the classical babystep-giantstep method \\cite{Sha-classnumber,SutherlandPHD}, assuming that the order is at most $D$. If for some element we fail to compute the order, then the order exceeds $D$ and we are done. If the order $m \\coloneqq \\ord_N(\\beta)$ is actually found, we try to use it to compute a nontrivial divisor of $N$, by examining $\\gcd(\\beta^{m/r} - 1, N)$ for prime divisors $r$ of $m$. If that fails, we deduce that $m$ divides $p-1$ for every prime factor $p$ of $N$. As we compute the exact orders of these small elements, and continue to fail to factorise $N$, we accumulate considerable information about the prime factors of $N$. Eventually, we recover sufficient information to proceed to the second stage of the algorithm, in which we attempt to factorise $N$ directly. If that fails too, we obtain a proof that $N$ is prime. It is noteworthy that the time complexity of this procedure is asymptotically the same as testing whether a single element has order at most $D$, namely\n\\[\nO\\left(\\frac{D^{1/2}}{(\\log \\log D)^{1/2}} \\log^2 N\\right).\n\\]\n\nThe 2018 algorithm for finding elements of large order was later used as a subroutine in a series of works \\cite{Hit-timespace,Har-onefifth,HH-onefifthloglog} that improved the runtime of deterministic factorisation to $N^{1/5+o(1)}$. The original assumption $D \\geqslant N^{2/5}$ was good enough to directly apply the original algorithm in these factorisation methods, but only just. The hypothesis on $D$ was subsequently improved on two occasions, first to $D \\geqslant N^{1/4+o(1)}$ by Gao, Feng, Hu and Pan \\cite[Lem.\\,3.5]{GFHP-factoringlattices} and then to $D \\geqslant N^{1/6}$ by Oznovich and Volk \\cite[Thm.\\,1.1]{OV-highorder}. These improvements are based on a better bound for the number of consecutive elements with the same order $m$ (namely, $O(\\sqrt{m})$ instead of $m$), leading to a faster progression in the loop over the elements $\\beta = 2,3,\\ldots$ in the first stage of the algorithm. Both of these papers (and additionally \\cite{HH-lehmangeneralization}) also considered the case of $N$ having an $r$-power divisor, which allows the assumption to be relaxed further to $D \\geqslant N^{1/6r}$ \\cite[Thm.\\,4.2]{OV-highorder}. Furthermore, both \\cite{GFHP-factoringlattices} and \\cite{OV-highorder} present improved techniques for the second stage, in which $N$ is factorised based on a large known factor of $p-1$ for all primes $p \\mathrel{|} N$.\n\nAll of the results mentioned so far impose a fairly restrictive (in fact, fully exponential) lower bound hypothesis on $D$. The main result of this paper is an algorithm that solves the same problem within the same amount of time, but without any restriction on $D$ at all.\n\nThe algorithm has two stages. In the first stage, we search for the order of small elements $\\beta=2,3,\\ldots$ via Sutherland's optimisation of the classical babystep-giantstep method \\cite{Sha-classnumber,SutherlandPHD}, assuming that the order is at most $D$. If for some element we fail to compute the order, then the order exceeds $D$ and we are done. If the order $m \\coloneqq \\ord_N(\\beta)$ is actually found, we try to use it to compute a nontrivial divisor of $N$, by examining $\\gcd(\\beta^{m/r} - 1, N)$ for prime divisors $r$ of $m$. If that fails, we deduce that $m$ divides $p-1$ for every prime factor $p$ of $N$. As we compute the exact orders of these small elements, and continue to fail to factorise $N$, we accumulate considerable information about the prime factors of $N$. Eventually, we recover sufficient information to proceed to the second stage of the algorithm, in which we attempt to factorise $N$ directly. If that fails too, we obtain a proof that $N$ is prime. It is noteworthy that the time complexity of this procedure is asymptotically the same as testing whether a single element has order at most $D$, namely\n\\[\nO\\left(\\frac{D^{1/2}}{(\\log \\log D)^{1/2}} \\log^2 N\\right).\n\\]\n\nThe 2018 algorithm for finding elements of large order was later used as a subroutine in a series of works \\cite{Hit-timespace,Har-onefifth,HH-onefifthloglog} that improved the runtime of deterministic factorisation to $N^{1/5+o(1)}$. The original assumption $D \\geq N^{2/5}$ was good enough to directly apply the original algorithm in these factorisation methods, but only just. The hypothesis on $D$ was subsequently improved on two occasions, first to $D \\geq N^{1/4+o(1)}$ by Gao, Feng, Hu and Pan \\cite[Lem.\\,3.5]{GFHP-factoringlattices} and then to $D \\geq N^{1/6}$ by Oznovich and Volk \\cite[Thm.\\,1.1]{OV-highorder}. These improvements are based on a better bound for the number of consecutive elements with the same order $m$ (namely, $O(\\sqrt{m})$ instead of $m$), leading to a faster progression in the loop over the elements $\\beta = 2,3,\\ldots$ in the first stage of the algorithm. Both of these papers (and additionally \\cite{HH-lehmangeneralization}) also considered the case of $N$ having an $r$-power divisor, which allows the assumption to be relaxed further to $D \\geq N^{1/6r}$ \\cite[Thm.\\,4.2]{OV-highorder}. Furthermore, both \\cite{GFHP-factoringlattices} and \\cite{OV-highorder} present improved techniques for the second stage, in which $N$ is factorised based on a large known factor of $p-1$ for all primes $p \\divides N$.\n\nAn important consequence of this theorem is that in the context of \\emph{any} deterministic factoring algorithm that runs in exponential time, finding elements of large order should no longer be considered a bottleneck, regardless of the exponent.\n\nAt the heart of all our algorithms is a minor variant of Sutherland's well-known method for computing the order of an element of a group.\n\\begin{lem}\n\\label{lem:bsgs}\nGiven as input integers $N \\geq 2$, $D \\geq 1$ and an element $\\alpha \\in \\Z_N^*$, we may determine if $\\ord_N(\\alpha) \\leq D$,\nand if so compute the exact order $m \\geq 1$, in time\n\\[\nO\\left(\\frac{r^{1/2}}{(\\log \\log r)^{1/2}} \\log^2 N\\right),\n\\qquad \\text{where } r \\coloneqq \\min(m,D).\n\\]\n\\end{lem}\n\\begin{proof}\nAccording to \\cite[Alg.\\,4.1]{SutherlandPHD}, we may search for the order of $\\alpha$ up to a given $T \\geq 1$ in time $O(T^{1/2}(\\log\\log T)^{-1/2} \\log^2 N)$. (See \\cite[Thm.\\,6.1]{Hit-BSGS} for more details of the complexity analysis of this algorithm in the Turing model.)\nWe apply this result successively for $T=1,2,4,\\ldots, 2^{\\lceil\\log_2 D\\rceil}$, terminating immediately if we find $m$. If $m$ is not found, we must have $m > 2^{\\lceil\\log_2 D\\rceil} \\geq D$, and we return ``$\\ord_N(\\alpha) > D$''. The overall time complexity bound follows from the estimate\n\\begin{align*}\n\\sum_{i=0}^{\\lceil \\log_2 r\\rceil} 2^{i/2}(\\log\\log(2^i))^{-1/2} & \\ll \\sum_{i=0}^{\\lceil (\\log_2 r)/2\\rceil}2^{i/2}+\\sum_{i=\\lceil (\\log_2 r)/2\\rceil+1}^{\\lceil \\log_2 r\\rceil}2^{i/2}(\\log i)^{-1/2} \\\\\n& \\ll r^{1/4}+ r^{1/2}(\\log\\log r)^{-1/2}. \\qedhere\n\\end{align*}\n\\end{proof}\n\n\\begin{lem}\n\\label{lem:compord}\nLet $N \\geq 2$.\nGiven as input elements $\\alpha, \\beta \\in \\Z_N^*$,\nand their orders $u \\coloneqq \\ord_N(\\alpha)$ and $v \\coloneqq \\ord_N(\\beta)$,\ntogether with the prime factorisations of $u$ and~$v$,\nwe may compute an element $\\gamma \\in \\Z_N^*$ of order $w \\coloneqq \\lcm(u,v)$,\ntogether with the prime factorisation of $w$,\nin time\n\\[\nO((\\log u + \\log v) \\log N \\log \\log N).\n\\]\n\\end{lem}\n\nTo prove \\eqref{eq:tildeZ-interval}, let\n\\begin{equation}\n\\label{eq:z-formula}\nz \\coloneqq \\log \\tilde Z = \\left(1 + \\frac{\\log \\log 2M}{-1 + \\log B}\\right) \\log 2M.\n\\end{equation}\nSince $M \\leq D$ and $\\log B \\asymp \\log D$ we have\n\\begin{equation}\n\\label{eq:loglogM-logB-bound}\n\\frac{\\log \\log 2M}{-1 + \\log B} \\ll \\frac{\\log \\log D}{\\log D}.\n\\end{equation}\nSo by using the estimate $\\log(1+x) = x + O(x^2)$, for sufficiently large $D$ we obtain\n\\begin{align*}\n\\log z\n& = \\frac{\\log \\log 2M}{-1 + \\log B} + \\log \\log 2M + O\\left(\\frac{(\\log \\log D)^2}{\\log^2 D}\\right) \\\\\n& = \\frac{\\log B}{-1 + \\log B} \\cdot \\log \\log 2M + O\\left(\\frac{(\\log \\log D)^2}{\\log^2 D}\\right).\n\\end{align*}\nThus\n\\[\n1 - \\frac{\\log z}{\\log B} = 1 - \\frac{\\log \\log 2M}{-1 + \\log B} + O\\left(\\frac{(\\log \\log D)^2}{\\log^3 D}\\right),\n\\]\nand it follows that\n\\begin{align*}\nz \\left(1 - \\frac{\\log z}{\\log B}\\right)\n& = \\left(1 + \\frac{\\log \\log 2M}{-1 + \\log B}\\right) \\left(1 - \\frac{\\log \\log 2M}{-1 + \\log B} + O\\left(\\frac{(\\log \\log D)^2}{\\log^3 D}\\right)\\right) \\log 2M \\\\\n& = \\left(1 - \\frac{(\\log \\log 2M)^2}{(-1 + \\log B)^2} + O\\left(\\frac{(\\log \\log D)^2}{\\log^3 D}\\right)\\right) \\log 2M \\\\\n& = \\left(1 + O\\left(\\frac{(\\log \\log D)^2}{\\log^2 D}\\right)\\right) \\log 2M \\\\\n& = \\log 2M + O\\left(\\frac{(\\log \\log D)^2}{\\log D}\\right).\n\\end{align*}\nFor sufficiently large $D$, this implies that\n\\[\n\\log M < z \\left(1 - \\frac{\\log z}{\\log B}\\right) < \\log 4M,\n\\]\nwhich is equivalent to \\eqref{eq:tildeZ-interval}.\nThis completes the proof of \\eqref{eq:p-bound}.\n\nWe next consider the complexity of an iteration that reaches the search for $m$ in line \\ref{line:m-search}.\nIf $m>D$, \\Cref{lem:bsgs} implies that the algorithm terminates in line \\ref{line:m-large} within the cost given by \\eqref{eq:main-complexity}.\nWe hence assume that $m \\leq D$. The complexity of line \\ref{line:m-search} is then\n\\begin{equation} \n\\label{eq:iter}\nO\\left(\\frac{m^{1/2}}{(\\log \\log m)^{1/2}} \\log^2 N\\right).\n\\end{equation}\nIn line \\ref{line:m-factor}, we find all divisors of $m$ via trial division in time $O(m^{1/2}\\log N \\log\\log N)$, which is negligible compared to \\eqref{eq:iter}.\nIn line \\ref{line:gcds}, the cost of computing each power $\\beta^{m/r} \\pmod N$ is $O(\\log m \\log N \\log \\log N)$, and each GCD requires time $O(\\log N \\, (\\log \\log N)^2)$.\nThe number of primes $r \\divides m$ is $O(\\log m)$, so the cost of line~\\ref{line:gcds} is at most $O(\\log^2 m \\log N \\, (\\log \\log N)^2)$, which is negligible compared to \\eqref{eq:iter}.\nAccording to \\Cref{lem:compord}, line \\ref{line:lcm} runs in time $O((\\log m +\\log M) \\log N \\log \\log N)$.\nThe first summand is negligible compared to \\eqref{eq:iter}.\nThe combined cost of the second summand over all iterations is $O(B \\log M \\log N \\log \\log N)$, which as we saw earlier is negligible.\nSo we may focus on \\eqref{eq:iter} as an upper bound for the complexity of any iteration of the for-loop that reaches line~\\ref{line:m-search}. We are left with the task of estimating the sum of \\eqref{eq:iter} over all such iterations.", "post_theorem_intro_text_len": 3131, "post_theorem_intro_text": "Note that the expression $\\log \\log D$ does not literally make sense for $D \\leqslant 2$; in cases like this, the reader should mentally substitute $\\log \\log \\max(D, 3)$.\nIt can be shown that the space complexity in \\Cref{thm:main} is given by\n\\[\nO\\left(\\frac{D^{1/2}}{(\\log \\log D)^{1/2}} \\log N\\right),\n\\]\nbut we will not carry out the details of this analysis.\n\nAn important consequence of this theorem is that in the context of \\emph{any} deterministic factoring algorithm that runs in exponential time, finding elements of large order should no longer be considered a bottleneck, regardless of the exponent.\n\nAn interesting feature of this algorithm is that, unlike its predecessors, it never returns ``$N$ is prime''. In particular, if $N$ is prime, it \\emph{always} returns $\\alpha \\in \\Z_N^*$ with $\\ord_N(\\alpha) > D$. This is interesting from a theoretical point of view, as it permits rigorously finding not only primitive roots but also elements of order exceeding $D$, even for small values of~$D$. (When $D$ is sufficiently large, namely for $D \\geqslant N^{1/2+\\varepsilon}$, it was already known how to find elements of order at least $D$ in time $O(D^{1/2+\\varepsilon})$, as one can find a primitive root in time $O(N^{1/4+\\varepsilon})$ \\cite{Shp-primitive}.) It is also worth mentioning that when $D$ is very small, the runtime may be even less than the cost of testing $N$ for primality via a deterministic polynomial time primality test, such as the AKS test \\cite{AKS-primes}.\n\nThe main idea behind the improved algorithm is based on estimates for the density of integers that factor completely into primes $q \\leqslant B$ for a given bound $B$, the so-called \\emph{$B$-smooth numbers}.\nThe algorithm has a broadly similar structure to its predecessors.\nIn the first stage, we attempt to compute the order of all elements $\\beta = 2,3,\\ldots, B$, for a suitable choice of $B$.\nIf we have not yet found an element of large order,\nor a factor of $N$ via GCDs as before,\nthen we have computed some $M \\leqslant D$ such that all these elements satisfy the congruence $\\beta^M \\equiv 1 \\pmod p$ for every prime factor $p$ of $N$, and such that $M \\mathrel{|} p-1$ for every such~$p$.\nThe key observation is that not only these $\\beta$, but \\emph{all $B$-smooth integers} in the interval $1 \\leqslant n \\leqslant p$ satisfy $n^M \\equiv 1 \\pmod p$.\nBut the latter congruence has at most $M$ solutions modulo $p$,\nso by applying well-known lower bounds for the density of $B$-smooth numbers, we obtain drastically improved estimates for how fast $M$ grows as we progress through the main loop over $\\beta$. This strategy incidentally leads to a considerable simplification in the second stage of the algorithm: it suffices to use trial division to check for divisors along the arithmetic progression $kM+1$ instead of applying more elaborate techniques.\n\nThe rest of the paper is structured as follows. \\Cref{sec:prelim} discusses preliminaries such as the babystep-giantstep method for finding orders of elements, and bounds for the density of $B$-smooth numbers. Then the main theorem is proved in \\Cref{sec:main}.", "sketch": "To prove Theorem~\\ref{thm:main}, the algorithm uses “estimates for the density of integers that factor completely into primes $q \\leqslant B$,” i.e. “$B$-smooth numbers.” It has “a broadly similar structure to its predecessors.”\n\n- **Stage 1 (order computations for small bases):** “In the first stage, we attempt to compute the order of all elements $\\beta = 2,3,\\ldots, B$, for a suitable choice of $B$.” If this does not already yield “an element of large order, or a factor of $N$ via GCDs as before,” then one has computed “some $M \\leqslant D$” such that for every prime factor $p$ of $N$ all these $\\beta$ satisfy $\\beta^M \\equiv 1 \\pmod p$, and moreover “$M \\mathrel{|} p-1$ for every such $p$.”\n\n- **Key observation via smooth numbers:** “Not only these $\\beta$, but \\emph{all $B$-smooth integers} in the interval $1 \\leqslant n \\leqslant p$ satisfy $n^M \\equiv 1 \\pmod p$.” Since “the latter congruence has at most $M$ solutions modulo $p$,” applying “well-known lower bounds for the density of $B$-smooth numbers” gives “drastically improved estimates for how fast $M$ grows as we progress through the main loop over $\\beta$.”\n\n- **Stage 2 (finding factors):** This density-based strategy “leads to a considerable simplification in the second stage of the algorithm: it suffices to use trial division to check for divisors along the arithmetic progression $kM+1$ instead of applying more elaborate techniques.”", "expanded_sketch": "To prove the main theorem, the algorithm uses “estimates for the density of integers that factor completely into primes $q \\leqslant B$,” i.e. “$B$-smooth numbers.” It has “a broadly similar structure to its predecessors.”\n\n- **Stage 1 (order computations for small bases):** “In the first stage, we attempt to compute the order of all elements $\\beta = 2,3,\\ldots, B$, for a suitable choice of $B$.” If this does not already yield “an element of large order, or a factor of $N$ via GCDs as before,” then one has computed “some $M \\leqslant D$” such that for every prime factor $p$ of $N$ all these $\\beta$ satisfy $\\beta^M \\equiv 1 \\pmod p$, and moreover “$M \\mathrel{|} p-1$ for every such $p$.”\n\n- **Key observation via smooth numbers:** “Not only these $\\beta$, but \\emph{all $B$-smooth integers} in the interval $1 \\leqslant n \\leqslant p$ satisfy $n^M \\equiv 1 \\pmod p$.” Since “the latter congruence has at most $M$ solutions modulo $p$,” applying “well-known lower bounds for the density of $B$-smooth numbers” gives “drastically improved estimates for how fast $M$ grows as we progress through the main loop over $\\beta$.”\n\n- **Stage 2 (finding factors):** This density-based strategy “leads to a considerable simplification in the second stage of the algorithm: it suffices to use trial division to check for divisors along the arithmetic progression $kM+1$ instead of applying more elaborate techniques.”", "expanded_theorem": "\\label{thm:main}\nThere exists an algorithm with the following properties. \nIt takes as input integers $N \\geqslant 3$ and $D \\geqslant 1$ with $D < N-1$.\nIt outputs either some $\\alpha\\in\\Z_N^*$ with $\\ord_N(\\alpha) > D$\nor a nontrivial divisor of $N$.\nIts time complexity is\n\\begin{equation}\n\\label{eq:main-complexity}\nO\\left(\\frac{D^{1/2}}{(\\log \\log D)^{1/2}} \\log^2 N\\right).\n\\end{equation}.", "theorem_type": ["Algorithmic or Constructive", "Existence"], "mcq": {"question": "Let \\(N\\ge 3\\) and \\(D\\ge 1\\) be integers with \\(DD\\) or a nontrivial divisor of \\(N\\), and its time complexity is \\(O\\!\\left(\\dfrac{D^{1/2}}{(\\log\\log D)^{1/2}}\\log^2 N\\right)\\)."}, "choices": [{"label": "B", "text": "There exists an algorithm that, given \\(N\\) and \\(D\\), outputs either an element \\(\\alpha\\in \\mathbb Z_N^*\\) with \\(\\operatorname{ord}_N(\\alpha)\\ge D\\) or a nontrivial divisor of \\(N\\), and its time complexity is \\(O\\!\\left(\\dfrac{D^{1/2}}{(\\log\\log D)^{1/2}}\\log^2 N\\right)\\)."}, {"label": "C", "text": "There exists an algorithm that, given \\(N\\) and \\(D\\), outputs either an element \\(\\alpha\\in \\mathbb Z_N^*\\) with \\(\\operatorname{ord}_N(\\alpha)>D\\) or a nontrivial divisor of \\(N\\)."}, {"label": "D", "text": "There exists an algorithm that, given \\(N\\) and \\(D\\), outputs either an element \\(\\alpha\\in \\mathbb Z_N^*\\) with \\(\\operatorname{ord}_N(\\alpha)>D\\) or a nontrivial divisor of \\(N\\), and its time complexity is \\(O\\!\\left(\\dfrac{D^{1/2}}{(\\log\\log D)^{1/2}}\\log N\\right)\\)."}, {"label": "E", "text": "There exists an algorithm that, given \\(N\\) and \\(D\\), outputs either an element \\(\\alpha\\in \\mathbb Z_N^*\\) with \\(\\operatorname{ord}_N(\\alpha)>D\\), or a nontrivial divisor of \\(N\\), or a proof that every \\(\\beta\\in \\mathbb Z_N^*\\) satisfies \\(\\operatorname{ord}_N(\\beta)\\le D\\), and its time complexity is \\(O\\!\\left(\\dfrac{D^{1/2}}{(\\log\\log D)^{1/2}}\\log^2 N\\right)\\)."}], "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": "strict_order_threshold_>D", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "explicit_complexity_bound_removed", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "computational_check", "tampered_component": "logarithmic_cost_from_order_search", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "smoothness", "tampered_component": "extra_global_certification_outcome", "template_used": "wildcard"}]}} +{"id": "2601.11131v1", "paper_link": "http://arxiv.org/abs/2601.11131v1", "theorems_cnt": 1, "theorem": {"env_name": "thm", "content": "\\label{thm:main}\nThere exists an algorithm with the following properties. \nIt takes as input integers $N \\geqslant 3$ and $D \\geqslant 1$ with $D < N-1$.\nIt outputs either some $\\alpha\\in\\Z_N^*$ with $\\ord_N(\\alpha) > D$\nor a nontrivial divisor of $N$.\nIts time complexity is\n\\begin{equation}\n\\label{eq:main-complexity}\nO\\left(\\frac{D^{1/2}}{(\\log \\log D)^{1/2}} \\log^2 N\\right).\n\\end{equation}", "start_pos": 9745, "end_pos": 10151, "label": "thm:main"}, "ref_dict": {"thm:main": "\\begin{thm}\n\\label{thm:main}\nThere exists an algorithm with the following properties. \nIt takes as input integers $N \\geq 3$ and $D \\geq 1$ with $D < N-1$.\nIt outputs either some $\\alpha\\in\\Z_N^*$ with $\\ord_N(\\alpha) > D$\nor a nontrivial divisor of $N$.\nIts time complexity is\n\\begin{equation}\n\\label{eq:main-complexity}\nO\\left(\\frac{D^{1/2}}{(\\log \\log D)^{1/2}} \\log^2 N\\right).\n\\end{equation}\n\\end{thm}"}, "pre_theorem_intro_text_len": 5670, "pre_theorem_intro_text": "\\label{sec:intro}\n\nLet $N \\geqslant 2$ be an integer and consider the multiplicative group $\\Z_N^*$ of invertible residues modulo $N$. The order of an element $\\alpha \\in \\Z_N^*$, denoted $\\ord_N(\\alpha)$, is defined to be the smallest positive integer $k$ such that $\\alpha^k = 1$. The problem of finding elements of large or maximal order in $\\Z_N^*$ occurs in various theoretical and applied contexts. Examples include the generation of pseudorandom numbers \\cite{BM-prng} and the discrete logarithm problem modulo a prime $N$, which is the basis of widely used cryptographic protocols \\cite{DH-dlp, E-elgamal}. Given $g, h \\in \\Z_N^*$, the task is to find an integer $x$ such that $g^x = h$, if such a solution exists. To maximise the difficulty of this problem, so as to increase the security of the resulting cryptographic scheme, we should choose $g$ to be a primitive root, i.e., an element of maximal order $\\ord_N(g)=N-1$. If the factorisation of $N-1$ is known, such an element can be found with high probability by testing a few randomly chosen candidates. However, the best known rigorous and unconditional upper bound \\cite{B-leastprimitiveroot} for the smallest primitive root modulo $N$ is still exponential in $\\log N$. So while it is easy to find an element of large order in $\\Z_N^*$ in randomised polynomial time (still assuming that the factorisation of $N-1$ is known), the same problem is much more difficult to solve deterministically and rigorously. \n\nAnother problem in which finding elements of large multiplicative order plays an important role is deterministic integer factorisation. Until recently, the best complexity bounds for rigorously factoring an integer $N$ were of the form $N^{1/4+o(1)}$, based on methods going back to Pollard \\cite{Pol-factoring}, Strassen \\cite{Str-factoring} and Coppersmith \\cite{Cop-lowexponent}. (Here, and for the rest of the paper, time complexity always refers to the number of steps performed by a multitape Turing machine. For a detailed description of this computational model, see \\cite[Ch.\\,2]{Pap-complexity} or \\cite[\\S1.6]{AHU-algorithms}.) The first improvement on these bounds by a superpolynomial factor (i.e., larger than any fixed power of $\\log N$) appeared in \\cite{Hit-BSGS}, and was based on the idea of solving the discrete logarithm problem $\\alpha^x = \\alpha^{N+1}$ in $\\Z_N^*$ for a given element $\\alpha$. If $N$ is a product of two distinct primes, say $N=pq$, then the solution $x=p+q$ immediately reveals the two factors of $N$. However, we can only find this solution if $\\ord_N(\\alpha)$ is sufficiently large. Therefore, the paper also included an algorithm that, for $D\\geqslant N^{2/5}$, either finds an element of order at least $D$, or a nontrivial divisor of $N$, or proves $N$ prime \\cite[Alg.\\,6.2]{Hit-BSGS}.\n\nThe algorithm has two stages. In the first stage, we search for the order of small elements $\\beta=2,3,\\ldots$ via Sutherland's optimisation of the classical babystep-giantstep method \\cite{Sha-classnumber,SutherlandPHD}, assuming that the order is at most $D$. If for some element we fail to compute the order, then the order exceeds $D$ and we are done. If the order $m \\coloneqq \\ord_N(\\beta)$ is actually found, we try to use it to compute a nontrivial divisor of $N$, by examining $\\gcd(\\beta^{m/r} - 1, N)$ for prime divisors $r$ of $m$. If that fails, we deduce that $m$ divides $p-1$ for every prime factor $p$ of $N$. As we compute the exact orders of these small elements, and continue to fail to factorise $N$, we accumulate considerable information about the prime factors of $N$. Eventually, we recover sufficient information to proceed to the second stage of the algorithm, in which we attempt to factorise $N$ directly. If that fails too, we obtain a proof that $N$ is prime. It is noteworthy that the time complexity of this procedure is asymptotically the same as testing whether a single element has order at most $D$, namely\n\\[\nO\\left(\\frac{D^{1/2}}{(\\log \\log D)^{1/2}} \\log^2 N\\right).\n\\]\n\nThe 2018 algorithm for finding elements of large order was later used as a subroutine in a series of works \\cite{Hit-timespace,Har-onefifth,HH-onefifthloglog} that improved the runtime of deterministic factorisation to $N^{1/5+o(1)}$. The original assumption $D \\geqslant N^{2/5}$ was good enough to directly apply the original algorithm in these factorisation methods, but only just. The hypothesis on $D$ was subsequently improved on two occasions, first to $D \\geqslant N^{1/4+o(1)}$ by Gao, Feng, Hu and Pan \\cite[Lem.\\,3.5]{GFHP-factoringlattices} and then to $D \\geqslant N^{1/6}$ by Oznovich and Volk \\cite[Thm.\\,1.1]{OV-highorder}. These improvements are based on a better bound for the number of consecutive elements with the same order $m$ (namely, $O(\\sqrt{m})$ instead of $m$), leading to a faster progression in the loop over the elements $\\beta = 2,3,\\ldots$ in the first stage of the algorithm. Both of these papers (and additionally \\cite{HH-lehmangeneralization}) also considered the case of $N$ having an $r$-power divisor, which allows the assumption to be relaxed further to $D \\geqslant N^{1/6r}$ \\cite[Thm.\\,4.2]{OV-highorder}. Furthermore, both \\cite{GFHP-factoringlattices} and \\cite{OV-highorder} present improved techniques for the second stage, in which $N$ is factorised based on a large known factor of $p-1$ for all primes $p \\mathrel{|} N$. \n\nAll of the results mentioned so far impose a fairly restrictive (in fact, fully exponential) lower bound hypothesis on $D$. The main result of this paper is an algorithm that solves the same problem within the same amount of time, but without any restriction on $D$ at all.", "context": "\\label{sec:intro}\n\nLet $N \\geqslant 2$ be an integer and consider the multiplicative group $\\Z_N^*$ of invertible residues modulo $N$. The order of an element $\\alpha \\in \\Z_N^*$, denoted $\\ord_N(\\alpha)$, is defined to be the smallest positive integer $k$ such that $\\alpha^k = 1$. The problem of finding elements of large or maximal order in $\\Z_N^*$ occurs in various theoretical and applied contexts. Examples include the generation of pseudorandom numbers \\cite{BM-prng} and the discrete logarithm problem modulo a prime $N$, which is the basis of widely used cryptographic protocols \\cite{DH-dlp, E-elgamal}. Given $g, h \\in \\Z_N^*$, the task is to find an integer $x$ such that $g^x = h$, if such a solution exists. To maximise the difficulty of this problem, so as to increase the security of the resulting cryptographic scheme, we should choose $g$ to be a primitive root, i.e., an element of maximal order $\\ord_N(g)=N-1$. If the factorisation of $N-1$ is known, such an element can be found with high probability by testing a few randomly chosen candidates. However, the best known rigorous and unconditional upper bound \\cite{B-leastprimitiveroot} for the smallest primitive root modulo $N$ is still exponential in $\\log N$. So while it is easy to find an element of large order in $\\Z_N^*$ in randomised polynomial time (still assuming that the factorisation of $N-1$ is known), the same problem is much more difficult to solve deterministically and rigorously.\n\nAnother problem in which finding elements of large multiplicative order plays an important role is deterministic integer factorisation. Until recently, the best complexity bounds for rigorously factoring an integer $N$ were of the form $N^{1/4+o(1)}$, based on methods going back to Pollard \\cite{Pol-factoring}, Strassen \\cite{Str-factoring} and Coppersmith \\cite{Cop-lowexponent}. (Here, and for the rest of the paper, time complexity always refers to the number of steps performed by a multitape Turing machine. For a detailed description of this computational model, see \\cite[Ch.\\,2]{Pap-complexity} or \\cite[\\S1.6]{AHU-algorithms}.) The first improvement on these bounds by a superpolynomial factor (i.e., larger than any fixed power of $\\log N$) appeared in \\cite{Hit-BSGS}, and was based on the idea of solving the discrete logarithm problem $\\alpha^x = \\alpha^{N+1}$ in $\\Z_N^*$ for a given element $\\alpha$. If $N$ is a product of two distinct primes, say $N=pq$, then the solution $x=p+q$ immediately reveals the two factors of $N$. However, we can only find this solution if $\\ord_N(\\alpha)$ is sufficiently large. Therefore, the paper also included an algorithm that, for $D\\geqslant N^{2/5}$, either finds an element of order at least $D$, or a nontrivial divisor of $N$, or proves $N$ prime \\cite[Alg.\\,6.2]{Hit-BSGS}.\n\nThe algorithm has two stages. In the first stage, we search for the order of small elements $\\beta=2,3,\\ldots$ via Sutherland's optimisation of the classical babystep-giantstep method \\cite{Sha-classnumber,SutherlandPHD}, assuming that the order is at most $D$. If for some element we fail to compute the order, then the order exceeds $D$ and we are done. If the order $m \\coloneqq \\ord_N(\\beta)$ is actually found, we try to use it to compute a nontrivial divisor of $N$, by examining $\\gcd(\\beta^{m/r} - 1, N)$ for prime divisors $r$ of $m$. If that fails, we deduce that $m$ divides $p-1$ for every prime factor $p$ of $N$. As we compute the exact orders of these small elements, and continue to fail to factorise $N$, we accumulate considerable information about the prime factors of $N$. Eventually, we recover sufficient information to proceed to the second stage of the algorithm, in which we attempt to factorise $N$ directly. If that fails too, we obtain a proof that $N$ is prime. It is noteworthy that the time complexity of this procedure is asymptotically the same as testing whether a single element has order at most $D$, namely\n\\[\nO\\left(\\frac{D^{1/2}}{(\\log \\log D)^{1/2}} \\log^2 N\\right).\n\\]\n\nThe 2018 algorithm for finding elements of large order was later used as a subroutine in a series of works \\cite{Hit-timespace,Har-onefifth,HH-onefifthloglog} that improved the runtime of deterministic factorisation to $N^{1/5+o(1)}$. The original assumption $D \\geqslant N^{2/5}$ was good enough to directly apply the original algorithm in these factorisation methods, but only just. The hypothesis on $D$ was subsequently improved on two occasions, first to $D \\geqslant N^{1/4+o(1)}$ by Gao, Feng, Hu and Pan \\cite[Lem.\\,3.5]{GFHP-factoringlattices} and then to $D \\geqslant N^{1/6}$ by Oznovich and Volk \\cite[Thm.\\,1.1]{OV-highorder}. These improvements are based on a better bound for the number of consecutive elements with the same order $m$ (namely, $O(\\sqrt{m})$ instead of $m$), leading to a faster progression in the loop over the elements $\\beta = 2,3,\\ldots$ in the first stage of the algorithm. Both of these papers (and additionally \\cite{HH-lehmangeneralization}) also considered the case of $N$ having an $r$-power divisor, which allows the assumption to be relaxed further to $D \\geqslant N^{1/6r}$ \\cite[Thm.\\,4.2]{OV-highorder}. Furthermore, both \\cite{GFHP-factoringlattices} and \\cite{OV-highorder} present improved techniques for the second stage, in which $N$ is factorised based on a large known factor of $p-1$ for all primes $p \\mathrel{|} N$.\n\nAll of the results mentioned so far impose a fairly restrictive (in fact, fully exponential) lower bound hypothesis on $D$. The main result of this paper is an algorithm that solves the same problem within the same amount of time, but without any restriction on $D$ at all.", "full_context": "\\label{sec:intro}\n\nLet $N \\geqslant 2$ be an integer and consider the multiplicative group $\\Z_N^*$ of invertible residues modulo $N$. The order of an element $\\alpha \\in \\Z_N^*$, denoted $\\ord_N(\\alpha)$, is defined to be the smallest positive integer $k$ such that $\\alpha^k = 1$. The problem of finding elements of large or maximal order in $\\Z_N^*$ occurs in various theoretical and applied contexts. Examples include the generation of pseudorandom numbers \\cite{BM-prng} and the discrete logarithm problem modulo a prime $N$, which is the basis of widely used cryptographic protocols \\cite{DH-dlp, E-elgamal}. Given $g, h \\in \\Z_N^*$, the task is to find an integer $x$ such that $g^x = h$, if such a solution exists. To maximise the difficulty of this problem, so as to increase the security of the resulting cryptographic scheme, we should choose $g$ to be a primitive root, i.e., an element of maximal order $\\ord_N(g)=N-1$. If the factorisation of $N-1$ is known, such an element can be found with high probability by testing a few randomly chosen candidates. However, the best known rigorous and unconditional upper bound \\cite{B-leastprimitiveroot} for the smallest primitive root modulo $N$ is still exponential in $\\log N$. So while it is easy to find an element of large order in $\\Z_N^*$ in randomised polynomial time (still assuming that the factorisation of $N-1$ is known), the same problem is much more difficult to solve deterministically and rigorously.\n\nAnother problem in which finding elements of large multiplicative order plays an important role is deterministic integer factorisation. Until recently, the best complexity bounds for rigorously factoring an integer $N$ were of the form $N^{1/4+o(1)}$, based on methods going back to Pollard \\cite{Pol-factoring}, Strassen \\cite{Str-factoring} and Coppersmith \\cite{Cop-lowexponent}. (Here, and for the rest of the paper, time complexity always refers to the number of steps performed by a multitape Turing machine. For a detailed description of this computational model, see \\cite[Ch.\\,2]{Pap-complexity} or \\cite[\\S1.6]{AHU-algorithms}.) The first improvement on these bounds by a superpolynomial factor (i.e., larger than any fixed power of $\\log N$) appeared in \\cite{Hit-BSGS}, and was based on the idea of solving the discrete logarithm problem $\\alpha^x = \\alpha^{N+1}$ in $\\Z_N^*$ for a given element $\\alpha$. If $N$ is a product of two distinct primes, say $N=pq$, then the solution $x=p+q$ immediately reveals the two factors of $N$. However, we can only find this solution if $\\ord_N(\\alpha)$ is sufficiently large. Therefore, the paper also included an algorithm that, for $D\\geqslant N^{2/5}$, either finds an element of order at least $D$, or a nontrivial divisor of $N$, or proves $N$ prime \\cite[Alg.\\,6.2]{Hit-BSGS}.\n\nThe algorithm has two stages. In the first stage, we search for the order of small elements $\\beta=2,3,\\ldots$ via Sutherland's optimisation of the classical babystep-giantstep method \\cite{Sha-classnumber,SutherlandPHD}, assuming that the order is at most $D$. If for some element we fail to compute the order, then the order exceeds $D$ and we are done. If the order $m \\coloneqq \\ord_N(\\beta)$ is actually found, we try to use it to compute a nontrivial divisor of $N$, by examining $\\gcd(\\beta^{m/r} - 1, N)$ for prime divisors $r$ of $m$. If that fails, we deduce that $m$ divides $p-1$ for every prime factor $p$ of $N$. As we compute the exact orders of these small elements, and continue to fail to factorise $N$, we accumulate considerable information about the prime factors of $N$. Eventually, we recover sufficient information to proceed to the second stage of the algorithm, in which we attempt to factorise $N$ directly. If that fails too, we obtain a proof that $N$ is prime. It is noteworthy that the time complexity of this procedure is asymptotically the same as testing whether a single element has order at most $D$, namely\n\\[\nO\\left(\\frac{D^{1/2}}{(\\log \\log D)^{1/2}} \\log^2 N\\right).\n\\]\n\nThe 2018 algorithm for finding elements of large order was later used as a subroutine in a series of works \\cite{Hit-timespace,Har-onefifth,HH-onefifthloglog} that improved the runtime of deterministic factorisation to $N^{1/5+o(1)}$. The original assumption $D \\geqslant N^{2/5}$ was good enough to directly apply the original algorithm in these factorisation methods, but only just. The hypothesis on $D$ was subsequently improved on two occasions, first to $D \\geqslant N^{1/4+o(1)}$ by Gao, Feng, Hu and Pan \\cite[Lem.\\,3.5]{GFHP-factoringlattices} and then to $D \\geqslant N^{1/6}$ by Oznovich and Volk \\cite[Thm.\\,1.1]{OV-highorder}. These improvements are based on a better bound for the number of consecutive elements with the same order $m$ (namely, $O(\\sqrt{m})$ instead of $m$), leading to a faster progression in the loop over the elements $\\beta = 2,3,\\ldots$ in the first stage of the algorithm. Both of these papers (and additionally \\cite{HH-lehmangeneralization}) also considered the case of $N$ having an $r$-power divisor, which allows the assumption to be relaxed further to $D \\geqslant N^{1/6r}$ \\cite[Thm.\\,4.2]{OV-highorder}. Furthermore, both \\cite{GFHP-factoringlattices} and \\cite{OV-highorder} present improved techniques for the second stage, in which $N$ is factorised based on a large known factor of $p-1$ for all primes $p \\mathrel{|} N$.\n\nAll of the results mentioned so far impose a fairly restrictive (in fact, fully exponential) lower bound hypothesis on $D$. The main result of this paper is an algorithm that solves the same problem within the same amount of time, but without any restriction on $D$ at all.\n\nThe algorithm has two stages. In the first stage, we search for the order of small elements $\\beta=2,3,\\ldots$ via Sutherland's optimisation of the classical babystep-giantstep method \\cite{Sha-classnumber,SutherlandPHD}, assuming that the order is at most $D$. If for some element we fail to compute the order, then the order exceeds $D$ and we are done. If the order $m \\coloneqq \\ord_N(\\beta)$ is actually found, we try to use it to compute a nontrivial divisor of $N$, by examining $\\gcd(\\beta^{m/r} - 1, N)$ for prime divisors $r$ of $m$. If that fails, we deduce that $m$ divides $p-1$ for every prime factor $p$ of $N$. As we compute the exact orders of these small elements, and continue to fail to factorise $N$, we accumulate considerable information about the prime factors of $N$. Eventually, we recover sufficient information to proceed to the second stage of the algorithm, in which we attempt to factorise $N$ directly. If that fails too, we obtain a proof that $N$ is prime. It is noteworthy that the time complexity of this procedure is asymptotically the same as testing whether a single element has order at most $D$, namely\n\\[\nO\\left(\\frac{D^{1/2}}{(\\log \\log D)^{1/2}} \\log^2 N\\right).\n\\]\n\nThe 2018 algorithm for finding elements of large order was later used as a subroutine in a series of works \\cite{Hit-timespace,Har-onefifth,HH-onefifthloglog} that improved the runtime of deterministic factorisation to $N^{1/5+o(1)}$. The original assumption $D \\geq N^{2/5}$ was good enough to directly apply the original algorithm in these factorisation methods, but only just. The hypothesis on $D$ was subsequently improved on two occasions, first to $D \\geq N^{1/4+o(1)}$ by Gao, Feng, Hu and Pan \\cite[Lem.\\,3.5]{GFHP-factoringlattices} and then to $D \\geq N^{1/6}$ by Oznovich and Volk \\cite[Thm.\\,1.1]{OV-highorder}. These improvements are based on a better bound for the number of consecutive elements with the same order $m$ (namely, $O(\\sqrt{m})$ instead of $m$), leading to a faster progression in the loop over the elements $\\beta = 2,3,\\ldots$ in the first stage of the algorithm. Both of these papers (and additionally \\cite{HH-lehmangeneralization}) also considered the case of $N$ having an $r$-power divisor, which allows the assumption to be relaxed further to $D \\geq N^{1/6r}$ \\cite[Thm.\\,4.2]{OV-highorder}. Furthermore, both \\cite{GFHP-factoringlattices} and \\cite{OV-highorder} present improved techniques for the second stage, in which $N$ is factorised based on a large known factor of $p-1$ for all primes $p \\divides N$.\n\nAn important consequence of this theorem is that in the context of \\emph{any} deterministic factoring algorithm that runs in exponential time, finding elements of large order should no longer be considered a bottleneck, regardless of the exponent.\n\nAt the heart of all our algorithms is a minor variant of Sutherland's well-known method for computing the order of an element of a group.\n\\begin{lem}\n\\label{lem:bsgs}\nGiven as input integers $N \\geq 2$, $D \\geq 1$ and an element $\\alpha \\in \\Z_N^*$, we may determine if $\\ord_N(\\alpha) \\leq D$,\nand if so compute the exact order $m \\geq 1$, in time\n\\[\nO\\left(\\frac{r^{1/2}}{(\\log \\log r)^{1/2}} \\log^2 N\\right),\n\\qquad \\text{where } r \\coloneqq \\min(m,D).\n\\]\n\\end{lem}\n\\begin{proof}\nAccording to \\cite[Alg.\\,4.1]{SutherlandPHD}, we may search for the order of $\\alpha$ up to a given $T \\geq 1$ in time $O(T^{1/2}(\\log\\log T)^{-1/2} \\log^2 N)$. (See \\cite[Thm.\\,6.1]{Hit-BSGS} for more details of the complexity analysis of this algorithm in the Turing model.)\nWe apply this result successively for $T=1,2,4,\\ldots, 2^{\\lceil\\log_2 D\\rceil}$, terminating immediately if we find $m$. If $m$ is not found, we must have $m > 2^{\\lceil\\log_2 D\\rceil} \\geq D$, and we return ``$\\ord_N(\\alpha) > D$''. The overall time complexity bound follows from the estimate\n\\begin{align*}\n\\sum_{i=0}^{\\lceil \\log_2 r\\rceil} 2^{i/2}(\\log\\log(2^i))^{-1/2} & \\ll \\sum_{i=0}^{\\lceil (\\log_2 r)/2\\rceil}2^{i/2}+\\sum_{i=\\lceil (\\log_2 r)/2\\rceil+1}^{\\lceil \\log_2 r\\rceil}2^{i/2}(\\log i)^{-1/2} \\\\\n& \\ll r^{1/4}+ r^{1/2}(\\log\\log r)^{-1/2}. \\qedhere\n\\end{align*}\n\\end{proof}\n\n\\begin{lem}\n\\label{lem:compord}\nLet $N \\geq 2$.\nGiven as input elements $\\alpha, \\beta \\in \\Z_N^*$,\nand their orders $u \\coloneqq \\ord_N(\\alpha)$ and $v \\coloneqq \\ord_N(\\beta)$,\ntogether with the prime factorisations of $u$ and~$v$,\nwe may compute an element $\\gamma \\in \\Z_N^*$ of order $w \\coloneqq \\lcm(u,v)$,\ntogether with the prime factorisation of $w$,\nin time\n\\[\nO((\\log u + \\log v) \\log N \\log \\log N).\n\\]\n\\end{lem}\n\nTo prove \\eqref{eq:tildeZ-interval}, let\n\\begin{equation}\n\\label{eq:z-formula}\nz \\coloneqq \\log \\tilde Z = \\left(1 + \\frac{\\log \\log 2M}{-1 + \\log B}\\right) \\log 2M.\n\\end{equation}\nSince $M \\leq D$ and $\\log B \\asymp \\log D$ we have\n\\begin{equation}\n\\label{eq:loglogM-logB-bound}\n\\frac{\\log \\log 2M}{-1 + \\log B} \\ll \\frac{\\log \\log D}{\\log D}.\n\\end{equation}\nSo by using the estimate $\\log(1+x) = x + O(x^2)$, for sufficiently large $D$ we obtain\n\\begin{align*}\n\\log z\n& = \\frac{\\log \\log 2M}{-1 + \\log B} + \\log \\log 2M + O\\left(\\frac{(\\log \\log D)^2}{\\log^2 D}\\right) \\\\\n& = \\frac{\\log B}{-1 + \\log B} \\cdot \\log \\log 2M + O\\left(\\frac{(\\log \\log D)^2}{\\log^2 D}\\right).\n\\end{align*}\nThus\n\\[\n1 - \\frac{\\log z}{\\log B} = 1 - \\frac{\\log \\log 2M}{-1 + \\log B} + O\\left(\\frac{(\\log \\log D)^2}{\\log^3 D}\\right),\n\\]\nand it follows that\n\\begin{align*}\nz \\left(1 - \\frac{\\log z}{\\log B}\\right)\n& = \\left(1 + \\frac{\\log \\log 2M}{-1 + \\log B}\\right) \\left(1 - \\frac{\\log \\log 2M}{-1 + \\log B} + O\\left(\\frac{(\\log \\log D)^2}{\\log^3 D}\\right)\\right) \\log 2M \\\\\n& = \\left(1 - \\frac{(\\log \\log 2M)^2}{(-1 + \\log B)^2} + O\\left(\\frac{(\\log \\log D)^2}{\\log^3 D}\\right)\\right) \\log 2M \\\\\n& = \\left(1 + O\\left(\\frac{(\\log \\log D)^2}{\\log^2 D}\\right)\\right) \\log 2M \\\\\n& = \\log 2M + O\\left(\\frac{(\\log \\log D)^2}{\\log D}\\right).\n\\end{align*}\nFor sufficiently large $D$, this implies that\n\\[\n\\log M < z \\left(1 - \\frac{\\log z}{\\log B}\\right) < \\log 4M,\n\\]\nwhich is equivalent to \\eqref{eq:tildeZ-interval}.\nThis completes the proof of \\eqref{eq:p-bound}.\n\nWe next consider the complexity of an iteration that reaches the search for $m$ in line \\ref{line:m-search}.\nIf $m>D$, \\Cref{lem:bsgs} implies that the algorithm terminates in line \\ref{line:m-large} within the cost given by \\eqref{eq:main-complexity}.\nWe hence assume that $m \\leq D$. The complexity of line \\ref{line:m-search} is then\n\\begin{equation} \n\\label{eq:iter}\nO\\left(\\frac{m^{1/2}}{(\\log \\log m)^{1/2}} \\log^2 N\\right).\n\\end{equation}\nIn line \\ref{line:m-factor}, we find all divisors of $m$ via trial division in time $O(m^{1/2}\\log N \\log\\log N)$, which is negligible compared to \\eqref{eq:iter}.\nIn line \\ref{line:gcds}, the cost of computing each power $\\beta^{m/r} \\pmod N$ is $O(\\log m \\log N \\log \\log N)$, and each GCD requires time $O(\\log N \\, (\\log \\log N)^2)$.\nThe number of primes $r \\divides m$ is $O(\\log m)$, so the cost of line~\\ref{line:gcds} is at most $O(\\log^2 m \\log N \\, (\\log \\log N)^2)$, which is negligible compared to \\eqref{eq:iter}.\nAccording to \\Cref{lem:compord}, line \\ref{line:lcm} runs in time $O((\\log m +\\log M) \\log N \\log \\log N)$.\nThe first summand is negligible compared to \\eqref{eq:iter}.\nThe combined cost of the second summand over all iterations is $O(B \\log M \\log N \\log \\log N)$, which as we saw earlier is negligible.\nSo we may focus on \\eqref{eq:iter} as an upper bound for the complexity of any iteration of the for-loop that reaches line~\\ref{line:m-search}. We are left with the task of estimating the sum of \\eqref{eq:iter} over all such iterations.", "post_theorem_intro_text_len": 3131, "post_theorem_intro_text": "Note that the expression $\\log \\log D$ does not literally make sense for $D \\leqslant 2$; in cases like this, the reader should mentally substitute $\\log \\log \\max(D, 3)$.\nIt can be shown that the space complexity in \\Cref{thm:main} is given by\n\\[\nO\\left(\\frac{D^{1/2}}{(\\log \\log D)^{1/2}} \\log N\\right),\n\\]\nbut we will not carry out the details of this analysis.\n\nAn important consequence of this theorem is that in the context of \\emph{any} deterministic factoring algorithm that runs in exponential time, finding elements of large order should no longer be considered a bottleneck, regardless of the exponent.\n\nAn interesting feature of this algorithm is that, unlike its predecessors, it never returns ``$N$ is prime''. In particular, if $N$ is prime, it \\emph{always} returns $\\alpha \\in \\Z_N^*$ with $\\ord_N(\\alpha) > D$. This is interesting from a theoretical point of view, as it permits rigorously finding not only primitive roots but also elements of order exceeding $D$, even for small values of~$D$. (When $D$ is sufficiently large, namely for $D \\geqslant N^{1/2+\\varepsilon}$, it was already known how to find elements of order at least $D$ in time $O(D^{1/2+\\varepsilon})$, as one can find a primitive root in time $O(N^{1/4+\\varepsilon})$ \\cite{Shp-primitive}.) It is also worth mentioning that when $D$ is very small, the runtime may be even less than the cost of testing $N$ for primality via a deterministic polynomial time primality test, such as the AKS test \\cite{AKS-primes}.\n\nThe main idea behind the improved algorithm is based on estimates for the density of integers that factor completely into primes $q \\leqslant B$ for a given bound $B$, the so-called \\emph{$B$-smooth numbers}.\nThe algorithm has a broadly similar structure to its predecessors.\nIn the first stage, we attempt to compute the order of all elements $\\beta = 2,3,\\ldots, B$, for a suitable choice of $B$.\nIf we have not yet found an element of large order,\nor a factor of $N$ via GCDs as before,\nthen we have computed some $M \\leqslant D$ such that all these elements satisfy the congruence $\\beta^M \\equiv 1 \\pmod p$ for every prime factor $p$ of $N$, and such that $M \\mathrel{|} p-1$ for every such~$p$.\nThe key observation is that not only these $\\beta$, but \\emph{all $B$-smooth integers} in the interval $1 \\leqslant n \\leqslant p$ satisfy $n^M \\equiv 1 \\pmod p$.\nBut the latter congruence has at most $M$ solutions modulo $p$,\nso by applying well-known lower bounds for the density of $B$-smooth numbers, we obtain drastically improved estimates for how fast $M$ grows as we progress through the main loop over $\\beta$. This strategy incidentally leads to a considerable simplification in the second stage of the algorithm: it suffices to use trial division to check for divisors along the arithmetic progression $kM+1$ instead of applying more elaborate techniques.\n\nThe rest of the paper is structured as follows. \\Cref{sec:prelim} discusses preliminaries such as the babystep-giantstep method for finding orders of elements, and bounds for the density of $B$-smooth numbers. Then the main theorem is proved in \\Cref{sec:main}.", "sketch": "To prove Theorem~\\ref{thm:main}, the algorithm uses “estimates for the density of integers that factor completely into primes $q \\leqslant B$,” i.e. “$B$-smooth numbers.” It has “a broadly similar structure to its predecessors.”\n\n- **Stage 1 (order computations for small bases):** “In the first stage, we attempt to compute the order of all elements $\\beta = 2,3,\\ldots, B$, for a suitable choice of $B$.” If this does not already yield “an element of large order, or a factor of $N$ via GCDs as before,” then one has computed “some $M \\leqslant D$” such that for every prime factor $p$ of $N$ all these $\\beta$ satisfy $\\beta^M \\equiv 1 \\pmod p$, and moreover “$M \\mathrel{|} p-1$ for every such $p$.”\n\n- **Key observation via smooth numbers:** “Not only these $\\beta$, but \\emph{all $B$-smooth integers} in the interval $1 \\leqslant n \\leqslant p$ satisfy $n^M \\equiv 1 \\pmod p$.” Since “the latter congruence has at most $M$ solutions modulo $p$,” applying “well-known lower bounds for the density of $B$-smooth numbers” gives “drastically improved estimates for how fast $M$ grows as we progress through the main loop over $\\beta$.”\n\n- **Stage 2 (finding factors):** This density-based strategy “leads to a considerable simplification in the second stage of the algorithm: it suffices to use trial division to check for divisors along the arithmetic progression $kM+1$ instead of applying more elaborate techniques.”", "expanded_sketch": "To prove the main theorem, the algorithm uses “estimates for the density of integers that factor completely into primes $q \\leqslant B$,” i.e. “$B$-smooth numbers.” It has “a broadly similar structure to its predecessors.”\n\n- **Stage 1 (order computations for small bases):** “In the first stage, we attempt to compute the order of all elements $\\beta = 2,3,\\ldots, B$, for a suitable choice of $B$.” If this does not already yield “an element of large order, or a factor of $N$ via GCDs as before,” then one has computed “some $M \\leqslant D$” such that for every prime factor $p$ of $N$ all these $\\beta$ satisfy $\\beta^M \\equiv 1 \\pmod p$, and moreover “$M \\mathrel{|} p-1$ for every such $p$.”\n\n- **Key observation via smooth numbers:** “Not only these $\\beta$, but \\emph{all $B$-smooth integers} in the interval $1 \\leqslant n \\leqslant p$ satisfy $n^M \\equiv 1 \\pmod p$.” Since “the latter congruence has at most $M$ solutions modulo $p$,” applying “well-known lower bounds for the density of $B$-smooth numbers” gives “drastically improved estimates for how fast $M$ grows as we progress through the main loop over $\\beta$.”\n\n- **Stage 2 (finding factors):** This density-based strategy “leads to a considerable simplification in the second stage of the algorithm: it suffices to use trial division to check for divisors along the arithmetic progression $kM+1$ instead of applying more elaborate techniques.”", "expanded_theorem": "\\label{thm:main}\nThere exists an algorithm with the following properties. \nIt takes as input integers $N \\geqslant 3$ and $D \\geqslant 1$ with $D < N-1$.\nIt outputs either some $\\alpha\\in\\Z_N^*$ with $\\ord_N(\\alpha) > D$\nor a nontrivial divisor of $N$.\nIts time complexity is\n\\begin{equation}\n\\label{eq:main-complexity}\nO\\left(\\frac{D^{1/2}}{(\\log \\log D)^{1/2}} \\log^2 N\\right).\n\\end{equation}.", "theorem_type": ["Algorithmic or Constructive", "Existence"], "mcq": {"question": "Let \\(N\\ge 3\\) and \\(D\\ge 1\\) be integers with \\(DD\\), or a nontrivial divisor of \\(N\\), and whose time complexity is \\(O\\!\\left(\\dfrac{D^{1/2}}{(\\log\\log D)^{1/2}}\\log^2 N\\right)\\)."}, "choices": [{"label": "B", "text": "There exists an algorithm that, given integers \\(N\\ge 3\\) and \\(D\\ge 1\\) with \\(D\\le N-1\\), outputs either an element \\(\\alpha\\in \\mathbb Z_N^*\\) such that \\(\\operatorname{ord}_N(\\alpha)\\ge D\\), or a nontrivial divisor of \\(N\\), and whose time complexity is \\(O\\!\\left(\\dfrac{D^{1/2}}{(\\log\\log D)^{1/2}}\\log^2 N\\right)\\)."}, {"label": "C", "text": "There exists an algorithm that, given integers \\(N\\ge 3\\) and \\(D\\ge 1\\) with \\(DD\\), or an integer certifying that no nontrivial divisor was found, and whose time complexity is \\(O\\!\\left(\\dfrac{D^{1/2}}{(\\log\\log D)^{1/2}}\\log^2 N\\right)\\)."}, {"label": "D", "text": "There exists an algorithm that, given an integer \\(N\\ge 3\\), runs in time \\(O\\!\\left(\\dfrac{D^{1/2}}{(\\log\\log D)^{1/2}}\\log^2 N\\right)\\) for every integer \\(D\\) with \\(1\\le DD\\), or a nontrivial divisor of \\(N\\)."}, {"label": "E", "text": "There exists an algorithm that, given integers \\(N\\ge 3\\) and \\(D\\ge 1\\) with \\(DD\\), or a proof that \\(N\\) is prime, and whose time complexity is \\(O\\!\\left(\\dfrac{D^{1/2}}{(\\log\\log D)^{1/2}}\\log^2 N\\right)\\)."}], "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": "strict inequality in both input range and order threshold", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the guaranteed production of a nontrivial divisor of N", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "dependence of the algorithm on D as an input parameter", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "replaced factor output from stage 2 by a primality-proof outcome", "template_used": "wildcard"}]}} +{"id": "2601.11596v1", "paper_link": "http://arxiv.org/abs/2601.11596v1", "theorems_cnt": 1, "theorem": {"env_name": "prop", "content": "\\label{1}Let $e^{t A_n}$ and $e^{-y\\sqrt{-{A}_n}}$ be the heat and the Poisson semigroup operators on spaces of constant curvature, the following formulas hold:\\\\\ni) $e^{-y \\sqrt{-A_n}}=\\frac{y}{\\sqrt{\\pi}}L_u\\left[e^{A_n/4 u}u^{-1/2}\\right](y^2)$,\\\\\nii) $e^{t A_n}=(4 t)^{-1/2}L_{y^2}^{-1}\\left[\\frac{\\sqrt{\\pi}e^{-y\\sqrt{-A_n}}}{y}\\right](1/4 t)$,\\\\\nwhere $L_u(f(u))(v)$ is the classical Laplace transform computed with respect to $u$ with the variable in transform $v$\nand $[L_{v}^{-1}f(v)](u)$ is the Laplace inverse transform computed with respect to $v$ with\n variable in inverse transform $u$.\\\\", "start_pos": 4957, "end_pos": 5582, "label": "1"}, "ref_dict": {"stri2": "\\begin{equation}\\label{stri2}e^{-y \\lambda}=\\frac{y}{\\sqrt{\\pi}}L_u\n\\left[u^{-1/2}e^{-\\lambda^2/4u}\\right]\n(y^2),\\end{equation}", "Poisson-Equation": "\\begin{equation}\\label{Poisson-Equation}\\begin{cases} A_n U(y, x)+\\frac{\\partial^{2}}{\\partial y^{2}}U(y, x)=0, (y, x)\\in \\R^+\\times\\Omega_n\\\\ U(0, x) = U_0(x);~~~~U_0\\in C^{\\infty}_{0}(\\Omega_n)\n\\end{cases},\\end{equation}", "Heat-Equation": "\\begin{equation}\\label{Heat-Equation}\\begin{cases} A_n u(t, x)=\\frac{\\partial }{\\partial t}u(t, x)=0, (t, x)\\in \\R^+\\times \\Omega_n\\\\ u(0, x) = u_0(x);~~~~u_0\\in C^{\\infty}_{0}(\\Omega_n)\n\\end{cases}\\end{equation}"}, "pre_theorem_intro_text_len": 3014, "pre_theorem_intro_text": "The most important partial differential equations in physics and mathematics are Poisson equation, heat equation and the wave equation.\nIn the last centenary the solutions of the heat, Poisson and wave equations on spaces of constant curvature\nhave been studied and computed explicitly and there are many interesting articles published in the mathematics and physics literature.\nThe wave equation on spaces of constant curvature is considered by Intissar and Ould Moustapha \\cite{Intissar-Ould Moustapha1},\\cite{Intissar-Ould Moustapha2}, Bunk et al.\\cite{Bunk et al.}, Lax-Phillips \\cite{Lax-Phillips}, Ould Moustapha \\cite{Ould-Moustapha3}, Abdelhay et al.\\cite{A-B-M}. The heat equation on the hyperbolic space is studied by Davies-Mandovalos \\cite{Davies-Mandouvalos}, Grigor'yan-Noguchi \\cite{Grigoryan-Noguchi}, Ikeda-Matsumoto\n\\cite{Ikeda-Matsumoto} and Gruet\\cite{Gruet}. For the heat equation on spaces of constant curvature see Camporesi\\cite{Camporesi}, Taylor\\cite{Taylor}, Lohoue-Richener\\cite{Lohoue-Rychener} , Matsumoto \\cite{Matsumoto} and Anker-Ostellari \\cite{Anker-Ostellari}.\nFor a recent work on Poisson equation on the spherical and hyperbolic space the reader can consult Adam et al. \\cite{Adam et al.} and Husam-Ould Moustapha \\cite{Husam-Ould Moustapha}, Betancor et al. \\cite{Betancor et al.}. For recent work on Gruet formulas (see Nizar \\cite{Nizar}).\\\\\nThe heat and Poisson kernels are the integral kernels of the heat and Poisson semigroups and thus provides solutions\nto the heat and Poisson equation based on the Laplace-Beltrami operators on spaces of constant curvature.\nThe heat kernel on spaces of constant curvature is a transition\nprobability density of the Brownian motion on theses spaces.\nLet $X_t\\in I\\!\\!R^n$ the Brownian motion started\nat $x$, then the function\n$u(t, x) = Exp[u_0(X_t)]$ solves the Cauchy problem \\eqref{Heat-Equation}.\nLet $X_y\\in I\\!\\!R^n$ be the random variable with Cauchy distribution then the function\n$U(y, x) = Exp[U_0(X_y)]$ solves the Cauchy problem \\eqref{Poisson-Equation}.\nThese facts lead to physical\nsignificance and applications.\nThis article deals with the Poisson and heat semigroups $e^{t A_n}$ and $e^{-y\\sqrt{-A_n}}$ associated to the Laplace Beltrami operators on spaces of constant curvature(Euclidean spherical and hyperbolic spaces), solving the problems:\n\\begin{equation}\\label{Heat-Equation}\\begin{cases} A_n u(t, x)=\\frac{\\partial }{\\partial t}u(t, x)=0, (t, x)\\in I\\!\\!R^+\\times \\Omega_n\\\\ u(0, x) = u_0(x);~~~~u_0\\in C^{\\infty}_{0}(\\Omega_n)\n\\end{cases}\\end{equation}\nand\n\\begin{equation}\\label{Poisson-Equation}\\begin{cases} A_n U(y, x)+\\frac{\\partial^{2}}{\\partial y^{2}}U(y, x)=0, (y, x)\\in I\\!\\!R^+\\times\\Omega_n\\\\ U(0, x) = U_0(x);~~~~U_0\\in C^{\\infty}_{0}(\\Omega_n)\n\\end{cases},\\end{equation}\nwhere $\\Omega_n=I\\!\\!R^n$, $\\mathbb S^n$ or $I\\!\\!H^n$, is the Euclidean, spherical and the hyperbolic space respectively and $A_n=\\Delta_n^E, \\Delta_n^S$ or $\\Delta_n^H $ is the corresponding Laplace Beltrami operator.", "context": "The most important partial differential equations in physics and mathematics are Poisson equation, heat equation and the wave equation.\nIn the last centenary the solutions of the heat, Poisson and wave equations on spaces of constant curvature\nhave been studied and computed explicitly and there are many interesting articles published in the mathematics and physics literature.\nThe wave equation on spaces of constant curvature is considered by Intissar and Ould Moustapha \\cite{Intissar-Ould Moustapha1},\\cite{Intissar-Ould Moustapha2}, Bunk et al.\\cite{Bunk et al.}, Lax-Phillips \\cite{Lax-Phillips}, Ould Moustapha \\cite{Ould-Moustapha3}, Abdelhay et al.\\cite{A-B-M}. The heat equation on the hyperbolic space is studied by Davies-Mandovalos \\cite{Davies-Mandouvalos}, Grigor'yan-Noguchi \\cite{Grigoryan-Noguchi}, Ikeda-Matsumoto\n\\cite{Ikeda-Matsumoto} and Gruet\\cite{Gruet}. For the heat equation on spaces of constant curvature see Camporesi\\cite{Camporesi}, Taylor\\cite{Taylor}, Lohoue-Richener\\cite{Lohoue-Rychener} , Matsumoto \\cite{Matsumoto} and Anker-Ostellari \\cite{Anker-Ostellari}.\nFor a recent work on Poisson equation on the spherical and hyperbolic space the reader can consult Adam et al. \\cite{Adam et al.} and Husam-Ould Moustapha \\cite{Husam-Ould Moustapha}, Betancor et al. \\cite{Betancor et al.}. For recent work on Gruet formulas (see Nizar \\cite{Nizar}).\\\\\nThe heat and Poisson kernels are the integral kernels of the heat and Poisson semigroups and thus provides solutions\nto the heat and Poisson equation based on the Laplace-Beltrami operators on spaces of constant curvature.\nThe heat kernel on spaces of constant curvature is a transition\nprobability density of the Brownian motion on theses spaces.\nLet $X_t\\in I\\!\\!R^n$ the Brownian motion started\nat $x$, then the function\n$u(t, x) = Exp[u_0(X_t)]$ solves the Cauchy problem \\eqref{Heat-Equation}.\nLet $X_y\\in I\\!\\!R^n$ be the random variable with Cauchy distribution then the function\n$U(y, x) = Exp[U_0(X_y)]$ solves the Cauchy problem \\eqref{Poisson-Equation}.\nThese facts lead to physical\nsignificance and applications.\nThis article deals with the Poisson and heat semigroups $e^{t A_n}$ and $e^{-y\\sqrt{-A_n}}$ associated to the Laplace Beltrami operators on spaces of constant curvature(Euclidean spherical and hyperbolic spaces), solving the problems:\n\\begin{equation}\\label{Heat-Equation}\\begin{cases} A_n u(t, x)=\\frac{\\partial }{\\partial t}u(t, x)=0, (t, x)\\in I\\!\\!R^+\\times \\Omega_n\\\\ u(0, x) = u_0(x);~~~~u_0\\in C^{\\infty}_{0}(\\Omega_n)\n\\end{cases}\\end{equation}\nand\n\\begin{equation}\\label{Poisson-Equation}\\begin{cases} A_n U(y, x)+\\frac{\\partial^{2}}{\\partial y^{2}}U(y, x)=0, (y, x)\\in I\\!\\!R^+\\times\\Omega_n\\\\ U(0, x) = U_0(x);~~~~U_0\\in C^{\\infty}_{0}(\\Omega_n)\n\\end{cases},\\end{equation}\nwhere $\\Omega_n=I\\!\\!R^n$, $\\mathbb S^n$ or $I\\!\\!H^n$, is the Euclidean, spherical and the hyperbolic space respectively and $A_n=\\Delta_n^E, \\Delta_n^S$ or $\\Delta_n^H $ is the corresponding Laplace Beltrami operator.\n\n\\begin{equation}\\label{Heat-Equation}\\begin{cases} A_n u(t, x)=\\frac{\\partial }{\\partial t}u(t, x)=0, (t, x)\\in \\R^+\\times \\Omega_n\\\\ u(0, x) = u_0(x);~~~~u_0\\in C^{\\infty}_{0}(\\Omega_n)\n\\end{cases}\\end{equation}\n\n\\begin{equation}\\label{Poisson-Equation}\\begin{cases} A_n U(y, x)+\\frac{\\partial^{2}}{\\partial y^{2}}U(y, x)=0, (y, x)\\in \\R^+\\times\\Omega_n\\\\ U(0, x) = U_0(x);~~~~U_0\\in C^{\\infty}_{0}(\\Omega_n)\n\\end{cases},\\end{equation}", "full_context": "The most important partial differential equations in physics and mathematics are Poisson equation, heat equation and the wave equation.\nIn the last centenary the solutions of the heat, Poisson and wave equations on spaces of constant curvature\nhave been studied and computed explicitly and there are many interesting articles published in the mathematics and physics literature.\nThe wave equation on spaces of constant curvature is considered by Intissar and Ould Moustapha \\cite{Intissar-Ould Moustapha1},\\cite{Intissar-Ould Moustapha2}, Bunk et al.\\cite{Bunk et al.}, Lax-Phillips \\cite{Lax-Phillips}, Ould Moustapha \\cite{Ould-Moustapha3}, Abdelhay et al.\\cite{A-B-M}. The heat equation on the hyperbolic space is studied by Davies-Mandovalos \\cite{Davies-Mandouvalos}, Grigor'yan-Noguchi \\cite{Grigoryan-Noguchi}, Ikeda-Matsumoto\n\\cite{Ikeda-Matsumoto} and Gruet\\cite{Gruet}. For the heat equation on spaces of constant curvature see Camporesi\\cite{Camporesi}, Taylor\\cite{Taylor}, Lohoue-Richener\\cite{Lohoue-Rychener} , Matsumoto \\cite{Matsumoto} and Anker-Ostellari \\cite{Anker-Ostellari}.\nFor a recent work on Poisson equation on the spherical and hyperbolic space the reader can consult Adam et al. \\cite{Adam et al.} and Husam-Ould Moustapha \\cite{Husam-Ould Moustapha}, Betancor et al. \\cite{Betancor et al.}. For recent work on Gruet formulas (see Nizar \\cite{Nizar}).\\\\\nThe heat and Poisson kernels are the integral kernels of the heat and Poisson semigroups and thus provides solutions\nto the heat and Poisson equation based on the Laplace-Beltrami operators on spaces of constant curvature.\nThe heat kernel on spaces of constant curvature is a transition\nprobability density of the Brownian motion on theses spaces.\nLet $X_t\\in I\\!\\!R^n$ the Brownian motion started\nat $x$, then the function\n$u(t, x) = Exp[u_0(X_t)]$ solves the Cauchy problem \\eqref{Heat-Equation}.\nLet $X_y\\in I\\!\\!R^n$ be the random variable with Cauchy distribution then the function\n$U(y, x) = Exp[U_0(X_y)]$ solves the Cauchy problem \\eqref{Poisson-Equation}.\nThese facts lead to physical\nsignificance and applications.\nThis article deals with the Poisson and heat semigroups $e^{t A_n}$ and $e^{-y\\sqrt{-A_n}}$ associated to the Laplace Beltrami operators on spaces of constant curvature(Euclidean spherical and hyperbolic spaces), solving the problems:\n\\begin{equation}\\label{Heat-Equation}\\begin{cases} A_n u(t, x)=\\frac{\\partial }{\\partial t}u(t, x)=0, (t, x)\\in I\\!\\!R^+\\times \\Omega_n\\\\ u(0, x) = u_0(x);~~~~u_0\\in C^{\\infty}_{0}(\\Omega_n)\n\\end{cases}\\end{equation}\nand\n\\begin{equation}\\label{Poisson-Equation}\\begin{cases} A_n U(y, x)+\\frac{\\partial^{2}}{\\partial y^{2}}U(y, x)=0, (y, x)\\in I\\!\\!R^+\\times\\Omega_n\\\\ U(0, x) = U_0(x);~~~~U_0\\in C^{\\infty}_{0}(\\Omega_n)\n\\end{cases},\\end{equation}\nwhere $\\Omega_n=I\\!\\!R^n$, $\\mathbb S^n$ or $I\\!\\!H^n$, is the Euclidean, spherical and the hyperbolic space respectively and $A_n=\\Delta_n^E, \\Delta_n^S$ or $\\Delta_n^H $ is the corresponding Laplace Beltrami operator.\n\n\\begin{equation}\\label{Heat-Equation}\\begin{cases} A_n u(t, x)=\\frac{\\partial }{\\partial t}u(t, x)=0, (t, x)\\in \\R^+\\times \\Omega_n\\\\ u(0, x) = u_0(x);~~~~u_0\\in C^{\\infty}_{0}(\\Omega_n)\n\\end{cases}\\end{equation}\n\n\\begin{equation}\\label{Poisson-Equation}\\begin{cases} A_n U(y, x)+\\frac{\\partial^{2}}{\\partial y^{2}}U(y, x)=0, (y, x)\\in \\R^+\\times\\Omega_n\\\\ U(0, x) = U_0(x);~~~~U_0\\in C^{\\infty}_{0}(\\Omega_n)\n\\end{cases},\\end{equation}\n\n\\begin{equation}\\label{pre-gruet-s}H^S_n(t, w, w')=c_n(4t)^{-1/2} \\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}\\frac{\\exp{\\left(\\frac{y^2}{4t}\\right)}\\, \\sinh y}{(\\cosh y-\\cos \\varphi)^{(n+1)/2}}dy,\\end{equation}\n\\begin{align} \\label{gruet-s}H_n^S(t, \\varphi)=2c_n(4t)^{-1/2}\\int_0^{+\\infty}{\\cal R}e\\left[\\frac{e^{(\\sigma-i\\xi)^2/4 t} \\sinh(\\sigma-i\\xi)}{(\\cosh (\\sigma-i\\xi)-\\cos\\varphi)^{(n+1)/2}}\\right] d\\xi,\n\\end{align}\nwhere $c_n=\\frac{\\Gamma((n+1)/2)}{2^{(n+3)/2}i\\pi^{(n/2+1)}}$.\n\\end{thm}\n \\begin{proof} The proof of \\eqref{pre-gruet-s} uses essentially ii) of Proposition \\ref{1}. To see \\eqref{gruet-s}\nSet $y=\\sigma+i\\xi$ in the formula \\eqref{pre-gruet-s} and\nsplit the integral into two integrals over $\\R^+$ and $\\R^-$.\nNotice that the formula \\eqref{gruet-s} is the spherical analogous of the Gruet formula on the hyperbolic space Gruet \\cite{Gruet} and we call it\nthe spherical Gruet formula.\n\\end{proof}\nThe integrand on the right hand side of \\eqref{pre-gruet-s}is\na meromorphic function in $s$ when $n$ is odd and we can apply the residue\ncalculus to obtain,\nfor n=2k+1,\n$$H_{2k+1}^{\\S}(t, x, x')=\n\\left(\\frac{\\partial}{-2\\pi \\sin r\\partial r}\\right)^k \\sum_{n\\in \\Z}\\frac{e^{-(\\varphi+2\\pi n)^2/4t}}{\\sqrt{4\\pi t}}.$$\n\n\\begin{thm}\\label{Heat-Hyperbolic} The heat kernel on the hyperbolic space is given by \\\\\n \\begin{equation}\\label{pre-gruet-h}H^H_n(t, w, w')=C_n (4t)^{-1/2}\\int_{\\sigma-i\\infty}^{\\sigma+i\\infty}\\frac{\\exp{\\left(\\frac{y^2}{4t}\\right)}\\, \\sin y}{(\\cosh\\rho(w, w')-\\cos y)^{(n+1)/2}}dy,\\end{equation}\n with $C_n=\\frac{\\Gamma((n+1)/2)}{2^{(n+3)/2}i\\pi^{(n/2+1)}}.$\n\\end{thm}\n\\begin{proof} The proof of \\eqref{pre-gruet-h} uses essentially ii) of Proposition \\ref{1}.\n\\end{proof}\n\\begin{cor} (Gruet formula)\n\\begin{align} \\label{gruet-h1}H^H_n(t, w, w')=\\frac{K_n}{(4t)^{1/2}}\\int_0^{+\\infty}{\\cal R}e\\left[\\frac{e^{(\\sigma-i\\xi)^2/4 t} \\sin(\\sigma-i\\xi)}{(\\cosh \\rho-\\cos(\\sigma-i\\xi))^{(n+1)/2}}\\right] d\\xi,\n\\end{align}\n$K_n=\\frac{\\Gamma((n+1)/2)}{2^{(n+3)/2}\\pi^{(n/2+1)}}$.\n\\begin{align} \\label{gruet-h2}H_n(t, z, z')=K_n t^{-1/2}\\int_0^{+\\infty}\\frac{e^{(\\pi^2-\\xi^2)/2 t} \\sinh\\xi \\sin \\pi \\xi/t}{(\\cosh \\rho +\\cosh \\xi\n)^{(n+1)/2}} d\\xi,\n\\end{align}\n$K_n=\\frac{\\Gamma((n+1)/2)}{2^{n/2}\\pi^{n/2+1}}$.\n\\end{cor}\n\\begin{proof}\nTo see \\eqref{gruet-h1},\nset $y=\\sigma+i\\xi$ in the formula \\eqref{pre-gruet-h}, we get after\nsplitting the integral into two integrals over $\\R^+$ and $\\R^-$\nwe obtain the result.\nSet $\\sigma=\\pi$ and replace $t$ by $t/2$ in \\eqref{gruet-h1} we obtain the \\eqref{gruet-h2}.\nNotice that the formula \\eqref{gruet-h2} is the Gruet formula on the hyperbolic space(Gruet \\cite{Gruet}).\nWhile the classical expressions for $H^H_n(t, r)$ have different forms for odd and even\ndimensions, but the Gruet's formula below holds for every $n$.\n\\end{proof}\nThe integrand on the right hand side of\\eqref{pre-gruet-h} is\na meromorphic function in $s$ when $n$ is odd and we can apply the residue\ntheorem to obtain,\nfor$ n=2k+1 $,\n$$H_{2k+1}^{\\H}(t, x, x')=\\left(\\frac{\\partial}{-2\\pi \\sinh r\\partial r}\\right)^k \\frac{e^{-r^2/4t}}{\\sqrt{4\\pi t}}.$$\n\nbegin{prop}\\label{Recurrence-Heat}\nLet $e^{-y\\sqrt{-{\\cal L}_n}}$ and $e^{t {\\cal L}_n}$ be the Poisson and heat semigroups on the hyperbolic space $\\H^n$ then we have\\\\\n i) $e^{t {\\cal L}_n}=(4t)^{-1/2}L_{y^2}^{-1}\\left[\\frac{\\sqrt{\\pi}e^{-y\\sqrt{-{\\cal L}_n}}}{y}\\right](1/4 t)$, where $L_{y^2}^{-1}$ is the Laplace inverse transform with respect to $y^2$.\\\\\n $\\frac{e^{-y \\lambda}}{y}=\\frac{1}{\\sqrt{\\pi}}\\int_0^\\infty e^{-u y^2/2}u^{-1/2}e^{A_n/4u}du$\nii) $ \\left(-\\frac{\\partial}{2\\pi\\sinh \\rho\\partial \\rho}\\right)K^{\\H}_n\\left(t, \\rho(w, w')\\right)=K^{\\H}_{n+2}\\left(t, \\rho(w, w')\\right),$\\\\\niii) $ \\int_r^\\infty \\frac{K^{\\H}_{n+1}\\left(t, \\rho\\right)}{\\sqrt{\\cosh^2\\rho/2-\\cosh^2 r/2}}\\sinh\\rho d\\rho =K^{\\H}_{n}\\left(t, r\\right).$\\\\\n\\end{prop}\n\\begin{proof}\nTo prove i) use the subordination formula (Strichartz \\cite{STRICHARTZ} $p.\\, 50$).\\\\\n$$\\frac{e^{-y \\lambda}}{y}=\\frac{1}{\\sqrt{\\pi}}\\int_0^\\infty e^{-u y^2}u^{-1/2}e^{-\\lambda^2/4u}du$$ or\n$\\frac{\\sqrt{\\pi}e^{-y \\lambda}}{y}=L\\left(u^{-1/2}e^{-\\lambda^2/4u}\\right)(y^2),$\nwhere $(Lf)(p)$ is the Laplace transform\ntaking $\\lambda=\\sqrt{-A/2}$ we can write\n$$e^{-y \\sqrt{-A/2}}=\\frac{y}{\\sqrt{2\\pi}}\\int_0^\\infty e^{-u y^2/2}u^{-1/2}e^{A/4u} du$$\nand\n$$e^{\\frac{-\\lambda^2}{4 u}}=u^{1/2}L^{-1}_{y^2}\\left(\\sqrt{\\pi}\\frac{e^{-y \\lambda}}{y}\\right)(u).$$\nSet $\\lambda=\\sqrt{-A/2}$ an $\\frac{1}{4u}=t $ in the last formula we can write\n$$ e^{t A/2}=(2t)^{-1/2}L_{y^2}^{-1}\\left[\\frac{\\sqrt{\\pi}e^{-y\\sqrt{-A/2}}}{y}\\right](1/2 t),\n$$\nwhere $L^{-1}$ is the inverse Laplace transform.\n\n\\\n \\\\\nThe parts ii) and iii) are consequence of i) and Proposition \\ref{Recurrence-Poisson}\n\\end{proof}\nThe Laplace-Beltrami operators on Riemannian manifold are known as very\nimportant operators in mathematics and its applications. This paper deals\nwith the Poisson and heat semigroups associated to these second order di\u000berential\noperators on Euclidean and non Euclidean spaces. In the last centenary\nthe heat and Poisson semigroups associated to the Laplace Beltrami operators\nhave been computed explicitly and there are many interesting studies\npublished in the mathematics and Physics literature (see for example Betancor\net al. [5], Isolda Cardoso [7], Keles and Bayrakci [?], Stein [?] and\nthe references theirin). The purpose of this article is to provide new method\nand explicit formulas for the heat and Poisson semigroups on the Euclidean\nand non Euclidean spaces. That is using the explicit formula connecting the\nclassical heat and Poisson semigroups we give a new formula for the Poisson\nsemigroups on these spaces and we prove that it is equivalent to the classical\none. We obtain the Gruet formula in Euclidean space and we provide a\nnew method to derive the classical Gruet formula for the kernel of the heat\nsemigroup on non Euclidean space. The main objective of this paper is to\ngive new explicit formulas for the kernels of the heat and Poisson semigroups\noperators etAn and e..y\np\nAn solving\nThe Laplace-Beltrami operators on Riemannian manifold are known as very\nimportant operators in mathematics and its applications. This paper deals\nwith the Poisson and heat semigroups associated to these second order di\u000berential\noperators on Euclidean and non Euclidean spaces. In the last centenary\nthe heat and Poisson semigroups associated to the Laplace Beltrami operators\nhave been computed explicitly and there are many interesting studies\npublished in the mathematics and Physics literature (see for example Betancor\net al. [5], Isolda Cardoso [7], Keles and Bayrakci [?], Stein [?] and\nthe references theirin). The purpose of this article is to provide new method\nand explicit formulas for the heat and Poisson semigroups on the Euclidean\nand non Euclidean spaces. That is using the explicit formula connecting the\nclassical heat and Poisson semigroups we give a new formula for the Poisson\nsemigroups on these spaces and we prove that it is equivalent to the classical\none. We obtain the Gruet formula in Euclidean space and we provide a\nnew method to derive the classical Gruet formula for the kernel of the heat\nsemigroup on non Euclidean space. The main objective of this paper is to\ngive new explicit formulas for the kernels of the heat and Poisson semigroups\noperators etAn and e..y\n..An solving", "post_theorem_intro_text_len": 1359, "post_theorem_intro_text": "\\begin{proof}\nTo prove i)we use the subordination formula in Strichartz \\cite{Strichartz} $p. 50$.\\\\\n\\begin{equation}\\label{stri1}\\frac{e^{-y \\lambda}}{y}=\\frac{1}{\\sqrt{\\pi}}\\int_0^\\infty e^{-u y^2}u^{-1/2}e^{-\\lambda^2/4u}du,\\end{equation}\nor\n\\begin{equation}\\label{stri2}e^{-y \\lambda}=\\frac{y}{\\sqrt{\\pi}}L_u\n\\left[u^{-1/2}e^{-\\lambda^2/4u}\\right]\n(y^2),\\end{equation}\ntaking $\\lambda=\\sqrt{-A_n}$ in the formula \\eqref{stri2}we obtain i).\nTo see ii) in view of the formula \\eqref{stri2} we can write $e^{\\frac{-\\lambda^2}{4 u}}=u^{1/2}L^{-1}_{y^2}\\left(\\sqrt{\\pi}\\frac{e^{-y \\lambda}}{y}\\right)(u)$,\nsetting $\\lambda=\\sqrt{-A_n}$ and $\\frac{1}{4u}=t$ in the last formula we have ii).\\\\\nNote that this proposition is contained in the work of Dettman \\cite{Dettman} formulas $(1.3)$ and $(1.4)$, see also Brag-Dettman \\cite{Brag-Dettman}.\n\\end{proof}\nThe remaining of the paper is organized as follows: in section 2, we study the Poisson and heat semigroups, and we give an analogous of the Gruet formulas on the Euclidean space $I\\!\\!R^n$. Section 3 is devoted to the Poisson and heat semigroups on spherical space and the Gruet formula for heat kernel on spherical space $\\mathbb S^n$ is obtained. In section 4 we consider the Poisson and heat equations and we give an elementary proof of the classical Gruet formula on the hyperbolic space $I\\!\\!H^n$ .", "sketch": "To prove i) the argument uses the subordination formula in Strichartz \\cite{Strichartz} (p. 50):\n\\begin{equation}\\label{stri1}\\frac{e^{-y \\lambda}}{y}=\\frac{1}{\\sqrt{\\pi}}\\int_0^\\infty e^{-u y^2}u^{-1/2}e^{-\\lambda^2/4u}du,\\end{equation}\nwhich is rewritten as the Laplace-transform identity\n\\begin{equation}\\label{stri2}e^{-y \\lambda}=\\frac{y}{\\sqrt{\\pi}}L_u\\left[u^{-1/2}e^{-\\lambda^2/4u}\\right](y^2).\\end{equation}\nThen, taking $\\lambda=\\sqrt{-A_n}$ in \\eqref{stri2} yields i).\n\nTo see ii), “in view of the formula \\eqref{stri2}” one writes\n\\[e^{-\\lambda^2/(4u)}=u^{1/2}L^{-1}_{y^2}\\left(\\sqrt{\\pi}\\frac{e^{-y \\lambda}}{y}\\right)(u).\\]\nSetting $\\lambda=\\sqrt{-A_n}$ and $\\frac{1}{4u}=t$ in this last formula gives ii).", "expanded_sketch": "To prove i) the argument uses the subordination formula in Strichartz \\cite{Strichartz} (p. 50):\n\\begin{equation}\\label{stri1}\\frac{e^{-y \\lambda}}{y}=\\frac{1}{\\sqrt{\\pi}}\\int_0^\\infty e^{-u y^2}u^{-1/2}e^{-\\lambda^2/4u}du,\\end{equation}\nwhich is rewritten as the Laplace-transform identity\n\\begin{equation}\\label{stri2}e^{-y \\lambda}=\\frac{y}{\\sqrt{\\pi}}L_u\n\\left[u^{-1/2}e^{-\\lambda^2/4u}\\right]\n(y^2),\\end{equation}\nThen, taking $\\lambda=\\sqrt{-A_n}$ in\n\\begin{equation}\\label{stri2}e^{-y \\lambda}=\\frac{y}{\\sqrt{\\pi}}L_u\n\\left[u^{-1/2}e^{-\\lambda^2/4u}\\right]\n(y^2),\\end{equation}\nyields i).\n\nTo see ii), in view of the equation above one writes\n\\[e^{-\\lambda^2/(4u)}=u^{1/2}L^{-1}_{y^2}\\left(\\sqrt{\\pi}\\frac{e^{-y \\lambda}}{y}\\right)(u).\\]\nSetting $\\lambda=\\sqrt{-A_n}$ and $\\frac{1}{4u}=t$ in this last formula gives ii).", "expanded_theorem": "\\label{1}Let $e^{t A_n}$ and $e^{-y\\sqrt{-{A}_n}}$ be the heat and the Poisson semigroup operators on spaces of constant curvature, the following formulas hold:\\\\\ni) $e^{-y \\sqrt{-A_n}}=\\frac{y}{\\sqrt{\\pi}}L_u\\left[e^{A_n/4 u}u^{-1/2}\\right](y^2)$,\\\\\nii) $e^{t A_n}=(4 t)^{-1/2}L_{y^2}^{-1}\\left[\\frac{\\sqrt{\\pi}e^{-y\\sqrt{-A_n}}}{y}\\right](1/4 t)$,\\\\\nwhere $L_u(f(u))(v)$ is the classical Laplace transform computed with respect to $u$ with the variable in transform $v$\nand $[L_{v}^{-1}f(v)](u)$ is the Laplace inverse transform computed with respect to $v$ with\n variable in inverse transform $u$.\\\\,", "theorem_type": ["Universal", "Equivalence"], "mcq": {"question": "Let $\\Omega_n$ be a space of constant curvature, namely $\\mathbb{R}^n$, $\\mathbb{S}^n$, or $\\mathbb{H}^n$, and let $A_n$ denote the corresponding Laplace--Beltrami operator. Write $e^{tA_n}$ for the heat semigroup and $e^{-y\\sqrt{-A_n}}$ for the Poisson semigroup. If $L_u(f(u))(v)$ denotes the classical Laplace transform of $f$ with respect to $u$ and transform variable $v$, and $[L_v^{-1}f(v)](u)$ denotes the inverse Laplace transform with respect to $v$ evaluated at $u$, which statement holds for these semigroups?", "correct_choice": {"label": "A", "text": "They satisfy both identities\n$$e^{-y\\sqrt{-A_n}}=\\frac{y}{\\sqrt{\\pi}}\\,L_u\\!\\left[e^{A_n/4u}u^{-1/2}\\right](y^2),$$\nand\n$$e^{tA_n}=(4t)^{-1/2}\\,L_{y^2}^{-1}\\!\\left[\\frac{\\sqrt{\\pi}\\,e^{-y\\sqrt{-A_n}}}{y}\\right]\\!(1/4t).$$"}, "choices": [{"label": "B", "text": "They satisfy both identities\n$$e^{-y\\sqrt{-A_n}}=\\frac{1}{\\sqrt{\\pi}}\\,L_u\\!\\left[e^{A_n/4u}u^{-1/2}\\right](y^2),$$\nand\n$$e^{tA_n}=(4t)^{-1/2}\\,L_{y^2}^{-1}\\!\\left[\\sqrt{\\pi}\\,e^{-y\\sqrt{-A_n}}\\right]\\!(1/4t).$$"}, {"label": "C", "text": "They satisfy the identity\n$$e^{-y\\sqrt{-A_n}}=\\frac{y}{\\sqrt{\\pi}}\\,L_u\\!\\left[e^{A_n/4u}u^{-1/2}\\right](y^2).$$"}, {"label": "D", "text": "They satisfy both identities\n$$e^{-y\\sqrt{-A_n}}=\\frac{y}{\\sqrt{\\pi}}\\,L_u\\!\\left[e^{uA_n/4}u^{-1/2}\\right](y^2),$$\nand\n$$e^{tA_n}=(4t)^{-1/2}\\,L_{y^2}^{-1}\\!\\left[\\frac{\\sqrt{\\pi}\\,e^{-y\\sqrt{-A_n}}}{y}\\right]\\!(1/4t).$$"}, {"label": "E", "text": "They satisfy both identities\n$$e^{-y\\sqrt{-A_n}}=\\frac{y}{\\sqrt{\\pi}}\\,L_u\\!\\left[e^{A_n/4u}u^{-1/2}\\right](y),$$\nand\n$$e^{tA_n}=t^{-1/2}\\,L_{y^2}^{-1}\\!\\left[\\frac{\\sqrt{\\pi}\\,e^{-y\\sqrt{-A_n}}}{y}\\right]\\!(1/4t).$$"}], "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": "prefactor-and-y-denominator from subordination identity", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the inverse-Laplace reconstruction formula for the heat semigroup", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence on 1/u inside the exponent inherited from e^{-\\lambda^2/(4u)}", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "evaluation at y^2 and normalization after the substitution 1/(4u)=t", "template_used": "boundary_range"}]}} +{"id": "2601.12140v1", "paper_link": "http://arxiv.org/abs/2601.12140v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm1}\nLet $n \\geqslant 2$, $0 < s < 1$. Suppose that $u\\in L^q(\\mathbb{H}^n), q>p$ is a nonnegative weak solution of \\eqref{main_equation_1}. Then\n\\begin{enumerate}\n \\item If $1 < p < \\frac{n+2s}{n-2s}$ and $u\\in L^{p+1}(\\mathbb{H}^n)$ , then $u\\equiv 0$. \n \\item If $p = \\frac{n+2s}{n-2s}$ and $u\\in L^q$, where $q>p$, then there exists a point $x_0 \\in \\mathbb{H}^n$ such that $u$ is radially symmetric and nondecreasing about $x_0$.\n\\end{enumerate}", "start_pos": 17666, "end_pos": 18132, "label": "thm1"}, "ref_dict": {"main_equation_1": "\\begin{equation}\\label{main_equation_1}\n(-\\Delta_{\\hn})^s u = u^p(x), \\quad x \\in \\hn.\n\\end{equation}", "lemma-Bessel": "\\begin{lemma}[\\cite{AS64}, or Lemma 2.3 of \\cite{BGS15}]\\label{lemma-Bessel}\nThe solution of the ODE\n\\begin{equation*}\n\\label{Bessel1}\\partial_{ss} \\varphi+ \\frac{\\alpha}{s}\\, \\partial_s \\varphi -\\varphi = 0. \\end{equation*}\nmay be written as $\\varphi(s)=s^\\nu \\psi(s)$, for $\\alpha=1-2\\nu$, where $\\psi$ solves the is the well known Bessel equation\n\\begin{equation}\\label{Bessel2}\ns^2\\psi''+s\\psi'-(s^2+\\nu^2)\\psi=0.\n\\end{equation}\nIn addition, \\eqref{Bessel2} has two linearly independent solutions, $I_\\nu,K_\\nu$, which are the modified Bessel functions; their asymptotic behavior is given precisely by\n\\begin{align*}\nI_\\nu(s)&\\sim \\frac{1}{\\Gamma(\\nu+1)}\\left(\\frac{s}{2}\\right)^\\nu\\left( 1+\\frac{s^2}{4(\\nu+1)}+\\frac{s^4}{32(\\nu+1)(\\nu+2)}+\\ldots\\right),\\\\\nK_\\nu(s)&\\sim \\frac{\\Gamma(\\nu)}{2}\\left(\\frac{2}{s}\\right)^{\\nu}\n\\left( 1+\\frac{s^2}{4(1-\\nu)}+\\frac{s^4}{32(1-\\nu)(2-\\nu)}+\\ldots\\right)\n\\\\&\\quad+\\frac{\\Gamma(-\\nu)}{2}\\left(\\frac{s}{2}\\right)^\\nu\\left( 1+\\frac{s^2}{4(\\nu+1)}+\\frac{s^4}{32(\\nu+1)(\\nu+2)}+\\ldots\\right),\n\\end{align*}\n\nfor $s\\to 0^+$, $\\nu\\not\\in\\mathbb Z$. And when $s\\to +\\infty$,\n\\begin{align*}\\label{asymptotic2}\n I_\\nu(s)\\sim \\frac{1}{\\sqrt{2\\pi s}}e^s\\left(1-\\frac{4\\nu^2-1}{8s}+\\frac{(4\\nu^2-1)(4\\nu^2-9)}{2!(8s)^2}-\\ldots \\right),\\\\\n K_\\nu(s)\\sim \\sqrt{\\frac{\\pi}{2s}}e^{-s}\\left(1+\\frac{4\\nu^2-1}{8s}+\\frac{(4\\nu^2-1)(4\\nu^2-9)}{2!(8s)^2}+\\ldots \\right).\n\\end{align*}\n\\end{lemma}", "thm1": "\\begin{theorem}\\label{thm1}\nLet $n \\geq 2$, $0 < s < 1$. Suppose that $u\\in L^q(\\hn), q>p$ is a nonnegative weak solution of \\eqref{main_equation_1}. Then\n\\begin{enumerate}\n \\item If $1 < p < \\frac{n+2s}{n-2s}$ and $u\\in L^{p+1}(\\hn)$ , then $u\\equiv 0$. \n \\item If $p = \\frac{n+2s}{n-2s}$ and $u\\in L^q$, where $q>p$, then there exists a point $x_0 \\in \\mathbb{H}^n$ such that $u$ is radially symmetric and nondecreasing about $x_0$.\n\\end{enumerate}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 12726, "pre_theorem_intro_text": "The fractional Laplacians arise in various fields of mathematics and physics, such as probability, finance, physics, and fluid dynamics. For instance, they arise naturally in the study of stochastic processes with jumps, and more precisely in L\\'evy processes \\cite{Levy}, which extend the concept of Brownian motion. Fractional order operators also appear in the study of conformal geometry and partial differential equations. In $\\mathbb{R}^n$, the fractional Laplacian is a nonlocal pseudo-differential operator, assuming the form\n\\[\n(-\\Delta)^s u(x) = c_{n,s} \\, \\text{P.V.} \\int_{\\mathbb{R}^n} \\frac{u(x) - u(y)}{|x-y|^{n+2s}} \\, dy, \\quad 0 < s < 1,\n\\]\nwhere P.V. stands for the Cauchy principal value and $c_{n,s}$ is a normalization constant. Equivalently, $(-\\Delta)^{\\frac{\\alpha}{2}}$ can be defined in terms of Fourier transform:\n\\begin{equation*}\n (-\\Delta)^{s}u(x)=\\mathcal{F}^{-1}[|\\xi|^{2s} \\mathcal{F}{u}(\\xi)](x).\n\\end{equation*}\n\nThe non-locality of the fractional Laplacian makes it usually difficult to investigate. To circumvent this difficulty, Caffarelli and Silvestre \\cite{CS07} introduced the extension method that reduced this nonlocal problem into a local one in higher dimensions. For a function $u : \\mathbb{R}^n \\to \\mathbb{R}$, we construct the extension $U : \\mathbb{R}^n \\times [0, +\\infty) \\to \\mathbb{R}, U = U (x, y)$, as the solution of the equation\n\\begin{equation}\n\\begin{cases}\n\\operatorname{div}(y^{1-2s} \\nabla U(x,y)) = 0, & (x,y) \\in \\mathbb{R}^n \\times (0,\\infty), \\\\\nU(x,0) = u(x), & x \\in \\mathbb{R}^n.\n\\end{cases}\n\\end{equation}\n\nThe fractional Laplacian is then recovered by:\n\\begin{equation}\n(-\\Delta)^s u(x) = - C_s \\lim_{y \\to 0^+} y^{1-2s} \\frac{\\partial U}{\\partial y}(x,y),\n\\end{equation}\nwhere $C_s$ is a normalization constant depending on $s$.\n\nThe fractional (as well as integer) powers of Laplacian that enjoy the conformal property can also be defined on Riemannian manifolds. Graham and Zworski \\cite{GZ03} studied the connection between scattering matrices on conformally compact asymptotically Einstein manifolds and conformally invariant objects on their boundaries at infinity. Let $(X^{n+1}, g^+)$ be a conformally compact Einstein manifold with conformal infinity $(M, [\\hat{g}])$. For any defining function $\\rho$ of $M$, we can write $\\bar{g} = \\rho^2 g^+$, which extends to a metric on $\\bar{X} = X \\cup M$. Given $f \\in C^\\infty(M)$, consider the generalized eigenvalue problem\n\\begin{equation*}\n -\\Delta_{g^+} u - s(n-s) u = 0 \\quad \\text{ in } X,\n\\end{equation*}\nwith the asymptotic expansion near $M$:\n\\begin{equation*}\n u = F \\rho^{n-s} + G \\rho^s, \\quad F, G \\in C^\\infty(\\bar{X}), \\quad F|_{\\rho=0} = f.\n\\end{equation*}\nThe scattering operator $S(\\lambda)$ is defined by $S(\\lambda) F = G$.\nIn \\cite{CG11}, Chang and Gonz\\'alez showed that the fractional order operators defined via the scattering operator can be realized as the Dirichlet-to-Neumann map of a degenerate elliptic equation on $X$, generalizing the extension method of Caffarelli-Silvestre to the setting of conformally compact Einstein manifolds. It is also worth mentioning that the fractional GJMS operators on hyperbolic spaces were explicitly calculated by Lu and Yang in \\cite{LY23}. They also pointed out in \\cite{FLY2} that these fractional GJMS operators are not conformally to the fractional Laplacians on the upper half space $\\mathbb{R}^n_+$ nor on the unit ball $B^n$ in $\\mathbb{R}^n$, which one may expect for the integer cases. \n\nAlthough the fractional order operators on general manifolds are rather sophisticated, see e.g. \\cites{GZ03,CG11} and the references therein, there have been numerous accomplishments in the past decades when one considers the integer powers of Laplacians, which we will discuss in the following briefly. In the celebrated work of Gidas-Ni-Nirenberg \\cite{GNN1}, they considered the following boundary value problem on the ball $B_R(0)\\subset{\\mathbb{R}^n}$.\n\\begin{equation*}\n \\begin{cases}\n -\\Delta u = f(u) &\\text{ in } B_R(0)\\\\\n u=0 &\\text{ on } \\partial B_R(0),\n \\end{cases}\n\\end{equation*}\nwhere $f$ is of class $C^1$. They proved that any positive solution $u$ in $C^2(\\overline{B_R(0)})$ is radially symmetric and decreasing.\nTheir approach is the so-called moving plane method, which was initiated by Alexandrov \\cite{Alexandrov} in the 1950s, and further developed by Serrin \\cite{serrin}.\nLater, Gidas-Ni-Nirenberg \\cite{GNN2} studied the equation $-\\Delta u = f(u)$ in the entire space $\\mathbb{R}^n\\setminus\\{0,\\infty\\}, n\\geqslant 3$\nwith two singularities located at the origin and infinity:\n\\begin{align}\n u(x)\\to+\\infty \\quad &\\text{ as } \\quad x\\to0. \\nonumber\\\\\n |x|^{n-2}u(x)\\to+\\infty \\quad &\\text{ as } \\quad x\\to\\infty.\n\\end{align}\nThey showed that the solution is radially symmetric about the origin and decreasing.\nSubsequently, Caffarelli-Gidas-Spruck \\cite{CGS} considered this equation in a punctured ball when $f$ has critical growth.\nTo be more precise, they require the nonlinearity $f(t)$ to be a locally nondecreasing Lipschitz function with $f(0)=0$, and to satisfy the following condition:\nfor sufficiently large $t$, the function $t^{-\\frac{n+2}{n-2}}f(t)$ is nonincreasing and $f(t)\\geqslant ct^p$ for some $p\\geqslant \\frac{n}{n-2}$.\nThey showed that the solution $u$ is radially symmetric and decreasing.\n\nThe moving plane method was later extended to the study of higher order equations on $\\mathbb{R}^n$.\nThe main challenge here is that the moving plane method heavily depends on the maximum principle, which does not always hold for higher order operators.\nTo overcome this obstacle, Chen, Li and Ou \\cites{CLO,ChenLiOu2} developed a powerful moving plane method for integral equations, obtaining the symmetry of solutions for higher order and even fractional order equations. More precisely, they proved that for the following equation.\n$$\n (- \\Delta )^{\\frac{\\alpha}{2}} u = u^{\\frac{n+\\alpha}{n - \\alpha}}\n$$\non $\\mathbb{R}^n$, every positive regular solution (i.e. locally $L^{\\frac{2n}{n-\\alpha}}$ solutions) $u$\nis radially symmetric and decreasing about some point $x_0$ and therefore assumes the form (up to some dilations):\n$$\n u(x) = \\frac{1}{\\left( a + b |x - x_0|^2 \\right)^{\\frac{n-\\alpha}{2}}}.\n$$\nTheir work completely classifies all the critical points of the functional corresponding to the Hardy-Littlewood-Sobolev\ninequalities of order $\\alpha$, whose sharp constant was previously obtained by Lieb \\cite{Lieb1}.\n\nThe method of moving planes is a powerful tool for proving symmetry and monotonicity properties of solutions to partial differential equations. It was first introduced by Alexandrov \\cite{Alexandrov} in the context of geometric problems and later adapted by Serrin, Gidas, Ni, and Nirenberg \\cites{K-P1, K-P2, CGS} to study symmetry properties of solutions to elliptic PDEs in Euclidean spaces. The method has since been extended to various settings, including Riemannian manifolds. In recent years, a direct method of moving planes was developed by Chen, Li and Li \\cite{CLL} to deal with fractional Laplacians and other nonlocal operators. This method has been successfully applied to various problems involving fractional Laplacians in $\\mathbb{R}^n$ and other settings (see, e.g., \\cites{ChenWu-ANS, ChenHuMa-ANS, LiY, Liao-ANS, LiuM-ANS, GuoMaZhang-ANS}).\n\nIn the present paper, we aim to study positive solutions to the following fractional order equations on $\\mathbb{H}^n$.\n\\begin{equation}\n(-\\Delta_{\\mathbb{H}^n})^s u = u^p(x), \\quad x \\in \\mathbb{H}^n.\n\\end{equation}\nHere $0 < s < 1$ and $p > 1$. We start by reviewing some work in the Euclidean space. In \\cite{BCPS}, among other results, the authors considered the properties of the positive solutions for \n\\begin{equation*}\n (-\\Delta)^s u = u^p, \\quad x \\in \\mathbb{R}^n,\n\\end{equation*}\nand first used the above extension method to reduce the nonlocal problem into a local one for $U(x,y)$ in one higher dimensional half space $\\mathbb{R}^n \\times [0, \\infty)$, then applied the method of moving planes to show the symmetry of $U(x, y)$ in $x$, and hence derived the non-existence in the subcritical case.\n\nLater, Chen, Li and Li \\cite{CLL} developed a direct method of moving planes for the fractional Laplacian in $\\mathbb{R}^n$, without using the extension method. They established several maximum principles for antisymmetric functions and applied the method of moving planes to obtain symmetry and nonexistence results for positive solutions of semilinear equations involving the fractional Laplacian in $\\mathbb{R}^n$. \nIn \\cite{CLZ}, Chen, Li and Zhang introduced another direct method - the method of moving spheres for the fractional Laplacian, which is more powerful than the method of moving planes. The method of moving spheres can be used to capture the solutions directly rather than going through the usual procedure of proving radial symmetry of solutions and then classifying radial solutions. It is worth noticing that both their approaches relies on the Kelvin transform. However, on hyperbolic spaces, the Kelvin transform for the fractional Laplacian is not available due to the lack of conformal invariance. To overcome this difficulty, we will make use of the asymptotic behavior of solutions and the Hardy-Littlewood-Sobolev inequality on hyperbolic spaces to start the moving plane process.\n\nWe point out here that the moving plane method on $\\mathbb{H}^n$ was first seen in the work of Kumaresan-Prajapat \\cites{K-P1,K-P2}, where they established the analog result of Gidas-Ni-Nireneberg type, as well as solved an overdetermined problem on $\\mathbb{H}^n$, which is an analogue of the problem on $\\mathbb{R}^n$ initiated by Serrin \\cite{serrin} (see also \\cite{HLZ}, \\cite{L-Z2}). \nThe hyperbolic space serves as one of the most important models of Riemannian manifolds with constant curvature. For definitions and basic properties of hyperbolic spaces, we refer to Section \\ref{sec:prelim}. The axial symmetry of solutions to a class of integral equations on half spaces was studied by Lu-Zhu in \\cite{L-Z}. More recently, the overdetermined problem of fractional order equations on hyperbolic spaces was studied by Li-Lu-Wang \\cite{LLW2}. The concept of the moving plane method in \\cites{K-P1,K-P2} was further developed in the work of Almeida-Ge \\cite{ADG2} and Almeida-Damascelli-Ge \\cite{ADG1},\nwhere they took advantage of the foliation structure of $\\mathbb{H}^n$ (for details see Section \\ref{sec:prelim}). \nLi, Lu and Yang \\cite{LLY} obtained a higher order symmetry result of the Brezis-Nirenberg problem on the hyperbolic spaces \\cite{LLY}. Recently. with Li and Lu, the author studied general higher order equations \\cite{LLW} by using the moving plane method, and later obtained the classification result using the moving sphere method \\cite{LLW25} on the hyperbolic space.\n\nThe definition of the fractional Laplacian on hyperbolic spaces via the singular integral representation was constructed in \\cites{BGS15}.\n\\begin{definition}\nLet $n \\geqslant 2$ and $0 < s < 1$. The fractional Laplacian on $\\mathbb{H}^n$ is defined by\n\\begin{equation}\n (-\\Delta_{\\mathbb{H}^n})^s u(x) = c_{n,s} \\, \\text{P.V.} \\int_{\\mathbb{H}^n} (u(x) - u(\\xi)) \\, \\mathcal{K}_{n,s}(d(x,\\xi)) \\, d\\xi\n\\end{equation}\nwith the kernel $\\mathcal{K}_{n,s}$ given by\n\\begin{equation}\\label{kernel}\n \\mathcal{K}_{n,s}(\\rho) = C_1 \\left( -\\frac{\\partial_\\rho}{\\sinh \\rho} \\right)^{\\frac{n-1}{2}} \\left( \\rho^{-\\frac{1+2s}{2}} K_{\\frac{1+2s}{2}} \\left( \\frac{n-1}{2} \\rho \\right) \\right)\n\\end{equation}\nwhen $n \\geqslant 3$ is odd and\n\\[\n\\mathcal{K}_{n,s}(\\rho) = C_1 \\int_{\\rho}^{\\infty} \\frac{\\sinh r}{\\sqrt{\\pi}\\sqrt{\\cosh r - \\cosh\\rho}} \n\\left( -\\frac{\\partial_r}{\\sinh r} \\right)^{n/2}\n\\left( r^{-\\frac{1+2s}{2}} K_{\\frac{1+2s}{2}} \\left( \\tfrac{n-1}{2}r \\right) \\right) dr\n\\]\nwhen $n \\geqslant 2$ is even, where\n\\[\nc_{n,s} = 2 \\, \\frac{4\\sqrt{2} \\,\\Gamma(n+s)}{3\\,\\Gamma(n/2)\\,\\Gamma(-s)}, \\quad\nC_1 = \\frac{1}{2^{n-2+2s}\\,\\Gamma\\!\\left( \\frac{n-1}{2} \\right) \\Gamma\\!\\left( \\frac{1+2s}{2} \\right)},\n\\]\nand $K_\\nu$ is the modified Bessel function of the second kind.\n\\end{definition}\nWe consider the following classic semilinear equation with power nonlinearity involving the fractional Laplacian:\n\\begin{equation}\\label{main_equation_1}\n(-\\Delta_{\\mathbb{H}^n})^s u = u^p(x), \\quad x \\in \\mathbb{H}^n.\n\\end{equation}\nWe say that a nonnegative function $u \\in H^s(\\mathbb{H}^n)$ is a weak solution of \\eqref{main_equation_1} if for any test function $\\varphi \\in C_c^\\infty(\\mathbb{H}^n)$, there holds\n\\begin{align*}\n \\int_{\\mathbb{H}^n} u(x) (-\\Delta_{\\mathbb{H}^n})^s \\varphi(x) dV_x &= \\int_{\\mathbb{H}^n} u(x)^p \\varphi(x) dV_x.\n\\end{align*}", "context": "The fractional Laplacians arise in various fields of mathematics and physics, such as probability, finance, physics, and fluid dynamics. For instance, they arise naturally in the study of stochastic processes with jumps, and more precisely in L\\'evy processes \\cite{Levy}, which extend the concept of Brownian motion. Fractional order operators also appear in the study of conformal geometry and partial differential equations. In $\\mathbb{R}^n$, the fractional Laplacian is a nonlocal pseudo-differential operator, assuming the form\n\\[\n(-\\Delta)^s u(x) = c_{n,s} \\, \\text{P.V.} \\int_{\\mathbb{R}^n} \\frac{u(x) - u(y)}{|x-y|^{n+2s}} \\, dy, \\quad 0 < s < 1,\n\\]\nwhere P.V. stands for the Cauchy principal value and $c_{n,s}$ is a normalization constant. Equivalently, $(-\\Delta)^{\\frac{\\alpha}{2}}$ can be defined in terms of Fourier transform:\n\\begin{equation*}\n (-\\Delta)^{s}u(x)=\\mathcal{F}^{-1}[|\\xi|^{2s} \\mathcal{F}{u}(\\xi)](x).\n\\end{equation*}\n\nThe moving plane method was later extended to the study of higher order equations on $\\mathbb{R}^n$.\nThe main challenge here is that the moving plane method heavily depends on the maximum principle, which does not always hold for higher order operators.\nTo overcome this obstacle, Chen, Li and Ou \\cites{CLO,ChenLiOu2} developed a powerful moving plane method for integral equations, obtaining the symmetry of solutions for higher order and even fractional order equations. More precisely, they proved that for the following equation.\n$$\n (- \\Delta )^{\\frac{\\alpha}{2}} u = u^{\\frac{n+\\alpha}{n - \\alpha}}\n$$\non $\\mathbb{R}^n$, every positive regular solution (i.e. locally $L^{\\frac{2n}{n-\\alpha}}$ solutions) $u$\nis radially symmetric and decreasing about some point $x_0$ and therefore assumes the form (up to some dilations):\n$$\n u(x) = \\frac{1}{\\left( a + b |x - x_0|^2 \\right)^{\\frac{n-\\alpha}{2}}}.\n$$\nTheir work completely classifies all the critical points of the functional corresponding to the Hardy-Littlewood-Sobolev\ninequalities of order $\\alpha$, whose sharp constant was previously obtained by Lieb \\cite{Lieb1}.\n\nWe point out here that the moving plane method on $\\mathbb{H}^n$ was first seen in the work of Kumaresan-Prajapat \\cites{K-P1,K-P2}, where they established the analog result of Gidas-Ni-Nireneberg type, as well as solved an overdetermined problem on $\\mathbb{H}^n$, which is an analogue of the problem on $\\mathbb{R}^n$ initiated by Serrin \\cite{serrin} (see also \\cite{HLZ}, \\cite{L-Z2}). \nThe hyperbolic space serves as one of the most important models of Riemannian manifolds with constant curvature. For definitions and basic properties of hyperbolic spaces, we refer to Section \\ref{sec:prelim}. The axial symmetry of solutions to a class of integral equations on half spaces was studied by Lu-Zhu in \\cite{L-Z}. More recently, the overdetermined problem of fractional order equations on hyperbolic spaces was studied by Li-Lu-Wang \\cite{LLW2}. The concept of the moving plane method in \\cites{K-P1,K-P2} was further developed in the work of Almeida-Ge \\cite{ADG2} and Almeida-Damascelli-Ge \\cite{ADG1},\nwhere they took advantage of the foliation structure of $\\mathbb{H}^n$ (for details see Section \\ref{sec:prelim}). \nLi, Lu and Yang \\cite{LLY} obtained a higher order symmetry result of the Brezis-Nirenberg problem on the hyperbolic spaces \\cite{LLY}. Recently. with Li and Lu, the author studied general higher order equations \\cite{LLW} by using the moving plane method, and later obtained the classification result using the moving sphere method \\cite{LLW25} on the hyperbolic space.\n\nThe definition of the fractional Laplacian on hyperbolic spaces via the singular integral representation was constructed in \\cites{BGS15}.\n\\begin{definition}\nLet $n \\geqslant 2$ and $0 < s < 1$. The fractional Laplacian on $\\mathbb{H}^n$ is defined by\n\\begin{equation}\n (-\\Delta_{\\mathbb{H}^n})^s u(x) = c_{n,s} \\, \\text{P.V.} \\int_{\\mathbb{H}^n} (u(x) - u(\\xi)) \\, \\mathcal{K}_{n,s}(d(x,\\xi)) \\, d\\xi\n\\end{equation}\nwith the kernel $\\mathcal{K}_{n,s}$ given by\n\\begin{equation}\\label{kernel}\n \\mathcal{K}_{n,s}(\\rho) = C_1 \\left( -\\frac{\\partial_\\rho}{\\sinh \\rho} \\right)^{\\frac{n-1}{2}} \\left( \\rho^{-\\frac{1+2s}{2}} K_{\\frac{1+2s}{2}} \\left( \\frac{n-1}{2} \\rho \\right) \\right)\n\\end{equation}\nwhen $n \\geqslant 3$ is odd and\n\\[\n\\mathcal{K}_{n,s}(\\rho) = C_1 \\int_{\\rho}^{\\infty} \\frac{\\sinh r}{\\sqrt{\\pi}\\sqrt{\\cosh r - \\cosh\\rho}} \n\\left( -\\frac{\\partial_r}{\\sinh r} \\right)^{n/2}\n\\left( r^{-\\frac{1+2s}{2}} K_{\\frac{1+2s}{2}} \\left( \\tfrac{n-1}{2}r \\right) \\right) dr\n\\]\nwhen $n \\geqslant 2$ is even, where\n\\[\nc_{n,s} = 2 \\, \\frac{4\\sqrt{2} \\,\\Gamma(n+s)}{3\\,\\Gamma(n/2)\\,\\Gamma(-s)}, \\quad\nC_1 = \\frac{1}{2^{n-2+2s}\\,\\Gamma\\!\\left( \\frac{n-1}{2} \\right) \\Gamma\\!\\left( \\frac{1+2s}{2} \\right)},\n\\]\nand $K_\\nu$ is the modified Bessel function of the second kind.\n\\end{definition}\nWe consider the following classic semilinear equation with power nonlinearity involving the fractional Laplacian:\n\\begin{equation}\\label{main_equation_1}\n(-\\Delta_{\\mathbb{H}^n})^s u = u^p(x), \\quad x \\in \\mathbb{H}^n.\n\\end{equation}\nWe say that a nonnegative function $u \\in H^s(\\mathbb{H}^n)$ is a weak solution of \\eqref{main_equation_1} if for any test function $\\varphi \\in C_c^\\infty(\\mathbb{H}^n)$, there holds\n\\begin{align*}\n \\int_{\\mathbb{H}^n} u(x) (-\\Delta_{\\mathbb{H}^n})^s \\varphi(x) dV_x &= \\int_{\\mathbb{H}^n} u(x)^p \\varphi(x) dV_x.\n\\end{align*}", "full_context": "The fractional Laplacians arise in various fields of mathematics and physics, such as probability, finance, physics, and fluid dynamics. For instance, they arise naturally in the study of stochastic processes with jumps, and more precisely in L\\'evy processes \\cite{Levy}, which extend the concept of Brownian motion. Fractional order operators also appear in the study of conformal geometry and partial differential equations. In $\\mathbb{R}^n$, the fractional Laplacian is a nonlocal pseudo-differential operator, assuming the form\n\\[\n(-\\Delta)^s u(x) = c_{n,s} \\, \\text{P.V.} \\int_{\\mathbb{R}^n} \\frac{u(x) - u(y)}{|x-y|^{n+2s}} \\, dy, \\quad 0 < s < 1,\n\\]\nwhere P.V. stands for the Cauchy principal value and $c_{n,s}$ is a normalization constant. Equivalently, $(-\\Delta)^{\\frac{\\alpha}{2}}$ can be defined in terms of Fourier transform:\n\\begin{equation*}\n (-\\Delta)^{s}u(x)=\\mathcal{F}^{-1}[|\\xi|^{2s} \\mathcal{F}{u}(\\xi)](x).\n\\end{equation*}\n\nThe moving plane method was later extended to the study of higher order equations on $\\mathbb{R}^n$.\nThe main challenge here is that the moving plane method heavily depends on the maximum principle, which does not always hold for higher order operators.\nTo overcome this obstacle, Chen, Li and Ou \\cites{CLO,ChenLiOu2} developed a powerful moving plane method for integral equations, obtaining the symmetry of solutions for higher order and even fractional order equations. More precisely, they proved that for the following equation.\n$$\n (- \\Delta )^{\\frac{\\alpha}{2}} u = u^{\\frac{n+\\alpha}{n - \\alpha}}\n$$\non $\\mathbb{R}^n$, every positive regular solution (i.e. locally $L^{\\frac{2n}{n-\\alpha}}$ solutions) $u$\nis radially symmetric and decreasing about some point $x_0$ and therefore assumes the form (up to some dilations):\n$$\n u(x) = \\frac{1}{\\left( a + b |x - x_0|^2 \\right)^{\\frac{n-\\alpha}{2}}}.\n$$\nTheir work completely classifies all the critical points of the functional corresponding to the Hardy-Littlewood-Sobolev\ninequalities of order $\\alpha$, whose sharp constant was previously obtained by Lieb \\cite{Lieb1}.\n\nWe point out here that the moving plane method on $\\mathbb{H}^n$ was first seen in the work of Kumaresan-Prajapat \\cites{K-P1,K-P2}, where they established the analog result of Gidas-Ni-Nireneberg type, as well as solved an overdetermined problem on $\\mathbb{H}^n$, which is an analogue of the problem on $\\mathbb{R}^n$ initiated by Serrin \\cite{serrin} (see also \\cite{HLZ}, \\cite{L-Z2}). \nThe hyperbolic space serves as one of the most important models of Riemannian manifolds with constant curvature. For definitions and basic properties of hyperbolic spaces, we refer to Section \\ref{sec:prelim}. The axial symmetry of solutions to a class of integral equations on half spaces was studied by Lu-Zhu in \\cite{L-Z}. More recently, the overdetermined problem of fractional order equations on hyperbolic spaces was studied by Li-Lu-Wang \\cite{LLW2}. The concept of the moving plane method in \\cites{K-P1,K-P2} was further developed in the work of Almeida-Ge \\cite{ADG2} and Almeida-Damascelli-Ge \\cite{ADG1},\nwhere they took advantage of the foliation structure of $\\mathbb{H}^n$ (for details see Section \\ref{sec:prelim}). \nLi, Lu and Yang \\cite{LLY} obtained a higher order symmetry result of the Brezis-Nirenberg problem on the hyperbolic spaces \\cite{LLY}. Recently. with Li and Lu, the author studied general higher order equations \\cite{LLW} by using the moving plane method, and later obtained the classification result using the moving sphere method \\cite{LLW25} on the hyperbolic space.\n\nThe definition of the fractional Laplacian on hyperbolic spaces via the singular integral representation was constructed in \\cites{BGS15}.\n\\begin{definition}\nLet $n \\geqslant 2$ and $0 < s < 1$. The fractional Laplacian on $\\mathbb{H}^n$ is defined by\n\\begin{equation}\n (-\\Delta_{\\mathbb{H}^n})^s u(x) = c_{n,s} \\, \\text{P.V.} \\int_{\\mathbb{H}^n} (u(x) - u(\\xi)) \\, \\mathcal{K}_{n,s}(d(x,\\xi)) \\, d\\xi\n\\end{equation}\nwith the kernel $\\mathcal{K}_{n,s}$ given by\n\\begin{equation}\\label{kernel}\n \\mathcal{K}_{n,s}(\\rho) = C_1 \\left( -\\frac{\\partial_\\rho}{\\sinh \\rho} \\right)^{\\frac{n-1}{2}} \\left( \\rho^{-\\frac{1+2s}{2}} K_{\\frac{1+2s}{2}} \\left( \\frac{n-1}{2} \\rho \\right) \\right)\n\\end{equation}\nwhen $n \\geqslant 3$ is odd and\n\\[\n\\mathcal{K}_{n,s}(\\rho) = C_1 \\int_{\\rho}^{\\infty} \\frac{\\sinh r}{\\sqrt{\\pi}\\sqrt{\\cosh r - \\cosh\\rho}} \n\\left( -\\frac{\\partial_r}{\\sinh r} \\right)^{n/2}\n\\left( r^{-\\frac{1+2s}{2}} K_{\\frac{1+2s}{2}} \\left( \\tfrac{n-1}{2}r \\right) \\right) dr\n\\]\nwhen $n \\geqslant 2$ is even, where\n\\[\nc_{n,s} = 2 \\, \\frac{4\\sqrt{2} \\,\\Gamma(n+s)}{3\\,\\Gamma(n/2)\\,\\Gamma(-s)}, \\quad\nC_1 = \\frac{1}{2^{n-2+2s}\\,\\Gamma\\!\\left( \\frac{n-1}{2} \\right) \\Gamma\\!\\left( \\frac{1+2s}{2} \\right)},\n\\]\nand $K_\\nu$ is the modified Bessel function of the second kind.\n\\end{definition}\nWe consider the following classic semilinear equation with power nonlinearity involving the fractional Laplacian:\n\\begin{equation}\\label{main_equation_1}\n(-\\Delta_{\\mathbb{H}^n})^s u = u^p(x), \\quad x \\in \\mathbb{H}^n.\n\\end{equation}\nWe say that a nonnegative function $u \\in H^s(\\mathbb{H}^n)$ is a weak solution of \\eqref{main_equation_1} if for any test function $\\varphi \\in C_c^\\infty(\\mathbb{H}^n)$, there holds\n\\begin{align*}\n \\int_{\\mathbb{H}^n} u(x) (-\\Delta_{\\mathbb{H}^n})^s \\varphi(x) dV_x &= \\int_{\\mathbb{H}^n} u(x)^p \\varphi(x) dV_x.\n\\end{align*}\n\n\\begin{theorem} \\label{thm-asymptotics}\nFor $s\\in(0,1)\\setminus\\mathbb{N}$, there holds that\n\\begin{itemize}\n\\item For $n\\geq 3$ odd,\n\\begin{equation}\\label{G1}\n G_s(\\rho)=\\alpha_\\gamma \\left(\\frac{\\partial_\\rho}{\\sinh\\rho}\\right)^\\frac{n-1}{2}\\rho^{-\\frac{1}{2}+s}K_{-\\frac{1}{2}+s}\\left(\\tfrac{n-1}{2} \\rho\\right),\\end{equation}\n\n\\textbf{Theorem.} \\textit{Let $0<\\lambdap$.\nThen the weak solution to \n\\begin{equation}\\label{diff_eq}\n (-\\Delta_{\\mathbb{H}^n})^s u = u^p \n\\end{equation}\nsatisfies the integral equation\n\\begin{equation}\\label{int_eq}\n u(x) = \\int_{\\mathbb{H}^n} \\mathcal K_s(\\rho(x,y)) u(y)^p dV_y,\n\\end{equation}\nwhere $\\mathcal K_s(\\rho)$ is the fractional kernel on $\\mathbb{H}^n$ defined in Section \\eqref{kernel}.\nConversely, if $u$ solves the integral equation \\eqref{int_eq}, then $u$ is a solution of the \\eqref{diff_eq}.\n\\end{proposition}\n\n\\begin{proof}\nBy Helgason-Fourier transform, we have for any $\\varphi\\in C^\\infty_c(\\hn)$:\n\\begin{align*}\n \\widehat{(-\\Delta_{\\mathbb{H}^n})^s \\varphi}(\\lambda,\\theta) &= \\left(\\lambda^2 + \\tfrac{(n-1)^2}{4}\\right)^s \\hat{\\varphi}(\\lambda,\\theta),\n\\end{align*}\nand \n\\begin{align}\\label{test_laplacian}\n (-\\Delta_{\\mathbb H^n})^s\\varphi(x)=\\int_{-\\infty}^\\infty\\int_{\\mathbb S^{n-1}}\n\\Bigl(\\lambda^2+\\frac{(n-1)^2}{4}\\Bigr)^s\\widehat{\\varphi}(\\lambda,\\theta)\n\\,e_{\\lambda,\\theta}(x)\n\\,|c(\\lambda)|^{-2}\\,d\\theta\\,d\\lambda.\n\\end{align}\nRecall the definition of weak solution of \\eqref{diff_eq}, we have\n\\begin{align*}\n \\int_{\\mathbb{H}^n} u(x) (-\\Delta_{\\mathbb{H}^n})^s \\varphi(x) dV_x &= \\int_{\\mathbb{H}^n} u(x)^p \\varphi(x) dV_x.\n\\end{align*}\nInserting \\eqref{test_laplacian} to the left side, we obtain\n\\begin{align*}\n \\int_{\\mathbb H^n} u(x)(-\\Delta_{\\mathbb H^n})^s\\varphi(x)\\,dV_x\n&=\n\\int_{\\mathbb H^n} u(x)\\int_{-\\infty}^\\infty \\int_{\\mathbb S^{n-1}}\n\\Bigl(\\lambda^2+\\tfrac{(n-1)^2}{4}\\Bigr)^s\n\\widehat{\\varphi}(\\lambda,\\theta)e_{\\lambda,\\theta}(x)\n|c(\\lambda)|^{-2}\\,d\\theta\\,d\\lambda\n\\,dV_x\\\\\n&=\n\\int_{-\\infty}^\\infty \\int_{\\mathbb S^{n-1}}\n\\Bigl(\\lambda^2+\\tfrac{(n-1)^2}{4}\\Bigr)^s\n\\widehat{\\varphi}(\\lambda,\\theta)\n\\left[\n\\int_{\\mathbb H^n}\nu(x)e_{\\lambda,\\theta}(x)\\,dV_x\n\\right]\n|c(\\lambda)|^{-2}\\,d\\theta\\,d\\lambda.\n\\end{align*}\n\nFor the far away part, we choose $a\\in(1,\\frac{q}{p})$, and let $a'$ denote the conjugate of $a$. By H\\\"older's inequality,\n\\[\n\\left|\\int_{\\mathbb H^n\\setminus B_R(o)}G(\\rho(x,y))u(y)^p\\,dV_y\\right| \\leq \\|G(\\rho(x,\\cdot))\\|_{L^{a'}(\\mathbb H^n\\setminus B_R)}\\;\\|u^p\\|_{L^{a}(\\mathbb H^n\\setminus B_R)}.\n\\]\n$\\|u^p\\|_{L^{a}(\\mathbb H^n\\setminus B_R)}$ is clearly finite since $ap0$. Thus, $\\|G(\\rho(x,\\cdot))\\|_{L^{a'}(\\mathbb H^n\\setminus B_R)}$ can be made arbitrarily small for large enough $R$. The proof is now complete.\n\\end{proof}\nFrom the above estimate at infinity, the moving plane argument in the subcritical case applies here as well, and we conclude that $u$ is radially symmetric about some point in $\\hn$.\n\nThen we introduce a maximum principle for anti-symmetric functions.\n\\begin{theorem}\n\\label{MP_anti}\n Let $\\Omega$ be a bounded domain in $\\hn$. Assume that $u \\in L^\\alpha \\cap C^{1,1}_{loc}(\\Omega)$ and is lower semi-continuous on $\\bar{\\Omega}$ . If\n \\begin{equation*}\n \\begin{cases}\n (-\\Delta_{\\hn})^s u + c(x)u \\geq 0, &\\text{ in } \\Omega,\\\\\n u \\geq 0, &\\text{ in } \\Sigma,\\\\\n u(x) = -u(x^\\lambda), &\\text{ in } \\Sigma,\n \\end{cases}\n \\end{equation*}\n where $c(x)$ is bounded from below in $\\Omega$. Then \n \\begin{equation}\\label{mp_anti_u}\n u \\geq 0 \\text{ in } \\Omega.\n \\end{equation}\n If $u = 0$ at some point in $\\Omega$, then $u(x) = 0$ almost everywhere in $\\hn$.\n\\end{theorem}\nThe next result is a maximum principle in narrow regions.\n\\begin{theorem}\n\\label{narrow_region}\n Let $U_\\lambda,\\, \\Sigma_\\lambda, x^\\lambda$ be defined as in Section \\ref{sec:prelim}. Assume that $u \\in L^\\alpha \\cap C^{1,1}_{loc}(\\Omega)$ and is lower semi-continuous on $\\bar{\\Omega}$ . If\n Let $\\Omega$ be an bounded narrow region in $\\Sigma_\\lambda$, which is contained in $\\{x\\mid \\lambda-lp$ is a nonnegative weak solution of\n\\begin{equation}\\label{main_equation_1}\n(-\\Delta_{\\hn})^s u = u^p(x), \\quad x \\in \\hn.\n\\end{equation}\nThen\n\\begin{enumerate}\n \\item If $1 < p < \\frac{n+2s}{n-2s}$ and $u\\in L^{p+1}(\\mathbb{H}^n)$ , then $u\\equiv 0$. \n \\item If $p = \\frac{n+2s}{n-2s}$ and $u\\in L^q$, where $q>p$, then there exists a point $x_0 \\in \\mathbb{H}^n$ such that $u$ is radially symmetric and nondecreasing about $x_0$.\n\\end{enumerate}", "theorem_type": ["Implication", "Nonexistence"], "mcq": {"question": "Let $n\\ge 2$ and $0p$ is a nonnegative weak solution of\n\\[\n(-\\Delta_{\\mathbb H^n})^s u = u^p \\quad \\text{in } \\mathbb H^n.\n\\]\nWhich of the following conclusions is valid under these hypotheses? (Here, \"radially symmetric and nondecreasing about $x_0$\" means that $u(x)$ depends only on $d(x,x_0)$ and is a nondecreasing function of that distance.)", "correct_choice": {"label": "A", "text": "Two cases occur: (i) if $1p$ satisfies $u\\equiv 0$ on $\\mathbb H^n$; (ii) if $p=\\frac{n+2s}{n-2s}$ and $u\\in L^{p+1}(\\mathbb H^n)$, then there exists a point $x_0\\in\\mathbb H^n$ such that $u$ is radially symmetric and nondecreasing about $x_0$."}, {"label": "E", "text": "Two cases occur: (i) if $1p$, then necessarily $u\\equiv 0$ on $\\mathbb H^n$."}], "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 boundary in the vanishing statement", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the nondecreasing monotonicity conclusion in the critical case", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "extra integrability assumptions assigned to the wrong case", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "interchanged the conclusions of the subcritical and critical cases", "template_used": "wildcard"}]}} +{"id": "2601.12140v1", "paper_link": "http://arxiv.org/abs/2601.12140v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "\\label{thm1}\nLet $n \\geqslant 2$, $0 < s < 1$. Suppose that $u\\in L^q(\\mathbb{H}^n), q>p$ is a nonnegative weak solution of \\eqref{main_equation_1}. Then\n\\begin{enumerate}\n \\item If $1 < p < \\frac{n+2s}{n-2s}$ and $u\\in L^{p+1}(\\mathbb{H}^n)$ , then $u\\equiv 0$. \n \\item If $p = \\frac{n+2s}{n-2s}$ and $u\\in L^q$, where $q>p$, then there exists a point $x_0 \\in \\mathbb{H}^n$ such that $u$ is radially symmetric and nondecreasing about $x_0$.\n\\end{enumerate}", "start_pos": 17666, "end_pos": 18132, "label": "thm1"}, "ref_dict": {"main_equation_1": "\\begin{equation}\\label{main_equation_1}\n(-\\Delta_{\\hn})^s u = u^p(x), \\quad x \\in \\hn.\n\\end{equation}", "lemma-Bessel": "\\begin{lemma}[\\cite{AS64}, or Lemma 2.3 of \\cite{BGS15}]\\label{lemma-Bessel}\nThe solution of the ODE\n\\begin{equation*}\n\\label{Bessel1}\\partial_{ss} \\varphi+ \\frac{\\alpha}{s}\\, \\partial_s \\varphi -\\varphi = 0. \\end{equation*}\nmay be written as $\\varphi(s)=s^\\nu \\psi(s)$, for $\\alpha=1-2\\nu$, where $\\psi$ solves the is the well known Bessel equation\n\\begin{equation}\\label{Bessel2}\ns^2\\psi''+s\\psi'-(s^2+\\nu^2)\\psi=0.\n\\end{equation}\nIn addition, \\eqref{Bessel2} has two linearly independent solutions, $I_\\nu,K_\\nu$, which are the modified Bessel functions; their asymptotic behavior is given precisely by\n\\begin{align*}\nI_\\nu(s)&\\sim \\frac{1}{\\Gamma(\\nu+1)}\\left(\\frac{s}{2}\\right)^\\nu\\left( 1+\\frac{s^2}{4(\\nu+1)}+\\frac{s^4}{32(\\nu+1)(\\nu+2)}+\\ldots\\right),\\\\\nK_\\nu(s)&\\sim \\frac{\\Gamma(\\nu)}{2}\\left(\\frac{2}{s}\\right)^{\\nu}\n\\left( 1+\\frac{s^2}{4(1-\\nu)}+\\frac{s^4}{32(1-\\nu)(2-\\nu)}+\\ldots\\right)\n\\\\&\\quad+\\frac{\\Gamma(-\\nu)}{2}\\left(\\frac{s}{2}\\right)^\\nu\\left( 1+\\frac{s^2}{4(\\nu+1)}+\\frac{s^4}{32(\\nu+1)(\\nu+2)}+\\ldots\\right),\n\\end{align*}\n\nfor $s\\to 0^+$, $\\nu\\not\\in\\mathbb Z$. And when $s\\to +\\infty$,\n\\begin{align*}\\label{asymptotic2}\n I_\\nu(s)\\sim \\frac{1}{\\sqrt{2\\pi s}}e^s\\left(1-\\frac{4\\nu^2-1}{8s}+\\frac{(4\\nu^2-1)(4\\nu^2-9)}{2!(8s)^2}-\\ldots \\right),\\\\\n K_\\nu(s)\\sim \\sqrt{\\frac{\\pi}{2s}}e^{-s}\\left(1+\\frac{4\\nu^2-1}{8s}+\\frac{(4\\nu^2-1)(4\\nu^2-9)}{2!(8s)^2}+\\ldots \\right).\n\\end{align*}\n\\end{lemma}", "thm1": "\\begin{theorem}\\label{thm1}\nLet $n \\geq 2$, $0 < s < 1$. Suppose that $u\\in L^q(\\hn), q>p$ is a nonnegative weak solution of \\eqref{main_equation_1}. Then\n\\begin{enumerate}\n \\item If $1 < p < \\frac{n+2s}{n-2s}$ and $u\\in L^{p+1}(\\hn)$ , then $u\\equiv 0$. \n \\item If $p = \\frac{n+2s}{n-2s}$ and $u\\in L^q$, where $q>p$, then there exists a point $x_0 \\in \\mathbb{H}^n$ such that $u$ is radially symmetric and nondecreasing about $x_0$.\n\\end{enumerate}\n\\end{theorem}"}, "pre_theorem_intro_text_len": 12726, "pre_theorem_intro_text": "The fractional Laplacians arise in various fields of mathematics and physics, such as probability, finance, physics, and fluid dynamics. For instance, they arise naturally in the study of stochastic processes with jumps, and more precisely in L\\'evy processes \\cite{Levy}, which extend the concept of Brownian motion. Fractional order operators also appear in the study of conformal geometry and partial differential equations. In $\\mathbb{R}^n$, the fractional Laplacian is a nonlocal pseudo-differential operator, assuming the form\n\\[\n(-\\Delta)^s u(x) = c_{n,s} \\, \\text{P.V.} \\int_{\\mathbb{R}^n} \\frac{u(x) - u(y)}{|x-y|^{n+2s}} \\, dy, \\quad 0 < s < 1,\n\\]\nwhere P.V. stands for the Cauchy principal value and $c_{n,s}$ is a normalization constant. Equivalently, $(-\\Delta)^{\\frac{\\alpha}{2}}$ can be defined in terms of Fourier transform:\n\\begin{equation*}\n (-\\Delta)^{s}u(x)=\\mathcal{F}^{-1}[|\\xi|^{2s} \\mathcal{F}{u}(\\xi)](x).\n\\end{equation*}\n\nThe non-locality of the fractional Laplacian makes it usually difficult to investigate. To circumvent this difficulty, Caffarelli and Silvestre \\cite{CS07} introduced the extension method that reduced this nonlocal problem into a local one in higher dimensions. For a function $u : \\mathbb{R}^n \\to \\mathbb{R}$, we construct the extension $U : \\mathbb{R}^n \\times [0, +\\infty) \\to \\mathbb{R}, U = U (x, y)$, as the solution of the equation\n\\begin{equation}\n\\begin{cases}\n\\operatorname{div}(y^{1-2s} \\nabla U(x,y)) = 0, & (x,y) \\in \\mathbb{R}^n \\times (0,\\infty), \\\\\nU(x,0) = u(x), & x \\in \\mathbb{R}^n.\n\\end{cases}\n\\end{equation}\n\nThe fractional Laplacian is then recovered by:\n\\begin{equation}\n(-\\Delta)^s u(x) = - C_s \\lim_{y \\to 0^+} y^{1-2s} \\frac{\\partial U}{\\partial y}(x,y),\n\\end{equation}\nwhere $C_s$ is a normalization constant depending on $s$.\n\nThe fractional (as well as integer) powers of Laplacian that enjoy the conformal property can also be defined on Riemannian manifolds. Graham and Zworski \\cite{GZ03} studied the connection between scattering matrices on conformally compact asymptotically Einstein manifolds and conformally invariant objects on their boundaries at infinity. Let $(X^{n+1}, g^+)$ be a conformally compact Einstein manifold with conformal infinity $(M, [\\hat{g}])$. For any defining function $\\rho$ of $M$, we can write $\\bar{g} = \\rho^2 g^+$, which extends to a metric on $\\bar{X} = X \\cup M$. Given $f \\in C^\\infty(M)$, consider the generalized eigenvalue problem\n\\begin{equation*}\n -\\Delta_{g^+} u - s(n-s) u = 0 \\quad \\text{ in } X,\n\\end{equation*}\nwith the asymptotic expansion near $M$:\n\\begin{equation*}\n u = F \\rho^{n-s} + G \\rho^s, \\quad F, G \\in C^\\infty(\\bar{X}), \\quad F|_{\\rho=0} = f.\n\\end{equation*}\nThe scattering operator $S(\\lambda)$ is defined by $S(\\lambda) F = G$.\nIn \\cite{CG11}, Chang and Gonz\\'alez showed that the fractional order operators defined via the scattering operator can be realized as the Dirichlet-to-Neumann map of a degenerate elliptic equation on $X$, generalizing the extension method of Caffarelli-Silvestre to the setting of conformally compact Einstein manifolds. It is also worth mentioning that the fractional GJMS operators on hyperbolic spaces were explicitly calculated by Lu and Yang in \\cite{LY23}. They also pointed out in \\cite{FLY2} that these fractional GJMS operators are not conformally to the fractional Laplacians on the upper half space $\\mathbb{R}^n_+$ nor on the unit ball $B^n$ in $\\mathbb{R}^n$, which one may expect for the integer cases. \n\nAlthough the fractional order operators on general manifolds are rather sophisticated, see e.g. \\cites{GZ03,CG11} and the references therein, there have been numerous accomplishments in the past decades when one considers the integer powers of Laplacians, which we will discuss in the following briefly. In the celebrated work of Gidas-Ni-Nirenberg \\cite{GNN1}, they considered the following boundary value problem on the ball $B_R(0)\\subset{\\mathbb{R}^n}$.\n\\begin{equation*}\n \\begin{cases}\n -\\Delta u = f(u) &\\text{ in } B_R(0)\\\\\n u=0 &\\text{ on } \\partial B_R(0),\n \\end{cases}\n\\end{equation*}\nwhere $f$ is of class $C^1$. They proved that any positive solution $u$ in $C^2(\\overline{B_R(0)})$ is radially symmetric and decreasing.\nTheir approach is the so-called moving plane method, which was initiated by Alexandrov \\cite{Alexandrov} in the 1950s, and further developed by Serrin \\cite{serrin}.\nLater, Gidas-Ni-Nirenberg \\cite{GNN2} studied the equation $-\\Delta u = f(u)$ in the entire space $\\mathbb{R}^n\\setminus\\{0,\\infty\\}, n\\geqslant 3$\nwith two singularities located at the origin and infinity:\n\\begin{align}\n u(x)\\to+\\infty \\quad &\\text{ as } \\quad x\\to0. \\nonumber\\\\\n |x|^{n-2}u(x)\\to+\\infty \\quad &\\text{ as } \\quad x\\to\\infty.\n\\end{align}\nThey showed that the solution is radially symmetric about the origin and decreasing.\nSubsequently, Caffarelli-Gidas-Spruck \\cite{CGS} considered this equation in a punctured ball when $f$ has critical growth.\nTo be more precise, they require the nonlinearity $f(t)$ to be a locally nondecreasing Lipschitz function with $f(0)=0$, and to satisfy the following condition:\nfor sufficiently large $t$, the function $t^{-\\frac{n+2}{n-2}}f(t)$ is nonincreasing and $f(t)\\geqslant ct^p$ for some $p\\geqslant \\frac{n}{n-2}$.\nThey showed that the solution $u$ is radially symmetric and decreasing.\n\nThe moving plane method was later extended to the study of higher order equations on $\\mathbb{R}^n$.\nThe main challenge here is that the moving plane method heavily depends on the maximum principle, which does not always hold for higher order operators.\nTo overcome this obstacle, Chen, Li and Ou \\cites{CLO,ChenLiOu2} developed a powerful moving plane method for integral equations, obtaining the symmetry of solutions for higher order and even fractional order equations. More precisely, they proved that for the following equation.\n$$\n (- \\Delta )^{\\frac{\\alpha}{2}} u = u^{\\frac{n+\\alpha}{n - \\alpha}}\n$$\non $\\mathbb{R}^n$, every positive regular solution (i.e. locally $L^{\\frac{2n}{n-\\alpha}}$ solutions) $u$\nis radially symmetric and decreasing about some point $x_0$ and therefore assumes the form (up to some dilations):\n$$\n u(x) = \\frac{1}{\\left( a + b |x - x_0|^2 \\right)^{\\frac{n-\\alpha}{2}}}.\n$$\nTheir work completely classifies all the critical points of the functional corresponding to the Hardy-Littlewood-Sobolev\ninequalities of order $\\alpha$, whose sharp constant was previously obtained by Lieb \\cite{Lieb1}.\n\nThe method of moving planes is a powerful tool for proving symmetry and monotonicity properties of solutions to partial differential equations. It was first introduced by Alexandrov \\cite{Alexandrov} in the context of geometric problems and later adapted by Serrin, Gidas, Ni, and Nirenberg \\cites{K-P1, K-P2, CGS} to study symmetry properties of solutions to elliptic PDEs in Euclidean spaces. The method has since been extended to various settings, including Riemannian manifolds. In recent years, a direct method of moving planes was developed by Chen, Li and Li \\cite{CLL} to deal with fractional Laplacians and other nonlocal operators. This method has been successfully applied to various problems involving fractional Laplacians in $\\mathbb{R}^n$ and other settings (see, e.g., \\cites{ChenWu-ANS, ChenHuMa-ANS, LiY, Liao-ANS, LiuM-ANS, GuoMaZhang-ANS}).\n\nIn the present paper, we aim to study positive solutions to the following fractional order equations on $\\mathbb{H}^n$.\n\\begin{equation}\n(-\\Delta_{\\mathbb{H}^n})^s u = u^p(x), \\quad x \\in \\mathbb{H}^n.\n\\end{equation}\nHere $0 < s < 1$ and $p > 1$. We start by reviewing some work in the Euclidean space. In \\cite{BCPS}, among other results, the authors considered the properties of the positive solutions for \n\\begin{equation*}\n (-\\Delta)^s u = u^p, \\quad x \\in \\mathbb{R}^n,\n\\end{equation*}\nand first used the above extension method to reduce the nonlocal problem into a local one for $U(x,y)$ in one higher dimensional half space $\\mathbb{R}^n \\times [0, \\infty)$, then applied the method of moving planes to show the symmetry of $U(x, y)$ in $x$, and hence derived the non-existence in the subcritical case.\n\nLater, Chen, Li and Li \\cite{CLL} developed a direct method of moving planes for the fractional Laplacian in $\\mathbb{R}^n$, without using the extension method. They established several maximum principles for antisymmetric functions and applied the method of moving planes to obtain symmetry and nonexistence results for positive solutions of semilinear equations involving the fractional Laplacian in $\\mathbb{R}^n$. \nIn \\cite{CLZ}, Chen, Li and Zhang introduced another direct method - the method of moving spheres for the fractional Laplacian, which is more powerful than the method of moving planes. The method of moving spheres can be used to capture the solutions directly rather than going through the usual procedure of proving radial symmetry of solutions and then classifying radial solutions. It is worth noticing that both their approaches relies on the Kelvin transform. However, on hyperbolic spaces, the Kelvin transform for the fractional Laplacian is not available due to the lack of conformal invariance. To overcome this difficulty, we will make use of the asymptotic behavior of solutions and the Hardy-Littlewood-Sobolev inequality on hyperbolic spaces to start the moving plane process.\n\nWe point out here that the moving plane method on $\\mathbb{H}^n$ was first seen in the work of Kumaresan-Prajapat \\cites{K-P1,K-P2}, where they established the analog result of Gidas-Ni-Nireneberg type, as well as solved an overdetermined problem on $\\mathbb{H}^n$, which is an analogue of the problem on $\\mathbb{R}^n$ initiated by Serrin \\cite{serrin} (see also \\cite{HLZ}, \\cite{L-Z2}). \nThe hyperbolic space serves as one of the most important models of Riemannian manifolds with constant curvature. For definitions and basic properties of hyperbolic spaces, we refer to Section \\ref{sec:prelim}. The axial symmetry of solutions to a class of integral equations on half spaces was studied by Lu-Zhu in \\cite{L-Z}. More recently, the overdetermined problem of fractional order equations on hyperbolic spaces was studied by Li-Lu-Wang \\cite{LLW2}. The concept of the moving plane method in \\cites{K-P1,K-P2} was further developed in the work of Almeida-Ge \\cite{ADG2} and Almeida-Damascelli-Ge \\cite{ADG1},\nwhere they took advantage of the foliation structure of $\\mathbb{H}^n$ (for details see Section \\ref{sec:prelim}). \nLi, Lu and Yang \\cite{LLY} obtained a higher order symmetry result of the Brezis-Nirenberg problem on the hyperbolic spaces \\cite{LLY}. Recently. with Li and Lu, the author studied general higher order equations \\cite{LLW} by using the moving plane method, and later obtained the classification result using the moving sphere method \\cite{LLW25} on the hyperbolic space.\n\nThe definition of the fractional Laplacian on hyperbolic spaces via the singular integral representation was constructed in \\cites{BGS15}.\n\\begin{definition}\nLet $n \\geqslant 2$ and $0 < s < 1$. The fractional Laplacian on $\\mathbb{H}^n$ is defined by\n\\begin{equation}\n (-\\Delta_{\\mathbb{H}^n})^s u(x) = c_{n,s} \\, \\text{P.V.} \\int_{\\mathbb{H}^n} (u(x) - u(\\xi)) \\, \\mathcal{K}_{n,s}(d(x,\\xi)) \\, d\\xi\n\\end{equation}\nwith the kernel $\\mathcal{K}_{n,s}$ given by\n\\begin{equation}\\label{kernel}\n \\mathcal{K}_{n,s}(\\rho) = C_1 \\left( -\\frac{\\partial_\\rho}{\\sinh \\rho} \\right)^{\\frac{n-1}{2}} \\left( \\rho^{-\\frac{1+2s}{2}} K_{\\frac{1+2s}{2}} \\left( \\frac{n-1}{2} \\rho \\right) \\right)\n\\end{equation}\nwhen $n \\geqslant 3$ is odd and\n\\[\n\\mathcal{K}_{n,s}(\\rho) = C_1 \\int_{\\rho}^{\\infty} \\frac{\\sinh r}{\\sqrt{\\pi}\\sqrt{\\cosh r - \\cosh\\rho}} \n\\left( -\\frac{\\partial_r}{\\sinh r} \\right)^{n/2}\n\\left( r^{-\\frac{1+2s}{2}} K_{\\frac{1+2s}{2}} \\left( \\tfrac{n-1}{2}r \\right) \\right) dr\n\\]\nwhen $n \\geqslant 2$ is even, where\n\\[\nc_{n,s} = 2 \\, \\frac{4\\sqrt{2} \\,\\Gamma(n+s)}{3\\,\\Gamma(n/2)\\,\\Gamma(-s)}, \\quad\nC_1 = \\frac{1}{2^{n-2+2s}\\,\\Gamma\\!\\left( \\frac{n-1}{2} \\right) \\Gamma\\!\\left( \\frac{1+2s}{2} \\right)},\n\\]\nand $K_\\nu$ is the modified Bessel function of the second kind.\n\\end{definition}\nWe consider the following classic semilinear equation with power nonlinearity involving the fractional Laplacian:\n\\begin{equation}\\label{main_equation_1}\n(-\\Delta_{\\mathbb{H}^n})^s u = u^p(x), \\quad x \\in \\mathbb{H}^n.\n\\end{equation}\nWe say that a nonnegative function $u \\in H^s(\\mathbb{H}^n)$ is a weak solution of \\eqref{main_equation_1} if for any test function $\\varphi \\in C_c^\\infty(\\mathbb{H}^n)$, there holds\n\\begin{align*}\n \\int_{\\mathbb{H}^n} u(x) (-\\Delta_{\\mathbb{H}^n})^s \\varphi(x) dV_x &= \\int_{\\mathbb{H}^n} u(x)^p \\varphi(x) dV_x.\n\\end{align*}", "context": "The fractional Laplacians arise in various fields of mathematics and physics, such as probability, finance, physics, and fluid dynamics. For instance, they arise naturally in the study of stochastic processes with jumps, and more precisely in L\\'evy processes \\cite{Levy}, which extend the concept of Brownian motion. Fractional order operators also appear in the study of conformal geometry and partial differential equations. In $\\mathbb{R}^n$, the fractional Laplacian is a nonlocal pseudo-differential operator, assuming the form\n\\[\n(-\\Delta)^s u(x) = c_{n,s} \\, \\text{P.V.} \\int_{\\mathbb{R}^n} \\frac{u(x) - u(y)}{|x-y|^{n+2s}} \\, dy, \\quad 0 < s < 1,\n\\]\nwhere P.V. stands for the Cauchy principal value and $c_{n,s}$ is a normalization constant. Equivalently, $(-\\Delta)^{\\frac{\\alpha}{2}}$ can be defined in terms of Fourier transform:\n\\begin{equation*}\n (-\\Delta)^{s}u(x)=\\mathcal{F}^{-1}[|\\xi|^{2s} \\mathcal{F}{u}(\\xi)](x).\n\\end{equation*}\n\nThe moving plane method was later extended to the study of higher order equations on $\\mathbb{R}^n$.\nThe main challenge here is that the moving plane method heavily depends on the maximum principle, which does not always hold for higher order operators.\nTo overcome this obstacle, Chen, Li and Ou \\cites{CLO,ChenLiOu2} developed a powerful moving plane method for integral equations, obtaining the symmetry of solutions for higher order and even fractional order equations. More precisely, they proved that for the following equation.\n$$\n (- \\Delta )^{\\frac{\\alpha}{2}} u = u^{\\frac{n+\\alpha}{n - \\alpha}}\n$$\non $\\mathbb{R}^n$, every positive regular solution (i.e. locally $L^{\\frac{2n}{n-\\alpha}}$ solutions) $u$\nis radially symmetric and decreasing about some point $x_0$ and therefore assumes the form (up to some dilations):\n$$\n u(x) = \\frac{1}{\\left( a + b |x - x_0|^2 \\right)^{\\frac{n-\\alpha}{2}}}.\n$$\nTheir work completely classifies all the critical points of the functional corresponding to the Hardy-Littlewood-Sobolev\ninequalities of order $\\alpha$, whose sharp constant was previously obtained by Lieb \\cite{Lieb1}.\n\nWe point out here that the moving plane method on $\\mathbb{H}^n$ was first seen in the work of Kumaresan-Prajapat \\cites{K-P1,K-P2}, where they established the analog result of Gidas-Ni-Nireneberg type, as well as solved an overdetermined problem on $\\mathbb{H}^n$, which is an analogue of the problem on $\\mathbb{R}^n$ initiated by Serrin \\cite{serrin} (see also \\cite{HLZ}, \\cite{L-Z2}). \nThe hyperbolic space serves as one of the most important models of Riemannian manifolds with constant curvature. For definitions and basic properties of hyperbolic spaces, we refer to Section \\ref{sec:prelim}. The axial symmetry of solutions to a class of integral equations on half spaces was studied by Lu-Zhu in \\cite{L-Z}. More recently, the overdetermined problem of fractional order equations on hyperbolic spaces was studied by Li-Lu-Wang \\cite{LLW2}. The concept of the moving plane method in \\cites{K-P1,K-P2} was further developed in the work of Almeida-Ge \\cite{ADG2} and Almeida-Damascelli-Ge \\cite{ADG1},\nwhere they took advantage of the foliation structure of $\\mathbb{H}^n$ (for details see Section \\ref{sec:prelim}). \nLi, Lu and Yang \\cite{LLY} obtained a higher order symmetry result of the Brezis-Nirenberg problem on the hyperbolic spaces \\cite{LLY}. Recently. with Li and Lu, the author studied general higher order equations \\cite{LLW} by using the moving plane method, and later obtained the classification result using the moving sphere method \\cite{LLW25} on the hyperbolic space.\n\nThe definition of the fractional Laplacian on hyperbolic spaces via the singular integral representation was constructed in \\cites{BGS15}.\n\\begin{definition}\nLet $n \\geqslant 2$ and $0 < s < 1$. The fractional Laplacian on $\\mathbb{H}^n$ is defined by\n\\begin{equation}\n (-\\Delta_{\\mathbb{H}^n})^s u(x) = c_{n,s} \\, \\text{P.V.} \\int_{\\mathbb{H}^n} (u(x) - u(\\xi)) \\, \\mathcal{K}_{n,s}(d(x,\\xi)) \\, d\\xi\n\\end{equation}\nwith the kernel $\\mathcal{K}_{n,s}$ given by\n\\begin{equation}\\label{kernel}\n \\mathcal{K}_{n,s}(\\rho) = C_1 \\left( -\\frac{\\partial_\\rho}{\\sinh \\rho} \\right)^{\\frac{n-1}{2}} \\left( \\rho^{-\\frac{1+2s}{2}} K_{\\frac{1+2s}{2}} \\left( \\frac{n-1}{2} \\rho \\right) \\right)\n\\end{equation}\nwhen $n \\geqslant 3$ is odd and\n\\[\n\\mathcal{K}_{n,s}(\\rho) = C_1 \\int_{\\rho}^{\\infty} \\frac{\\sinh r}{\\sqrt{\\pi}\\sqrt{\\cosh r - \\cosh\\rho}} \n\\left( -\\frac{\\partial_r}{\\sinh r} \\right)^{n/2}\n\\left( r^{-\\frac{1+2s}{2}} K_{\\frac{1+2s}{2}} \\left( \\tfrac{n-1}{2}r \\right) \\right) dr\n\\]\nwhen $n \\geqslant 2$ is even, where\n\\[\nc_{n,s} = 2 \\, \\frac{4\\sqrt{2} \\,\\Gamma(n+s)}{3\\,\\Gamma(n/2)\\,\\Gamma(-s)}, \\quad\nC_1 = \\frac{1}{2^{n-2+2s}\\,\\Gamma\\!\\left( \\frac{n-1}{2} \\right) \\Gamma\\!\\left( \\frac{1+2s}{2} \\right)},\n\\]\nand $K_\\nu$ is the modified Bessel function of the second kind.\n\\end{definition}\nWe consider the following classic semilinear equation with power nonlinearity involving the fractional Laplacian:\n\\begin{equation}\\label{main_equation_1}\n(-\\Delta_{\\mathbb{H}^n})^s u = u^p(x), \\quad x \\in \\mathbb{H}^n.\n\\end{equation}\nWe say that a nonnegative function $u \\in H^s(\\mathbb{H}^n)$ is a weak solution of \\eqref{main_equation_1} if for any test function $\\varphi \\in C_c^\\infty(\\mathbb{H}^n)$, there holds\n\\begin{align*}\n \\int_{\\mathbb{H}^n} u(x) (-\\Delta_{\\mathbb{H}^n})^s \\varphi(x) dV_x &= \\int_{\\mathbb{H}^n} u(x)^p \\varphi(x) dV_x.\n\\end{align*}", "full_context": "The fractional Laplacians arise in various fields of mathematics and physics, such as probability, finance, physics, and fluid dynamics. For instance, they arise naturally in the study of stochastic processes with jumps, and more precisely in L\\'evy processes \\cite{Levy}, which extend the concept of Brownian motion. Fractional order operators also appear in the study of conformal geometry and partial differential equations. In $\\mathbb{R}^n$, the fractional Laplacian is a nonlocal pseudo-differential operator, assuming the form\n\\[\n(-\\Delta)^s u(x) = c_{n,s} \\, \\text{P.V.} \\int_{\\mathbb{R}^n} \\frac{u(x) - u(y)}{|x-y|^{n+2s}} \\, dy, \\quad 0 < s < 1,\n\\]\nwhere P.V. stands for the Cauchy principal value and $c_{n,s}$ is a normalization constant. Equivalently, $(-\\Delta)^{\\frac{\\alpha}{2}}$ can be defined in terms of Fourier transform:\n\\begin{equation*}\n (-\\Delta)^{s}u(x)=\\mathcal{F}^{-1}[|\\xi|^{2s} \\mathcal{F}{u}(\\xi)](x).\n\\end{equation*}\n\nThe moving plane method was later extended to the study of higher order equations on $\\mathbb{R}^n$.\nThe main challenge here is that the moving plane method heavily depends on the maximum principle, which does not always hold for higher order operators.\nTo overcome this obstacle, Chen, Li and Ou \\cites{CLO,ChenLiOu2} developed a powerful moving plane method for integral equations, obtaining the symmetry of solutions for higher order and even fractional order equations. More precisely, they proved that for the following equation.\n$$\n (- \\Delta )^{\\frac{\\alpha}{2}} u = u^{\\frac{n+\\alpha}{n - \\alpha}}\n$$\non $\\mathbb{R}^n$, every positive regular solution (i.e. locally $L^{\\frac{2n}{n-\\alpha}}$ solutions) $u$\nis radially symmetric and decreasing about some point $x_0$ and therefore assumes the form (up to some dilations):\n$$\n u(x) = \\frac{1}{\\left( a + b |x - x_0|^2 \\right)^{\\frac{n-\\alpha}{2}}}.\n$$\nTheir work completely classifies all the critical points of the functional corresponding to the Hardy-Littlewood-Sobolev\ninequalities of order $\\alpha$, whose sharp constant was previously obtained by Lieb \\cite{Lieb1}.\n\nWe point out here that the moving plane method on $\\mathbb{H}^n$ was first seen in the work of Kumaresan-Prajapat \\cites{K-P1,K-P2}, where they established the analog result of Gidas-Ni-Nireneberg type, as well as solved an overdetermined problem on $\\mathbb{H}^n$, which is an analogue of the problem on $\\mathbb{R}^n$ initiated by Serrin \\cite{serrin} (see also \\cite{HLZ}, \\cite{L-Z2}). \nThe hyperbolic space serves as one of the most important models of Riemannian manifolds with constant curvature. For definitions and basic properties of hyperbolic spaces, we refer to Section \\ref{sec:prelim}. The axial symmetry of solutions to a class of integral equations on half spaces was studied by Lu-Zhu in \\cite{L-Z}. More recently, the overdetermined problem of fractional order equations on hyperbolic spaces was studied by Li-Lu-Wang \\cite{LLW2}. The concept of the moving plane method in \\cites{K-P1,K-P2} was further developed in the work of Almeida-Ge \\cite{ADG2} and Almeida-Damascelli-Ge \\cite{ADG1},\nwhere they took advantage of the foliation structure of $\\mathbb{H}^n$ (for details see Section \\ref{sec:prelim}). \nLi, Lu and Yang \\cite{LLY} obtained a higher order symmetry result of the Brezis-Nirenberg problem on the hyperbolic spaces \\cite{LLY}. Recently. with Li and Lu, the author studied general higher order equations \\cite{LLW} by using the moving plane method, and later obtained the classification result using the moving sphere method \\cite{LLW25} on the hyperbolic space.\n\nThe definition of the fractional Laplacian on hyperbolic spaces via the singular integral representation was constructed in \\cites{BGS15}.\n\\begin{definition}\nLet $n \\geqslant 2$ and $0 < s < 1$. The fractional Laplacian on $\\mathbb{H}^n$ is defined by\n\\begin{equation}\n (-\\Delta_{\\mathbb{H}^n})^s u(x) = c_{n,s} \\, \\text{P.V.} \\int_{\\mathbb{H}^n} (u(x) - u(\\xi)) \\, \\mathcal{K}_{n,s}(d(x,\\xi)) \\, d\\xi\n\\end{equation}\nwith the kernel $\\mathcal{K}_{n,s}$ given by\n\\begin{equation}\\label{kernel}\n \\mathcal{K}_{n,s}(\\rho) = C_1 \\left( -\\frac{\\partial_\\rho}{\\sinh \\rho} \\right)^{\\frac{n-1}{2}} \\left( \\rho^{-\\frac{1+2s}{2}} K_{\\frac{1+2s}{2}} \\left( \\frac{n-1}{2} \\rho \\right) \\right)\n\\end{equation}\nwhen $n \\geqslant 3$ is odd and\n\\[\n\\mathcal{K}_{n,s}(\\rho) = C_1 \\int_{\\rho}^{\\infty} \\frac{\\sinh r}{\\sqrt{\\pi}\\sqrt{\\cosh r - \\cosh\\rho}} \n\\left( -\\frac{\\partial_r}{\\sinh r} \\right)^{n/2}\n\\left( r^{-\\frac{1+2s}{2}} K_{\\frac{1+2s}{2}} \\left( \\tfrac{n-1}{2}r \\right) \\right) dr\n\\]\nwhen $n \\geqslant 2$ is even, where\n\\[\nc_{n,s} = 2 \\, \\frac{4\\sqrt{2} \\,\\Gamma(n+s)}{3\\,\\Gamma(n/2)\\,\\Gamma(-s)}, \\quad\nC_1 = \\frac{1}{2^{n-2+2s}\\,\\Gamma\\!\\left( \\frac{n-1}{2} \\right) \\Gamma\\!\\left( \\frac{1+2s}{2} \\right)},\n\\]\nand $K_\\nu$ is the modified Bessel function of the second kind.\n\\end{definition}\nWe consider the following classic semilinear equation with power nonlinearity involving the fractional Laplacian:\n\\begin{equation}\\label{main_equation_1}\n(-\\Delta_{\\mathbb{H}^n})^s u = u^p(x), \\quad x \\in \\mathbb{H}^n.\n\\end{equation}\nWe say that a nonnegative function $u \\in H^s(\\mathbb{H}^n)$ is a weak solution of \\eqref{main_equation_1} if for any test function $\\varphi \\in C_c^\\infty(\\mathbb{H}^n)$, there holds\n\\begin{align*}\n \\int_{\\mathbb{H}^n} u(x) (-\\Delta_{\\mathbb{H}^n})^s \\varphi(x) dV_x &= \\int_{\\mathbb{H}^n} u(x)^p \\varphi(x) dV_x.\n\\end{align*}\n\n\\begin{theorem} \\label{thm-asymptotics}\nFor $s\\in(0,1)\\setminus\\mathbb{N}$, there holds that\n\\begin{itemize}\n\\item For $n\\geq 3$ odd,\n\\begin{equation}\\label{G1}\n G_s(\\rho)=\\alpha_\\gamma \\left(\\frac{\\partial_\\rho}{\\sinh\\rho}\\right)^\\frac{n-1}{2}\\rho^{-\\frac{1}{2}+s}K_{-\\frac{1}{2}+s}\\left(\\tfrac{n-1}{2} \\rho\\right),\\end{equation}\n\n\\textbf{Theorem.} \\textit{Let $0<\\lambdap$.\nThen the weak solution to \n\\begin{equation}\\label{diff_eq}\n (-\\Delta_{\\mathbb{H}^n})^s u = u^p \n\\end{equation}\nsatisfies the integral equation\n\\begin{equation}\\label{int_eq}\n u(x) = \\int_{\\mathbb{H}^n} \\mathcal K_s(\\rho(x,y)) u(y)^p dV_y,\n\\end{equation}\nwhere $\\mathcal K_s(\\rho)$ is the fractional kernel on $\\mathbb{H}^n$ defined in Section \\eqref{kernel}.\nConversely, if $u$ solves the integral equation \\eqref{int_eq}, then $u$ is a solution of the \\eqref{diff_eq}.\n\\end{proposition}\n\n\\begin{proof}\nBy Helgason-Fourier transform, we have for any $\\varphi\\in C^\\infty_c(\\hn)$:\n\\begin{align*}\n \\widehat{(-\\Delta_{\\mathbb{H}^n})^s \\varphi}(\\lambda,\\theta) &= \\left(\\lambda^2 + \\tfrac{(n-1)^2}{4}\\right)^s \\hat{\\varphi}(\\lambda,\\theta),\n\\end{align*}\nand \n\\begin{align}\\label{test_laplacian}\n (-\\Delta_{\\mathbb H^n})^s\\varphi(x)=\\int_{-\\infty}^\\infty\\int_{\\mathbb S^{n-1}}\n\\Bigl(\\lambda^2+\\frac{(n-1)^2}{4}\\Bigr)^s\\widehat{\\varphi}(\\lambda,\\theta)\n\\,e_{\\lambda,\\theta}(x)\n\\,|c(\\lambda)|^{-2}\\,d\\theta\\,d\\lambda.\n\\end{align}\nRecall the definition of weak solution of \\eqref{diff_eq}, we have\n\\begin{align*}\n \\int_{\\mathbb{H}^n} u(x) (-\\Delta_{\\mathbb{H}^n})^s \\varphi(x) dV_x &= \\int_{\\mathbb{H}^n} u(x)^p \\varphi(x) dV_x.\n\\end{align*}\nInserting \\eqref{test_laplacian} to the left side, we obtain\n\\begin{align*}\n \\int_{\\mathbb H^n} u(x)(-\\Delta_{\\mathbb H^n})^s\\varphi(x)\\,dV_x\n&=\n\\int_{\\mathbb H^n} u(x)\\int_{-\\infty}^\\infty \\int_{\\mathbb S^{n-1}}\n\\Bigl(\\lambda^2+\\tfrac{(n-1)^2}{4}\\Bigr)^s\n\\widehat{\\varphi}(\\lambda,\\theta)e_{\\lambda,\\theta}(x)\n|c(\\lambda)|^{-2}\\,d\\theta\\,d\\lambda\n\\,dV_x\\\\\n&=\n\\int_{-\\infty}^\\infty \\int_{\\mathbb S^{n-1}}\n\\Bigl(\\lambda^2+\\tfrac{(n-1)^2}{4}\\Bigr)^s\n\\widehat{\\varphi}(\\lambda,\\theta)\n\\left[\n\\int_{\\mathbb H^n}\nu(x)e_{\\lambda,\\theta}(x)\\,dV_x\n\\right]\n|c(\\lambda)|^{-2}\\,d\\theta\\,d\\lambda.\n\\end{align*}\n\nFor the far away part, we choose $a\\in(1,\\frac{q}{p})$, and let $a'$ denote the conjugate of $a$. By H\\\"older's inequality,\n\\[\n\\left|\\int_{\\mathbb H^n\\setminus B_R(o)}G(\\rho(x,y))u(y)^p\\,dV_y\\right| \\leq \\|G(\\rho(x,\\cdot))\\|_{L^{a'}(\\mathbb H^n\\setminus B_R)}\\;\\|u^p\\|_{L^{a}(\\mathbb H^n\\setminus B_R)}.\n\\]\n$\\|u^p\\|_{L^{a}(\\mathbb H^n\\setminus B_R)}$ is clearly finite since $ap0$. Thus, $\\|G(\\rho(x,\\cdot))\\|_{L^{a'}(\\mathbb H^n\\setminus B_R)}$ can be made arbitrarily small for large enough $R$. The proof is now complete.\n\\end{proof}\nFrom the above estimate at infinity, the moving plane argument in the subcritical case applies here as well, and we conclude that $u$ is radially symmetric about some point in $\\hn$.\n\nThen we introduce a maximum principle for anti-symmetric functions.\n\\begin{theorem}\n\\label{MP_anti}\n Let $\\Omega$ be a bounded domain in $\\hn$. Assume that $u \\in L^\\alpha \\cap C^{1,1}_{loc}(\\Omega)$ and is lower semi-continuous on $\\bar{\\Omega}$ . If\n \\begin{equation*}\n \\begin{cases}\n (-\\Delta_{\\hn})^s u + c(x)u \\geq 0, &\\text{ in } \\Omega,\\\\\n u \\geq 0, &\\text{ in } \\Sigma,\\\\\n u(x) = -u(x^\\lambda), &\\text{ in } \\Sigma,\n \\end{cases}\n \\end{equation*}\n where $c(x)$ is bounded from below in $\\Omega$. Then \n \\begin{equation}\\label{mp_anti_u}\n u \\geq 0 \\text{ in } \\Omega.\n \\end{equation}\n If $u = 0$ at some point in $\\Omega$, then $u(x) = 0$ almost everywhere in $\\hn$.\n\\end{theorem}\nThe next result is a maximum principle in narrow regions.\n\\begin{theorem}\n\\label{narrow_region}\n Let $U_\\lambda,\\, \\Sigma_\\lambda, x^\\lambda$ be defined as in Section \\ref{sec:prelim}. Assume that $u \\in L^\\alpha \\cap C^{1,1}_{loc}(\\Omega)$ and is lower semi-continuous on $\\bar{\\Omega}$ . If\n Let $\\Omega$ be an bounded narrow region in $\\Sigma_\\lambda$, which is contained in $\\{x\\mid \\lambda-lp$ is a nonnegative weak solution of\n\\begin{equation}\\label{main_equation_1}\n(-\\Delta_{\\hn})^s u = u^p(x), \\quad x \\in \\hn.\n\\end{equation}\nThen\n\\begin{enumerate}\n \\item If $1 < p < \\frac{n+2s}{n-2s}$ and $u\\in L^{p+1}(\\mathbb{H}^n)$ , then $u\\equiv 0$. \n \\item If $p = \\frac{n+2s}{n-2s}$ and $u\\in L^q$, where $q>p$, then there exists a point $x_0 \\in \\mathbb{H}^n$ such that $u$ is radially symmetric and nondecreasing about $x_0$.\n\\end{enumerate}", "theorem_type": ["Implication", "Nonexistence"], "mcq": {"question": "Let $n\\ge 2$ and $0p$ is a nonnegative weak solution of\n\\[\n(-\\Delta_{\\mathbb H^n})^s u=u^p\\quad \\text{in }\\mathbb H^n.\n\\]\nWhich statement is valid under these assumptions?", "correct_choice": {"label": "A", "text": "Two cases occur: (i) if $1\\frac{n+2s}{n-2s}$, then there exists a point $x_0\\in\\mathbb H^n$ such that $u$ is radially symmetric and nondecreasing about $x_0$, equivalently $u(x)$ depends only on the hyperbolic distance $d_{\\mathbb H^n}(x,x_0)$ and is nondecreasing as that distance increases."}, {"label": "C", "text": "If $1 0$, and suppose that\n\\[\n |A + A^*| \\leqslant K\\sqrt{p|A|}.\n\\]\nThen there exist sets $P, Q \\subseteq \\F_p$ such that, after removing $O_{K,\\alpha}(1)$ elements from $A$, the following holds:\n\\begin{itemize}[itemsep=0.5em]\n \\item $A \\subseteq P \\cap Q^*$;\n \\item $|P \\cap Q^*| = \\tfrac{1 + o(1)}{p}|P||Q|$;\n \\item $|P + Q| \\leqslant (1 + o(1)) K\\sqrt{p|A|}$.\n\\end{itemize}\n\\end{conjecture}\n\nThe first two conditions imply\n\\[\np|A| \\leqslant (1 + o(1))|P||Q|.\n\\]\nCombined with the third condition, this gives\n\\[\n|P + Q| \\leqslant (1 + o(1))K\\sqrt{|P||Q|}, \n\\]\nand sets $P$ and $Q$ can be further characterized using Freiman’s Theorem~\\cite{Freiman_book}.\n\n\\subsection{Main results}\n\nWe verify several variants of this conjecture for dense sets $A$. \nThe methods we develop for this conjecture turn out to be applicable in a more general setting, where the exponent configuration $(1, -1)$ in \\eqref{as:A+A*_intr} is replaced by a tuple $(r_1, \\ldots, r_k)$ satisfying certain arithmetic conditions. Some definitions are therefore to be introduced. \n\nLet $r$ be an exponent (non--zero integer), and $A \\subseteq \\F_p$. \nDefine $A^r := \\{a^r : a \\in A\\}$ (note that this notation is not standard, and should not be confused with the $r$-fold product set or the set of $r$-tuples). Since we work in $\\F_p$, an exponent $r$ might be treated as an element of $\\Z_{p - 1}$. \n\\begin{definition}\nExponent $r$ is called \\emph{generic} if $(r, p - 1) = O(1)$, and \\emph{coprime} if $(r, p - 1) = 1$. More generally, a tuple $(r_1, \\ldots, r_k)$ of distinct exponents satisfying $(r_i - r_j, p - 1) < p^{1 - \\delta}$ for some $\\delta > 0$ is \\emph{generic} or \\emph{coprime} if all individual $r_i$'s are generic or coprime, respectively.\n\\end{definition}\n\nDefine $\\sqrt[r]{A} := \\{x: x^r \\in A\\}$. We will work only with generic $r$, implying that always $|\\sqrt[r]{A}| \\leqslant (r, p - 1)|A| \\ll |A|$.\n\nThe following result generalizes the lower bound~\\eqref{eq:lower-bound-a-a*}.", "context": "Given an abelian group ${\\mathbf G}$ and two sets $A, B\\subseteq {\\mathbf G}$, define \nthe {\\it sumset} of $A$ and $B$ as \n\\begin{equation}\\label{def:A+B_intr}\n A+B:=\\{a+b ~:~ a\\in{A}, b\\in{B}\\}.\n\\end{equation}\nIn a similar way we define the \\emph{difference sets} and \\emph{higher sumsets}, e.g., $A-A$, $A+A-A$, and so on.\nThe \\emph{doubling constant} of a finite set $A$ is \n\\begin{equation}\\label{def:doubling}\n \\mathcal{D} [A] := \\frac{|A+A|}{|A|}, \n\\end{equation}\nan important additive--combinatorial characteristic of $A$.\n\n\\subsection{Main conjecture}\nIn this paper we limit ourselves to considering the field $\\F_p$.\nLet $p$ be a prime number and let $A \\subseteq \\F_p$. \nDenote by\n\\[\n A^* := A^{-1} = \\{1/a \\bmod p : a \\in A, a \\neq 0\\}\n\\]\nits inverse set. In~\\cite{Semchankau_wrappers}, we proved the lower bound\n\\begin{equation}\\label{eq:lower-bound-a-a*}\n |A + A^*| \\geqslant (1 - o(1)) \\min\\{ 2\\sqrt{p|A|}, p \\},\n\\end{equation}\nvalid whenever $|A| \\gg p(\\log p)^{-1/2 + o(1)}$. \nThe archetypal \nquestion of the present work is the corresponding \\emph{inverse problem} for a simple rational function \\eqref{def:rational_sp}:\ngiven that\n\\begin{equation}\\label{as:A+A*_intr}\n |A + A^*| \\leqslant K\\sqrt{p|A|},\n\\end{equation}\nfor a fixed parameter $K\\geqslant 2$, what structural information can we deduce about $A$?\n\nThis heuristic suggests the following: \n\\begin{conjecture}\nLet $A \\subseteq \\F_p$ be a set of size at least $p^{1 - c}$ for some absolute constant $c > 0$, and suppose that\n\\[\n |A + A^*| \\leqslant K\\sqrt{p|A|}.\n\\]\nThen there exist sets $P, Q \\subseteq \\F_p$ such that, after removing $O_{K,\\alpha}(1)$ elements from $A$, the following holds:\n\\begin{itemize}[itemsep=0.5em]\n \\item $A \\subseteq P \\cap Q^*$;\n \\item $|P \\cap Q^*| = \\tfrac{1 + o(1)}{p}|P||Q|$;\n \\item $|P + Q| \\leqslant (1 + o(1)) K\\sqrt{p|A|}$.\n\\end{itemize}\n\\end{conjecture}\n\nLet $r$ be an exponent (non--zero integer), and $A \\subseteq \\F_p$. \nDefine $A^r := \\{a^r : a \\in A\\}$ (note that this notation is not standard, and should not be confused with the $r$-fold product set or the set of $r$-tuples). Since we work in $\\F_p$, an exponent $r$ might be treated as an element of $\\Z_{p - 1}$. \n\\begin{definition}\nExponent $r$ is called \\emph{generic} if $(r, p - 1) = O(1)$, and \\emph{coprime} if $(r, p - 1) = 1$. More generally, a tuple $(r_1, \\ldots, r_k)$ of distinct exponents satisfying $(r_i - r_j, p - 1) < p^{1 - \\delta}$ for some $\\delta > 0$ is \\emph{generic} or \\emph{coprime} if all individual $r_i$'s are generic or coprime, respectively.\n\\end{definition}\n\nDefine $\\sqrt[r]{A} := \\{x: x^r \\in A\\}$. We will work only with generic $r$, implying that always $|\\sqrt[r]{A}| \\leqslant (r, p - 1)|A| \\ll |A|$.\n\nThe following result generalizes the lower bound~\\eqref{eq:lower-bound-a-a*}.\n\n\\begin{equation}\\label{def:rational_sp}\n \\left\\{ \\frac{f(a_1,\\dots,a_n)}{g(a_1,\\dots,a_n)} ~:~ a_1,\\dots, a_n \\in A \\right\\}.\n\\end{equation}\n\n\\begin{equation}\\label{eq:lower-bound-a-a*}\n |A + A^*| \\ge (1 - o(1)) \\min\\{ 2\\sqrt{p|A|}, p \\},\n\\end{equation}", "full_context": "Given an abelian group ${\\mathbf G}$ and two sets $A, B\\subseteq {\\mathbf G}$, define \nthe {\\it sumset} of $A$ and $B$ as \n\\begin{equation}\\label{def:A+B_intr}\n A+B:=\\{a+b ~:~ a\\in{A}, b\\in{B}\\}.\n\\end{equation}\nIn a similar way we define the \\emph{difference sets} and \\emph{higher sumsets}, e.g., $A-A$, $A+A-A$, and so on.\nThe \\emph{doubling constant} of a finite set $A$ is \n\\begin{equation}\\label{def:doubling}\n \\mathcal{D} [A] := \\frac{|A+A|}{|A|}, \n\\end{equation}\nan important additive--combinatorial characteristic of $A$.\n\n\\subsection{Main conjecture}\nIn this paper we limit ourselves to considering the field $\\F_p$.\nLet $p$ be a prime number and let $A \\subseteq \\F_p$. \nDenote by\n\\[\n A^* := A^{-1} = \\{1/a \\bmod p : a \\in A, a \\neq 0\\}\n\\]\nits inverse set. In~\\cite{Semchankau_wrappers}, we proved the lower bound\n\\begin{equation}\\label{eq:lower-bound-a-a*}\n |A + A^*| \\geqslant (1 - o(1)) \\min\\{ 2\\sqrt{p|A|}, p \\},\n\\end{equation}\nvalid whenever $|A| \\gg p(\\log p)^{-1/2 + o(1)}$. \nThe archetypal \nquestion of the present work is the corresponding \\emph{inverse problem} for a simple rational function \\eqref{def:rational_sp}:\ngiven that\n\\begin{equation}\\label{as:A+A*_intr}\n |A + A^*| \\leqslant K\\sqrt{p|A|},\n\\end{equation}\nfor a fixed parameter $K\\geqslant 2$, what structural information can we deduce about $A$?\n\nThis heuristic suggests the following: \n\\begin{conjecture}\nLet $A \\subseteq \\F_p$ be a set of size at least $p^{1 - c}$ for some absolute constant $c > 0$, and suppose that\n\\[\n |A + A^*| \\leqslant K\\sqrt{p|A|}.\n\\]\nThen there exist sets $P, Q \\subseteq \\F_p$ such that, after removing $O_{K,\\alpha}(1)$ elements from $A$, the following holds:\n\\begin{itemize}[itemsep=0.5em]\n \\item $A \\subseteq P \\cap Q^*$;\n \\item $|P \\cap Q^*| = \\tfrac{1 + o(1)}{p}|P||Q|$;\n \\item $|P + Q| \\leqslant (1 + o(1)) K\\sqrt{p|A|}$.\n\\end{itemize}\n\\end{conjecture}\n\nLet $r$ be an exponent (non--zero integer), and $A \\subseteq \\F_p$. \nDefine $A^r := \\{a^r : a \\in A\\}$ (note that this notation is not standard, and should not be confused with the $r$-fold product set or the set of $r$-tuples). Since we work in $\\F_p$, an exponent $r$ might be treated as an element of $\\Z_{p - 1}$. \n\\begin{definition}\nExponent $r$ is called \\emph{generic} if $(r, p - 1) = O(1)$, and \\emph{coprime} if $(r, p - 1) = 1$. More generally, a tuple $(r_1, \\ldots, r_k)$ of distinct exponents satisfying $(r_i - r_j, p - 1) < p^{1 - \\delta}$ for some $\\delta > 0$ is \\emph{generic} or \\emph{coprime} if all individual $r_i$'s are generic or coprime, respectively.\n\\end{definition}\n\nDefine $\\sqrt[r]{A} := \\{x: x^r \\in A\\}$. We will work only with generic $r$, implying that always $|\\sqrt[r]{A}| \\leqslant (r, p - 1)|A| \\ll |A|$.\n\nThe following result generalizes the lower bound~\\eqref{eq:lower-bound-a-a*}.\n\n\\begin{equation}\\label{def:rational_sp}\n \\left\\{ \\frac{f(a_1,\\dots,a_n)}{g(a_1,\\dots,a_n)} ~:~ a_1,\\dots, a_n \\in A \\right\\}.\n\\end{equation}\n\n\\begin{equation}\\label{eq:lower-bound-a-a*}\n |A + A^*| \\ge (1 - o(1)) \\min\\{ 2\\sqrt{p|A|}, p \\},\n\\end{equation}\n\nThe following result generalizes the lower bound~\\eqref{eq:lower-bound-a-a*}.\n\nThe following theorem confirms the conjecture for almost all, or `99\\%' elements of $A$, that is, after removing only $o(p)$ of them.\n\n\\begin{theorem}[$99\\%$ Characterization of $A$]\\label{thm:99-percent}\nLet $A \\subseteq \\F_p$ be a set with $|A| \\gg p$, and let $(r_1, \\ldots, r_k)$ be a generic tuple. If\n\\[\n |A^{r_1} + \\cdots + A^{r_k}| \n \\le K p^{\\frac{k-1}{k}}|A|^{1/k},\n\\]\nthen there exist sets $P_1, \\ldots, P_k \\subseteq \\F_p$ such that, \nafter removing $o(p)$ elements from~$A$, the following holds:\n\\begin{itemize}[itemsep=0.5em]\n \\item $A \\subseteq \\sqrt[r_1]{P_1} \\cap \\cdots \\cap \\sqrt[r_k]{P_k}$;\n \\item $\\big|\\sqrt[r_1]{P_1} \\cap \\cdots \\cap \\sqrt[r_k]{P_k}\\big|\n = \\frac{1 + o(1)}{p^{k-1}}|P_1|\\cdots|P_k|$;\n \\item $|P_1 + \\cdots + P_k|\n \\le (1 + o(1)) K p^{\\frac{k-1}{k}}|A|^{1/k}$.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{theorem}[$100\\%$ Characterization of $A$]\\label{thm:100-percent}\nLet $A \\subseteq \\F_p$ be a set of size $|A| = \\alpha p$ with $\\alpha \\gg 1$, \nand let tuple $(r_1, \\ldots, r_k)$ be coprime and bounded such that\n\\[\n |A^{r_1} + \\cdots + A^{r_k}|\n \\le K p^{\\frac{k-1}{k}}|A|^{1/k}.\n\\]\nThen there exist sets $P_1, \\ldots, P_k \\subseteq \\F_p$ such that, \nafter removing $O(1/\\alpha)$ elements from~$A$, the following holds:\n\\begin{itemize}[itemsep=0.5em]\n \\item $A \\subseteq \\sqrt[r_1]{P_1} \\cap \\cdots \\cap \\sqrt[r_k]{P_k}$;\n \\item $\\big|\\sqrt[r_1]{P_1} \\cap \\cdots \\cap \\sqrt[r_k]{P_k}\\big|\n = \\frac{1 + o(1)}{p^{k-1}}|P_1|\\cdots|P_k|$;\n \\item $|P_1 + \\cdots + P_k| \\le f(K) p^{\\frac{k-1}{k}}|A|^{1/k}$,\n\\end{itemize}\nwhere $f(K) \\ll K^{O(1)}$.\n\\end{theorem}\n\n\\begin{lemma}\\label{lemma:aip}\nLet $W_1, \\ldots, W_k$ be sets with Wiener norms bounded by $M$. Let tuple $(r_1, \\ldots, r_k)$ be generic. Then\n\\[\n\\bigl|\\sqrt[r_1]{W_1} \\cap \\cdots \\cap \\sqrt[r_k]{W_k}\\bigr|\n= \\frac{1}{p^{\\,k-1}}\\,|W_1|\\cdots|W_k|\n + O\\left(M^{k} p^{1 - \\eps}\\right),\n\\]\nwhere $\\eps = \\eps(k) > 0$.\n\\end{lemma}\n\\begin{proof}\nIdentifying sets $W_i$'s with their indicator functions, we write\n\\begin{equation}\\label{eq:aip}\n\\bigl|\\sqrt[r_1]{W_1} \\cap \\cdots \\cap \\sqrt[r_k]{W_k}\\bigr|\n= \\sum_{x \\in \\F_p} W_1(x^{r_1}) \\cdots W_k(x^{r_k}).\n\\end{equation}\n\nThe family $\\mathcal{F}$ provides a much broader source of sets with AIP:\n\\begin{proposition}\n\\label{proposition:aip}\n Any collection of sets $P_1, \\ldots, P_k$ from the family $\\mathcal{F}$ has AIP.\n\\end{proposition}\n\\begin{proof}\nLet $(r_1, \\ldots, r_k)$ be an arbitrary generic tuple, and let $W_1, \\ldots, W_k$ be sets approximating $P_1, \\ldots, P_k$. Then, for all $i$, the symmetric difference $\\sqrt[r]{P_i} \\triangle \\sqrt[r]{W_i}$ is a subset of $\\sqrt[r]{P_i \\triangle W_i}$, and therefore its size does not execeed $(r, p - 1)|P_i \\triangle W_i| = o(p)$. Thus,\n\\begin{multline*}\n\\bigl|\\sqrt[r_1]{P_1} \\cap \\cdots \\cap \\sqrt[r_k]{P_k}\\bigr| \n\\oversetgap{\\text{Proposition~\\ref{proposition:set-triangles}}}{=}\n\\bigl|\\sqrt[r_1]{W_1} \\cap \\cdots \\cap \\sqrt[r_k]{W_k}\\bigr| + o(p) = \\\\\n\\oversetgap{\\text{Lemma~\\ref{lemma:aip}}}{=} \n\\frac{1}{p^{k-1}}|W_1|\\ldots|W_k| + o(p) \n\\oversetgap{\\text{Proposition~\\ref{proposition:prods-and-diffs}}}{=} \n\\frac{1}{p^{k - 1}}|P_1|\\ldots|P_k| + o(p) = \\\\\n\\oversetgap{|P_i| \\gg p}{=}\n\\frac{1 + o(1)}{p^{k - 1}}|P_1|\\ldots|P_k|.\n\\end{multline*}\n\\end{proof}\n\n\\begin{authortheorem}[Green, 2005]\\label{thm:green}\nLet $k \\geqslant 3$, and let dense sets $A_1, A_2, \\ldots, A_k \\subseteq \\F_p$ be such that there are $o(p^{k-1})$ solutions to the equation\n\\[\n a_1 + \\cdots + a_k = 0.\n\\]\nThen we may remove $o(p)$ elements from each $A_i$, leaving subsets $A_i' \\subseteq A_i$ with the property that the equation\n\\[\n a_1' + \\cdots + a_k' = 0, \n \\qquad a_i' \\in A_i',\n\\]\nhas no solutions.\n\\end{authortheorem}\n\n\\begin{lemma}\n\\label{lemma:X+T-growth-structure}\nLet $W, W_1, \\ldots, W_k \\subseteq \\F_p$ be sets with Wiener norms $p^{o(1)}$\nand densities $\\omega, \\omega_1, \\ldots, \\omega_k$, respectively.\nSet $\\omega_\\times := \\omega_1 \\cdots \\omega_k$.\nLet tuple $(r_1, \\ldots, r_k)$ be coprime and bounded.\n\n\\begin{equation}\\label{eq:lower-bound-a-a*}\n |A + A^*| \\ge (1 - o(1)) \\min\\{ 2\\sqrt{p|A|}, p \\},\n\\end{equation}", "post_theorem_intro_text_len": 5700, "post_theorem_intro_text": "The following theorem confirms the conjecture for almost all, or `99\\%' elements of $A$, that is, after removing only $o(p)$ of them.\n\n\\begin{theorem}[$99\\%$ Characterization of $A$]\\label{thm:99-percent}\nLet $A \\subseteq \\F_p$ be a set with $|A| \\gg p$, and let $(r_1, \\ldots, r_k)$ be a generic tuple. If\n\\[\n |A^{r_1} + \\cdots + A^{r_k}| \n \\leqslant K p^{\\frac{k-1}{k}}|A|^{1/k},\n\\]\nthen there exist sets $P_1, \\ldots, P_k \\subseteq \\F_p$ such that, \nafter removing $o(p)$ elements from~$A$, the following holds:\n\\begin{itemize}[itemsep=0.5em]\n \\item $A \\subseteq \\sqrt[r_1]{P_1} \\cap \\cdots \\cap \\sqrt[r_k]{P_k}$;\n \\item $\\big|\\sqrt[r_1]{P_1} \\cap \\cdots \\cap \\sqrt[r_k]{P_k}\\big|\n = \\frac{1 + o(1)}{p^{k-1}}|P_1|\\cdots|P_k|$;\n \\item $|P_1 + \\cdots + P_k|\n \\leqslant (1 + o(1)) K p^{\\frac{k-1}{k}}|A|^{1/k}$.\n\\end{itemize}\n\\end{theorem}\n\nIn fact, the sets $P_1, \\ldots, P_k$ obtained in Theorem~\\ref{thm:99-percent} (and in Theorem~\\ref{thm:100-percent} below) are quite specific and possess a good additive structure. More precisely, they are well approximated by sets of small Wiener norm; see the proofs and Section~\\ref{sec:comb-wiener}.\n\n\\bigskip \n\nWe can also obtain a structural characterization for the \\emph{entire} set~$A$,\nafter removing only $O(p/|A|)$ elements, although with a suboptimal dependence of $f(K)$ on $K$, see Theorem \\ref{thm:100-percent} below. We also have to make additional assumptions on the exponents.\n\n\\begin{definition}\nA generic tuple $(r_1, \\ldots, r_k)$ is called \\emph{bounded} \nif there exists a scaling parameter $L = O(1)$, coprime to $p - 1$, \nsuch that all residues $r_i L \\bmod (p - 1)$ have order $O(1)$. \nIn other words, there is an absolute constant $C > 0$ for which\n\\[\n |r_i L \\bmod (p - 1)| \\leqslant C,\n \\qquad i = 1, \\ldots, k,\n\\]\nwhere residues modulo $p - 1$ are identified with the interval \n$[-\\frac{p-1}{2}, \\frac{p-1}{2}]$.\n\\end{definition}\n\n\\begin{theorem}[$100\\%$ Characterization of $A$]\\label{thm:100-percent}\nLet $A \\subseteq \\F_p$ be a set of size $|A| = \\alpha p$ with $\\alpha \\gg 1$, \nand let tuple $(r_1, \\ldots, r_k)$ be coprime and bounded such that\n\\[\n |A^{r_1} + \\cdots + A^{r_k}|\n \\leqslant K p^{\\frac{k-1}{k}}|A|^{1/k}.\n\\]\nThen there exist sets $P_1, \\ldots, P_k \\subseteq \\F_p$ such that, \nafter removing $O(1/\\alpha)$ elements from~$A$, the following holds:\n\\begin{itemize}[itemsep=0.5em]\n \\item $A \\subseteq \\sqrt[r_1]{P_1} \\cap \\cdots \\cap \\sqrt[r_k]{P_k}$;\n \\item $\\big|\\sqrt[r_1]{P_1} \\cap \\cdots \\cap \\sqrt[r_k]{P_k}\\big|\n = \\frac{1 + o(1)}{p^{k-1}}|P_1|\\cdots|P_k|$;\n \\item $|P_1 + \\cdots + P_k| \\leqslant f(K) p^{\\frac{k-1}{k}}|A|^{1/k}$,\n\\end{itemize}\nwhere $f(K) \\ll K^{O(1)}$.\n\\end{theorem}\n\nAlthough both the density $\\alpha$ and the parameter $K$ are assumed to be `constants', they have different scales --- one may think the case $K = 10$ and $\\alpha = 10^{-100}$.\n\nThe main ingredient in the proof of Theorem~\\ref{thm:lower-bound} is the observation that algebraic transformations $\\sqrt[r]{W}$ of sets with small Wiener norm intersect in a manner similar to \\emph{independent} sets. \nThe main ingredient in Theorem~\\ref{thm:99-percent} is Green's Arithmetic Regularity Lemma. \nFinally, the main ingredient in the proof of Theorem~\\ref{thm:100-percent} is that, under additional assumptions, the intersections\n\\[\nW \\cap \\sqrt[r_1]{W_1} \\cap \\ldots \\cap \\sqrt[r_k]{W_k}\n\\]\nbehave similarly to \\emph{pseudorandom} subsets of $W$.\n\n\\newpage \n\n\\noindent\n\\textbf{Organization of the paper}\n\n\\begin{itemize}\n \\item \\textbf{Section~\\ref{sec:prelims}} recalls the basic properties of the Fourier transform, the Wiener norm, exponential sums, sumsets, and proves various auxiliary propositions.\n\n \\item \\textbf{Section~\\ref{sec:comb-wiener}} develops combinatorial tools based on the Wiener norm. We introduce the \\emph{algebraic intersection property} and establish it for sets, approximated by sets of small Wiener norm: their algebraic transformations intersect as independent sets.\n We also recall the concept of \\emph{wrappers} from~\\cite{Semchankau_wrappers}, \n which serve as a source of such sets and help to extend the algebraic intersection property to sets of popular differences.\n\n \\item \\textbf{Sections~\\ref{sec:lower-bound}} and~\\textbf{\\ref{sec:99-percent}} contain the proofs of Theorems~\\ref{thm:lower-bound} and~\\ref{thm:99-percent}, respectively. These proofs are relatively short and use only a subset of tools introduced earlier.\n\n \\item \\textbf{Section~\\ref{sec:new_norm}} defines an additive distance $\\rho(f,g)$ between functions and shows that intersection of algebraic transformations of sets with algebraic intersection property not only has `expected' cardinality, but also exhibits strong pseudorandom properties.\n\n \\item \\textbf{Section~\\ref{sec:sumsets}} studies sumsets of the form $Y + T$, where $T$ is a dense subset of intersections such as $P \\cap Q^*$. \n We first show that $|Y + T| \\gg |P|$, and then derive a Ruzsa-type structural description of sets $Y$ for which $|Y + T| \\ll |P|$. The first result heavily relies on the previous section. \n\n \\item \\textbf{Section~\\ref{sec:100-percent}} concludes with the proof of the \n \\emph{$100\\%$ Characterization Theorem}~\\ref{thm:100-percent}. \n This argument builds on the $99\\%$ case and relies on the results of the preceding section.\n\n \\item \\textbf{Appendix~\\ref{sec:appendix_dist}} includes some further facts on the \n additive distance introduced in Subsection \\ref{ssec:new_normI} and other metrics. \n\\end{itemize}\n\n\\noindent\n\\textbf{Acknowledgements}. The first author would like to thank Boris Bukh and Prasad Tetali for useful discussions. The first author was supported in part by Prasad Tetali's NSF grant DMS-2151283.", "sketch": "The post-theorem introduction gives only a high-level ingredient for proving Theorem~\\ref{thm:lower-bound}: it says that the main ingredient is “the observation that algebraic transformations $\\sqrt[r]{W}$ of sets with small Wiener norm intersect in a manner similar to \\emph{independent} sets.” It also indicates where the full proof appears: “Sections~\\ref{sec:lower-bound} ... contain the proofs of Theorems~\\ref{thm:lower-bound}...”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "[Lower Bound]\\label{thm:lower-bound}\nLet $A \\subseteq \\F_p$ be a set with $|A| \\gg p$, and let $(r_1, \\ldots, r_k)$ be a generic tuple. Then\n\\[\n \\big|A^{r_1} + \\cdots + A^{r_k}\\big|\n \\geqslant (1 - o(1)) \\min \\big(k p^{\\frac{k-1}{k}}|A|^{1/k}, p\\big).\n\\]", "theorem_type": ["Inequality or Bound", "Universal"], "mcq": {"question": "Let $p$ be a prime, let $A\\subseteq \\mathbb F_p$ satisfy $|A|\\gg p$ (that is, $|A|\\ge c p$ for some absolute constant $c>0$), and for a nonzero integer exponent $r$ define $A^r:=\\{a^r:a\\in A\\}$. Suppose $(r_1,\\dots,r_k)$ is a generic tuple of exponents, meaning the $r_i$ are distinct, each satisfies $(r_i,p-1)=O(1)$, and $(r_i-r_j,p-1)0$ and all $i\\neq j$. Which quantitative lower bound is valid for the sumset $A^{r_1}+\\cdots+A^{r_k}:=\\{x_1+\\cdots+x_k:x_i\\in A^{r_i}\\}$?", "correct_choice": {"label": "A", "text": "\\[\\big|A^{r_1}+\\cdots+A^{r_k}\\big|\\ge (1-o(1))\\min\\!\\left(k\\,p^{\\frac{k-1}{k}}|A|^{1/k},\\,p\\right).\\]"}, "choices": [{"label": "B", "text": "\\[\\big|A^{r_1}+\\cdots+A^{r_k}\\big|\\ge (1-o(1))\\min\\!\\left(k\\,p^{\\frac{k-1}{k}}|A|,\\,p\\right).\\]"}, {"label": "C", "text": "\\[\\big|A^{r_1}+\\cdots+A^{r_k}\\big|\\ge (1-o(1))\\min\\!\\left(p^{\\frac{k-1}{k}}|A|^{1/k},\\,p\\right).\\]"}, {"label": "D", "text": "\\[\\big|A^{r_1}+\\cdots+A^{r_k}\\big|\\ge (1-o(1))\\min\\!\\left(2\\,p^{\\frac{k-1}{k}}|A|^{1/k},\\,p\\right).\\]"}, {"label": "E", "text": "\\[\\big|A^{r_1}+\\cdots+A^{r_k}\\big|\\ge (1-o(1))\\,k\\,p^{\\frac{k-1}{k}}|A|^{1/k}.\\]"}], "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": "power_of_|A|", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "characteristic", "tampered_component": "leading_factor_k", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "characteristic", "tampered_component": "sharp_constant_k_replaced_by_2", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "missing_min_with_p", "template_used": "boundary_range"}]}} +{"id": "2601.13212v1", "paper_link": "http://arxiv.org/abs/2601.13212v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{intro:maintheo}\nLet $\\O$ be an open bounded domain in $\\mathbb{R}^2$ which contains the origin, $\\{0\\}\\subset \\O$. Assume that\n$v_n$ is a sequence of solutions of \\eqref{intro:eq} satisfying \\eqref{intro:alphan} and \\eqref{intro: Vn} with\n$$\n\\alpha_\\infty\\in (-1,1).\n$$\nThen, for any\n $$ C_1>\\max\\left\\{1,\\tfrac{1+\\alpha_\\infty}{1-\\alpha_\\infty}\\right\\}$$\nand for any compact set $K\\subset\\O$, there exists a constant $C_2>0$, which depends only by $a,b,dist(K,\\partial\\O), \\alpha_\\infty$ and by the\nuniform modulus of continuity of $V$ on $K$, such that,\n\\begin{equation}\n \\label{intro:sup+Cinf}\n \\underset{K}\\sup\\, v_n + C_1\\underset{\\O}\\inf\\, v_n\\leq C_2.\n\\end{equation}", "start_pos": 13157, "end_pos": 13855, "label": "intro:maintheo"}, "ref_dict": {"intro: Vn": "\\begin{equation}\\label{intro: Vn}\n 00)$\nvortex at the origin. In this context (see Lemma 3.2 in \\cite{bcwyzOnsager}) a delicate point arise in the analysis of \\eqref{intro:eq} which requires\nan estimate of ``sup+C inf\" type, in the same spirit of analogous results about the cases $\\alpha_n \\equiv 0$ (\\cite{bls},\\cite{CL}, \\cite{S}, \\cite{yy}) or\n$\\epsilon_n \\equiv 0$ (\\cite{b0}, \\cite{b1}, \\cite{b2}, \\cite{b8}, \\cite{BCLT}, \\cite{cosentino2025}, \\cite{Tar3}, \\cite{T1}). Remark that\nthe examples of blow up without concentration in \\cite{llty} are built upon the assumption $\\al_n\\rightarrow \\alpha>1$. Also, blow up implies concentration as far as $\\al_n\\leq 1$ (see either Theorem 2.2 in \\cite{bcwyzOnsager} or the general results in \\cite{os}). \nThus, by analogy with the ``regular\" case $\\al_n=0$ (\\cite{S}), one may wonder about the validity of a ``sup+C inf\" Harnack-type inequality as far as $\\al_n\\leq 1$. We point out that this class of ``sup+C inf\" inequalities are the two-dimensional singular analogue of the classical ``sup x inf\" inequalities first established in dimension $N\\geq3$ in the context of the Yamabe problem, see \\cite{lz}, \\cite{Schoen} and references quoted therein. \\\\ \\\\ \nLet us recall some well known facts about this problem. \nIt was first conjectured in \\cite{bm} that if $v_n$ is a sequence of solution of \\eqref{intro:eq} with $\\alpha_n\\equiv0$ and such that\n\\[\n 01$, for a suitable $C_2$. Assuming that,\n\\begin{equation}\\label{intro:KLip}\n ||\\nabla V_n||_\\infty\\leq C_3,\n\\end{equation}\nit has been proved in \\cite{bls} that \\eqref{sup+inf.intro} holds with $C_1=1$ and $C_2$ depending by $C_3$ as well.\nFinally it has been proved in \\cite{CL} that \\eqref{sup+inf.intro} holds with $C_1=1$ under even weaker assumptions on $V_n$, such as\nthe existence of a logarithmic uniform modulus of continuity for $V_n$. Remark that in \\cite{CL} one also finds that $C_1=\\sqrt{\\frac{b}{a}}$ is\nthe sharp constant for $V_n$ just satisfying $ 00$ and $\\epsilon_n\\equiv0$) is more\ndelicate. It has been proved in \\cite{T1} (see also \\cite{tar-sd}) that a weaker but still sharp inequality holds true: for any $\\alpha_\\infty>0$,\nthere exists a constant $C>0$ such that\n\\begin{equation}\\label{u0+inf}\n v_n(0)+\\underset{\\O}\\inf\\, v_n\\leq C\n\\end{equation}\nfor any sequence of solutions to \n\\eqref{intro:eq}, \\eqref{intro:alphan} and \\eqref{intro:KLip}. See \n\\cite{b0}, \\cite{b1}, \\cite{b8} for other\npartial results about this problem. In particular in \\cite{BCLT} the sharp inequality \\eqref{sup+inf.intro} was obtained with $C_1=1$ but under stronger assumptions, \nsee Theorem 1.3 in \\cite{BCLT}.\\\\\nMotivated by the Onsager vortex model pursued in \\cite{bcwyzOnsager}, we are interested in a generalization of the result in \\cite{b1}\nto solutions of \\eqref{intro:eq}, satisfying \\eqref{intro:alphan} and \\eqref{intro: Vn}, with $\\epsilon_n\\to0^+$ and $\\alpha_\\infty\\in (-1,1)$.\nRemark that any inequality of the form \\eqref{sup+inf.intro} implies that blow up implies concentration in the sense of \\cite{bm} and \\cite{bt}. Therefore,\nin view of the examples of blow up without concentration in \\cite{llty}, it cannot hold in general as far as for $\\al_\\infty>1$.\nHere we prove the following,", "context": "{equation}\n \\label{intro:alphan}\n \\a_n\\rightarrow\\a_\\infty\\in(-1,+\\infty),\n\\end{equation}\n$\\O$ is a bounded domain in $\\mathbb{R}^2$ that contains the origin $x=0\\in \\Omega$ and $V_n$ satisfies,\n\\begin{equation}\\label{intro: Vn}\n 00)$\nvortex at the origin. In this context (see Lemma 3.2 in \\cite{bcwyzOnsager}) a delicate point arise in the analysis of \\eqref{intro:eq} which requires\nan estimate of ``sup+C inf\" type, in the same spirit of analogous results about the cases $\\alpha_n \\equiv 0$ (\\cite{bls},\\cite{CL}, \\cite{S}, \\cite{yy}) or\n$\\epsilon_n \\equiv 0$ (\\cite{b0}, \\cite{b1}, \\cite{b2}, \\cite{b8}, \\cite{BCLT}, \\cite{cosentino2025}, \\cite{Tar3}, \\cite{T1}). Remark that\nthe examples of blow up without concentration in \\cite{llty} are built upon the assumption $\\al_n\\rightarrow \\alpha>1$. Also, blow up implies concentration as far as $\\al_n\\leq 1$ (see either Theorem 2.2 in \\cite{bcwyzOnsager} or the general results in \\cite{os}). \nThus, by analogy with the ``regular\" case $\\al_n=0$ (\\cite{S}), one may wonder about the validity of a ``sup+C inf\" Harnack-type inequality as far as $\\al_n\\leq 1$. We point out that this class of ``sup+C inf\" inequalities are the two-dimensional singular analogue of the classical ``sup x inf\" inequalities first established in dimension $N\\geq3$ in the context of the Yamabe problem, see \\cite{lz}, \\cite{Schoen} and references quoted therein. \\\\ \\\\ \nLet us recall some well known facts about this problem. \nIt was first conjectured in \\cite{bm} that if $v_n$ is a sequence of solution of \\eqref{intro:eq} with $\\alpha_n\\equiv0$ and such that\n\\[\n 01$, for a suitable $C_2$. Assuming that,\n\\begin{equation}\\label{intro:KLip}\n ||\\nabla V_n||_\\infty\\leq C_3,\n\\end{equation}\nit has been proved in \\cite{bls} that \\eqref{sup+inf.intro} holds with $C_1=1$ and $C_2$ depending by $C_3$ as well.\nFinally it has been proved in \\cite{CL} that \\eqref{sup+inf.intro} holds with $C_1=1$ under even weaker assumptions on $V_n$, such as\nthe existence of a logarithmic uniform modulus of continuity for $V_n$. Remark that in \\cite{CL} one also finds that $C_1=\\sqrt{\\frac{b}{a}}$ is\nthe sharp constant for $V_n$ just satisfying $ 00$ and $\\epsilon_n\\equiv0$) is more\ndelicate. It has been proved in \\cite{T1} (see also \\cite{tar-sd}) that a weaker but still sharp inequality holds true: for any $\\alpha_\\infty>0$,\nthere exists a constant $C>0$ such that\n\\begin{equation}\\label{u0+inf}\n v_n(0)+\\underset{\\O}\\inf\\, v_n\\leq C\n\\end{equation}\nfor any sequence of solutions to \n\\eqref{intro:eq}, \\eqref{intro:alphan} and \\eqref{intro:KLip}. See \n\\cite{b0}, \\cite{b1}, \\cite{b8} for other\npartial results about this problem. In particular in \\cite{BCLT} the sharp inequality \\eqref{sup+inf.intro} was obtained with $C_1=1$ but under stronger assumptions, \nsee Theorem 1.3 in \\cite{BCLT}.\\\\\nMotivated by the Onsager vortex model pursued in \\cite{bcwyzOnsager}, we are interested in a generalization of the result in \\cite{b1}\nto solutions of \\eqref{intro:eq}, satisfying \\eqref{intro:alphan} and \\eqref{intro: Vn}, with $\\epsilon_n\\to0^+$ and $\\alpha_\\infty\\in (-1,1)$.\nRemark that any inequality of the form \\eqref{sup+inf.intro} implies that blow up implies concentration in the sense of \\cite{bm} and \\cite{bt}. Therefore,\nin view of the examples of blow up without concentration in \\cite{llty}, it cannot hold in general as far as for $\\al_\\infty>1$.\nHere we prove the following,\n\n\\begin{equation}\\label{intro: Vn}\n 00)$\nvortex at the origin. In this context (see Lemma 3.2 in \\cite{bcwyzOnsager}) a delicate point arise in the analysis of \\eqref{intro:eq} which requires\nan estimate of ``sup+C inf\" type, in the same spirit of analogous results about the cases $\\alpha_n \\equiv 0$ (\\cite{bls},\\cite{CL}, \\cite{S}, \\cite{yy}) or\n$\\epsilon_n \\equiv 0$ (\\cite{b0}, \\cite{b1}, \\cite{b2}, \\cite{b8}, \\cite{BCLT}, \\cite{cosentino2025}, \\cite{Tar3}, \\cite{T1}). Remark that\nthe examples of blow up without concentration in \\cite{llty} are built upon the assumption $\\al_n\\rightarrow \\alpha>1$. Also, blow up implies concentration as far as $\\al_n\\leq 1$ (see either Theorem 2.2 in \\cite{bcwyzOnsager} or the general results in \\cite{os}). \nThus, by analogy with the ``regular\" case $\\al_n=0$ (\\cite{S}), one may wonder about the validity of a ``sup+C inf\" Harnack-type inequality as far as $\\al_n\\leq 1$. We point out that this class of ``sup+C inf\" inequalities are the two-dimensional singular analogue of the classical ``sup x inf\" inequalities first established in dimension $N\\geq3$ in the context of the Yamabe problem, see \\cite{lz}, \\cite{Schoen} and references quoted therein. \\\\ \\\\ \nLet us recall some well known facts about this problem. \nIt was first conjectured in \\cite{bm} that if $v_n$ is a sequence of solution of \\eqref{intro:eq} with $\\alpha_n\\equiv0$ and such that\n\\[\n 01$, for a suitable $C_2$. Assuming that,\n\\begin{equation}\\label{intro:KLip}\n ||\\nabla V_n||_\\infty\\leq C_3,\n\\end{equation}\nit has been proved in \\cite{bls} that \\eqref{sup+inf.intro} holds with $C_1=1$ and $C_2$ depending by $C_3$ as well.\nFinally it has been proved in \\cite{CL} that \\eqref{sup+inf.intro} holds with $C_1=1$ under even weaker assumptions on $V_n$, such as\nthe existence of a logarithmic uniform modulus of continuity for $V_n$. Remark that in \\cite{CL} one also finds that $C_1=\\sqrt{\\frac{b}{a}}$ is\nthe sharp constant for $V_n$ just satisfying $ 00$ and $\\epsilon_n\\equiv0$) is more\ndelicate. It has been proved in \\cite{T1} (see also \\cite{tar-sd}) that a weaker but still sharp inequality holds true: for any $\\alpha_\\infty>0$,\nthere exists a constant $C>0$ such that\n\\begin{equation}\\label{u0+inf}\n v_n(0)+\\underset{\\O}\\inf\\, v_n\\leq C\n\\end{equation}\nfor any sequence of solutions to \n\\eqref{intro:eq}, \\eqref{intro:alphan} and \\eqref{intro:KLip}. See \n\\cite{b0}, \\cite{b1}, \\cite{b8} for other\npartial results about this problem. In particular in \\cite{BCLT} the sharp inequality \\eqref{sup+inf.intro} was obtained with $C_1=1$ but under stronger assumptions, \nsee Theorem 1.3 in \\cite{BCLT}.\\\\\nMotivated by the Onsager vortex model pursued in \\cite{bcwyzOnsager}, we are interested in a generalization of the result in \\cite{b1}\nto solutions of \\eqref{intro:eq}, satisfying \\eqref{intro:alphan} and \\eqref{intro: Vn}, with $\\epsilon_n\\to0^+$ and $\\alpha_\\infty\\in (-1,1)$.\nRemark that any inequality of the form \\eqref{sup+inf.intro} implies that blow up implies concentration in the sense of \\cite{bm} and \\cite{bt}. Therefore,\nin view of the examples of blow up without concentration in \\cite{llty}, it cannot hold in general as far as for $\\al_\\infty>1$.\nHere we prove the following,\n\n\\begin{equation}\\label{intro: Vn}\n 00$ constant, since in fact we have,\n\\begin{align*}\n \\frac{\\epsilon_n}{\\delta_n}=\\frac{\\epsilon_n}{t_n}\\frac{t_n}{|x_n|}\\frac{|x_n|}{\\delta_n}=\\begin{cases}\n \\frac{\\epsilon_n}{t_n},&\\quad\\text{if}\\,\\,\\delta_n\\geq|x_n|,\\\\\n \\frac{\\epsilon_n}{t_n}\\frac{|x_n|}{\\delta_n}, &\\quad\\text{if}\\,\\,\\delta_n\\leq|x_n|. \\end{cases}\n\\end{align*}\nLet us define\n$$\nD_n=B_{\\frac{r}{\\delta_n}}(0),\n$$\nand possibly along a subsequence, assume without loss of generality that,\n\\begin{equation*}\n \\frac{x_n}{\\delta_n}\\to y_0, \\,\\,\\text{as}\\,\\,n\\to+\\infty\n\\end{equation*}\n and\n\\begin{equation*}\n \\frac{\\epsilon_n}{\\delta_n}\\to \\epsilon_0, \\,\\,\\text{as}\\,\\,n\\to+\\infty,\n\\end{equation*}\nfor some $y_0\\in\\R^2$ and $\\epsilon_0\\geq 0$. At this point, we define,\n\\begin{equation*}\n \\widetilde w_n(y)=v_n(\\delta_n y)+2(1+\\alpha_n)\\log \\delta_n=v_n(\\delta_n y)-v_n(x_n)\\quad\\text{in}\\,\\, D_n,\n\\end{equation*}\nwhich satisfies,\n\\begin{align}\\label{equationgoodcase}\n\\begin{cases}\n -\\D \\widetilde w_n=\\left({\\frac{\\epsilon_n^2}{\\delta_n^2}+\\left|y\\right|^2}\\right)^{\\alpha_n}\n V_n(\\delta_n y)e^{\\dis \\widetilde w_n}:=f_n \\qquad\\text{in}\\,\\,D_n, \\\\\n\\int_{D_n}\\left({\\frac{\\epsilon_n^2}{\\delta_n^2}+|y|^2}\\right)^{\\alpha_n}e^{\\dis \\widetilde w_n}\\leq C,\\\\\n\\widetilde w_n(y)\\leq\\widetilde w_n(\\tfrac{x_n}{\\delta_n})=\\max\\limits_{D_n}\\widetilde w_n=0.\n\\end{cases}\n\\end{align}\nRemark that $f_n$ is uniformly bounded in $L^{p}(B_R)$, for $R$ large enough where $p=p(|\\sigma|)>1$, if $\\sigma<0$, while $p=\\infty$, if $\\sigma\\geq0$.\nMoreover we see that, by the Harnack inequality, for every $R\\geq 1+ 2|y_0|$ there exists a constant $C_R>0$ such that\n\\begin{equation}\n \\label{Estimateboundary}\n \\sup_{\\p B_{R}}|\\widetilde w_n|\\leq C_R.\n\\end{equation}\nIndeed, since $f_n$ is uniformly bounded in $L^p(B_{2R})$, for some $p>1$, and\n$$\\sup_{\\p B_{2R}}\\widetilde w_n\\leq \\sup_{ B_{4R}}\\widetilde w_n =\\widetilde w_n(\\tfrac{x_n}{\\delta_n})=0,$$\nthen there exist $\\tau\\in(0,1)$ and $C_0>0$, which does not depend by $n$, such that\n\\[\n \\sup_{ B_{R}} \\widetilde w_n\\leq \\tau\\inf_{B_{R}}\\widetilde w_n +C_0\n\\]\nand this implies that\n\\[\n \\inf_{\\p B_{R}}\\widetilde w_n=\\inf_{ B_{R}}\\widetilde w_n\\geq -\\tau^{-1}C_0,\n\\]\nwhere we used the fact $\\sup\\limits_{ B_{R}} \\widetilde w_n=0$, for any $n$ large enough, and the superharmonicity of $\\widetilde w_n$.\\\\\nTherefore, by \\eqref{equationgoodcase}, \\eqref{Estimateboundary} and standard elliptic estimates, we conclude that $\\widetilde w_n$ is uniformly\nbounded in $C^{t}_{\\rm loc}(\\R^2)$, for some $t\\in(0,1)$ and then, possibly along a subsequence, $\\widetilde w_n\\to \\widetilde w$\nuniformly on compact sets of $\\R^2$, where $\\widetilde w$ satisfies\n\\begin{align*}\n\\begin{cases}\n -\\D \\widetilde w=(\\epsilon_0^2+|y|^2)^{\\alpha_\\infty}V(0)e^{\\dis \\widetilde w} \\quad\\text{in}\\quad\\R^2 \\\\\n\\int_{\\R^2}(\\epsilon_0^2+|y|^2)^{\\alpha_\\infty}e^{\\dis \\widetilde w}\\leq C\n\\end{cases}\n\\end{align*}\nSince $\\widetilde w$ is bounded from above, then necessarily $V(0)\\neq0$. At this point, by Lemma \\ref{massstrange} below, we have that either,\n\n\\section{The inequality \\eqref{sup+inf.intro} is almost sharp.}\\label{sec4}\nIt has ben proved in \\cite{bcwyzOnsager}\nthat if $v_n$ is a sequence of solutions of \\eqref{intro:eq} in $\\om=B_1$, such that \n$$\n\\al_n=\\frac{\\lm_n}{4\\pi}\\sg,\\quad\\lm_n\\to \\lm_\\ii\\in \\left(0,\\frac{4\\pi}{\\sg}\\right),\\quad \\al_\\ii=\\frac{\\lm_\\ii}{4\\pi}\\sg<1,\\quad \\sg\\in \\left(0,\\frac12\\right),\n$$\nwhich also satisfy, \n$$\n\\lm_n=\\int\\limits_{\\om}\\left({\\epsilon_n^2+|x|^2}\\right)^{\\al_n} V_n e^{v_n},\\quad \\epsilon_n\\to 0^+,\n$$\n$$\nV_n\\geq 0, \\,\\, V_n\\to V \\text{ uniformly in }\\overline B_1\\,\\,\\mbox{and in } C^{1}_{\\rm loc}(B_1),\n$$\nand\n$$\n \\underset{\\p B_1}\\max\\,v_n-\\underset{\\p B_1}\\min\\,v_n\\leq C,\n$$ \nand that $x=0$ is the unique blow up point for $v_n$ in $B_1$, that is,\n\\[\n \\text{for any}\\,\\,r\\in(0,1),\\exists \\, C_r>0, \\,\\,\\text{such that}:\n\\]\n\\begin{equation*}\n \\label{bounded}\n\\underset{\\overline{B_1\\backslash B_r}}{\\max}\\, v_n\\leq C_r,\n\\end{equation*}\n\\begin{equation*}\n \\label{explosion}\n\\underset{\\overline{B_r}}\\max\\, v_n\\to\\infty,\n\\end{equation*}\nthen $V(0)>0$ and a new kind of blow up phenomenon takes place. We called it ``blow-up and concentration without quantization\", where the lack of compactness of solutions comes with a concentration phenomenon but, unlike the classical regular (\\cite{yy}) and singular cases (\\cite{bt}), the corresponding mass is not quantized, being free to take values in the full interval $\\lm_\\ii\\in (8\\pi,\\min\\{\\frac{8\\pi}{1-2\\sg},\\tfrac{4\\pi}{\\sg}\\})$. More exactly (see Theorem 3.3 in \\cite{bcwyzOnsager}), we proved that there are only three possibilities, corresponding in fact to the three situations already discussed in the minimal mass Lemma above, in which cases we obtain the following global profiles. \nLet $r\\leq \\frac12$,\n$$\nv_n(x_n)=\\underset{\\overline{B_r}}\n\\max\\, v_n\\to+\\infty, \\quad |x_n|\\to 0^+, \n$$\n$$\n\\delta_n^{2(1+\\al_n)}=e^{\\dis -v_n(x_n)}\\to 0^+,\n$$\nand \n\\[\n t_n=\\max\\{\\delta_n,|x_n|\\}\\to0^+.\n\\]\nThen, either,\n\\begin{itemize}\n \\item \\mbox{\\rm (I):} there exists a subsequence such that $\\tfrac{\\epsilon_n}{t_n}\\to+\\infty$,\n\\end{itemize}\nin which case we have $\\lm_\\ii=8\\pi$ and\n\\begin{equation}\\label{profilev:H}\nv_n(x)=\\log\\left(\\dfrac{e^{\\dis v_n(x_n)}}\n{\\left(1+\\gamma_n\\theta_n^{2(1+\\bns)}\\epsilon_n^{-\\frac{\\lm_n}{4\\pi}}|x-{x_n}|^{\\frac{\\lm_n}{4\\pi}}\\right)^2}\\right)+O(1),\\quad z\\in B_{r}(0);\n\\end{equation}\nwhere\n$\\theta_n^{2(1+\\al_{n})}=\n\\left(\\tfrac{\\epsilon_n}{\\delta_n}\\right)^{2(1+\\al_{n})}\\to +\\ii$, $8 \\gamma_n={(1+|\\frac{x_n}{\\epsilon_n}|^2)^{\\bns} V_n({x_n})}$, or\n\\begin{itemize}\n \\item \\mbox{\\rm (II):} there exists a subsequence such that $\\tfrac{\\epsilon_n}{t_n}\\leq C$ and $\\tfrac{|x_n|}{\\delta_n}\\to+\\infty$,\n\\end{itemize}\nin which case we have $\\lm_\\ii=8\\pi$ and\n\\begin{equation}\\label{profilev:H1}\nv_n(x)=\\log\\left(\\dfrac{e^{\\dis v_n(x_n)}}\n{\\left(1+\\bar\\gamma_n\\bar \\theta_n^{2(1+\\al_{n})}|x_n|^{-\\frac{\\lm_n}{4\\pi}} |x-{x_n}|^{\\frac{\\lm_n}{4\\pi}}\\right)^2}\\right)+O(1),\\quad z\\in B_{r}(0),\n\\end{equation}\nwhere $\\bar \\theta_n^{2(1+\\al_{n})}=\\left(\\tfrac{|x_{n}|}{\\delta_n}\\right)^{2(1+\\al_{n})}\\to +\\ii$,\n$8\\bar\\gamma_n={((\\tfrac{\\epsilon_n}{|x_n|})^2+1)^{\\bns}}V_n(x_n)$, or\n\\begin{itemize}\n \\item \\mbox{\\rm (III):} there exists a subsequence such that $\\tfrac{\\epsilon_n}{t_n}\\leq C$ and\n $\\tfrac{|x_n|}{\\delta_n}\\leq C$,\n\\end{itemize}\nin which case, along a further subsequence if necessary, we have \n$$\n\\tfrac{\\epsilon_n}{\\delta_n}\\to \\epsilon_0\\geq 0, \\quad\\tfrac{x_n}{\\delta_n}\\to y_0\\in\\R^2, $$ \nand $\\lm_\\ii\\in (8\\pi,\\tfrac{8\\pi}{1-2\\sg}]$, if $\\sigma\\in(0,\\tfrac{1}{4})$, or $\\lm_\\ii\\in(8\\pi,\\tfrac{4\\pi}{\\sigma})$, if $\\sg\\in[\\tfrac{1}{4},\\tfrac12)$ and\n\\begin{equation}\\label{profilevtilde:H1-IIb}\n v_n(x)=v_n(x_n)+ \\widetilde U_n(\\delta_n^{-1}x)+O(1), \\quad x\\in B_{r}(0),\n\\end{equation}", "post_theorem_intro_text_len": 2276, "post_theorem_intro_text": "Interestingly enough, in the more subtle case $\\al_\\infty\\in (0,1)$, the refined profiles obtained in \\cite{bcwyzOnsager} suggest that \\eqref{intro:sup+Cinf} is almost sharp, see section \\ref{sec4} for further details. In particular those profiles provide examples \nof sequences of solutions of \\eqref{intro:eq}, \\eqref{intro:alphan}, \\eqref{intro: Vn} that, for any fixed $C_1<\\tfrac{1+\\alpha_\\infty}{1-\\alpha_\\infty}$, satisfy \n$$\n\\sup\\limits_{K} v_n+C_1\\inf\\limits_{\\Omega}v_n\\rightarrow +\\infty.\n$$\nTherefore, we see that, even in the case $\\al_\\infty\\in (0,1)$, if the homogeneous classical expression of a conical singularity at the origin as $|x|^{2\\alpha}$, is replaced by some uniform approximating sequence as $(\\epsilon_n^2+|x|^2)^{\\al_n}$ in \\eqref{intro:eq}, there is no chance to come up with an inequality neither of the form \\eqref{u0+inf}. It is an interesting open problem to check whether or not, possibly under stronger assumptions as in \\cite{BCLT}, the sharp constant in \\eqref{intro:sup+Cinf} is $C_1=\\tfrac{1+\\alpha_\\infty}{1-\\alpha_\\infty}$. We observe that a similar open problem was formulated in \\cite{cosentino2025} in the case $\\alpha_\\infty\\in(-1,0)$ and $\\epsilon_n\\equiv0$, under weaker assumptions on $V_n$. \n\n\\bigskip \n\nConcerning the proof, we argue as in \\cite{b1} (see also \\cite{S}) via known blow-up arguments (\\cite{bt}, \\cite{det}, \\cite{ls}, \\cite{llty}), where one has\nto carefully handle the various possibilities arising due to the perturbation (i.e. $\\epsilon_n\\rightarrow 0^+$) of the homogeneous conical singularity.\nThe contradiction argument requires, in each one of the blow up scenarios at hand, to find at least one bubble coming with some large enough amount of mass.\nThis is also why we need a preliminary result, which is a ``minimal mass\" Lemma for solutions of \\eqref{intro:eq}, see Section \\ref{sec2}.\n\n\\bigskip\n\nThis paper is organized as follows: in Section 2 we give a proof of the minimal mass Lemma, while the proof of the ``$\\sup+C\\inf$''\ninequality is done in Section 3. We discuss the sharpness of the ``$\\sup+C\\inf$'' inequality in section \\ref{sec4}. Lastly, in the appendix, we state some well-known properties of the solutions of some perturbed singular\nLiouville equations in $\\mathbb{R}^2$.\n\n\\medskip", "sketch": "Concerning the proof (of Theorem~\\ref{intro:maintheo}), the authors \"argue as in \\cite{b1} (see also \\cite{S}) via known blow-up arguments\" (\\cite{bt}, \\cite{det}, \\cite{ls}, \\cite{llty}), and they \"carefully handle the various possibilities arising due to the perturbation (i.e. $\\epsilon_n\\rightarrow 0^+$) of the homogeneous conical singularity.\" The proof is by \"contradiction\" and, \"in each one of the blow up scenarios at hand,\" it is required \"to find at least one bubble coming with some large enough amount of mass.\" For this reason they first prove a preliminary \"minimal mass\" lemma for solutions of \\eqref{intro:eq} (Section~\\ref{sec2}), and then prove the ``$\\sup+C\\inf$'' inequality in Section~3.", "expanded_sketch": "Concerning the proof (of the main theorem), the authors “argue as in \\cite{b1} (see also \\cite{S}) via known blow-up arguments” (\\cite{bt}, \\cite{det}, \\cite{ls}, \\cite{llty}), and they “carefully handle the various possibilities arising due to the perturbation (i.e. $\\epsilon_n\\rightarrow 0^+$) of the homogeneous conical singularity.” The proof is by “contradiction” and, “in each one of the blow up scenarios at hand,” it is required “to find at least one bubble coming with some large enough amount of mass.” For this reason they first prove a preliminary “minimal mass” lemma for solutions of\n\\begin{equation}\\label{intro:eq}\n -\\D v_n=\\left({\\epsilon_n^2+|x|^2}\\right)^{\\alpha_n}V_n(x)e^{\\displaystyle v_n} \\qquad\\text{in} \\,\\,\\, \\Omega,\n\\end{equation}\nand then prove the ``$\\sup+C\\inf$'' inequality later.", "expanded_theorem": "\\label{intro:maintheo}\nLet $\\O$ be an open bounded domain in $\\mathbb{R}^2$ which contains the origin, $\\{0\\}\\subset \\O$. Assume that\n$v_n$ is a sequence of solutions of\n\\begin{equation}\\label{intro:eq}\n -\\D v_n=\\left({\\epsilon_n^2+|x|^2}\\right)^{\\alpha_n}V_n(x)e^{\\displaystyle v_n} \\qquad\\text{in} \\,\\,\\, \\Omega,\n\\end{equation}\nsatisfying\n\\begin{equation}\n \\label{intro:alphan}\n \\a_n\\to\\a_\\infty\\in(-1,+\\ii),\n\\end{equation}\nand\n\\begin{equation}\\label{intro: Vn}\n 0\\max\\left\\{1,\\tfrac{1+\\alpha_\\infty}{1-\\alpha_\\infty}\\right\\}$$\nand for any compact set $K\\subset\\O$, there exists a constant $C_2>0$, which depends only by $a,b,dist(K,\\partial\\O), \\alpha_\\infty$ and by the\nuniform modulus of continuity of $V$ on $K$, such that,\n\\begin{equation}\n \\label{intro:sup+Cinf}\n \\underset{K}\\sup\\, v_n + C_1\\underset{\\O}\\inf\\, v_n\\leq C_2.\n\\end{equation}", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let \\(\\Omega\\subset \\mathbb{R}^2\\) be a bounded open domain containing the origin. Suppose \\((v_n)\\) is a sequence of solutions of\n\\[\n-\\Delta v_n=(\\epsilon_n^2+|x|^2)^{\\alpha_n}V_n(x)e^{v_n}\\qquad\\text{in }\\Omega,\n\\]\nwhere \\(\\alpha_n\\to \\alpha_\\infty\\in(-1,\\infty)\\) with in fact \\(\\alpha_\\infty\\in(-1,1)\\), and the coefficients satisfy\n\\[\n0\\max\\!\\left\\{1,\\dfrac{1+\\alpha_\\infty}{1-\\alpha_\\infty}\\right\\}\\) and every compact set \\(K\\subset \\Omega\\), there exists a constant \\(C_2>0\\), depending only on \\(a\\), \\(b\\), \\(\\operatorname{dist}(K,\\partial\\Omega)\\), \\(\\alpha_\\infty\\), and the uniform modulus of continuity of \\(V\\) on \\(K\\), such that\n\\[\n\\sup_K v_n + C_1\\inf_\\Omega v_n\\le C_2\n\\]\nfor all \\(n\\)."}, "choices": [{"label": "B", "text": "For every \\(C_1\\ge \\max\\!\\left\\{1,\\dfrac{1+\\alpha_\\infty}{1-\\alpha_\\infty}\\right\\}\\) and every compact set \\(K\\subset \\Omega\\), there exists a constant \\(C_2>0\\), depending only on \\(a\\), \\(b\\), \\(\\operatorname{dist}(K,\\partial\\Omega)\\), \\(\\alpha_\\infty\\), and the uniform modulus of continuity of \\(V\\) on \\(K\\), such that\n\\[\n\\sup_K v_n + C_1\\inf_\\Omega v_n\\le C_2\n\\]\nfor all \\(n\\)."}, {"label": "C", "text": "For every compact set \\(K\\subset \\Omega\\), there exist constants \\(C_1>\\max\\!\\left\\{1,\\dfrac{1+\\alpha_\\infty}{1-\\alpha_\\infty}\\right\\}\\) and \\(C_2>0\\), depending only on \\(a\\), \\(b\\), \\(\\operatorname{dist}(K,\\partial\\Omega)\\), \\(\\alpha_\\infty\\), and the uniform modulus of continuity of \\(V\\) on \\(K\\), such that\n\\[\n\\sup_K v_n + C_1\\inf_\\Omega v_n\\le C_2\n\\]\nfor all \\(n\\)."}, {"label": "D", "text": "For every \\(C_1>\\max\\!\\left\\{1,\\dfrac{1+\\alpha_\\infty}{1-\\alpha_\\infty}\\right\\}\\) and every compact set \\(K\\subset \\Omega\\), there exists a constant \\(C_2>0\\), depending only on \\(a\\), \\(b\\), \\(\\operatorname{dist}(K,\\partial\\Omega)\\), \\(\\alpha_\\infty\\), and the uniform modulus of continuity of \\(V_n\\) on \\(K\\), such that\n\\[\n\\sup_K v_n + C_1\\inf_\\Omega v_n\\le C_2\n\\]\nfor all \\(n\\)."}, {"label": "E", "text": "For every \\(C_1>1\\) and every compact set \\(K\\subset \\Omega\\), there exists a constant \\(C_2>0\\), depending only on \\(a\\), \\(b\\), \\(\\operatorname{dist}(K,\\partial\\Omega)\\), \\(\\alpha_\\infty\\), and the uniform modulus of continuity of \\(V\\) on \\(K\\), such that\n\\[\n\\sup_K v_n + C_1\\inf_\\Omega v_n\\le C_2\n\\]\nfor all \\(n\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "minimal_mass", "tampered_component": "strict lower bound on C_1", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "minimal_mass", "tampered_component": "universal quantification over all admissible C_1", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "dependence on modulus of continuity of V instead of the sequence V_n", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "minimal_mass", "tampered_component": "threshold C_1>(1+alpha_infty)/(1-alpha_infty)", "template_used": "stronger_trap"}]}} +{"id": "2601.13552v1", "paper_link": "http://arxiv.org/abs/2601.13552v1", "theorems_cnt": 4, "theorem": {"env_name": "thm", "content": "[Main Theorem]\\label{main 1}\n\tFor every integer $k\\ge 6$, let $G$ be a graph with minimum degree at least $k$.\n Then exactly one of the following holds:\n \\begin{itemize}\n \\item[(1)] $G$ contains a cycle of length $r \\pmod k$ for every even integer $r$;\n \\item[(2)] $k$ is odd, every end-block of $G$ is isomorphic to a graph in $\\{K_{k+1}, K_{k,k}\\}\\cup \\mathcal{H}_{k}$, and every non-end-block contains no cycle of length $2 \\pmod k$;\n \\item[(3)] $k$ is even, every end-block of $G$ is isomorphic to $K_{k+1}$, and every non-end-block contains no cycle of length $2 \\pmod k$.\n \\end{itemize}", "start_pos": 5833, "end_pos": 6476, "label": "main 1"}, "ref_dict": {"conj:Dean": "\\begin{conjecture}[Dean's conjecture]\\label{conj:Dean}\n For every integer $k \\geq 3$, every graph with minimum degree at least $k$ contains a cycle of length divisible by $k$.\n\\end{conjecture}", "thm:exist admis": "\\begin{thm}\\label{thm:exist admis}\n Let $k\\ge 7$ and let $G$ be a graph with minimum degree at least $k$.\n Then $G$ contains $k$ admissible cycles, unless every end-block of $G$ is isomorphic to a graph in $\\{K_{k+1}, K_{k,k}\\}\\cup \\mathcal{H}_{k}$.\n\\end{thm}", "main 1": "\\begin{thm}[Main Theorem]\\label{main 1}\n\tFor every integer $k\\ge 6$, let $G$ be a graph with minimum degree at least $k$.\n Then exactly one of the following holds:\n \\begin{itemize}\n \\item[(1)] $G$ contains a cycle of length $r \\pmod k$ for every even integer $r$;\n \\item[(2)] $k$ is odd, every end-block of $G$ is isomorphic to a graph in $\\{K_{k+1}, K_{k,k}\\}\\cup \\mathcal{H}_{k}$, and every non-end-block contains no cycle of length $2 \\pmod k$;\n \\item[(3)] $k$ is even, every end-block of $G$ is isomorphic to $K_{k+1}$, and every non-end-block contains no cycle of length $2 \\pmod k$.\n \\end{itemize}\n\\end{thm}", "thm:weak graph": "\\begin{thm}\\label{thm:weak graph}\nLet $k\\geq 6$ be an integer. If $G$ is a $k$-weak graph not isomorphic to any graph in $\\{K_{k+1}, K_{k,k}\\}\\cup \\mathcal{H}_{k}$, then $G$ contains a cycle of length $r\\pmod k$ for every even integer $r$.\n\\end{thm}"}, "pre_theorem_intro_text_len": 1792, "pre_theorem_intro_text": "\\label{sec: intro}\n\\noindent The study of cycle lengths in graphs is a central and classical theme in graph theory; see \\cite{bondy1996basic,verstraete2016extremal} for comprehensive treatments.\nA particularly interesting problem in this area concerns the existence of cycles whose lengths are divisible by a given integer~$k$; see, for example, \\cite{alon1989cycles,dean1988graphs,thomassen1988presence,thomassen1992even}.\nThe present work is motivated by the following beautiful conjecture of Dean (see Conjecture~7.4 in \\cite{bondy1996basic}), which has remained open for three decades.\n\n\\vskip -1mm\n\n\\begin{conjecture}[Dean's conjecture]\\label{conj:Dean}\n For every integer $k \\geq 3$, every graph with minimum degree at least $k$ contains a cycle of length divisible by $k$.\n\\end{conjecture}\n\n\\vskip -1mm\n\nThe minimum degree condition in Conjecture~\\ref{conj:Dean} is best possible, as complete bipartite graphs $K_{k-1,n}$ for odd $k$ and $n\\geq k-1$ show that it cannot be weakened to $k-1$.\nThe conjecture has been verified for $k=3$ and $k=4$ by Chen and Saito \\cite{chen1994graphs} and by Dean, Lesniak, and Saito \\cite{dean1993cycles}, respectively, but remains open for all $k \\ge 5$.\nA weaker version, also proposed by Dean \\cite{dean1988graphs}, asserting that every $k$-connected graph contains a cycle of length divisible by $k$, was resolved by Gao, Huo, Liu, and Ma \\cite{gao2022unified} via a unified approach to related cycle problems.\n\nIn this paper, we prove Conjecture~\\ref{conj:Dean} for all $k \\geq 6$. \nTo state our main result, let $\\mathcal{H}_k$ denote the family of all graphs $H_{k,n;t}$, where $2 \\le t \\le k < n$, obtained from the complete bipartite graph $K_{k,n}$ by deleting $k-t$ edges incident to a single vertex in the part of size~$n$.\n\n\\vskip -1mm", "context": "\\label{sec: intro}\n\\noindent The study of cycle lengths in graphs is a central and classical theme in graph theory; see \\cite{bondy1996basic,verstraete2016extremal} for comprehensive treatments.\nA particularly interesting problem in this area concerns the existence of cycles whose lengths are divisible by a given integer~$k$; see, for example, \\cite{alon1989cycles,dean1988graphs,thomassen1988presence,thomassen1992even}.\nThe present work is motivated by the following beautiful conjecture of Dean (see Conjecture~7.4 in \\cite{bondy1996basic}), which has remained open for three decades.\n\n\\begin{conjecture}[Dean's conjecture]\\label{conj:Dean}\n For every integer $k \\geq 3$, every graph with minimum degree at least $k$ contains a cycle of length divisible by $k$.\n\\end{conjecture}\n\nThe minimum degree condition in Conjecture~\\ref{conj:Dean} is best possible, as complete bipartite graphs $K_{k-1,n}$ for odd $k$ and $n\\geq k-1$ show that it cannot be weakened to $k-1$.\nThe conjecture has been verified for $k=3$ and $k=4$ by Chen and Saito \\cite{chen1994graphs} and by Dean, Lesniak, and Saito \\cite{dean1993cycles}, respectively, but remains open for all $k \\ge 5$.\nA weaker version, also proposed by Dean \\cite{dean1988graphs}, asserting that every $k$-connected graph contains a cycle of length divisible by $k$, was resolved by Gao, Huo, Liu, and Ma \\cite{gao2022unified} via a unified approach to related cycle problems.\n\nIn this paper, we prove Conjecture~\\ref{conj:Dean} for all $k \\geq 6$. \nTo state our main result, let $\\mathcal{H}_k$ denote the family of all graphs $H_{k,n;t}$, where $2 \\le t \\le k < n$, obtained from the complete bipartite graph $K_{k,n}$ by deleting $k-t$ edges incident to a single vertex in the part of size~$n$.\n\n\\vskip -1mm\n\n\\begin{conjecture}[Dean's conjecture]\\label{conj:Dean}\n For every integer $k \\geq 3$, every graph with minimum degree at least $k$ contains a cycle of length divisible by $k$.\n\\end{conjecture}", "full_context": "\\label{sec: intro}\n\\noindent The study of cycle lengths in graphs is a central and classical theme in graph theory; see \\cite{bondy1996basic,verstraete2016extremal} for comprehensive treatments.\nA particularly interesting problem in this area concerns the existence of cycles whose lengths are divisible by a given integer~$k$; see, for example, \\cite{alon1989cycles,dean1988graphs,thomassen1988presence,thomassen1992even}.\nThe present work is motivated by the following beautiful conjecture of Dean (see Conjecture~7.4 in \\cite{bondy1996basic}), which has remained open for three decades.\n\n\\begin{conjecture}[Dean's conjecture]\\label{conj:Dean}\n For every integer $k \\geq 3$, every graph with minimum degree at least $k$ contains a cycle of length divisible by $k$.\n\\end{conjecture}\n\nThe minimum degree condition in Conjecture~\\ref{conj:Dean} is best possible, as complete bipartite graphs $K_{k-1,n}$ for odd $k$ and $n\\geq k-1$ show that it cannot be weakened to $k-1$.\nThe conjecture has been verified for $k=3$ and $k=4$ by Chen and Saito \\cite{chen1994graphs} and by Dean, Lesniak, and Saito \\cite{dean1993cycles}, respectively, but remains open for all $k \\ge 5$.\nA weaker version, also proposed by Dean \\cite{dean1988graphs}, asserting that every $k$-connected graph contains a cycle of length divisible by $k$, was resolved by Gao, Huo, Liu, and Ma \\cite{gao2022unified} via a unified approach to related cycle problems.\n\nIn this paper, we prove Conjecture~\\ref{conj:Dean} for all $k \\geq 6$. \nTo state our main result, let $\\mathcal{H}_k$ denote the family of all graphs $H_{k,n;t}$, where $2 \\le t \\le k < n$, obtained from the complete bipartite graph $K_{k,n}$ by deleting $k-t$ edges incident to a single vertex in the part of size~$n$.\n\n\\vskip -1mm\n\n\\begin{conjecture}[Dean's conjecture]\\label{conj:Dean}\n For every integer $k \\geq 3$, every graph with minimum degree at least $k$ contains a cycle of length divisible by $k$.\n\\end{conjecture}\n\n\\vskip -1mm\n\nObserve that every graph in $\\{K_{k+1}, K_{k,k}\\} \\cup \\mathcal{H}_k$ contains cycles of all even lengths modulo~$k$, with the sole exception of length $2 \\pmod k$.\nConsequently, we obtain the following immediate corollary, \nwhich resolves Conjecture~\\ref{conj:Dean} affirmatively for all $k \\ge 6$.\\footnote{The case $k=5$ requires separate arguments and we will address this special case in forthcoming work.}\n\n\\begin{thm}\\label{thm:exist admis}\n Let $k\\ge 7$ and let $G$ be a graph with minimum degree at least $k$.\n Then $G$ contains $k$ admissible cycles, unless every end-block of $G$ is isomorphic to a graph in $\\{K_{k+1}, K_{k,k}\\}\\cup \\mathcal{H}_{k}$.\n\\end{thm}\n\n\\begin{thm}\\label{thm:weak graph}\nLet $k\\geq 6$ be an integer. If $G$ is a $k$-weak graph not isomorphic to any graph in $\\{K_{k+1}, K_{k,k}\\}\\cup \\mathcal{H}_{k}$, then $G$ contains a cycle of length $r\\pmod k$ for every even integer $r$.\n\\end{thm}\n\n\\begin{proof}[\\bf{Proof of Theorem~\\ref{main 1} (assuming Theorem~\\ref{thm:weak graph}).}]\nIt suffices to establish the following stability result for 2-connected graphs: for $k\\geq 6$, every 2-connected graph $G$ with $\\delta_2(G) \\ge k$ contains a cycle of length $r \\pmod k$ for every even integer $r$, unless $G$ is isomorphic to a graph in $\\{K_{k+1}, K_{k,k}\\}\\cup \\mathcal{H}_{k}$.\nObserve that every graph in $\\{K_{k+1}, K_{k,k}\\}\\cup \\mathcal{H}_{k}$ contains cycles of all even lengths modulo $k$ with the only exception of $2 \\pmod k$. \nThus, Theorem~\\ref{main 1} follows directly by applying this result to each end-block of the graph (since every end-block $B$ of a graph with $\\delta(G) \\ge k$ satisfies $\\delta_2(B) \\ge k$).\n\n\\begin{thm}\\label{main}\n Let $k\\ge 6$ be an integer, and let $G^\\star$ be a $k$-weak graph not isomorphic to $K_{k+1}$. If $G$ is non-bipartite, then $G^\\star$ contains a cycle of length $r\\pmod k$ for every even integer $r$.\n\\end{thm}\n\nThe following lemma characterizes the set of path lengths $\\Pset_{u,v}^{T^*}$.\n\\begin{lem}\\label{lem:Q-admis}\n Let $u_1, u_2$ be two distinct vertices in $V(G-T^*)$ with positive degrees $d_1:=\\deg_{T^*}(u_1)$ and $d_2:=\\deg_{T^*}(u_2)$. Then one of the following statements holds:\n \\begin{itemize}\n \\item[(1)] $\\mathcal{P}_{u_1,u_2}^{T^*}$ contains at least $\\min\\{d_1+d_2-1,m\\}$ admissible paths.\n \\item[(2)] $\\Pset_{u_1,u_2}^{T^*} \\supseteq \\{2\\} \\cup \\{2+d, 4+d, \\dots, 2m+2-d\\}$ for some even $d \\leq \\max\\{2, 2m/\\max\\{d_1,d_2\\}, \\\\m-d_1-d_2+3\\}$.\n \\end{itemize}\n\\end{lem}\n\nUsing an argument analogous to Lemma~\\ref{lem: B}, we deduce that $(u,v)$ is $(k-d_{\\min})$-valid.\n In fact, the proof of Lemma~\\ref{lem: B} relied on the fact that $\\deg_{T^*}(w) \\le \\deg_{T^*}(u)$ for every $w\\in V(B_M)\\setminus\\big(\\{u\\}\\cup \\mathrm{Cut}(B_M)\\big)$. \n Here, by the maximality of $\\deg_{T^*}(v)$, every $w\\in V(B_M)\\setminus\\big(\\{u\\}\\cup \\mathrm{Cut}(B_M)\\big)$ satisfies $\\deg_{T^*}(w) \\le \\deg_{T^*}(v)$, and thus $\\deg_{T^*}(w) \\le d_{\\min}$.\n Substituting this stronger inequality into the proof of Lemma~\\ref{lem: B} confirms that $(u, v)$ is $(k-d_{\\min})$-valid.\n By Observation~\\ref{obs:valid}, $\\mathcal{P}^M_{u,v}$ contains $k-d_{\\min}-1$ admissible paths.\n Let $\\Pset_1$ denote the set of lengths of these admissible\n paths.\n According to Lemma~\\ref{lem:Q-admis}, $\\Pset^{T^*}_{u,v}$ contains a subset $\\Pset_2$ with one of the following forms:\n\\begin{itemize}\n \\item An admissible set of size at least $\\min\\{d_{\\min}+d_{\\max}-1,m\\}$;\n \\item $\\{2\\}\\cup\\{2+d,4+d,\\cdots,2m+2-d\\}$ for some even $d\\leq\\max\\{2,2m/d_{\\max},m-d_{\\min}-d_{\\max}+3\\}$.\n\\end{itemize}\nWe verify that $\\Pset_1+\\Pset_2$ is an admissible set of size at least $k$.\nIf $\\Pset_2$ is of the former case, since $3\\le d_{\\max}\\le m-2$, the sum $\\Pset_1+\\Pset_2$ is an admissible set of length at least $(k-d_{\\min}-1)+\\min\\{d_{\\min}+d_{\\max}-1,m\\}-1\\geq k$.\nIn the latter case, the condition for $\\Pset_1+\\Pset_2$ to form an admissible set is equivalent to:\n\\begin{align*}\n d\\leq \\max\\{\\Pset_{1}\\}-\\min\\{\\Pset_{1}\\}+2\\nonumber\n \\iff d\\leq 2(k-d_{\\min}-1)\\nonumber\n\\end{align*}\nThis is guaranteed by the following strict inequality, since $d$ is even. \n\\begin{align*}\n &d\\leq \\max\\{2,2m/d_{\\max},m-d_{\\min}-d_{\\max}+3\\}\n \\overset{\\text{($\\ast$)}}{<}2(m-d_{\\min})\\le 2(k-d_{\\min}),\n\\end{align*}\nwhere $(\\ast)$ is implied by the following three inequalities (as $m\\geq 5$):\n\\begin{align*}\n 2(m-d_{\\min})&\\geq 4>2,\\\\ \n 2(m-d_{\\min})&\\geq 2(m-d_{\\max})>2m/d_{\\max}, ~~~\\mbox{and}\\\\\n 2(m-d_{\\min})&>m-2d_{\\min}+3\\geq m-d_{\\min}-d_{\\max}+3.\n\\end{align*}\nTherefore, $\\Pset_1+\\Pset_2$ is an admissible set with minimum element $\\min\\{\\Pset_1\\}+2$ and maximum element $\\max\\{\\Pset_1\\}+\\max\\{\\Pset_2\\}$, thus has size $$\\frac{\\max\\{\\Pset_1\\}-\\min\\{\\Pset_1\\}+\\max\\{\\Pset_2\\}}{2}=k+m-d_{\\min}-d/2-1.$$\nNote that $k+m-d_{\\min}-d/2-1\\geq k$ is equivalent to $d\\leq 2(m-d_{\\min}-1)$, which we have already established via inequality $(\\ast)$.\nHence, $\\Pset_1+\\Pset_2$ is an admissible set of size at least $k$, and the union of paths in $\\mathcal{P}^M_{u,v}$ and $\\mathcal{P}^{T^*}_{u,v}$ produces $k$ admissible cycles in $G$.\nThis completes the proof of Lemma~\\ref{lem:one comp deg}.\n\\end{proof}\n\n\\begin{conjecture}[Dean's conjecture]\\label{conj:Dean}\n For every integer $k \\geq 3$, every graph with minimum degree at least $k$ contains a cycle of length divisible by $k$.\n\\end{conjecture}\n\n\\begin{thm}[Main Theorem]\\label{main 1}\n\tFor every integer $k\\ge 6$, let $G$ be a graph with minimum degree at least $k$.\n Then exactly one of the following holds:\n \\begin{itemize}\n \\item[(1)] $G$ contains a cycle of length $r \\pmod k$ for every even integer $r$;\n \\item[(2)] $k$ is odd, every end-block of $G$ is isomorphic to a graph in $\\{K_{k+1}, K_{k,k}\\}\\cup \\mathcal{H}_{k}$, and every non-end-block contains no cycle of length $2 \\pmod k$;\n \\item[(3)] $k$ is even, every end-block of $G$ is isomorphic to $K_{k+1}$, and every non-end-block contains no cycle of length $2 \\pmod k$.\n \\end{itemize}\n\\end{thm}", "post_theorem_intro_text_len": 7011, "post_theorem_intro_text": "Observe that every graph in $\\{K_{k+1}, K_{k,k}\\} \\cup \\mathcal{H}_k$ contains cycles of all even lengths modulo~$k$, with the sole exception of length $2 \\pmod k$.\nConsequently, we obtain the following immediate corollary, \nwhich resolves Conjecture~\\ref{conj:Dean} affirmatively for all $k \\ge 6$.\\footnote{The case $k=5$ requires separate arguments and we will address this special case in forthcoming work.}\n\n\\begin{corollary}\n Let $k \\ge 6$ be an integer. Then for every even integer $r \\not\\equiv 2 \\pmod k$, every graph with minimum degree at least $k$ contains a cycle of length $r \\pmod k$.\n\\end{corollary}\n\nThe general study of cycle lengths modulo~$k$ dates back to the 1970s, initiated by the work of Burr and Erd\\H{o}s \\cite{Erdos1976}.\nFor integers $\\ell$ and $k$ with even integers in the residue class $\\ell \\pmod k$, let $c_{\\ell,k}$ denote the smallest constant $c$ such that every $n$-vertex graph with at least $cn$ edges contains a cycle of length $\\ell \\pmod k$.\nErd\\H{o}s \\cite{Erdos1976} conjectured that $c_{\\ell,k}$ exists for all $\\ell$ when $k$ is odd,\nand this was later confirmed by Bollob\\'as \\cite{bollobas1977cycles}.\nThomassen \\cite{thomassen1983graph} further improved the bound to $c_{\\ell, k} \\le 4k(k+1)$ for all integers $k$ and even $\\ell$.\nIn a subsequent work \\cite{thomassen1988presence}, Thomassen provided a polynomial-time algorithm for finding a cycle of length divisible by $k$.\nResolving a conjecture of Thomassen \\cite{thomassen1983graph}, Gao, Huo, Liu, and Ma \\cite{gao2022unified} showed that for any $k \\ge 3$, every graph with minimum degree at least $k+1$ contains cycles of all even lengths modulo~$k$ (the case of even $k$ was previously proved in \\cite{liu2018cycle}).\nFrom an extremal perspective, Sudakov and Verstra\\\"ete \\cite{sudakov2017extremal} established a striking relation that for all $3\\leq \\ell0$: (a) “$H$ contains $t$ admissible paths between many pairs of vertices” (a “robust spreading property”); and (b) after “contracting or deleting $V(H)$,” the remaining graph “satisfies suitable minimum degree conditions and yields $k+1-t$ admissible paths between prescribed pairs of vertices.” “By concatenating these paths appropriately, one obtains the desired $k$ admissible paths in $G$.”\n\nA main new tool is that, unlike earlier works where core subgraphs were “typically dense structures” (complete or complete bipartite, or close to them), this work introduces “two new families of core graphs that can be very sparse … while still satisfying properties (a) and (b),” called “\\textit{trigonal graphs} and \\textit{tetragonal graphs},” which “handle the non-bipartite and bipartite cases, respectively.” Because of their sparsity, “these core subgraphs can be found in graphs with lower connectivity, providing the crucial ingredient” for removing connectivity assumptions.\n\nOrganizationally, the proof is reduced in Section~4 by introducing “\\textit{$k$-weak} graphs” and “reduc[ing] the proof of Theorem~\\ref{main 1} to this class (see Theorem~\\ref{thm:weak graph}),” with the proof then “split between Sections~5 and~6, according to whether the host graph is non-bipartite or bipartite.”", "expanded_sketch": "To prove the main theorem, the paper “builds upon the approach of Gao et al.” by finding “a collection of $k$ admissible paths between any two given vertices $x,y$ in a suitable 2-connected graph $G$.” The “key step is to identify a specified subgraph $H$ (called the {\\it core} subgraph)” so that for some $t>0$: (a) “$H$ contains $t$ admissible paths between many pairs of vertices” (a “robust spreading property”); and (b) after “contracting or deleting $V(H)$,” the remaining graph “satisfies suitable minimum degree conditions and yields $k+1-t$ admissible paths between prescribed pairs of vertices.” “By concatenating these paths appropriately, one obtains the desired $k$ admissible paths in $G$.”\n\nA main new tool is that, unlike earlier works where core subgraphs were “typically dense structures” (complete or complete bipartite, or close to them), this work introduces “two new families of core graphs that can be very sparse … while still satisfying properties (a) and (b),” called “\\textit{trigonal graphs} and \\textit{tetragonal graphs},” which “handle the non-bipartite and bipartite cases, respectively.” Because of their sparsity, “these core subgraphs can be found in graphs with lower connectivity, providing the crucial ingredient” for removing connectivity assumptions.\n\nOrganizationally, the proof is reduced later by introducing “\\textit{$k$-weak} graphs” and reducing the proof of the main theorem to this class; specifically, we use the following theorem.\n\n\\begin{thm}\\label{thm:weak graph}\nLet $k\\geq 6$ be an integer. If $G$ is a $k$-weak graph not isomorphic to any graph in $\\{K_{k+1}, K_{k,k}\\}\\cup \\mathcal{H}_{k}$, then $G$ contains a cycle of length $r\\pmod k$ for every even integer $r$.\n\\end{thm}\n\nThe proof is then “split between” the subsequent parts, according to whether the host graph is non-bipartite or bipartite.", "expanded_theorem": "[Main Theorem]\\label{main 1}\n\tFor every integer $k\\ge 6$, let $G$ be a graph with minimum degree at least $k$.\n Then exactly one of the following holds:\n \\begin{itemize}\n \\item[(1)] $G$ contains a cycle of length $r \\pmod k$ for every even integer $r$;\n \\item[(2)] $k$ is odd, every end-block of $G$ is isomorphic to a graph in $\\{K_{k+1}, K_{k,k}\\}\\cup \\mathcal{H}_{k}$, and every non-end-block contains no cycle of length $2 \\pmod k$;\n \\item[(3)] $k$ is even, every end-block of $G$ is isomorphic to $K_{k+1}$, and every non-end-block contains no cycle of length $2 \\pmod k$.\n \\end{itemize}", "theorem_type": ["Universal", "Classification or Bijection"], "mcq": {"question": "Let $k\\ge 6$ be an integer, and let $G$ be a graph with minimum degree at least $k$. Define $\\mathcal H_k$ to be the family of all graphs $H_{k,n;t}$, where $2\\le t\\le k0$?\n\\begin{enumerate}\n\\item The answer is positive in case $K$ is finite. Essentially finite bundles are slope-semistable of degree $0$ with respect to an arbitrary polarization, and therefore essentially finite bundles of given rank form a bounded family. \n\\item\\label{line}The answer is positive in case $n=1$ and $K$ is a global or local field. We can apply Lang-Néron theorem in the global case and we refer to this post \\cite{148547} in the local case.\n\\item\\label{nori} Point \\ref{line} turns out to be misleading: the answer is negative in case $n\\geq 2$ and $K$ is infinite. Nilpotent bundles (i.e. iterated extensions of trivial bundles) are essentially finite (Proposition 3 Chapter \\uppercase\\expandafter{\\romannumeral4} Part \\uppercase\\expandafter{\\romannumeral2} \\cite{nori1982fundamental}) and nilpotent bundles of rank $2$ are classified by $\\mathrm{H}^{1}(X,\\mathcal{O}_{X})$, which is infinite whenever it is non-zero.\n\\item \\label{etale}We have seen some failure in point \\ref{nori}. The situation does not improve even if we restrict to those representations which factor through the maximal pro-étale quotient of Nori's fundamental group scheme. Let $(X,x)$ be a pointed geometrically connected smooth proper curve over an infinite field $K$ of characteristic $p>0$, which is assumed to be ordinary when we make a base change to $\\bar{K}$. Assume $X$ has genus $g>1$ so there are infinitely many isomorphism classes of nilpotent (and therefore essentially finite) bundles of rank $2$. We claim any nilpotent bundle $\\mathcal{E}$ of rank $2$ give rises a unipotent étale quotient $\\pi_{1}(\\langle\\mathcal{E}\\rangle,x)$ of $\\pi_{1}^{EF}(X,x)$. Indeed, it suffices to note that $\\pi_{1}(\\langle\\mathcal{E}\\rangle,x)_{\\bar{K}}$ is étale as $\\mathcal{E}_{\\bar{K}}$ has an infinite sequence of Frobenius descent. \n\\item Using the same proof as that for Theorem 3.3 \\cite{10.1215/S0012-7094-03-11723-8}, we can prove the following statement: let $(X,x)$ be a pointed geometrically connected smooth projective variety over a global field $K$ of characteristic $p>0$. For each $n$ there are only finitely many representations $\\pi_{1}^{EF}(X,x)\\rightarrow\\mathrm{GL}_{n}$ which factor through the maximal prime-to-$p$ quotient $\\pi_{1}^{EF}(X,x)^{p'}$ of $\\pi_{1}^{EF}(X,x)$. To pass to the prime-to-$p$ part seems necessary according to point \\ref{nori} and point \\ref{etale}. We also remark that our proof for Theorem \\ref{main_thm1} does not work in characteristic $p>0$. Indeed, the naive analogy of Proposition \\ref{lang} is wrong, as there may be infinitely many Artin–Schreier extensions, which are separable of degree $p$.\n\\end{enumerate}\n\\end{rmk}\n\n\\bibliographystyle{plain}\n \\bibliography{Finiteness_FiniteBundle}\n\\end{document}", "post_theorem_intro_text_len": 2130, "post_theorem_intro_text": "The technique will be different from that from \\cite{10.1215/S0012-7094-03-11723-8} and will be based on Jordan's theorem on finite groups embedded into the group of invertible matrices, and some standard arithmetic results over a $p$-adic field. \\par\n\\subsection*{Related work}\nLet $K$ be an arbitrary field of characteristic $0$ and $X$ be a geometrically connected smooth proper curve over $K$. It is well-known that essentially finite bundles are semistable of degree $0$. Assuming $X$ has genus $\\geq 2$ and $n\\geq2$, it is proved in \\cite{ghiasabadi2023essentially}\\cite{olsson2023moduli} that essentially finite bundles of rank $n$ are not Zariski dense in the moduli space of semistable bundles of rank $n$ and of degree $0$. Therefore our result in can be regarded as an improvement of theirs for particular $K$.\n\\subsection*{Notation and Convention}\nAll rings are assumed to be commutative and with a unit. \n\\begin{enumerate}\n\\item For morphisms $X\\rightarrow S$ and $S'\\rightarrow S$ between schemes, we write $X_{S'}$ for the fiber product $X\\times_{S}S'$. In case $S'=\\mathrm{Spec}\\,R$, we also write $X_{R}$ for $X_{\\mathrm{Spec}\\,R'}$. For a quasi-coherent sheaf $\\mathcal{E}$ on $X$, we use $\\mathcal{E}_{R}$ to denote the pull-back quasi-coherent sheaf on $X_{R}$.\n\\item Let $K$ be a field. By a variety over $K$ we mean an integral scheme which is of finite type and separated over $K$. A pointed variety over $K$ is a pair $(X,x)$, which is a variety $X$ over $K$ provided with $x\\in X(K)$. \n\\item Let $K$ be a field. We write $\\mathrm{GL}_{n,K}$ to be the general linear group scheme over $K$. We also write $\\mathrm{GL}_{n}$ when $K$ is clear from the context. By a rank $n$-representation of an affine group scheme $\\pi$ over $K$,we mean a morphism $\\pi_{1}\\rightarrow \\mathrm{GL}_{n}$ between affine group schemes. Two representations are said to be isomorphic if they differ up to a conjugation by an element from $\\mathrm{GL}_{n}(K)$.\n\\end{enumerate}\n\\subsection*{Acknowledgement}\nThe author would like to thank Professor C.Gasbarri for his encouragement to generalize his result in greater generality.", "sketch": "The post-theorem introduction indicates that the proof uses a different technique than \\cite{10.1215/S0012-7094-03-11723-8}, and is \"based on Jordan's theorem on finite groups embedded into the group of invertible matrices, and some standard arithmetic results over a $p$-adic field.\"", "expanded_sketch": "The post-theorem introduction indicates that the proof uses a different technique than \\cite{10.1215/S0012-7094-03-11723-8}, and is \"based on Jordan's theorem on finite groups embedded into the group of invertible matrices, and some standard arithmetic results over a $p$-adic field.\"", "expanded_theorem": "[To prove the main theorem, we proceed as follows.]\\label{intro_1}\nLet $(X,x)$ be a pointed geometrically connected smooth projective variety over a sub-$p$-adic field $K$. Then for a given rank $n$ there are only finitely many isomorphism classes of representations $\\pi_{1}^{EF}(X,x)\\rightarrow\\mathrm{GL}_{n}$. Equivalently, there are only finitely many isomorphism classes of essentially finite bundles of rank $n$.", "theorem_type": "unknown", "mcq": {"question": "Let $(X,x)$ be a pointed geometrically connected smooth projective variety over a sub-$p$-adic field $K$ (i.e. a subfield of a finitely generated extension of $\\mathbf{Q}_p$). Let $\\pi_1^{EF}(X,x)$ denote Nori’s essentially finite fundamental group scheme of $(X,x)$; its finite-dimensional representations are equivalent, via Tannakian duality, to essentially finite vector bundles on $X$. For a fixed positive integer $n$, which statement holds about rank-$n$ representations and essentially finite bundles?", "correct_choice": {"label": "A", "text": "There are only finitely many isomorphism classes of representations $\\pi_1^{EF}(X,x)\\to \\mathrm{GL}_n$, equivalently only finitely many isomorphism classes of essentially finite vector bundles of rank $n$ on $X$."}, "choices": [{"label": "B", "text": "There exists a finite extension $L/K$ such that, after base change to $L$, there are only finitely many isomorphism classes of representations $\\pi_1^{EF}(X_L,x_L)\\to \\mathrm{GL}_n$; equivalently, finiteness of essentially finite vector bundles of rank $n$ holds only after replacing $K$ by a suitable finite extension."}, {"label": "C", "text": "There are only finitely many isomorphism classes of representations $\\pi_1^{EF}(X,x)\\to \\mathrm{GL}_n$ whose image is abelian; equivalently, there are only finitely many isomorphism classes of rank-$n$ essentially finite vector bundles on $X$ arising from abelian finite quotient group schemes."}, {"label": "D", "text": "There is a constant $J=J(n)$, depending only on $n$, such that every representation $\\pi_1^{EF}(X,x)\\to \\mathrm{GL}_n$ factors through a finite quotient group scheme of order at most $J$; in particular, there are only finitely many isomorphism classes of rank-$n$ essentially finite vector bundles on $X$."}, {"label": "E", "text": "For a fixed positive integer $n$, there are only finitely many isomorphism classes of semisimple representations $\\pi_1^{EF}(X,x)\\to \\mathrm{GL}_n$; equivalently, only semisimple essentially finite vector bundles of rank $n$ occur up to isomorphism, while non-semisimple ones need not satisfy any finiteness statement."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "base-field hypothesis versus after-finite-extension finiteness", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the nonabelian generality by restricting to abelian image representations", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence of the Jordan bound on $(K,X,x)$, not just on $n$, and index bound versus order bound", "template_used": "uniformity_effectivity"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "misidentifying the theorem as only a semisimple-finiteness statement rather than full representation finiteness", "template_used": "wildcard"}]}} +{"id": "2601.14120v2", "paper_link": "http://arxiv.org/abs/2601.14120v2", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "[Universal Cord Theorem]\nFor a given length $\\ell$, the necessary and sufficient condition for a continuous function $f:[0,L]\\to \\mathbb R$ with $f(0)=f(L)$ to have a horizontal chord of length $\\ell$ is that $\\ell=\\frac{L}{m}$ for some positive integer $m$.", "start_pos": 3403, "end_pos": 3691, "label": null}, "ref_dict": {"eq1": "\\begin{equation}\\label{eq1}\n{\\cal P}=\\cap_{f\\in {\\cal F} } H(f).\n\\end{equation}"}, "pre_theorem_intro_text_len": 1799, "pre_theorem_intro_text": "The story of the Universal Chord Theorem (UCT) is a long one and full of twists and turns, and it has recently resurfaced in the literature (see \\cite{Diana2024}). \n\nIt goes back to 1806, when \nAndré-Marie Ampère \\cite{Ampere} proved the following equivalent statement: \n\n\\par \\vspace{0.2in} \n{\\it \\color{blue} ``The graph of every continuous function \non the compact interval $[0,1]$, with $f(0)=f(1)=0$, has at least one horizontal chord of length $\\frac{1}{n}$, i.e., given $\\bf n$ a natural number, there exist $a$ and $b$ in $[0,1]$ such that $|a-b|=\\frac{1}{\\bf n}$ and $f(a)=f(b)$.\" } \n\\par \\vspace{0.2in} \n$$\\underset{Figure\\ 1.\\ Chord \\ of \\ length\\ 0.233843 }{\n\\underset{f(x)=\\sin(2\\pi x) +\\sin(4\\pi x)+\\sin(6\\pi x) } {\\epsfig{file=fig1.eps,height=1in,width=2in}}\\ \\ \\underset{ y=f(x) \\ and\\ y=f(x-a) }{\\epsfig{file=fig1b.eps,height=1in,width=2in}}}$$ \n\nIn other words, the chords in the set ${\\cal P} :=\\{0\\}\\cup \\{\\frac{1}{n} : n\\in \\mathbb N\\}$ \nare always in the set of chords of every continuous function $f$ as described above. Our notation here for $\\mathbb N$ stands for all positive integers. This is only half of what the standard UCT says. \n\nFor some notations, we denote by ${\\cal F}$ the class of all continuous functions \non the compact interval $[0,1]$, with $f(0)=f(1)=0$, and \n\n $$H(f):=\\{ \\ell \\in [0,1]|\\ \\ \\ f(x-\\ell)=f(x) \\ \\ \\text{for some} \\ \\ x\\in [\\ell,1]\\}, $$\n\\noindent the set of chords for the function $f$. The UCT is known nowadays in a more precise form and it is attributed to Lévy (\\cite{Levy}), who in 1934, proved the other inclusion in the equality \n\n\\begin{equation}\\label{eq1}\n{\\cal P}=\\cap_{f\\in {\\cal F} } H(f).\n\\end{equation}\n\nBut let us include the equivalent version as stated in \\cite{Burns} (Theorem 4) which uses less notations.", "context": "\\par \\vspace{0.2in} \n{\\it \\color{blue} ``The graph of every continuous function \non the compact interval $[0,1]$, with $f(0)=f(1)=0$, has at least one horizontal chord of length $\\frac{1}{n}$, i.e., given $\\bf n$ a natural number, there exist $a$ and $b$ in $[0,1]$ such that $|a-b|=\\frac{1}{\\bf n}$ and $f(a)=f(b)$.\" } \n\\par \\vspace{0.2in} \n$$\\underset{Figure\\ 1.\\ Chord \\ of \\ length\\ 0.233843 }{\n\\underset{f(x)=\\sin(2\\pi x) +\\sin(4\\pi x)+\\sin(6\\pi x) } {\\epsfig{file=fig1.eps,height=1in,width=2in}}\\ \\ \\underset{ y=f(x) \\ and\\ y=f(x-a) }{\\epsfig{file=fig1b.eps,height=1in,width=2in}}}$$\n\nIn other words, the chords in the set ${\\cal P} :=\\{0\\}\\cup \\{\\frac{1}{n} : n\\in \\mathbb N\\}$ \nare always in the set of chords of every continuous function $f$ as described above. Our notation here for $\\mathbb N$ stands for all positive integers. This is only half of what the standard UCT says.\n\nFor some notations, we denote by ${\\cal F}$ the class of all continuous functions \non the compact interval $[0,1]$, with $f(0)=f(1)=0$, and\n\n$$H(f):=\\{ \\ell \\in [0,1]|\\ \\ \\ f(x-\\ell)=f(x) \\ \\ \\text{for some} \\ \\ x\\in [\\ell,1]\\}, $$\n\\noindent the set of chords for the function $f$. The UCT is known nowadays in a more precise form and it is attributed to Lévy (\\cite{Levy}), who in 1934, proved the other inclusion in the equality\n\n\\begin{equation}\\label{eq1}\n{\\cal P}=\\cap_{f\\in {\\cal F} } H(f).\n\\end{equation}\n\nBut let us include the equivalent version as stated in \\cite{Burns} (Theorem 4) which uses less notations.", "full_context": "\\par \\vspace{0.2in} \n{\\it \\color{blue} ``The graph of every continuous function \non the compact interval $[0,1]$, with $f(0)=f(1)=0$, has at least one horizontal chord of length $\\frac{1}{n}$, i.e., given $\\bf n$ a natural number, there exist $a$ and $b$ in $[0,1]$ such that $|a-b|=\\frac{1}{\\bf n}$ and $f(a)=f(b)$.\" } \n\\par \\vspace{0.2in} \n$$\\underset{Figure\\ 1.\\ Chord \\ of \\ length\\ 0.233843 }{\n\\underset{f(x)=\\sin(2\\pi x) +\\sin(4\\pi x)+\\sin(6\\pi x) } {\\epsfig{file=fig1.eps,height=1in,width=2in}}\\ \\ \\underset{ y=f(x) \\ and\\ y=f(x-a) }{\\epsfig{file=fig1b.eps,height=1in,width=2in}}}$$\n\nIn other words, the chords in the set ${\\cal P} :=\\{0\\}\\cup \\{\\frac{1}{n} : n\\in \\mathbb N\\}$ \nare always in the set of chords of every continuous function $f$ as described above. Our notation here for $\\mathbb N$ stands for all positive integers. This is only half of what the standard UCT says.\n\nFor some notations, we denote by ${\\cal F}$ the class of all continuous functions \non the compact interval $[0,1]$, with $f(0)=f(1)=0$, and\n\n$$H(f):=\\{ \\ell \\in [0,1]|\\ \\ \\ f(x-\\ell)=f(x) \\ \\ \\text{for some} \\ \\ x\\in [\\ell,1]\\}, $$\n\\noindent the set of chords for the function $f$. The UCT is known nowadays in a more precise form and it is attributed to Lévy (\\cite{Levy}), who in 1934, proved the other inclusion in the equality\n\n\\begin{equation}\\label{eq1}\n{\\cal P}=\\cap_{f\\in {\\cal F} } H(f).\n\\end{equation}\n\nBut let us include the equivalent version as stated in \\cite{Burns} (Theorem 4) which uses less notations.\n\n\\par \\vspace{0.2in} \n{\\it \\color{blue} ``The graph of every continuous function \non the compact interval $[0,1]$, with $f(0)=f(1)=0$, has at least one horizontal chord of length $\\frac{1}{n}$, i.e., given $\\bf n$ a natural number, there exist $a$ and $b$ in $[0,1]$ such that $|a-b|=\\frac{1}{\\bf n}$ and $f(a)=f(b)$.\" } \n\\par \\vspace{0.2in} \n$$\\underset{Figure\\ 1.\\ Chord \\ of \\ length\\ 0.233843 }{\n\\underset{f(x)=\\sin(2\\pi x) +\\sin(4\\pi x)+\\sin(6\\pi x) } {\\epsfig{file=fig1.eps,height=1in,width=2in}}\\ \\ \\underset{ y=f(x) \\ and\\ y=f(x-a) }{\\epsfig{file=fig1b.eps,height=1in,width=2in}}}$$\n\nIn other words, the chords in the set ${\\cal P} :=\\{0\\}\\cup \\{\\frac{1}{n} : n\\in \\mathbb N\\}$ \nare always in the set of chords of every continuous function $f$ as described above. Our notation here for $\\mathbb N$ stands for all positive integers. This is only half of what the standard UCT says.\n\nBut let us include the equivalent version as stated in \\cite{Burns} (Theorem 4) which uses less notations.\n\n$$\\underset{Figure\\ 2.\\ \\text{L\\'{e}vi's function, for} \\ \\ell=\\frac{3}{11}} {\\epsfig{file=fig2v3.eps,height=1in,width=2in}}$$\n\nLet us point out that Proposition~\\ref{openinterval} is more about the open additive sets of real numbers.\nIf we take $V=(1/2,1)$ we have a maximal Hopf set. A function $f$ which has $H(f)=[0,1]\\setminus U$ is $f(x)=\\sin (2\\pi x)$. So, we see that the point $1$ can be isolated in $H(f)$. \n\\begin{theorem}\\label{nonisolated}\n For $n\\in \\mathbb N$, $n\\ge 2$ and $f\\in \\cal F$ , the point $\\frac{1}{n}$ cannot be isolated in $H(f)$.\n\\end{theorem}\n\\begin{proof} Let us denote by $c=\\frac{1}{n}$. If $c$ is isolated then there is a set $J=(a,c)\\cup (c,b)$ in $H(f)^*$. Because we can write $n=p+q$ with $p$ and $q$ positive integers, \n $$1=nc=q(c-p\\epsilon)+p(c+q\\epsilon)$$ and for $\\epsilon$ small enough we have $c-p\\epsilon$ and $c+q\\epsilon$ in $J$. Since $H(f)^*$ is additive, we have $1$ written as linear combination with positive integer coefficients of two elements in $H(f)^*$, and so it follows that $1\\in H(f)^*$. This is in contradiction with $1\\not \\in H(f)^*$. \n\\end{proof}\nWe observe that we proved a little more here: if $\\ell \\in H(f)$ then $\\frac{\\ell}{n}$ cannot be isolated ($n\\ge 2$).\nWhat is interesting is that every point in $(0,1)\\setminus {\\cal P}$ can be isolated. Before we state and prove this let us introduce an important example of a maximal Hopf set. For every $n\\in \\mathbb N$, we denote by $J_n$ the open interval $(\\frac{1}{n+1},\\frac{1}{n})$.\n\n\\begin{proof}\nBy our assumption, there exists $n \\in \\mathbb{N}$ such that $\\frac{1}{n+1} < a < \\frac{1}{n}$. \nWe consider the Hopf set $U_n$ defined above, and define $W_n=U_n\\setminus \\{a\\}$. First we see that $W_n$ is still open. Taking a point off from $U_n$\nin general does not preserve the aditivity, but in this case, there is no problem since the equation $x+y=a$, with $x$, $y$ in $U_n$ is not possible (we are removing a point from the first set in the union defining $U_n$ in (\\ref{generichopf})). \nBy Hopf's theorem there exists a function $f$ which has $H(f)=[0,1]\\setminus W_n$. It is clear that $a$ is isolated in $H(f)$. \n\\end{proof}\n$$\\underset{Figure\\ 3,\\ a\\ \\in (\\frac{1}{3},\\frac{1}{2}) } {\\epsfig{file=th10fig.eps,height=1in,width=2in}}$$\n\n\\begin{theorem}\\label{Newman63}\nFor each $n \\in \\mathbb{N}$ and each $f \\in \\mathcal{F}$, \nthere exist at least $n$ distinct chords of $f$ whose lengths are integer multiples of $\\frac{1}{n}$.\n\\end{theorem}\n\nWe invite the reders to prove this by induction using the UCT. We observe that for each $\\ell \\in {\\cal P}$ we can define a vector $v(\\ell)=[m_1,m_2,...,m_n]$ with $m_i$ non-negative integers that represent the number of instances the chord $\\ell$ appears in the graph of $f$. Clearly, we have $m_1\\ge 1$ by UCT, $m_n=1$ and $\\sum_{k}m_k\\ge n$ by Theorem~\\ref{Newman63}. Knowing this information tells us more about the function $f$, so a natural question here is if this vector function on $\\cal P$, determines the function $f$ up to certain transformations that leave the set $H(f)$ invariant. \nSuch transformations include $f\\to h\\circ f$ and $f\\to f\\circ \\phi$, with $h:\\mathbb R \\to \\mathbb R$ continuous bijection, $h(0)=0$, and $\\phi :[0,1]\\to [0,1]$, the symmetry with respect to $1/2$ we used earlier, $\\phi(x)=1-x$, $x\\in [0,1]$.", "post_theorem_intro_text_len": 3467, "post_theorem_intro_text": "To prove the other inclusion in (\\ref{eq1}), one needs to show that if $\\ell\\not \\in {\\cal P}$ then there exists a function \n$f_{\\ell}$ for which $H(f_{\\ell})$ does not contain $\\ell$. Paul Levi's example was quite simple: take \n$$f_{\\ell}(x)=|\\sin (\\frac{\\pi x}{\\ell}) |-x |\\sin (\\frac{\\pi }{\\ell}) |$$\nand observe that because $|\\sin x|$ is periodic with period $\\pi$, we have for all $x\\in [\\ell , 1]$, the difference $f_{\\ell}(x-\\ell)-f_{\\ell}(x)=\\ell |\\sin (\\frac{\\pi }{\\ell}) |\\not =0$ given the hypothesis on $\\ell$ (see Figure~2).\n\n$$\\underset{Figure\\ 2.\\ \\text{L\\'{e}vi's function, for} \\ \\ell=\\frac{3}{11}} {\\epsfig{file=fig2v3.eps,height=1in,width=2in}}$$\n\nIn Figure~1 we have included an example and also a geometric way to think about the existence of a chord of a given length $c$, by translating the graph of $y=f(x)$ to the right by $c$ and look for the intersections of the two graphs. We observe that in this case we have multiple points of intersection.\n\n\\par\\vspace{0.1in} \nIn \\cite{Burns}, one can find an elegant account of UCT and a real world application of it in races. We include Proposition 2 to give the reader a flavour of this kind of applications. \n\n \\begin{proposition} [\\cite{Burns}] \\label{Burns2017}\n If the race distance is a whole number of miles, then some mile must be covered at exacly the average pace. \n\\end{proposition}\n\n The next important development in this subject was done at short time after Levy's work, by Heinz Hopf (\\cite{Hopf}) in 1937. His result is quite surprising. \nWe say that a set $U$ of real numbers is additive if it is closed under the addition, i.e., for every $a$ and $b$ in $U$, we have $a+b\\in U$. Let us denote by $H(f)^*$ the complement of $H(f)$ in $[0,\\infty)$.\n\\begin{theorem}[Hopf]\\label{hopf}\n The set $H(f)^*$ is open and additive. Given a set $U$ of positive numbers, which is open and additive, containing $(1,\\infty)$ as a maximal interval ($1\\not\\in U$), there exists a function \n$f\\in \\cal F$ such that $H(f)=[0,\\infty)\\setminus U$.\n\\end{theorem}\nFor a simple proof of Hopf's theorem and other results one can refer to \\cite{Diana2024}.\n\n\\par\\vspace{0.1in} \n\nLooking at the identity (\\ref{eq1}), one may think that for every $n\\in \\mathbb N$, $n\\ge 2$, there exists a function $f_n\\in {\\cal F}$ which has a chord set $H(f_n)$ with \n\n$$H(f_n) \\subset [0,\\frac{1}{n}] \\cup \\bigcup_{k=1}^{n-1} \\{\\frac{1}{k}\\}.$$\nBut this cannot happen for the several reasons. We will show that the points $\\frac{1}{n}$ cannot be isolated in $H(f)$. The second reason is shown in \\cite{Diana2024}. \n\\begin{theorem}[\\cite{Diana2024}]\nFor every $f\\in \\cal F$, the set $H(f)$ must have measure greater or equal to $\\frac{1}{2}$. \n\\end{theorem}\n\\begin{definition}\\label{hopfset} Let us call a set $U$ which is open, additive and of the form $V\\cup (1,\\infty)$ with $V\\subset (0,1)$ a {\\sf Hopf set}. If in addition, we have $m(V)=\\frac{1}{2}$, we will refer to $U$ as a {\\sf maximal Hopf set}. We used $m( A)$ for the Lebesgue measure of the set $A$. \n\\end{definition}\n\nIn this paper we are interested in the isolated points and the accumulation points of $H(f)$. In general the additive open sets of positive real numbers such as $H(f)^*$ are quite complicated. We also investigate some possible ways to construct such sets and address a question posed in \\cite{Diana2024}. \n\nIn what follows we will assume that $H(f)^*$ is $V\\cup (1,\\infty)$ for some open set in $(0,1)$ which can be empty.", "sketch": "To show the remaining inclusion in (\\ref{eq1}), it suffices to prove: if $\\ell\\not\\in {\\cal P}$ then there exists a function $f_{\\ell}$ such that $H(f_{\\ell})$ does not contain $\\ell$. L\\'{e}vi gives an explicit example\n\\[\n f_{\\ell}(x)=\\bigl|\\sin \\bigl(\\tfrac{\\pi x}{\\ell}\\bigr)\\bigr|-x\\bigl|\\sin \\bigl(\\tfrac{\\pi}{\\ell}\\bigr)\\bigr|.\n\\]\nUsing that $|\\sin x|$ is periodic with period $\\pi$, one observes that for all $x\\in[\\ell,1]$,\n\\[\n f_{\\ell}(x-\\ell)-f_{\\ell}(x)=\\ell\\,\\bigl|\\sin \\bigl(\\tfrac{\\pi}{\\ell}\\bigr)\\bigr|\\neq 0\n\\]\n(under the hypothesis on $\\ell$). Hence $f_{\\ell}(x-\\ell)\\neq f_{\\ell}(x)$ for such $x$, so there is no horizontal chord of length $\\ell$, i.e. $\\ell\\notin H(f_{\\ell})$.", "expanded_sketch": "To show the remaining inclusion in \\begin{equation}\\label{eq1}\n{\\cal P}=\\cap_{f\\in {\\cal F} } H(f).\n\\end{equation}\n, it suffices to prove: if $\\ell\\not\\in {\\cal P}$ then there exists a function $f_{\\ell}$ such that $H(f_{\\ell})$ does not contain $\\ell$. L\\'{e}vi gives an explicit example\n\\[\n f_{\\ell}(x)=\\bigl|\\sin \\bigl(\\tfrac{\\pi x}{\\ell}\\bigr)\\bigr|-x\\bigl|\\sin \\bigl(\\tfrac{\\pi}{\\ell}\\bigr)\\bigr|.\n\\]\nUsing that $|\\sin x|$ is periodic with period $\\pi$, one observes that for all $x\\in[\\ell,1]$,\n\\[\n f_{\\ell}(x-\\ell)-f_{\\ell}(x)=\\ell\\,\\bigl|\\sin \\bigl(\\tfrac{\\pi}{\\ell}\\bigr)\\bigr|\\neq 0\n\\]\n(under the hypothesis on $\\ell$). Hence $f_{\\ell}(x-\\ell)\\neq f_{\\ell}(x)$ for such $x$, so there is no horizontal chord of length $\\ell$, i.e. $\\ell\\notin H(f_{\\ell})$.", "expanded_theorem": "[Universal Cord Theorem]\nFor a given length $\\ell$, the necessary and sufficient condition for a continuous function $f:[0,L]\\to \\mathbb R$ with $f(0)=f(L)$ to have a horizontal chord of length $\\ell$ is that $\\ell=\\frac{L}{m}$ for some positive integer $m$.", "theorem_type": ["Biconditional or Equivalence"], "mcq": {"question": "Let \\(\\ell\\) be a fixed real number with \\(0\\le \\ell\\le L\\). A horizontal chord of length \\(\\ell\\) for a function \\(f:[0,L]\\to \\mathbb R\\) means that there exists \\(x\\in[0,L-\\ell]\\) such that \\(f(x)=f(x+\\ell)\\). Which of the following statements is equivalent to the assertion that every continuous function \\(f:[0,L]\\to \\mathbb R\\) satisfying \\(f(0)=f(L)\\) has a horizontal chord of length \\(\\ell\\)?", "correct_choice": {"label": "A", "text": "\\(\\ell=\\frac{L}{m}\\) for some positive integer \\(m\\)."}, "choices": [{"label": "B", "text": "\\(\\ell\\in(0,L]\\) is rationally related to \\(L\\), i.e. \\(\\ell=\\frac{p}{q}L\\) for some positive integers \\(p,q\\)."}, {"label": "C", "text": "\\(\\ell=\\frac{L}{m}\\) for some integer \\(m\\ge 2\\)."}, {"label": "D", "text": "For every continuous function \\(f:[0,L]\\to\\mathbb R\\) with \\(f(0)=f(L)\\), there exists a positive integer \\(m\\) such that \\(f\\) has a horizontal chord of length \\(\\frac{\\ell}{m}\\)."}, {"label": "E", "text": "\\(\\ell=\\frac{L}{m}\\) for some nonnegative integer \\(m\\)."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "trace_identity", "tampered_component": "excluded_case_not_in_P", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "trace_identity", "tampered_component": "drops_the_case_m=1_equivalently_ell=L", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "trace_identity", "tampered_component": "fixed_length_requirement", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "trace_identity", "tampered_component": "positivity_of_integer_parameter", "template_used": "boundary_range"}]}} +{"id": "2601.14805v1", "paper_link": "http://arxiv.org/abs/2601.14805v1", "theorems_cnt": 10, "theorem": {"env_name": "theorem", "content": "\\label{thm:minimizer}\nFor a positive integer $k$, let $\\mathcal{F}\\subseteq 2^V$ be the complement of a $k$-hierarchical lattice.\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nLet $X^*\\in \\mathcal{F}$ be a minimizer of $f$ over $\\mathcal{F}$.\nThen, there exist $S,T\\subseteq V$ with $S\\cap T=\\emptyset$ and $\\max\\{|S|,|T|\\}\\le k$ such that $X^*$ is a minimizer of $f$ over the lattice $\\mathcal{F}_{ST}$.", "start_pos": 8321, "end_pos": 8761, "label": "thm:minimizer"}, "ref_dict": {"thm:algorithm": "\\begin{theorem}\n\\label{thm:algorithm}\nFor a positive integer $k$, let $\\mathcal{F}\\subseteq 2^V$ be the complement of a $k$-hierarchical lattice.\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nThen, a minimizer of $f$ over $\\mathcal{F}$ can be obtained in $O(n^{2k}\\tau(n))$ time, where $n:=|V|$ and $\\tau(n)$ denotes the time required for minimizing a submodular function on a ground set of size $n$.\n\\end{theorem}", "thm:minimizer": "\\begin{theorem}\n\\label{thm:minimizer}\nFor a positive integer $k$, let $\\mathcal{F}\\subseteq 2^V$ be the complement of a $k$-hierarchical lattice.\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nLet $X^*\\in \\mathcal{F}$ be a minimizer of $f$ over $\\mathcal{F}$.\nThen, there exist $S,T\\subseteq V$ with $S\\cap T=\\emptyset$ and $\\max\\{|S|,|T|\\}\\le k$ such that $X^*$ is a minimizer of $f$ over the lattice $\\mathcal{F}_{ST}$.\n\\end{theorem}", "thm:minimal_minimizer": "\\begin{theorem}\n\\label{thm:minimal_minimizer}\nFor a positive integer $k$, let $\\mathcal{F}\\subseteq 2^V$ be the complement of a $k$-hierarchical lattice.\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nLet $X^*\\in \\mathcal{F}$ be a minimal minimizer of $f$ over $\\mathcal{F}$.\nThen, there exist $S,T\\subseteq V$ with $S\\cap T=\\emptyset$ and $\\max\\{|S|,|T|\\}\\le k$ such that $X^*$ is the unique minimal minimizer of $f$ over the lattice $\\mathcal{F}_{ST}$.\n\\end{theorem}", "cor:intersecting_crossing": "\\begin{corollary}\n\\label{cor:intersecting_crossing}\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nIf $\\mathcal{F}\\subseteq 2^V$ is an intersecting family or a crossing family, then a minimizer of $f$ over $2^V\\setminus \\mathcal{F}$ can be obtained in polynomial-time.\n\\end{corollary}"}, "pre_theorem_intro_text_len": 6204, "pre_theorem_intro_text": "\\label{sec:intro}\nLet $V$ be a finite set.\nA set function $f:2^V\\to \\mathbb{Z}$ is called \\textit{submodular} if $f(X)+f(Y)\\ge f(X\\cup Y)+f(X\\cap Y)$ holds for all $X,Y\\subseteq V$.\nIt is a central result in combinatorial optimization that submodular functions can be minimized in polynomial-time in the evaluation oracle model \\cite{grotschel1981,iwata2001,schrijver2000}.\n\nWhile submodular function minimization (SFM) is efficiently solvable, polynomial-solvability of SFM restricted to a set family is known only for a few classes of families.\nClassical examples of such families are intersecting and crossing families (see e.g., \\cite[Volume B]{schrijver2003}).\nAs another tractable case, Goemans and Ramakrishnan \\cite{goemans1995} showed that SFM can be solved in polynomial-time over parity families.\nTheir result generalizes polynomial-solvability of SFM over the families of all odd sets and all even sets, and more generally, over triple families \\cite{grotschel1981}. \nN\\\"{a}gele, Sudakov, and Zenklusen \\cite{nagele2019} pursued a different extension of the odd/even-cardinality constraint to the congruency constraint, showing that SFM over all sets of cardinality $r$ mod $m$, where $m$ is a constant prime power, can be solved in polynomial-time.\nIn this paper, we introduce a new class of families, which includes complements of various families such as intersecting families, crossing families, and the unions of lattices.\nWe show that SFM over such families can be solved in polynomial-time, which partially settles the open question posed by N\\\"{a}gele, Sudakov, and Zenklusen \\cite{nagele2019}.\n\nTo provide a new tractable class of families, we introduce some definitions.\nA set family $\\mathcal{F}\\subseteq 2^V$ is called a \\textit{lattice} if for all $X,Y\\in \\mathcal{F}$, we have $X\\cup Y,X\\cap Y\\in \\mathcal{F}$.\nFor a positive integer $k$, we call a family $\\mathcal{F}\\subseteq 2^V$ a \\textit{$k$-hierarchical lattice} if $\\mathcal{F}$ can be partitioned into $k$ families $\\mathcal{F}_1,\\mathcal{F}_2,\\ldots,\\mathcal{F}_k\\subseteq 2^V$ such that\n\\begin{itemize}\n \\item $\\mathcal{F}_1$ is a lattice,\n \\item for all $X,Y\\in \\mathcal{F}_i$ and $i=2,\\ldots,k$, we have that $X\\cup Y,X\\cap Y\\in \\mathcal{F}_i$ or at least one of $X\\cup Y$ and $X\\cap Y$ belongs to $\\mathcal{F}_1\\cup \\mathcal{F}_2\\cup \\cdots \\cup \\mathcal{F}_{i-1}$.\n\\end{itemize}\nNote that if $k=1$, then a $k$-hierarchical lattice coincides with a lattice.\nWe provide several examples of $k$-hierarchical lattices.\n\n\\begin{example}[Intersecting family]\nLet $\\mathcal{F}\\subseteq 2^V$ be an \\textit{intersecting family}, i.e., for all $X,Y\\in \\mathcal{F}$ with $X\\cap Y\\ne \\emptyset$, we have $X\\cup Y,X\\cap Y\\in \\mathcal{F}$.\nDefine $\\mathcal{F}_1:=\\{\\emptyset\\}$ and $\\mathcal{F}_2:=\\mathcal{F}\\setminus \\{\\emptyset\\}$.\nThen, $\\mathcal{F}_1$ is a lattice, and for all $X,Y\\in \\mathcal{F}_2$, we have $X\\cap Y=\\emptyset\\in \\mathcal{F}_1$ or $X\\cup Y,X\\cap Y\\in \\mathcal{F}_2$.\nHence, $\\mathcal{F}\\cup \\{\\emptyset\\}$ is a 2-hierarchical lattice.\n\\end{example}\n\n\\begin{example}[Crossing family]\nLet $\\mathcal{F}\\subseteq 2^V$ be a \\textit{crossing family}, i.e., for all $X,Y\\in \\mathcal{F}$ with $X\\cap Y\\ne \\emptyset$ and $X\\cup Y\\ne V$, we have $X\\cup Y,X\\cap Y\\in \\mathcal{F}$.\nDefine $\\mathcal{F}_1:=\\{\\emptyset,V\\}$ and $\\mathcal{F}_2:=\\mathcal{F}\\setminus \\{\\emptyset,V\\}$.\nThen, $\\mathcal{F}_1$ is a lattice, and for all $X,Y\\in \\mathcal{F}_2$, we have $X\\cap Y=\\emptyset\\in \\mathcal{F}_1$ or $X\\cup Y=V\\in \\mathcal{F}_1$ or $X\\cup Y,X\\cap Y\\in \\mathcal{F}_2$.\nHence, $\\mathcal{F}\\cup \\{\\emptyset,V\\}$ is a 2-hierarchical lattice.\n\\end{example}\n\n\\begin{example}[Union of $k$ lattices]\nLet $\\mathcal{L}_1,\\mathcal{L}_2,\\ldots,\\mathcal{L}_k\\subseteq 2^V$ be lattices.\nDefine $\\mathcal{F}_1:=\\mathcal{L}_1$ and $\\mathcal{F}_i:=\\mathcal{L}_i\\setminus (\\mathcal{L}_1\\cup \\mathcal{L}_2\\cup \\cdots \\cup\\mathcal{L}_{i-1})$ for $i=2,\\ldots,k$.\nThen, $\\mathcal{F}_1$ is a lattice, and for all $X,Y\\in \\mathcal{F}_i$ and $i=2,\\ldots,k$, we have that $X\\cup Y,X\\cap Y\\in \\mathcal{F}_i$ or at least one of $X\\cup Y$ and $X\\cap Y$ belongs to $\\mathcal{L}_1\\cup \\mathcal{L}_2\\cup \\cdots \\cup\\mathcal{L}_{i-1}=\\mathcal{F}_1\\cup \\mathcal{F}_2\\cup \\cdots \\cup\\mathcal{F}_{i-1}$.\nHence, $\\mathcal{L}_1\\cup \\mathcal{L}_2\\cup \\cdots \\cup\\mathcal{L}_{k}$ is a $k$-hierarchical lattice.\n\\end{example}\n\nIt is well known that the minimizers of a submodular function form a lattice (see e.g., \\cite{fujishige2005}).\nThe following example shows that the family of subsets whose function values are at most the $k$-th smallest value of a submodular function is a $k$-hierarchical lattice.\n\n\\begin{example}[$k$-th smallest value of a submodular function]\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nFor $i=1,2,\\ldots,k$, define $\\mathcal{F}_i$ as the family of subsets $X\\subseteq V$ whose values $f(X)$ are equal to the $i$-th smallest value of $f$.\nThen, since the minimizers of $f$ form a lattice, $\\mathcal{F}_1$ is a lattice.\nFor $i=2,\\ldots,k$, take $X,Y\\in \\mathcal{F}_i$.\nThen, the submodular inequality $f(X)+f(Y)\\ge f(X\\cup Y)+f(X\\cap Y)$ implies that if $X\\cup Y\\notin \\mathcal{F}_i$ or $X\\cap Y\\notin \\mathcal{F}_i$, then at least one of $X\\cup Y$ and $X\\cap Y$ belongs to $\\mathcal{F}_j$ for some $1\\le j\\le i-1$.\nHence, for all $X,Y\\in \\mathcal{F}_i$ and $i=2,\\ldots,k$, we have that $X\\cup Y,X\\cap Y\\in \\mathcal{F}_i$, or at least one of $X\\cup Y$ and $X\\cap Y$ belongs to $\\mathcal{F}_1\\cup \\mathcal{F}_2\\cup \\cdots \\cup \\mathcal{F}_{i-1}$.\nThus, $\\mathcal{F}_1\\cup \\mathcal{F}_2\\cup \\cdots \\cup \\mathcal{F}_{k}$ is a $k$-hierarchical lattice.\n\\end{example}\n\nWe call a set family $\\mathcal{F}\\subseteq 2^V$ the \\textit{complement of a $k$-hierarchical lattice} if $2^V\\setminus \\mathcal{F}$ is a $k$-hierarchical lattice.\nFor disjoint subsets $S,T\\subseteq V$, we define\n\\begin{align*}\n\\mathcal{F}_{ST}:=\\{X\\subseteq V\\mid S\\subseteq X\\subseteq V\\setminus T\\}.\n\\end{align*}\nNote that $\\mathcal{F}_{ST}$ is a lattice for every disjoint subsets $S,T\\subseteq V$.\nOur main result is the following structural theorem on the minimizers of a submodular function over the complement of a $k$-hierarchical lattice.", "context": "\\label{sec:intro}\nLet $V$ be a finite set.\nA set function $f:2^V\\to \\mathbb{Z}$ is called \\textit{submodular} if $f(X)+f(Y)\\ge f(X\\cup Y)+f(X\\cap Y)$ holds for all $X,Y\\subseteq V$.\nIt is a central result in combinatorial optimization that submodular functions can be minimized in polynomial-time in the evaluation oracle model \\cite{grotschel1981,iwata2001,schrijver2000}.\n\nTo provide a new tractable class of families, we introduce some definitions.\nA set family $\\mathcal{F}\\subseteq 2^V$ is called a \\textit{lattice} if for all $X,Y\\in \\mathcal{F}$, we have $X\\cup Y,X\\cap Y\\in \\mathcal{F}$.\nFor a positive integer $k$, we call a family $\\mathcal{F}\\subseteq 2^V$ a \\textit{$k$-hierarchical lattice} if $\\mathcal{F}$ can be partitioned into $k$ families $\\mathcal{F}_1,\\mathcal{F}_2,\\ldots,\\mathcal{F}_k\\subseteq 2^V$ such that\n\\begin{itemize}\n \\item $\\mathcal{F}_1$ is a lattice,\n \\item for all $X,Y\\in \\mathcal{F}_i$ and $i=2,\\ldots,k$, we have that $X\\cup Y,X\\cap Y\\in \\mathcal{F}_i$ or at least one of $X\\cup Y$ and $X\\cap Y$ belongs to $\\mathcal{F}_1\\cup \\mathcal{F}_2\\cup \\cdots \\cup \\mathcal{F}_{i-1}$.\n\\end{itemize}\nNote that if $k=1$, then a $k$-hierarchical lattice coincides with a lattice.\nWe provide several examples of $k$-hierarchical lattices.\n\n\\begin{example}[Intersecting family]\nLet $\\mathcal{F}\\subseteq 2^V$ be an \\textit{intersecting family}, i.e., for all $X,Y\\in \\mathcal{F}$ with $X\\cap Y\\ne \\emptyset$, we have $X\\cup Y,X\\cap Y\\in \\mathcal{F}$.\nDefine $\\mathcal{F}_1:=\\{\\emptyset\\}$ and $\\mathcal{F}_2:=\\mathcal{F}\\setminus \\{\\emptyset\\}$.\nThen, $\\mathcal{F}_1$ is a lattice, and for all $X,Y\\in \\mathcal{F}_2$, we have $X\\cap Y=\\emptyset\\in \\mathcal{F}_1$ or $X\\cup Y,X\\cap Y\\in \\mathcal{F}_2$.\nHence, $\\mathcal{F}\\cup \\{\\emptyset\\}$ is a 2-hierarchical lattice.\n\\end{example}\n\n\\begin{example}[Crossing family]\nLet $\\mathcal{F}\\subseteq 2^V$ be a \\textit{crossing family}, i.e., for all $X,Y\\in \\mathcal{F}$ with $X\\cap Y\\ne \\emptyset$ and $X\\cup Y\\ne V$, we have $X\\cup Y,X\\cap Y\\in \\mathcal{F}$.\nDefine $\\mathcal{F}_1:=\\{\\emptyset,V\\}$ and $\\mathcal{F}_2:=\\mathcal{F}\\setminus \\{\\emptyset,V\\}$.\nThen, $\\mathcal{F}_1$ is a lattice, and for all $X,Y\\in \\mathcal{F}_2$, we have $X\\cap Y=\\emptyset\\in \\mathcal{F}_1$ or $X\\cup Y=V\\in \\mathcal{F}_1$ or $X\\cup Y,X\\cap Y\\in \\mathcal{F}_2$.\nHence, $\\mathcal{F}\\cup \\{\\emptyset,V\\}$ is a 2-hierarchical lattice.\n\\end{example}\n\n\\begin{example}[$k$-th smallest value of a submodular function]\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nFor $i=1,2,\\ldots,k$, define $\\mathcal{F}_i$ as the family of subsets $X\\subseteq V$ whose values $f(X)$ are equal to the $i$-th smallest value of $f$.\nThen, since the minimizers of $f$ form a lattice, $\\mathcal{F}_1$ is a lattice.\nFor $i=2,\\ldots,k$, take $X,Y\\in \\mathcal{F}_i$.\nThen, the submodular inequality $f(X)+f(Y)\\ge f(X\\cup Y)+f(X\\cap Y)$ implies that if $X\\cup Y\\notin \\mathcal{F}_i$ or $X\\cap Y\\notin \\mathcal{F}_i$, then at least one of $X\\cup Y$ and $X\\cap Y$ belongs to $\\mathcal{F}_j$ for some $1\\le j\\le i-1$.\nHence, for all $X,Y\\in \\mathcal{F}_i$ and $i=2,\\ldots,k$, we have that $X\\cup Y,X\\cap Y\\in \\mathcal{F}_i$, or at least one of $X\\cup Y$ and $X\\cap Y$ belongs to $\\mathcal{F}_1\\cup \\mathcal{F}_2\\cup \\cdots \\cup \\mathcal{F}_{i-1}$.\nThus, $\\mathcal{F}_1\\cup \\mathcal{F}_2\\cup \\cdots \\cup \\mathcal{F}_{k}$ is a $k$-hierarchical lattice.\n\\end{example}\n\nWe call a set family $\\mathcal{F}\\subseteq 2^V$ the \\textit{complement of a $k$-hierarchical lattice} if $2^V\\setminus \\mathcal{F}$ is a $k$-hierarchical lattice.\nFor disjoint subsets $S,T\\subseteq V$, we define\n\\begin{align*}\n\\mathcal{F}_{ST}:=\\{X\\subseteq V\\mid S\\subseteq X\\subseteq V\\setminus T\\}.\n\\end{align*}\nNote that $\\mathcal{F}_{ST}$ is a lattice for every disjoint subsets $S,T\\subseteq V$.\nOur main result is the following structural theorem on the minimizers of a submodular function over the complement of a $k$-hierarchical lattice.", "full_context": "\\label{sec:intro}\nLet $V$ be a finite set.\nA set function $f:2^V\\to \\mathbb{Z}$ is called \\textit{submodular} if $f(X)+f(Y)\\ge f(X\\cup Y)+f(X\\cap Y)$ holds for all $X,Y\\subseteq V$.\nIt is a central result in combinatorial optimization that submodular functions can be minimized in polynomial-time in the evaluation oracle model \\cite{grotschel1981,iwata2001,schrijver2000}.\n\nTo provide a new tractable class of families, we introduce some definitions.\nA set family $\\mathcal{F}\\subseteq 2^V$ is called a \\textit{lattice} if for all $X,Y\\in \\mathcal{F}$, we have $X\\cup Y,X\\cap Y\\in \\mathcal{F}$.\nFor a positive integer $k$, we call a family $\\mathcal{F}\\subseteq 2^V$ a \\textit{$k$-hierarchical lattice} if $\\mathcal{F}$ can be partitioned into $k$ families $\\mathcal{F}_1,\\mathcal{F}_2,\\ldots,\\mathcal{F}_k\\subseteq 2^V$ such that\n\\begin{itemize}\n \\item $\\mathcal{F}_1$ is a lattice,\n \\item for all $X,Y\\in \\mathcal{F}_i$ and $i=2,\\ldots,k$, we have that $X\\cup Y,X\\cap Y\\in \\mathcal{F}_i$ or at least one of $X\\cup Y$ and $X\\cap Y$ belongs to $\\mathcal{F}_1\\cup \\mathcal{F}_2\\cup \\cdots \\cup \\mathcal{F}_{i-1}$.\n\\end{itemize}\nNote that if $k=1$, then a $k$-hierarchical lattice coincides with a lattice.\nWe provide several examples of $k$-hierarchical lattices.\n\n\\begin{example}[Intersecting family]\nLet $\\mathcal{F}\\subseteq 2^V$ be an \\textit{intersecting family}, i.e., for all $X,Y\\in \\mathcal{F}$ with $X\\cap Y\\ne \\emptyset$, we have $X\\cup Y,X\\cap Y\\in \\mathcal{F}$.\nDefine $\\mathcal{F}_1:=\\{\\emptyset\\}$ and $\\mathcal{F}_2:=\\mathcal{F}\\setminus \\{\\emptyset\\}$.\nThen, $\\mathcal{F}_1$ is a lattice, and for all $X,Y\\in \\mathcal{F}_2$, we have $X\\cap Y=\\emptyset\\in \\mathcal{F}_1$ or $X\\cup Y,X\\cap Y\\in \\mathcal{F}_2$.\nHence, $\\mathcal{F}\\cup \\{\\emptyset\\}$ is a 2-hierarchical lattice.\n\\end{example}\n\n\\begin{example}[Crossing family]\nLet $\\mathcal{F}\\subseteq 2^V$ be a \\textit{crossing family}, i.e., for all $X,Y\\in \\mathcal{F}$ with $X\\cap Y\\ne \\emptyset$ and $X\\cup Y\\ne V$, we have $X\\cup Y,X\\cap Y\\in \\mathcal{F}$.\nDefine $\\mathcal{F}_1:=\\{\\emptyset,V\\}$ and $\\mathcal{F}_2:=\\mathcal{F}\\setminus \\{\\emptyset,V\\}$.\nThen, $\\mathcal{F}_1$ is a lattice, and for all $X,Y\\in \\mathcal{F}_2$, we have $X\\cap Y=\\emptyset\\in \\mathcal{F}_1$ or $X\\cup Y=V\\in \\mathcal{F}_1$ or $X\\cup Y,X\\cap Y\\in \\mathcal{F}_2$.\nHence, $\\mathcal{F}\\cup \\{\\emptyset,V\\}$ is a 2-hierarchical lattice.\n\\end{example}\n\n\\begin{example}[$k$-th smallest value of a submodular function]\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nFor $i=1,2,\\ldots,k$, define $\\mathcal{F}_i$ as the family of subsets $X\\subseteq V$ whose values $f(X)$ are equal to the $i$-th smallest value of $f$.\nThen, since the minimizers of $f$ form a lattice, $\\mathcal{F}_1$ is a lattice.\nFor $i=2,\\ldots,k$, take $X,Y\\in \\mathcal{F}_i$.\nThen, the submodular inequality $f(X)+f(Y)\\ge f(X\\cup Y)+f(X\\cap Y)$ implies that if $X\\cup Y\\notin \\mathcal{F}_i$ or $X\\cap Y\\notin \\mathcal{F}_i$, then at least one of $X\\cup Y$ and $X\\cap Y$ belongs to $\\mathcal{F}_j$ for some $1\\le j\\le i-1$.\nHence, for all $X,Y\\in \\mathcal{F}_i$ and $i=2,\\ldots,k$, we have that $X\\cup Y,X\\cap Y\\in \\mathcal{F}_i$, or at least one of $X\\cup Y$ and $X\\cap Y$ belongs to $\\mathcal{F}_1\\cup \\mathcal{F}_2\\cup \\cdots \\cup \\mathcal{F}_{i-1}$.\nThus, $\\mathcal{F}_1\\cup \\mathcal{F}_2\\cup \\cdots \\cup \\mathcal{F}_{k}$ is a $k$-hierarchical lattice.\n\\end{example}\n\nWe call a set family $\\mathcal{F}\\subseteq 2^V$ the \\textit{complement of a $k$-hierarchical lattice} if $2^V\\setminus \\mathcal{F}$ is a $k$-hierarchical lattice.\nFor disjoint subsets $S,T\\subseteq V$, we define\n\\begin{align*}\n\\mathcal{F}_{ST}:=\\{X\\subseteq V\\mid S\\subseteq X\\subseteq V\\setminus T\\}.\n\\end{align*}\nNote that $\\mathcal{F}_{ST}$ is a lattice for every disjoint subsets $S,T\\subseteq V$.\nOur main result is the following structural theorem on the minimizers of a submodular function over the complement of a $k$-hierarchical lattice.\n\nWe call a set family $\\mathcal{F}\\subseteq 2^V$ the \\textit{complement of a $k$-hierarchical lattice} if $2^V\\setminus \\mathcal{F}$ is a $k$-hierarchical lattice.\nFor disjoint subsets $S,T\\subseteq V$, we define\n\\begin{align*}\n\\mathcal{F}_{ST}:=\\{X\\subseteq V\\mid S\\subseteq X\\subseteq V\\setminus T\\}.\n\\end{align*}\nNote that $\\mathcal{F}_{ST}$ is a lattice for every disjoint subsets $S,T\\subseteq V$.\nOur main result is the following structural theorem on the minimizers of a submodular function over the complement of a $k$-hierarchical lattice.\n\nTo obtain a polynomial-time algorithm for SFM over the complement of $k$-hierarchical lattice, we show the following slight refinement of Theorem \\ref{thm:minimizer}.\n\n\\begin{theorem}\n\\label{thm:minimal_minimizer}\nFor a positive integer $k$, let $\\mathcal{F}\\subseteq 2^V$ be the complement of a $k$-hierarchical lattice.\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nLet $X^*\\in \\mathcal{F}$ be a minimal minimizer of $f$ over $\\mathcal{F}$.\nThen, there exist $S,T\\subseteq V$ with $S\\cap T=\\emptyset$ and $\\max\\{|S|,|T|\\}\\le k$ such that $X^*$ is the unique minimal minimizer of $f$ over the lattice $\\mathcal{F}_{ST}$.\n\\end{theorem}\n\n\\begin{theorem}\n\\label{thm:algorithm}\nFor a positive integer $k$, let $\\mathcal{F}\\subseteq 2^V$ be the complement of a $k$-hierarchical lattice.\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nThen, a minimizer of $f$ over $\\mathcal{F}$ can be obtained in $O(n^{2k}\\tau(n))$ time, where $n:=|V|$ and $\\tau(n)$ denotes the time required for minimizing a submodular function on a ground set of size $n$.\n\\end{theorem}\n\n\\begin{corollary}\n\\label{cor:parity_intersection}\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nFor a constant positive integer $k$, if $\\mathcal{F}\\subseteq 2^V$ is the intersection of the complements of $k$ lattices, then a minimizer of $f$ over $\\mathcal{F}$ can be obtained in polynomial-time. \n\\end{corollary}\n\nWe are now ready to prove Theorems \\ref{thm:minimizer} and \\ref{thm:minimal_minimizer}.\n\\begin{proof}[Proof of Theorem \\ref{thm:minimizer}]\nTake a minimizer $X^*\\in \\mathcal{F}$ of $f$ over $\\mathcal{F}$.\nBy Lemmas \\ref{lem:key} and \\ref{lem:complement}, there exist $S,T\\subseteq V$ such that\n\\begin{itemize}\n \\item $S\\subseteq X^*$ and $T\\subseteq V\\setminus X^*$,\n \\item $\\max\\{|S|,|T|\\}\\le k$,\n \\item $Y\\in \\mathcal{F}$ for all $Y\\subseteq V$ with $S\\subseteq Y\\subseteq X^*$ or $X^*\\subseteq Y\\subseteq V\\setminus T$.\n\\end{itemize}\nSince $S\\cap T=\\emptyset$, it suffices to show that $X^*$ is a minimizer of $f$ over the lattice $\\mathcal{F}_{ST}$.\nLet $Z\\in \\mathcal{F}_{ST}$ be a minimizer of $f$ over $\\mathcal{F}_{ST}$.\nSince $f$ is submodular, we have\n\\begin{align}\n\\label{eq:xz}\n2f(X^*)\\ge f(X^*)+f(Z)\\ge f(X^*\\cup Z)+f(X^*\\cap Z).\n\\end{align}\nThis implies that we have at least one of $f(X^*)\\ge f(X^*\\cup Z)$ and $f(X^*)\\ge f(X^*\\cap Z)$.\nSince $X^*\\in \\mathcal{F}_{ST}$ and $Z\\in \\mathcal{F}_{ST}$, we have $X^*\\cup Z\\in \\mathcal{F}_{ST}$ and $X^*\\cap Z\\in \\mathcal{F}_{ST}$.\nHence, we have $S\\subseteq X^*\\cap Z\\subseteq X^*\\subseteq X^*\\cup Z\\subseteq V\\setminus T$, which implies $X^*\\cup Z\\in \\mathcal{F}$ and $X^*\\cap Z\\in \\mathcal{F}$.\nSince $X^*$ is a minimizer of $f$ over $\\mathcal{F}$, this implies $f(X^*\\cup Z)\\ge f(X^*)$ and $f(X^*\\cap Z)\\ge f(X^*)$.\nCombined with (\\ref{eq:xz}), this yields $f(X^*)=f(Z)$, as desired.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:minimal_minimizer}]\nTake a minimal minimizer $X^*\\in \\mathcal{F}$ of $f$ over $\\mathcal{F}$.\nBy Lemmas \\ref{lem:key} and \\ref{lem:complement}, there exist $S,T\\subseteq V$ such that\n\\begin{itemize}\n \\item $S\\subseteq X^*$ and $T\\subseteq V\\setminus X^*$,\n \\item $\\max\\{|S|,|T|\\}\\le k$,\n \\item $Y\\in \\mathcal{F}$ for all $Y\\subseteq V$ with $S\\subseteq Y\\subseteq X^*$ or $X^*\\subseteq Y\\subseteq V\\setminus T$.\n\\end{itemize}\nSince $S\\cap T=\\emptyset$, it suffices to show that $X^*$ is the unique minimal minimizer of $f$ over the lattice $\\mathcal{F}_{ST}$.\nBy Theorem \\ref{thm:minimizer}, $X^*$ is a minimizer of $f$ over $\\mathcal{F}_{ST}$.\nWe show that $X^*\\subseteq Z$ holds for every minimizer $Z\\in \\mathcal{F}_{ST}$ of $f$ over $\\mathcal{F}_{ST}$.\nBy the proof of Theorem \\ref{thm:minimizer}, we have $X^*\\cap Z\\in \\mathcal{F}$ and $f(X^*\\cap Z)=f(X^*)$ for every minimizer $Z$ of $f$ over $\\mathcal{F}_{ST}$.\nHence, if $X^*\\not\\subseteq Z$ for some minimizer $Z$ of $f$ over $\\mathcal{F}_{ST}$, then we have $X^*\\cap Z\\subsetneq X^*$, which contradicts that $X^*$ is a minimal minimizer of $f$ over $\\mathcal{F}$.\nThus, we have $X^*\\subseteq Z$ for every minimizer $Z$ of $f$ over $\\mathcal{F}_{ST}$, as desired.\n\\end{proof}\n\nTheorem \\ref{thm:algorithm} follows from Theorem \\ref{thm:minimal_minimizer} as follows.\nThe following proof is essentially the same as that in \\cite{nagele2019}; we include it for the reader’s convenience.\n\\begin{proof}[Proof of Theorem \\ref{thm:algorithm}]\nDefine $g:2^V\\to \\mathbb{Z}$ as $g(X):=(n+1)f(X)+|X|$ for every $X\\subseteq V$.\nNote that since $f$ is submodular, $g$ is also submodular.\nFor every pair of sets $S,T\\subseteq V$ with $S\\cap T=\\emptyset$ and $\\max\\{|S|,|T|\\}\\le k$, compute a minimizer of $g$ over $\\mathcal{F}_{ST}$ by polynomial-time algorithms for submodular function minimization \\cite{grotschel1981,iwata2001,schrijver2000}.\nThis procedure can be done in $O(n^{2k}\\tau (n))$ time.\nNote that since $f$ is integer-valued and minimizers of $f$ are closed under intersection, $g$ has the unique minimizer over $\\mathcal{F}_{ST}$, which is the unique minimal minimizer of $f$ over $\\mathcal{F}_{ST}$.\nHence, by Theorem~\\ref{thm:minimal_minimizer}, there exist $S,T\\subseteq V$ with $S\\cap T=\\emptyset$ and $\\max\\{|S|,|T|\\}\\le k$ such that the unique minimizer of $g$ over $\\mathcal{F}_{ST}$ minimizes $f$ over $\\mathcal{F}$.\nHence, a minimizer of $f$ over $\\mathcal{F}$ can be obtained in $O(n^{2k}\\tau (n))$ time.\n\\end{proof}\n\n\\begin{theorem}\n\\label{thm:minimizer}\nFor a positive integer $k$, let $\\mathcal{F}\\subseteq 2^V$ be the complement of a $k$-hierarchical lattice.\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nLet $X^*\\in \\mathcal{F}$ be a minimizer of $f$ over $\\mathcal{F}$.\nThen, there exist $S,T\\subseteq V$ with $S\\cap T=\\emptyset$ and $\\max\\{|S|,|T|\\}\\le k$ such that $X^*$ is a minimizer of $f$ over the lattice $\\mathcal{F}_{ST}$.\n\\end{theorem}", "post_theorem_intro_text_len": 5405, "post_theorem_intro_text": "To obtain a polynomial-time algorithm for SFM over the complement of $k$-hierarchical lattice, we show the following slight refinement of Theorem \\ref{thm:minimizer}.\n\n\\begin{theorem}\n\\label{thm:minimal_minimizer}\nFor a positive integer $k$, let $\\mathcal{F}\\subseteq 2^V$ be the complement of a $k$-hierarchical lattice.\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nLet $X^*\\in \\mathcal{F}$ be a minimal minimizer of $f$ over $\\mathcal{F}$.\nThen, there exist $S,T\\subseteq V$ with $S\\cap T=\\emptyset$ and $\\max\\{|S|,|T|\\}\\le k$ such that $X^*$ is the unique minimal minimizer of $f$ over the lattice $\\mathcal{F}_{ST}$.\n\\end{theorem}\n\nBy Theorem \\ref{thm:minimal_minimizer}, we can obtain a minimizer over the complement of a $k$-hierarchical lattice by solving SFM over $\\mathcal{F}_{ST}$ for all $S,T\\subseteq V$ with $S\\cap T=\\emptyset$ and $\\max\\{|S|,|T|\\}\\le k$.\nThis yields the following result:\n\n\\begin{theorem}\n\\label{thm:algorithm}\nFor a positive integer $k$, let $\\mathcal{F}\\subseteq 2^V$ be the complement of a $k$-hierarchical lattice.\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nThen, a minimizer of $f$ over $\\mathcal{F}$ can be obtained in $O(n^{2k}\\tau(n))$ time, where $n:=|V|$ and $\\tau(n)$ denotes the time required for minimizing a submodular function on a ground set of size $n$.\n\\end{theorem}\n\nSince SFM can be solved in polynomial-time \\cite{grotschel1981,iwata2001,schrijver2000}, $\\tau(n)$ in Theorem \\ref{thm:algorithm} is polynomial in $n$.\nHence, for a fixed positive integer $k$, SFM over the complement of a $k$-hierarchical lattice can be solved in polynomial-time by Theorem \\ref{thm:algorithm}.\n\nTheorem \\ref{thm:algorithm} implies a polynomial-time algorithm for minimizing submodular functions over the complement of an intersecting family or a crossing family.\nIndeed, for an intersecting family $\\mathcal{F}$, since $\\mathcal{F}\\cup \\{\\emptyset\\}$ is a 2-hierarchical lattice, we can find a minimizer of a submodular function $f$ over $2^V\\setminus (\\mathcal{F}\\cup \\{\\emptyset\\})$ in polynomial-time.\nHence, if $\\emptyset\\notin \\mathcal{F}$, by comparing the minimum value of $f$ over $2^V\\setminus (\\mathcal{F}\\cup \\{\\emptyset\\})$ with $f(\\emptyset)$, we obtain a minimizer over $2^V\\setminus \\mathcal{F}$; otherwise, a minimizer over $2^V\\setminus (\\mathcal{F}\\cup \\{\\emptyset\\})$ is already a minimizer over $2^V\\setminus \\mathcal{F}$.\nSimilarly, for a crossing family $\\mathcal{F}$, since $\\mathcal{F}\\cup \\{\\emptyset, V\\}$ is a 2-hierarchical lattice, we can find a minimizer of $f$ over $2^V\\setminus (\\mathcal{F}\\cup \\{\\emptyset, V\\})$ in polynomial-time.\nHence, by comparing the minimum value with $f(\\emptyset)$ and $f(V)$ as needed, we obtain a minimizer over $2^V\\setminus \\mathcal{F}$.\nTherefore, we have the following:\n\n\\begin{corollary}\n\\label{cor:intersecting_crossing}\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nIf $\\mathcal{F}\\subseteq 2^V$ is an intersecting family or a crossing family, then a minimizer of $f$ over $2^V\\setminus \\mathcal{F}$ can be obtained in polynomial-time.\n\\end{corollary}\n\nA family $\\mathcal{F}\\subseteq 2^V$ is called a \\textit{parity family} if for all $X,Y\\in 2^V\\setminus \\mathcal{F}$, we have\n\\begin{align*}\n(X\\cup Y\\in \\mathcal{F}) \\iff (X\\cap Y\\in \\mathcal{F}).\n\\end{align*}\nAn important example of a parity family is the complement of a lattice.\nAs mentioned above, Goemans and Ramakrishnan \\cite{goemans1995} showed that SFM over a parity family can be solved in polynomial-time; in particular, this implies polynomial-solvability of SFM over the complement of a lattice.\nSince lattices are a subclass of intersecting and crossing families, Corollary \\ref{cor:intersecting_crossing} can be viewed as a generalization of this tractability result for the complement of a lattice.\n\nAs a common generalization of SFM over parity families and congruency constraints, N\\\"{a}gele, Sudakov, and Zenklusen \\cite{nagele2019} posed the question of whether SFM over the intersection of a constant number of parity families can be solved efficiently.\nTheorem \\ref{thm:algorithm} settles this question in the case when each parity family in the intersection is the complement of a lattice.\nIndeed, the intersection of the complements of $k$ lattices is the complement of the union of these $k$ lattices; moreover, this union forms a $k$-hierarchical lattice.\nHence, Theorem \\ref{thm:algorithm} implies the following:\n\n\\begin{corollary}\n\\label{cor:parity_intersection}\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nFor a constant positive integer $k$, if $\\mathcal{F}\\subseteq 2^V$ is the intersection of the complements of $k$ lattices, then a minimizer of $f$ over $\\mathcal{F}$ can be obtained in polynomial-time. \n\\end{corollary}\n\nAs a corollary of polynomial-solvability of SFM over parity families, Goemans and Ramakrishnan \\cite{goemans1995} showed that the second smallest value of a submodular function can be obtained in polynomial-time.\nSince the family of subsets whose function values are at most the $k$-th smallest value of a submodular function forms a $k$-hierarchical lattice, Theorem \\ref{thm:algorithm} implies the following generalization to the $k$-th smallest value:\n\n\\begin{corollary}\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nFor a constant positive integer $k$, a set attaining the $k$-th smallest value of $f$ can be obtained in polynomial-time.\n\\end{corollary}", "sketch": "To obtain a polynomial-time algorithm for SFM over the complement of a $k$-hierarchical lattice, the text states a slight refinement of Theorem~\\ref{thm:minimizer} (Theorem~\\ref{thm:minimal_minimizer}). By Theorem~\\ref{thm:minimal_minimizer}, one can obtain a minimizer over the complement of a $k$-hierarchical lattice \"by solving SFM over $\\mathcal{F}_{ST}$ for all $S,T\\subseteq V$ with $S\\cap T=\\emptyset$ and $\\max\\{|S|,|T|\\}\\le k$.\" This enumeration yields Theorem~\\ref{thm:algorithm}, giving runtime $O(n^{2k}\\tau(n))$, and since \"SFM can be solved in polynomial-time,\" for fixed $k$ this implies a polynomial-time algorithm.", "expanded_sketch": "To obtain a polynomial-time algorithm for SFM over the complement of a $k$-hierarchical lattice, the text states a slight refinement of the main theorem (\\begin{theorem}\n\\label{thm:minimal_minimizer}\nFor a positive integer $k$, let $\\mathcal{F}\\subseteq 2^V$ be the complement of a $k$-hierarchical lattice.\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nLet $X^*\\in \\mathcal{F}$ be a minimal minimizer of $f$ over $\\mathcal{F}$.\nThen, there exist $S,T\\subseteq V$ with $S\\cap T=\\emptyset$ and $\\max\\{|S|,|T|\\}\\le k$ such that $X^*$ is the unique minimal minimizer of $f$ over the lattice $\\mathcal{F}_{ST}$.\n\\end{theorem}). By the preceding theorem, one can obtain a minimizer over the complement of a $k$-hierarchical lattice ``by solving SFM over $\\mathcal{F}_{ST}$ for all $S,T\\subseteq V$ with $S\\cap T=\\emptyset$ and $\\max\\{|S|,|T|\\}\\le k$.'' This enumeration yields\n\\begin{theorem}\n\\label{thm:algorithm}\nFor a positive integer $k$, let $\\mathcal{F}\\subseteq 2^V$ be the complement of a $k$-hierarchical lattice.\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nThen, a minimizer of $f$ over $\\mathcal{F}$ can be obtained in $O(n^{2k}\\tau(n))$ time, where $n:=|V|$ and $\\tau(n)$ denotes the time required for minimizing a submodular function on a ground set of size $n$.\n\\end{theorem}\nwhich gives runtime $O(n^{2k}\\tau(n))$, and since ``SFM can be solved in polynomial-time,'' for fixed $k$ this implies a polynomial-time algorithm.", "expanded_theorem": "\\label{thm:minimizer}\nFor a positive integer $k$, let $\\mathcal{F}\\subseteq 2^V$ be the complement of a $k$-hierarchical lattice.\nLet $f:2^V\\to \\mathbb{Z}$ be a submodular function.\nLet $X^*\\in \\mathcal{F}$ be a minimizer of $f$ over $\\mathcal{F}$.\nThen, there exist $S,T\\subseteq V$ with $S\\cap T=\\emptyset$ and $\\max\\{|S|,|T|\\}\\le k$ such that $X^*$ is a minimizer of $f$ over the lattice $\\mathcal{F}_{ST}$.,", "theorem_type": ["Existence", "Implication"], "mcq": {"question": "Let $V$ be a finite set. A set function $f:2^V\\to \\mathbb{Z}$ is called submodular if\n$$f(X)+f(Y)\\ge f(X\\cup Y)+f(X\\cap Y) \\quad \\text{for all } X,Y\\subseteq V.$$ \nA family $\\mathcal{L}\\subseteq 2^V$ is a lattice if $X\\cup Y, X\\cap Y\\in \\mathcal{L}$ for all $X,Y\\in \\mathcal{L}$. For a positive integer $k$, a family $\\mathcal{G}\\subseteq 2^V$ is a $k$-hierarchical lattice if $\\mathcal{G}$ can be partitioned into $\\mathcal{G}_1,\\dots,\\mathcal{G}_k$ such that $\\mathcal{G}_1$ is a lattice, and for each $i=2,\\dots,k$ and all $X,Y\\in \\mathcal{G}_i$, either $X\\cup Y$ and $X\\cap Y$ both belong to $\\mathcal{G}_i$, or at least one of $X\\cup Y$ and $X\\cap Y$ belongs to $\\mathcal{G}_1\\cup\\cdots\\cup\\mathcal{G}_{i-1}$. A family $\\mathcal{F}\\subseteq 2^V$ is called the complement of a $k$-hierarchical lattice if $2^V\\setminus \\mathcal{F}$ is a $k$-hierarchical lattice. For disjoint subsets $S,T\\subseteq V$, define\n$$\\mathcal{F}_{ST}:=\\{X\\subseteq V\\mid S\\subseteq X\\subseteq V\\setminus T\\}.$$ \nAssume $k$ is a positive integer, $\\mathcal{F}\\subseteq 2^V$ is the complement of a $k$-hierarchical lattice, $f:2^V\\to \\mathbb{Z}$ is submodular, and $X^*\\in \\mathcal{F}$ is a minimizer of $f$ over $\\mathcal{F}$. Which existence statement holds?", "correct_choice": {"label": "A", "text": "There exist subsets $S,T\\subseteq V$ with $S\\cap T=\\emptyset$ and $\\max\\{|S|,|T|\\}\\le k$ such that $X^*$ is a minimizer of $f$ over the lattice $\\mathcal{F}_{ST}$."}, "choices": [{"label": "B", "text": "There exist subsets $S,T\\subseteq V$ with $S\\cap T=\\emptyset$ and $|S|+|T|\\le k$ such that $X^*$ is a minimizer of $f$ over the lattice $\\mathcal{F}_{ST}$."}, {"label": "C", "text": "There exist subsets $S,T\\subseteq V$ with $S\\cap T=\\emptyset$ such that $X^*$ is a minimizer of $f$ over the lattice $\\mathcal{F}_{ST}$."}, {"label": "D", "text": "There exist subsets $S,T\\subseteq V$ with $S\\cap T=\\emptyset$ and $\\max\\{|S|,|T|\\}\\le k$ such that $X^*$ is the unique minimizer of $f$ over the lattice $\\mathcal{F}_{ST}$."}, {"label": "E", "text": "For every subsets $S,T\\subseteq V$ with $S\\cap T=\\emptyset$ and $\\max\\{|S|,|T|\\}\\le k$, the set $X^*$ is a minimizer of $f$ over the lattice $\\mathcal{F}_{ST}$."}], "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": "size bound uses max not sum", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the cardinality bound on $S,T$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "minimal/unique conclusion belongs only to the refinement for minimal minimizers", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "existential choice of $S,T$ replaced by universal quantification", "template_used": "quantifier_dependence"}]}} +{"id": "2601.15276v3", "paper_link": "http://arxiv.org/abs/2601.15276v3", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\nThere exists an absolute constant $c > 0$ such that the following holds. For every field $\\mathbb K$ of characteristic $0$, we have $|\\mathcal{S}(\\mathbf{a},\\mathbf{b})| \\geq c n^3$ for all $\\mathbf{a}, \\mathbf{b}\\in\\mathbb K^n$ each with distinct entries.", "start_pos": 4284, "end_pos": 4559, "label": "thm:main"}, "ref_dict": {"thm:main": "\\begin{theorem}\\label{thm:main}\nThere exists an absolute constant $c > 0$ such that the following holds. For every field $\\mathbb K$ of characteristic $0$, we have $|\\mathcal{S}(\\ba,\\bb)| \\geq c n^3$ for all $\\ba, \\bb\\in\\mathbb K^n$ each with distinct entries. \n\\end{theorem}"}, "pre_theorem_intro_text_len": 1593, "pre_theorem_intro_text": "Let $\\mathbf{a}=(a_1,\\ldots,a_n)$ and $\\mathbf{b}=(b_1,\\ldots,b_n)$ be $n$-tuples of elements of a field $\\mathbb K$ of characteristic $0$. For each permutation $\\pi$ in the symmetric group $\\mathfrak S_n$, define the quantity\n$$S(\\mathbf{a},\\mathbf{b};\\pi):=\\sum_{i=1}^n a_i b_{\\pi(i)},$$\nwhich is the ``dot product of $\\mathbf{a},\\mathbf{b}$ twisted by $\\pi$''. Motivated by Littlewood--Offord theory and a geometric problem of Pawlowski~\\cite{P24}, Do, Nguyen, Phan, Tran, and Vu \\cite{DNPTV25}, and independently Hunter, Pohoata, and Zhu \\cite{HPZ26} recently raised the problem of studying the anticoncentration properties of the random variable $S(\\mathbf{a},\\mathbf{b};\\pi)$ when $\\mathbf{a}$ and $\\mathbf{b}$ are fixed real vectors and $\\pi$ is chosen uniformly at random. One of their main results states that if $a_1,\\ldots, a_n\\in\\mathbb R$ are distinct and $b_1,\\ldots,b_n\\in\\mathbb R$ are distinct, then each possible value of the sum is assumed with probability at most $O(n^{-5/2}\\log n)$; this bound is optimal up to the logarithm (which likely can be removed). The paper \\cite{AaronEtAl} has rounded out this small flurry of recent activity.\n\nWe are interested in the set\n$$\\mathcal{S}(\\mathbf{a}, \\mathbf{b}):=\\left\\{S(\\mathbf{a},\\mathbf{b}; \\pi) : \\pi \\in \\mathfrak S_n\\right\\}$$\nof all possible sums as $\\pi$ ranges; this is of course precisely the support of the random variable $S(\\mathbf{a},\\mathbf{b};\\pi)$. Our main result states that if $a_1,\\ldots,a_n$ are all distinct and $b_1,\\ldots,b_n$ are all distinct, then there are at least $\\Omega(n^3)$ distinct sums.", "context": "Let $\\mathbf{a}=(a_1,\\ldots,a_n)$ and $\\mathbf{b}=(b_1,\\ldots,b_n)$ be $n$-tuples of elements of a field $\\mathbb K$ of characteristic $0$. For each permutation $\\pi$ in the symmetric group $\\mathfrak S_n$, define the quantity\n$$S(\\mathbf{a},\\mathbf{b};\\pi):=\\sum_{i=1}^n a_i b_{\\pi(i)},$$\nwhich is the ``dot product of $\\mathbf{a},\\mathbf{b}$ twisted by $\\pi$''. Motivated by Littlewood--Offord theory and a geometric problem of Pawlowski~\\cite{P24}, Do, Nguyen, Phan, Tran, and Vu \\cite{DNPTV25}, and independently Hunter, Pohoata, and Zhu \\cite{HPZ26} recently raised the problem of studying the anticoncentration properties of the random variable $S(\\mathbf{a},\\mathbf{b};\\pi)$ when $\\mathbf{a}$ and $\\mathbf{b}$ are fixed real vectors and $\\pi$ is chosen uniformly at random. One of their main results states that if $a_1,\\ldots, a_n\\in\\mathbb R$ are distinct and $b_1,\\ldots,b_n\\in\\mathbb R$ are distinct, then each possible value of the sum is assumed with probability at most $O(n^{-5/2}\\log n)$; this bound is optimal up to the logarithm (which likely can be removed). The paper \\cite{AaronEtAl} has rounded out this small flurry of recent activity.\n\nWe are interested in the set\n$$\\mathcal{S}(\\mathbf{a}, \\mathbf{b}):=\\left\\{S(\\mathbf{a},\\mathbf{b}; \\pi) : \\pi \\in \\mathfrak S_n\\right\\}$$\nof all possible sums as $\\pi$ ranges; this is of course precisely the support of the random variable $S(\\mathbf{a},\\mathbf{b};\\pi)$. Our main result states that if $a_1,\\ldots,a_n$ are all distinct and $b_1,\\ldots,b_n$ are all distinct, then there are at least $\\Omega(n^3)$ distinct sums.", "full_context": "Let $\\mathbf{a}=(a_1,\\ldots,a_n)$ and $\\mathbf{b}=(b_1,\\ldots,b_n)$ be $n$-tuples of elements of a field $\\mathbb K$ of characteristic $0$. For each permutation $\\pi$ in the symmetric group $\\mathfrak S_n$, define the quantity\n$$S(\\mathbf{a},\\mathbf{b};\\pi):=\\sum_{i=1}^n a_i b_{\\pi(i)},$$\nwhich is the ``dot product of $\\mathbf{a},\\mathbf{b}$ twisted by $\\pi$''. Motivated by Littlewood--Offord theory and a geometric problem of Pawlowski~\\cite{P24}, Do, Nguyen, Phan, Tran, and Vu \\cite{DNPTV25}, and independently Hunter, Pohoata, and Zhu \\cite{HPZ26} recently raised the problem of studying the anticoncentration properties of the random variable $S(\\mathbf{a},\\mathbf{b};\\pi)$ when $\\mathbf{a}$ and $\\mathbf{b}$ are fixed real vectors and $\\pi$ is chosen uniformly at random. One of their main results states that if $a_1,\\ldots, a_n\\in\\mathbb R$ are distinct and $b_1,\\ldots,b_n\\in\\mathbb R$ are distinct, then each possible value of the sum is assumed with probability at most $O(n^{-5/2}\\log n)$; this bound is optimal up to the logarithm (which likely can be removed). The paper \\cite{AaronEtAl} has rounded out this small flurry of recent activity.\n\nWe are interested in the set\n$$\\mathcal{S}(\\mathbf{a}, \\mathbf{b}):=\\left\\{S(\\mathbf{a},\\mathbf{b}; \\pi) : \\pi \\in \\mathfrak S_n\\right\\}$$\nof all possible sums as $\\pi$ ranges; this is of course precisely the support of the random variable $S(\\mathbf{a},\\mathbf{b};\\pi)$. Our main result states that if $a_1,\\ldots,a_n$ are all distinct and $b_1,\\ldots,b_n$ are all distinct, then there are at least $\\Omega(n^3)$ distinct sums.\n\n\\begin{abstract}\nLet $\\mathbb{K}$ be a field of characteristic $0$. For each choice of distinct $a_1, \\ldots, a_n\\in \\mathbb{K}$ and distinct $b_1, \\ldots, b_n\\in \\mathbb{K}$, consider the sum $S=\\sum_{i=1}^n a_i b_{\\pi(i)}$ as $\\pi$ ranges over the permutations of $[n]$. We show that this sum always assumes at least $\\Omega(n^3)$ distinct values. This ``support'' bound, which is optimal up to the value of the implicit constant, complements recent work of Do, Nguyen, Phan, Tran, and Vu, and of Hunter, Pohoata, and Zhu on the anticoncentration properties of $S$ when $a_1,\\ldots,a_n,b_1,\\ldots,b_n$ are real and $\\pi$ is chosen uniformly at random.\n\\end{abstract}\n\nWe are interested in the set\n$$\\mathcal{S}(\\ba, \\bb):=\\left\\{S(\\ba,\\bb; \\pi) : \\pi \\in \\mathfrak S_n\\right\\}$$\nof all possible sums as $\\pi$ ranges; this is of course precisely the support of the random variable $S(\\ba,\\bb;\\pi)$. Our main result states that if $a_1,\\ldots,a_n$ are all distinct and $b_1,\\ldots,b_n$ are all distinct, then there are at least $\\Omega(n^3)$ distinct sums.\n\nSome version of the distinctness condition on $a_1,\\ldots,a_n$ and $b_1,\\ldots,b_n$ is necessary since otherwise $S(\\ba,\\bb; \\pi)$ may not vary at all as $\\pi$ varies. In the case where $\\ba,\\bb$ have real coordinates, one can obtain the trivial lower bound $|\\mathcal S(\\ba,\\bb)| \\geq 1+\\binom{n}{2}$ by first ordering both tuples $\\ba,\\bb$ in increasing order and then iteratively swapping a pair of adjacent elements of $\\bb$ until all elements are in decreasing order. The bound in \\Cref{thm:main} is sharp up to the value of the constant $c$ since $|\\mathcal{S}(\\ba, \\bb)|=(1/6+o(1))n^3$ for $\\ba=\\bb=(1,2,\\ldots, n)$.\n\nTo prove \\cref{thm:main}, we first handle the special case where $\\ba$ and $\\bb$ are real vectors, which is already of interest. Our proof is elementary and fairly short. Pohoata \\cite{P26} has recently obtained an independent proof of this special case of \\cref{thm:main} using more complicated methods. \nWe then prove \\cref{thm:main} when $\\mathbb K$ is the field $\\mathbb C$ of complex numbers. To do so, we invoke Beck's theorem from discrete geometry in order to split into two cases. In the first case, each of $\\ba,\\bb$ contains a large collinear subset, and we can reduce the argument to the real setting. In the second case, one of $\\ba,\\bb$ defines many distinct lines, and we can run a different argument based on the $2$-dimensional case of the following theorem, which may be of independent interest. \n\\begin{theorem}\\label{thm:gamma} \nFor each positive integer $d$, there exists a constant $\\gamma_d>0$ such that the following holds. If $A \\subseteq \\mathbb{R}^d$ is an $m$-element set with no linearly dependent $d$-element subsets, then $A$ has at least $\\gamma_dm^{d+1}$ distinct subset sums. \n\\end{theorem}\n\n\\section{Real Numbers}\\label{sec:reals} \nIn this section, let $\\ba,\\bb \\in \\mathbb{R}^n$ be $n$-tuples each with distinct entries.\nWrite $S_0:=S(\\ba,\\bb; \\id)$, where $\\id$ denotes the identity permutation. For the permutation $\\pi=(j,k)$ that swaps $j,k$ and fixes all elements of $[n]\\setminus\\{j,k\\}$, we can compute\n$$S(\\ba,\\bb; (j,k))=S_0+a_j(b_k-b_j)+a_k(b_j-b_k)=S_0-(a_k-a_j)(b_k-b_j).$$\nIn general, if $\\pi=(j_1,k_1) \\cdots (j_r,k_r)$ is a product of disjoint transpositions, then we obtain\n$$S(\\ba,\\bb;(j_1,k_1) \\cdots (j_r,k_r))=S_0-\\sum_{i=1}^r (a_{k_i}-a_{j_i})(b_{k_i}-b_{j_i}).$$\nThus, to prove \\Cref{thm:main}, it suffices to find disjoint transpositions $(j_1,k_1), \\ldots, (j_r,k_r)$ with the property that the set\n\\begin{equation}\\label{eq:differences}\n\\{(a_{k_1}-a_{j_1})(b_{k_1}-b_{j_1}), \\ldots, (a_{k_r}-a_{j_r})(b_{k_r}-b_{j_r})\\}\n\\end{equation}\nhas $\\Omega(n^3)$ distinct subset sums. We are of course free to reorder the entries of $\\ba$ and of $\\bb$ (possibly in different ways) before specifying our family of disjoint transpositions.\n\nFix $m\\geq d$, and let $A\\subseteq \\mathbb R^d$ be an $m$-element set with no $d$-element linearly dependent subsets. The condition $m\\geq d$ ensures that no element of $A$ is a scalar multiple of any other, as otherwise any $d$-element subset containing both would be linearly dependent. For each $w\\in A$, let $\\mathcal A_w$ be the collection of $2$-element sets $\\{u,v\\}\\subseteq A\\setminus\\{w\\}$ such that $v-u$ is a scalar multiple of $w$. Note that no pair $\\{u,v\\}$ can belong to $\\mathcal A_w$ and $\\mathcal A_{w'}$ for distinct $w,w'$ (since $w$ and $w'$ are linearly independent). Hence, there is some $w^*\\in A$ such that $|\\mathcal A_{w^*}|\\leq \\binom{m}{2}/m=(m-1)/2$.\n\n\\section{Fields of characteristic $0$} \nWe have shown that there is a constant $c>0$ such that $|\\mathcal S(\\ba,\\bb)|\\geq cn^3$ for all $\\ba,\\bb\\in\\mathbb C^n$ with distinct entries. We claim that the full version of \\cref{thm:main} holds with the same value of $c$.\n\nLet $\\mathbb K$ be a field of characteristic $0$, and let $\\ba=(a_1,\\ldots,a_n)$ and $\\bb=(b_1,\\ldots,b_n)$ be tuples in $\\mathbb K^n$ each with distinct entries. Let $\\mathbb K'$ be the field extension of $\\mathbb Q$ generated by $a_1,\\ldots,a_n,b_1,\\ldots,b_n$. It is well known that every finitely generated field extension of $\\mathbb Q$ embeds into $\\mathbb C$. Thus, there exists a field embedding $\\iota\\colon\\mathbb K'\\to\\mathbb C$. Consider the tuples $\\iota(\\ba):=(\\iota(a_1),\\ldots,\\iota(a_n))$ and $\\iota(\\bb):=(\\iota(b_1),\\ldots,\\iota(b_n))$ in $\\mathbb{C}^n$. Then $|\\mathcal S(\\ba,\\bb)|=|\\mathcal S(\\iota(\\ba),\\iota(\\bb))|\\geq cn^3$ by the complex case of \\Cref{thm:main}.\n\n\\begin{theorem}\\label{thm:main}\nThere exists an absolute constant $c > 0$ such that the following holds. For every field $\\mathbb K$ of characteristic $0$, we have $|\\mathcal{S}(\\ba,\\bb)| \\geq c n^3$ for all $\\ba, \\bb\\in\\mathbb K^n$ each with distinct entries. \n\\end{theorem}", "post_theorem_intro_text_len": 2055, "post_theorem_intro_text": "Some version of the distinctness condition on $a_1,\\ldots,a_n$ and $b_1,\\ldots,b_n$ is necessary since otherwise $S(\\mathbf{a},\\mathbf{b}; \\pi)$ may not vary at all as $\\pi$ varies. In the case where $\\mathbf{a},\\mathbf{b}$ have real coordinates, one can obtain the trivial lower bound $|\\mathcal S(\\mathbf{a},\\mathbf{b})| \\geq 1+\\binom{n}{2}$ by first ordering both tuples $\\mathbf{a},\\mathbf{b}$ in increasing order and then iteratively swapping a pair of adjacent elements of $\\mathbf{b}$ until all elements are in decreasing order. The bound in \\Cref{thm:main} is sharp up to the value of the constant $c$ since $|\\mathcal{S}(\\mathbf{a}, \\mathbf{b})|=(1/6+o(1))n^3$ for $\\mathbf{a}=\\mathbf{b}=(1,2,\\ldots, n)$.\n\nTo prove \\cref{thm:main}, we first handle the special case where $\\mathbf{a}$ and $\\mathbf{b}$ are real vectors, which is already of interest. Our proof is elementary and fairly short. Pohoata \\cite{P26} has recently obtained an independent proof of this special case of \\cref{thm:main} using more complicated methods. \nWe then prove \\cref{thm:main} when $\\mathbb K$ is the field $\\mathbb C$ of complex numbers. To do so, we invoke Beck's theorem from discrete geometry in order to split into two cases. In the first case, each of $\\mathbf{a},\\mathbf{b}$ contains a large collinear subset, and we can reduce the argument to the real setting. In the second case, one of $\\mathbf{a},\\mathbf{b}$ defines many distinct lines, and we can run a different argument based on the $2$-dimensional case of the following theorem, which may be of independent interest. \n\\begin{theorem}\\label{thm:gamma} \nFor each positive integer $d$, there exists a constant $\\gamma_d>0$ such that the following holds. If $A \\subseteq \\mathbb{R}^d$ is an $m$-element set with no linearly dependent $d$-element subsets, then $A$ has at least $\\gamma_dm^{d+1}$ distinct subset sums. \n\\end{theorem}\n\nWe deduce the full version of \\cref{thm:main} from the complex case using the fact that every finitely generated field extension of $\\mathbb Q$ embeds into $\\mathbb C$.", "sketch": "To prove \\cref{thm:main}, the introduction outlines these steps:\n\n1. **Real case first:** “we first handle the special case where $\\mathbf{a}$ and $\\mathbf{b}$ are real vectors.” The proof there is said to be “elementary and fairly short.”\n\n2. **Complex case via Beck’s theorem:** Next, “we then prove \\cref{thm:main} when $\\mathbb K$ is the field $\\mathbb C$ of complex numbers.” This uses “Beck's theorem from discrete geometry in order to split into two cases.”\n - **Case 1:** “each of $\\mathbf{a},\\mathbf{b}$ contains a large collinear subset,” allowing one to “reduce the argument to the real setting.”\n - **Case 2:** “one of $\\mathbf{a},\\mathbf{b}$ defines many distinct lines,” and then one “run[s] a different argument based on the $2$-dimensional case” of \\Cref{thm:gamma} (a theorem giving many distinct subset sums under a linear-independence hypothesis).\n\n3. **General characteristic 0 fields from the complex case:** Finally, “We deduce the full version of \\cref{thm:main} from the complex case using the fact that every finitely generated field extension of $\\mathbb Q$ embeds into $\\mathbb C$.”", "expanded_sketch": "To prove the main theorem, the introduction outlines these steps:\n\n1. **Real case first:** “we first handle the special case where $\\mathbf{a}$ and $\\mathbf{b}$ are real vectors.” The proof there is said to be “elementary and fairly short.”\n\n2. **Complex case via Beck’s theorem:** Next, “we then prove the main theorem when $\\mathbb K$ is the field $\\mathbb C$ of complex numbers.” This uses “Beck's theorem from discrete geometry in order to split into two cases.”\n - **Case 1:** “each of $\\mathbf{a},\\mathbf{b}$ contains a large collinear subset,” allowing one to “reduce the argument to the real setting.”\n - **Case 2:** “one of $\\mathbf{a},\\mathbf{b}$ defines many distinct lines,” and then one “run[s] a different argument based on the $2$-dimensional case” of \\Cref{thm:gamma} (a theorem giving many distinct subset sums under a linear-independence hypothesis).\n\n3. **General characteristic 0 fields from the complex case:** Finally, “We deduce the full version of the main theorem from the complex case using the fact that every finitely generated field extension of $\\mathbb Q$ embeds into $\\mathbb C$.”,", "expanded_theorem": "\\label{thm:main}\nThere exists an absolute constant $c > 0$ such that the following holds. For every field $\\mathbb K$ of characteristic $0$, we have $|\\mathcal{S}(\\mathbf{a},\\mathbf{b})| \\geq c n^3$ for all $\\mathbf{a}, \\mathbf{b}\\in\\mathbb K^n$ each with distinct entries.", "theorem_type": ["Existential–Universal", "Inequality or Bound"], "mcq": {"question": "Let $n$ be a positive integer, let $\\mathbb K$ be a field of characteristic $0$, and let $\\mathbf a=(a_1,\\dots,a_n)$ and $\\mathbf b=(b_1,\\dots,b_n)$ be elements of $\\mathbb K^n$, with the entries of $\\mathbf a$ all distinct and the entries of $\\mathbf b$ all distinct. For each permutation $\\pi\\in \\mathfrak S_n$, define\n$$S(\\mathbf a,\\mathbf b;\\pi):=\\sum_{i=1}^n a_i b_{\\pi(i)},$$\nand let\n$$\\mathcal S(\\mathbf a,\\mathbf b):=\\{S(\\mathbf a,\\mathbf b;\\pi):\\pi\\in \\mathfrak S_n\\}.$$ \nWhich statement holds?", "correct_choice": {"label": "A", "text": "There exists an absolute constant $c>0$ such that for every positive integer $n$, every field $\\mathbb K$ of characteristic $0$, and all $\\mathbf a,\\mathbf b\\in \\mathbb K^n$ each with distinct entries, the set $\\mathcal S(\\mathbf a,\\mathbf b)$ satisfies\n$$|\\mathcal S(\\mathbf a,\\mathbf b)|\\ge c n^3.$$"}, "choices": [{"label": "B", "text": "There exists an absolute constant $c>0$ such that for every positive integer $n$, every field $\\mathbb K$, and all $\\mathbf a,\\mathbf b\\in \\mathbb K^n$ each with distinct entries, the set $\\mathcal S(\\mathbf a,\\mathbf b)$ satisfies\n$$|\\mathcal S(\\mathbf a,\\mathbf b)|\\ge c n^3.$$"}, {"label": "C", "text": "For every positive integer $n$, every field $\\mathbb K$ of characteristic $0$, and all $\\mathbf a,\\mathbf b\\in \\mathbb K^n$ each with distinct entries, the set $\\mathcal S(\\mathbf a,\\mathbf b)$ has cardinality at least on the order of $n^2$; equivalently, there exists an absolute constant $c>0$ such that\n$$|\\mathcal S(\\mathbf a,\\mathbf b)|\\ge c n^2.$$"}, {"label": "D", "text": "For every field $\\mathbb K$ of characteristic $0$ there exists a constant $c_{\\mathbb K}>0$ such that for every positive integer $n$ and all $\\mathbf a,\\mathbf b\\in \\mathbb K^n$ each with distinct entries, the set $\\mathcal S(\\mathbf a,\\mathbf b)$ satisfies\n$$|\\mathcal S(\\mathbf a,\\mathbf b)|\\ge c_{\\mathbb K} n^3.$$"}, {"label": "E", "text": "There exists an absolute constant $c>0$ such that for every positive integer $n$, every field $\\mathbb K$ of characteristic $0$, and all $\\mathbf a,\\mathbf b\\in \\mathbb K^n$ each with distinct entries, every value assumed by $S(\\mathbf a,\\mathbf b;\\pi)$ as $\\pi$ ranges over $\\mathfrak S_n$ is taken by at most $c^{-1}n^{-3}$ permutations; equivalently, the random variable $S(\\mathbf a,\\mathbf b;\\pi)$ is uniformly $O(n^{-3})$-anticoncentrated."}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "E"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "characteristic", "tampered_component": "characteristic_0_hypothesis", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "cubic_exponent_lowered_to_quadratic", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "absolute_constant_replaced_by_field_dependent_constant", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "support_bound_confused_with_anticoncentration_bound", "template_used": "wildcard"}]}} +{"id": "2601.15988v2", "paper_link": "http://arxiv.org/abs/2601.15988v2", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm1}Let $E$ be an elliptic curve given by equation $$y^2=x^3+ax+b$$ where $a,b$ are rational integers and $4a^3+27b^2\\neq 0$. Let the Mordell-Weil group $E(\\QQ)$ has positive rank and there exists a rational point on $E$, except the rank point. Then for an infinite number of prime numbers $p$, $p^3+ap+b$ is a square free positive integer and the class group of $\\QQ(\\sqrt{p^3+ap+b})$ contains a non-trivial element which is obtained by specializing a generator of the rank part of $E(\\QQ)$.", "start_pos": 6399, "end_pos": 6929, "label": "thm1"}, "ref_dict": {}, "pre_theorem_intro_text_len": 1288, "pre_theorem_intro_text": "The study of elliptic curves is an interesting area of research in number theory and arithmetic geometry. Elliptic curves are smooth plane curves given by the equation: \n$$y^2=x^3+ax+b$$\nwhere $a,b$ are rational numbers with the property that\n$$4a^3+27b^2\\neq 0\\;.$$\nThis ensures that we can draw a tangent at each and every point of the elliptic curve $E$ given by the above equation. Also, it should be noted that since it is a non-singular cubic curve, the curve is endowed with a group structure where there is a point at infinity serving as the identity of the group. \n\nOn the other hand, there is another interesting object to study in the context of number theory, namely the ideal class group. This is the free abelian group generated by all the prime ideals in the ring of integers of a number field modulo the ideals which are principal. The folklore conjecture says that there exist class groups of number fields with arbitrarily large order. The question is : is it somehow related to the torsions and rank (points of infinite order) of an elliptic curve? That is, given an elliptic curve, can we somehow construct a number field whose class group has an element of order $p$ for a large prime $p$.\n\nIn this paper, we address this question and the main theorem is as follows:", "context": "The study of elliptic curves is an interesting area of research in number theory and arithmetic geometry. Elliptic curves are smooth plane curves given by the equation: \n$$y^2=x^3+ax+b$$\nwhere $a,b$ are rational numbers with the property that\n$$4a^3+27b^2\\neq 0\\;.$$\nThis ensures that we can draw a tangent at each and every point of the elliptic curve $E$ given by the above equation. Also, it should be noted that since it is a non-singular cubic curve, the curve is endowed with a group structure where there is a point at infinity serving as the identity of the group.\n\nOn the other hand, there is another interesting object to study in the context of number theory, namely the ideal class group. This is the free abelian group generated by all the prime ideals in the ring of integers of a number field modulo the ideals which are principal. The folklore conjecture says that there exist class groups of number fields with arbitrarily large order. The question is : is it somehow related to the torsions and rank (points of infinite order) of an elliptic curve? That is, given an elliptic curve, can we somehow construct a number field whose class group has an element of order $p$ for a large prime $p$.\n\nIn this paper, we address this question and the main theorem is as follows:", "full_context": "The study of elliptic curves is an interesting area of research in number theory and arithmetic geometry. Elliptic curves are smooth plane curves given by the equation: \n$$y^2=x^3+ax+b$$\nwhere $a,b$ are rational numbers with the property that\n$$4a^3+27b^2\\neq 0\\;.$$\nThis ensures that we can draw a tangent at each and every point of the elliptic curve $E$ given by the above equation. Also, it should be noted that since it is a non-singular cubic curve, the curve is endowed with a group structure where there is a point at infinity serving as the identity of the group.\n\nOn the other hand, there is another interesting object to study in the context of number theory, namely the ideal class group. This is the free abelian group generated by all the prime ideals in the ring of integers of a number field modulo the ideals which are principal. The folklore conjecture says that there exist class groups of number fields with arbitrarily large order. The question is : is it somehow related to the torsions and rank (points of infinite order) of an elliptic curve? That is, given an elliptic curve, can we somehow construct a number field whose class group has an element of order $p$ for a large prime $p$.\n\nIn this paper, we address this question and the main theorem is as follows:\n\n\\section{Introduction}\nThe study of elliptic curves is an interesting area of research in number theory and arithmetic geometry. Elliptic curves are smooth plane curves given by the equation: \n$$y^2=x^3+ax+b$$\nwhere $a,b$ are rational numbers with the property that\n$$4a^3+27b^2\\neq 0\\;.$$\nThis ensures that we can draw a tangent at each and every point of the elliptic curve $E$ given by the above equation. Also, it should be noted that since it is a non-singular cubic curve, the curve is endowed with a group structure where there is a point at infinity serving as the identity of the group.\n\nOn the other hand, there is another interesting object to study in the context of number theory, namely the ideal class group. This is the free abelian group generated by all the prime ideals in the ring of integers of a number field modulo the ideals which are principal. The folklore conjecture says that there exist class groups of number fields with arbitrarily large order. The question is : is it somehow related to the torsions and rank (points of infinite order) of an elliptic curve? That is, given an elliptic curve, can we somehow construct a number field whose class group has an element of order $p$ for a large prime $p$.\n\nIn this paper, we address this question and the main theorem is as follows:\n\nHere, the existence of infinitely many primes such that $p^3+ap+b$ is a non-square exists by the Siegel's theorem that the number of integral points on an elliptic curve with integer coefficients is finite, so the number of solutions of the equation $y^2=p^3+ap+b$ is finite and hence there are infinitely many primes $p$ such that $p^3+ap+b$ is non-square.\n\n\\section{Proof of the theorem \\ref{thm1}}\nFor the proof of this theorem we consider the technique of Chow schemes or Hilbert schemes on a scheme or variety defined over $\\Spec(\\ZZ)$. Let $E_{\\ZZ}$ be a smooth integral model of the curve $E$ defined over $\\Spec(\\ZZ)$. Taking a point on the elliptic curve as a Weil divisor on the curve, that is, using an isomorphism $E\\cong \\Pic^0(E)$ we have such a realization, we spread the Weil divisor on $E_{\\ZZ}$ to obtain a divisor on the fixed integral model of $E$ and then specialize this divisor at a prime integer $p$ of $Spec(\\ZZ)$ to obtain an element of the ideal class group of a number field given by $\\QQ(\\sqrt{p^3+ap+b})$. The only thing is that if we start with an element of infinite order of the elliptic curve defined over $\\QQ$ we have to prove that the ideal class element that it produces in $\\QQ(\\sqrt{p^3+ap+b})$ is non-principal.\n\nLet us give some details on the spreading technique for Weil divisors on $E$. Let us take a point $(x,y)$ in $E(\\QQ)$ which is a rank point. Since $y^2=x^3+ax+b$, we have the following. Write $x=k/l, y=m/n$, and then substituting into the equation we have\n$$(\\frac{m}{n})^2=(\\frac{k}{l})^3+a\\frac{k}{l}+b$$\nThen by clearing the denominator we have \n$$l^3m^2=n^2k^3+an^2kl^2+bl^3n^2$$\nThe above equation is defined over the ring of integers and defines an arithmetic variety. Given the point $(x,y)$, it defines a point in this arithmetic variety and vise versa. Consider $Spec(\\bcO_K)$ where $K=\\QQ(\\sqrt{p^3+ap+b})$, $p\\geq 3$ is a prime number. There is a morphism of schemes from $\\Spec(\\bcO_K)$ to $\\Spec \\ZZ$, induced by the homomorphism $\\ZZ\\to \\bcO_K$. Now consider the arithmetic variety and consider its specialization at $x=p$, then we have the arithmetic variety \n$$m^2=n^2p^3+an^2p+bn^2$$\nand its normal closure , which is exactly $\\bcO_K$. Now given a rank point in $E(\\QQ)$ we can think of it as a Weil divisor on $E$ of infinite order, using the spread and the specialization we get a divisor on $\\bcO_K$ and hence an element in $cl(K)$.\n\nThe condition that the elements are in the rank part of $E(\\QQ)$ indicates (by the previous theorem) that they are parametrized by the complement $\\cup_{d,n}Z_d^n=Z$. This complement is a countable intersection of Zariski open subsets of the relative Chow variety. Now, we know that there is an infinite order element of the group $E(\\QQ)$, so the countable intersection is non-empty. Then if we spread it out over $E_{\\ZZ}$ we get a non-trivial element in the Chow group of co-dimension one cycles $CH^1(E_{\\ZZ})$. Suppose that for all number fields of the form $\\QQ(\\sqrt{p^3+ap+b})$ it is zero in the class group of the given number field for $p$ in a Zariski open set $U$ in $\\Spec(\\ZZ)$. In that case the element say $D$ is such that \n$$D_p=div(f)$$\nfor a rational function $f$ on the ring of integers of the number field. Since it happens for all such number fields we have a collection of rational functions on some open sets $U_i\\subset E_{\\ZZ}$, such that $(U_i,f_i)$ forms a Cartier divisor and we have a Line bundle corresponding to $D$ which is trivial on the union $\\cup_i U_i$, so that $D$ is rationally equivalent to zero on $CH^1(E_U)$.\n\nLet us give some details on this. We have $f$ is a rational function on the ring of integers, we have $D_p=div(f)$ which gives that $D$ is rationally equivalent to zero on some Zariski open $U$ in $\\bcO_K$. This is because $$cl(K)\\supset\\varprojlim CH^1(E_U)$$ where $E_U$ is the pull-back of $E_{\\ZZ}$ over $U\\subset Spec(\\bcO_K)$. The cycle $D_p$ is in $\\varprojlim CH^1(E_U)$. Now $\\pi:\\Spec(\\bcO_K)\\to \\Spec(\\ZZ)$ is a finite morphism. Hence, if we take a open set $V\\subset Spec(\\ZZ)$ and pull-it back to $U$, we have $D$ restricted to such an open set is rationally equivalent to zero and hence the push-down to $CH^1(E_V)$ is rationally equivalent to zero as well. Hence, we have, by the push-forward pullback formula, \n$$\\pi_*\\pi^*(D)=nD=0$$\nin $Pic^0(E)(\\QQ)\\cong E(\\QQ)$.\nHere we assume that the elliptic curve has a $\\QQ$ point. Since $D$ is of infinite order, this gives a contradiction as $nD=0$ and hence we have $D_p$ in $Cl(K)$ is non-trivial for infinitely many primes $p$. So, there exists infinitely many $p$ such that $\\QQ(\\sqrt{p^3+ap+b})$ has class number larger than one.", "post_theorem_intro_text_len": 1288, "post_theorem_intro_text": "Here, the existence of infinitely many primes such that $p^3+ap+b$ is a non-square exists by the Siegel's theorem that the number of integral points on an elliptic curve with integer coefficients is finite, so the number of solutions of the equation $y^2=p^3+ap+b$ is finite and hence there are infinitely many primes $p$ such that $p^3+ap+b$ is non-square.\n\nThis problem has previously been extensively studied from an algebraic geometry perspective by \\cite{GL}, \\cite{GL-18}, \\cite{AP}, \\cite{So}.\nIn their approach, they had used the notion on Picard group of Cartier divisors to deduce the result about the connection between line bundles and elements of class groups. In addition, the approach of \\cite{GI} is important for the hyperelliptic curves in the same context. Here the novelty is that we are using the Weil divisors in strak difference with Cartier divisors and this approach does extend to singular varieties and to produce class groups of large order from them as well. The other advantage that we have from the approach of Soleng \\cite{So}, where a similar study has been done in a concrete way, is that the homomorphism defined from the elliptic curve to the class group here is motivic and functorial in nature and can be generalized to higher dimensional varieties.", "sketch": "Using Siegel's theorem: since an elliptic curve with integer coefficients has only finitely many integral points, the Diophantine equation $y^2=p^3+ap+b$ has only finitely many solutions in integers. Hence there are infinitely many primes $p$ such that $p^3+ap+b$ is non-square.", "expanded_sketch": "Using Siegel's theorem: since an elliptic curve with integer coefficients has only finitely many integral points, the Diophantine equation $y^2=p^3+ap+b$ has only finitely many solutions in integers. Hence there are infinitely many primes $p$ such that $p^3+ap+b$ is non-square.", "expanded_theorem": "\\label{thm1}Let $E$ be an elliptic curve given by equation $$y^2=x^3+ax+b$$ where $a,b$ are rational integers and $4a^3+27b^2\\neq 0$. Let the Mordell-Weil group $E(\\QQ)$ has positive rank and there exists a rational point on $E$, except the rank point. Then for an infinite number of prime numbers $p$, $p^3+ap+b$ is a square free positive integer and the class group of $\\QQ(\\sqrt{p^3+ap+b})$ contains a non-trivial element which is obtained by specializing a generator of the rank part of $E(\\QQ)$.", "theorem_type": ["Existence", "Universal"], "mcq": {"question": "Let \n\\[\nE: y^2=x^3+ax+b\n\\]\nbe an elliptic curve with integers \\(a,b\\) satisfying \\(4a^3+27b^2\\neq 0\\). Assume the Mordell--Weil group \\(E(\\mathbb{Q})\\) has positive rank, and that \\(E\\) has a rational point in addition to a rank point (that is, in addition to a point of infinite order in \\(E(\\mathbb{Q})\\)). Here the rank part of \\(E(\\mathbb{Q})\\) means its free abelian part. Which statement holds for every such elliptic curve?", "correct_choice": {"label": "A", "text": "There exist infinitely many prime numbers \\(p\\) such that \\(p^3+ap+b\\) is a positive square-free integer, and the class group of the quadratic field \\(\\mathbb{Q}(\\sqrt{p^3+ap+b})\\) contains a non-trivial element obtained by specializing a generator of the rank part of \\(E(\\mathbb{Q})\\)."}, "choices": [{"label": "B", "text": "There exist infinitely many prime numbers \\(p\\) such that \\(p^3+ap+b\\) is a positive integer, and the class group of the quadratic field \\(\\mathbb{Q}(\\sqrt{p^3+ap+b})\\) contains a non-trivial element obtained by specializing a generator of the torsion part of \\(E(\\mathbb{Q})\\)."}, {"label": "C", "text": "There exist infinitely many prime numbers \\(p\\) such that the class group of the quadratic field \\(\\mathbb{Q}(\\sqrt{p^3+ap+b})\\) is non-trivial."}, {"label": "D", "text": "For every prime number \\(p\\) such that \\(p^3+ap+b\\) is a positive square-free integer, the class group of the quadratic field \\(\\mathbb{Q}(\\sqrt{p^3+ap+b})\\) contains a non-trivial element obtained by specializing a generator of the rank part of \\(E(\\mathbb{Q})\\)."}, {"label": "E", "text": "There exists a prime number \\(p\\) such that \\(p^3+ap+b\\) is a positive square-free integer, and for every generator of the rank part of \\(E(\\mathbb{Q})\\) the class group of the quadratic field \\(\\mathbb{Q}(\\sqrt{p^3+ap+b})\\) contains a non-trivial element obtained by specializing that generator."}], "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": "rank-generator source replaced by torsion specialization", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped square-free condition and dropped specialization-from-rank-generator clause", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "infinitely many primes weakened incorrectly to all admissible primes", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "quantifier order on prime versus generators of the rank part", "template_used": "quantifier_dependence"}]}} +{"id": "2601.16415v1", "paper_link": "http://arxiv.org/abs/2601.16415v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "The Chow ring of $\\bar\\M_{0,n}$ is generated by the classes $[D_I]$ modulo only the relations\n \\begin{align*}\n [D_I]=[D_{I^c}]&,\\\\\n [D_I][D_J]=0 &\\text{ if }I\\not\\subseteq J,J^c\\text{ and }J,J^c\\not\\subseteq I, \\text{ and }\\\\\n \\sum_{\\substack{i,j\\in I\\\\ k,\\ell \\not\\in I}} [D_I]=\\sum_{\\substack{i,k\\in I\\\\ j,\\ell \\not\\in I}} [D_I]& \\text{ for }i,j,k,\\ell\\in [n]\\text{ distinct (WDVV Relations).}\n \\end{align*}", "start_pos": 5265, "end_pos": 5731, "label": null}, "ref_dict": {"def": "\\begin{definition}\\label{def}\n The simplicially stable space $\\bar\\M_{g,\\mathcal K}$ parameterizes nodal genus $g$ curves with smooth marked points indexed by $S$, $(C,\\{p_s|s\\in S\\})$, with finitely many automorphisms, so that \n \\begin{itemize}\n \\item for each $x\\in C$, the set $\\{s \\in S| p_s=x\\}$ is in $\\mathcal K$ and\n \\item for each rational tail $Z\\subseteq C$, the set $\\{s\\in S| p_s\\in Z\\}$ is not in $\\mathcal K$. \n \\end{itemize}\n\\end{definition}", "main3": "\\begin{thm}\\label{main3}\n The Chow ring of $\\bar\\M_{0,\\mathcal K}$ is generated by the classes $[D_I]$ and $[E_{st}]$ modulo only the relations\n \\begin{align*}\n [D_I]=[D_{I^c}]&,\\\\\n [D_I][D_J]=0 &\\text{ if }I\\not\\subseteq J,J^c\\text{ and }J,J^c\\not\\subseteq I, \\\\\n [D_I][E_{st}]=0 & \\text{ if } s\\in I, t\\in I^c\\text{ or } t\\in I, s\\in I^c,\\\\\n [E_{st}][E_{tu}]=0 & \\text{ if } \\{s,t,u\\}\\notin \\mathcal K, \\text{ and}\\\\\n \\text{WDVV}(i,j,k,\\ell)& \\text{ for }i,j,k,\\ell\\in S\\text{ distinct,}\n \\end{align*}\n where the WDVV relation associated to $i,j,k,\\ell$ is \n \\[[E_{ij}]+[E_{k\\ell}]+\\sum_{\\substack{i,j\\in I\\\\ k,\\ell \\not\\in I}} [D_I]=[E_{ik}]+[E_{j\\ell}]+\\sum_{\\substack{i,k\\in I\\\\ j,\\ell \\not\\in I}} [D_I].\\]\n\\end{thm}", "main2": "\\begin{thm}[Restatement of Theorem~\\ref{main}]\\label{main2}\n The Chow ring of $\\bar\\M_{0,\\mathcal K}$ is generated by the classes $\\delta$ for codimension $1$ $\\mathcal K$-stable graphs $\\Delta$, subject to only the relations\n \\[\\delta\\cdot \\delta'=0\\text{ if } \\bar\\M_\\Delta\\cap \\bar\\M_{\\Delta'}=\\emptyset\\]\n \\[\\sum_{\\substack{\\text{codim $1$ }\\Delta\\\\ ij\\Delta k\\ell}} \\delta=\\sum_{\\substack{\\text{codim $1$ }\\Delta\\\\ ik\\Delta j\\ell}} \\delta\\text{ for distinct } i,j,k,\\ell\\in S.\\]\n\\end{thm}"}, "pre_theorem_intro_text_len": 1981, "pre_theorem_intro_text": "The simplicially stable spaces were introduced in by Blankers and Bozlee in \\cite{BB}. They are compactifications of $\\M_{g,S},$ the moduli space of smooth genus $g$ curves with markings in a finite set $S$, which are indexed by (abstract) simplicial complexes $\\mathcal K$ and are denoted $\\bar\\M_{g,\\mathcal K}$. The simplicial complex controls whether the marked points may ``collide'' in $\\bar\\M_{g,\\mathcal K}$ (see Definition~\\ref{def} for a precise definition). For example, given distinct $s,t\\in S$, if $\\{s,t\\}\\in \\mathcal K$ and $(C,\\{p_s: s\\in S\\})\\in \\bar\\M_{g,\\mathcal K}$, then we may have $p_s=p_t$, in contrast with the usual compactification $\\bar\\M_{g,S}$. \n\nSimplicially stable spaces generalize the earlier moduli spaces of weighted stable curves first studied by Hassett in \\cite{Hassett}. Given weight data $\\mathcal A=\\{a_s:s\\in S\\}$, one gets a simplicial complex $\\mathcal K_{\\mathcal A}:=\\{T\\subseteq S| \\sum_{s\\in T} a_s <1\\}$, and one has $\\bar\\M_{g,\\mathcal A}=\\bar\\M_{g,\\mathcal K_{\\mathcal A}}.$\n\nMuch recent work has focused on the Chow rings of $\\bar\\M_{g,n}$ \\cite{BDL2,CL2,DLPV,ELarson}. We study the Chow ring of these simplicially stable spaces in genus $0$. Notably, these spaces give all stable (in the sense of Smyth \\cite{Smyth}) smooth modular compactifications of $\\M_{0,n}$ (combine \\cite[Theorem 7.8]{MSvAX}] with \\cite[Lemma 4.15]{BB}). Previously, Keel computed the Chow ring of $\\bar\\M_{0,n}$ in \\cite{Keel}. This was generalized by Kannan, Karp, and Li in \\cite{KKL}, where the Chow rings of so-called ``heavy-light'' Hassett spaces were calculated in genus $0$ using tropical methods.\n\nBefore stating the main theorem, we recall Keel's presentation of $\\CH(\\bar\\M_{0,n})$ from \\cite{Keel}. Given a subset $I\\subseteq [n]$ so that $2\\leq \\# I\\leq n-2$, let $D_I$ be the closure of the set of curves in $\\bar\\M_{0,n}$ with two components, with markings from $I$ on one component and markings from the complement $I^c$ on the other.", "context": "The simplicially stable spaces were introduced in by Blankers and Bozlee in \\cite{BB}. They are compactifications of $\\M_{g,S},$ the moduli space of smooth genus $g$ curves with markings in a finite set $S$, which are indexed by (abstract) simplicial complexes $\\mathcal K$ and are denoted $\\bar\\M_{g,\\mathcal K}$. The simplicial complex controls whether the marked points may ``collide'' in $\\bar\\M_{g,\\mathcal K}$ (see Definition~\\ref{def} for a precise definition). For example, given distinct $s,t\\in S$, if $\\{s,t\\}\\in \\mathcal K$ and $(C,\\{p_s: s\\in S\\})\\in \\bar\\M_{g,\\mathcal K}$, then we may have $p_s=p_t$, in contrast with the usual compactification $\\bar\\M_{g,S}$.\n\nSimplicially stable spaces generalize the earlier moduli spaces of weighted stable curves first studied by Hassett in \\cite{Hassett}. Given weight data $\\mathcal A=\\{a_s:s\\in S\\}$, one gets a simplicial complex $\\mathcal K_{\\mathcal A}:=\\{T\\subseteq S| \\sum_{s\\in T} a_s <1\\}$, and one has $\\bar\\M_{g,\\mathcal A}=\\bar\\M_{g,\\mathcal K_{\\mathcal A}}.$\n\nMuch recent work has focused on the Chow rings of $\\bar\\M_{g,n}$ \\cite{BDL2,CL2,DLPV,ELarson}. We study the Chow ring of these simplicially stable spaces in genus $0$. Notably, these spaces give all stable (in the sense of Smyth \\cite{Smyth}) smooth modular compactifications of $\\M_{0,n}$ (combine \\cite[Theorem 7.8]{MSvAX}] with \\cite[Lemma 4.15]{BB}). Previously, Keel computed the Chow ring of $\\bar\\M_{0,n}$ in \\cite{Keel}. This was generalized by Kannan, Karp, and Li in \\cite{KKL}, where the Chow rings of so-called ``heavy-light'' Hassett spaces were calculated in genus $0$ using tropical methods.\n\nBefore stating the main theorem, we recall Keel's presentation of $\\CH(\\bar\\M_{0,n})$ from \\cite{Keel}. Given a subset $I\\subseteq [n]$ so that $2\\leq \\# I\\leq n-2$, let $D_I$ be the closure of the set of curves in $\\bar\\M_{0,n}$ with two components, with markings from $I$ on one component and markings from the complement $I^c$ on the other.\n\n\\begin{definition}\\label{def}\n The simplicially stable space $\\bar\\M_{g,\\mathcal K}$ parameterizes nodal genus $g$ curves with smooth marked points indexed by $S$, $(C,\\{p_s|s\\in S\\})$, with finitely many automorphisms, so that \n \\begin{itemize}\n \\item for each $x\\in C$, the set $\\{s \\in S| p_s=x\\}$ is in $\\mathcal K$ and\n \\item for each rational tail $Z\\subseteq C$, the set $\\{s\\in S| p_s\\in Z\\}$ is not in $\\mathcal K$. \n \\end{itemize}\n\\end{definition}", "full_context": "The simplicially stable spaces were introduced in by Blankers and Bozlee in \\cite{BB}. They are compactifications of $\\M_{g,S},$ the moduli space of smooth genus $g$ curves with markings in a finite set $S$, which are indexed by (abstract) simplicial complexes $\\mathcal K$ and are denoted $\\bar\\M_{g,\\mathcal K}$. The simplicial complex controls whether the marked points may ``collide'' in $\\bar\\M_{g,\\mathcal K}$ (see Definition~\\ref{def} for a precise definition). For example, given distinct $s,t\\in S$, if $\\{s,t\\}\\in \\mathcal K$ and $(C,\\{p_s: s\\in S\\})\\in \\bar\\M_{g,\\mathcal K}$, then we may have $p_s=p_t$, in contrast with the usual compactification $\\bar\\M_{g,S}$.\n\nSimplicially stable spaces generalize the earlier moduli spaces of weighted stable curves first studied by Hassett in \\cite{Hassett}. Given weight data $\\mathcal A=\\{a_s:s\\in S\\}$, one gets a simplicial complex $\\mathcal K_{\\mathcal A}:=\\{T\\subseteq S| \\sum_{s\\in T} a_s <1\\}$, and one has $\\bar\\M_{g,\\mathcal A}=\\bar\\M_{g,\\mathcal K_{\\mathcal A}}.$\n\nMuch recent work has focused on the Chow rings of $\\bar\\M_{g,n}$ \\cite{BDL2,CL2,DLPV,ELarson}. We study the Chow ring of these simplicially stable spaces in genus $0$. Notably, these spaces give all stable (in the sense of Smyth \\cite{Smyth}) smooth modular compactifications of $\\M_{0,n}$ (combine \\cite[Theorem 7.8]{MSvAX}] with \\cite[Lemma 4.15]{BB}). Previously, Keel computed the Chow ring of $\\bar\\M_{0,n}$ in \\cite{Keel}. This was generalized by Kannan, Karp, and Li in \\cite{KKL}, where the Chow rings of so-called ``heavy-light'' Hassett spaces were calculated in genus $0$ using tropical methods.\n\nBefore stating the main theorem, we recall Keel's presentation of $\\CH(\\bar\\M_{0,n})$ from \\cite{Keel}. Given a subset $I\\subseteq [n]$ so that $2\\leq \\# I\\leq n-2$, let $D_I$ be the closure of the set of curves in $\\bar\\M_{0,n}$ with two components, with markings from $I$ on one component and markings from the complement $I^c$ on the other.\n\n\\begin{definition}\\label{def}\n The simplicially stable space $\\bar\\M_{g,\\mathcal K}$ parameterizes nodal genus $g$ curves with smooth marked points indexed by $S$, $(C,\\{p_s|s\\in S\\})$, with finitely many automorphisms, so that \n \\begin{itemize}\n \\item for each $x\\in C$, the set $\\{s \\in S| p_s=x\\}$ is in $\\mathcal K$ and\n \\item for each rational tail $Z\\subseteq C$, the set $\\{s\\in S| p_s\\in Z\\}$ is not in $\\mathcal K$. \n \\end{itemize}\n\\end{definition}\n\nIn \\cite{Me1}, the author gave new proofs of presentations of the (integer coefficient) Chow rings $\\CH(\\bar\\M_{1,n})$ for $n\\leq 4$ using higher Chow groups and the motivic Kunneth property. The same techniques are used in this paper, giving a much shorter display of their usefulness.\n\nWe follow the notation of \\cite{Me1}: given a $\\mathcal K$-stable graph $\\Gamma$, always a capital Greek letter, the lower case $\\gamma$ denotes the class $[\\bar\\M_\\Gamma]$ inside the Chow ring of any space containing it, such as $\\bar\\M_\\Gamma$, $\\partial\\bar\\M_{0,\\mathcal K}$, and $\\bar\\M_{0,\\mathcal K}$. With this notation, we can write the relations on $\\bar\\M_{0,S}$ as\n\\[\\sum_{\\substack{\\text{codim $1$ }\\Delta\\\\ ij\\Delta k\\ell}} \\delta=\\sum_{\\substack{\\text{codim $1$ }\\Delta\\\\ ik\\Delta j\\ell}} \\delta.\\]\nRecall the morphsim\n\\[\\pi: \\bar\\M_{0,S}\\to \\bar\\M_{0,\\mathcal K}\\]\nfrom Theorem~\\ref{morphism}. For a geometric point $C\\in \\bar\\M_{0,S}$, the stable graph of $C$ separates $\\{i,j\\}$ from $\\{k,\\ell\\}$ if and only if the $\\mathcal K$-stable graph of $\\pi(C)$ separates $\\{i,j\\}$ from $\\{k,\\ell\\}$. Thus, the pushforward of the WDVV relations to $\\CH(\\bar\\M_{0,\\mathcal K})$ can also be written as \n\\[\\sum_{\\substack{\\text{codim $1$ }\\Delta\\\\ ij\\Delta k\\ell}} \\delta=\\sum_{\\substack{\\text{codim $1$ }\\Delta\\\\ ik\\Delta j\\ell}} \\delta.\\]\nFrom this, we can rephrase the main theorem in the following way.\n\\begin{thm}[Restatement of Theorem~\\ref{main}]\\label{main2}\n The Chow ring of $\\bar\\M_{0,\\mathcal K}$ is generated by the classes $\\delta$ for codimension $1$ $\\mathcal K$-stable graphs $\\Delta$, subject to only the relations\n \\[\\delta\\cdot \\delta'=0\\text{ if } \\bar\\M_\\Delta\\cap \\bar\\M_{\\Delta'}=\\emptyset\\]\n \\[\\sum_{\\substack{\\text{codim $1$ }\\Delta\\\\ ij\\Delta k\\ell}} \\delta=\\sum_{\\substack{\\text{codim $1$ }\\Delta\\\\ ik\\Delta j\\ell}} \\delta\\text{ for distinct } i,j,k,\\ell\\in S.\\]\n\\end{thm}\nWe will write $\\Delta\\wedge \\Delta'=\\emptyset$ as shorthand for $\\bar\\M_{\\Delta}\\cap \\bar\\M_{\\Delta'}=\\emptyset$. By Lemma~\\ref{leq} and Proposition~\\ref{strata}, we have $\\Delta\\wedge \\Delta'=\\emptyset$ if and only if there is no $\\mathcal K$-stable graph $\\Gamma$ with $\\Gamma\\leq \\Delta$ and $\\Gamma\\leq \\Delta'$.\n\n\\begin{definition}\n Define the ring $R_{\\mathcal K}$ be the free ring generated by symbols $\\tilde{\\delta}$ for genus $0$ $\\mathcal K$-stable graphs $\\Delta$ of codimension $1$, modulo the relations\n \\[\\tilde{\\delta}\\cdot \\tilde{\\delta'}=0 \\text{ if } \\Delta \\wedge \\Delta'=\\emptyset\\]\n and \n \\[\\sum_{\\substack{\\text{codim $1$ }\\Delta\\\\ ij\\Delta k\\ell}} \\tilde{\\delta}=\\sum_{\\substack{\\text{codim $1$ }\\Delta\\\\ ik\\Delta j\\ell}} \\tilde{\\delta} \\text{ for distinct } i,j,k,\\ell\\in S.\\]\n By the above, we have a ring homomorphism \n\\[h_{\\mathcal K}: R_{\\mathcal K}\\to \\CH(\\bar\\M_{0,\\mathcal K})\\]\n\\[\\tilde{\\delta}\\mapsto \\delta.\\]\nWe see that Theorem~\\ref{main2} is equivalent to saying that $h_{\\mathcal K}$ is an isomorphism.\n\\end{definition}\n\nWe start with localization exact sequence for $\\partial\\bar\\M_{0,\\mathcal K}\\subseteq \\bar\\M_{0,\\mathcal K}$:\n\\[\\CH(\\M_{0,S},1)\\xrightarrow{\\partial_1} \\CH(\\partial\\bar\\M_{0,\\mathcal K})\\to \\CH(\\bar\\M_{0,\\mathcal K})\\to \\CH(\\M_{0,E})\\to 0.\\]\nWe first compute the image of $\\partial_1$. \n\\begin{prop}\\label{higher}\n The image of $\\partial_1$ is generated by the WDVV relations on $\\bar\\M_{0,\\mathcal K}$:\n\\[\\sum_{\\substack{\\text{codim $1$ }\\Delta\\\\ ij\\Delta k\\ell}} \\delta=\\sum_{\\substack{\\text{codim $1$ }\\Delta\\\\ ik\\Delta j\\ell}} \\delta.\\]\n\\end{prop}\n\\begin{proof}\n Recall the morphism $\\pi:\\bar\\M_{0,S}\\to \\bar\\M_{0,\\mathcal K}$ from Theorem~\\ref{morphism}. Note that it sends $\\partial\\bar\\M_{0,S}$ to $\\partial\\bar\\M_{0,\\mathcal K}$. By the commutative diagram of localization exact sequences\n \\begin{center}\n \\begin{tikzcd}\n \\CH(\\M_{0,S},1)\\arrow[r,\"\\partial_1\"]\\arrow[d,\"\\operatorname{id}\"] & \\CH(\\partial\\bar\\M_{0,S})\\arrow[d,\"\\pi_*\"] \\\\\n \\CH(\\M_{0,S},1)\\arrow[r,\"\\partial_1\"] & \\CH(\\partial\\bar\\M_{0,\\mathcal K}),\n \\end{tikzcd}\n \\end{center}\n we see that the image of the bottom $\\partial_1$ is equal the pushforward along $\\pi_*$ of the image of the top $\\partial_1$. By \\cite[Theorem 7.8]{Me1} or \\cite[Proposition 2.36]{BS2}, this image is generated by the WDVV relations on $\\bar\\M_{0,S}$, so the result follows.\n\\end{proof}\n\nThe notation here means that the $R_\\mathcal K$-module $\\CH(\\bar\\M_{\\Delta})$ is generated by $\\delta=[\\bar\\M_{\\Delta}]$ subject to the relations $\\tilde \\gamma\\cdot \\delta=0$ for all codimension $1$ $\\mathcal K$-stable graphs $\\Gamma$ with $\\Delta\\wedge \\Gamma=\\emptyset$.\n\\begin{proof}\nWe will prove this by comparing generators and relations for both sides. The generator $\\tilde{\\delta}$ of $R_{\\mathcal K}$ can be removed from the generating set of $R_{\\mathcal K}$ using the relations by choosing a linear relation involving $\\tilde{\\delta}$, solving for $\\tilde{\\delta}$, and substituting it into all other relations involving $\\tilde{\\delta}$. This gives a different presentation for $R_{\\mathcal K}$. The induced presentation of $R_{\\mathcal K}/(\\tilde{\\gamma}: \\Gamma\\wedge \\Delta=\\emptyset)$ has generators $\\tilde{\\gamma}$ for codimension $1$ $\\mathcal K$-stable graphs $\\Gamma$ with $\\bar\\M_\\Gamma\\cap \\bar\\M_\\Delta$ of codimension $2$, and the relations are \n\\begin{align*}\n \\tilde{\\gamma} \\cdot \\tilde{\\lambda}&=0 \\text{ if }\\Gamma\\wedge \\Lambda=\\emptyset\\\\\n \\sum_{\\substack{\\text{codim $1$ }\\Gamma\\\\ ij\\Gamma k\\ell}} \\tilde{\\gamma}&=\\sum_{\\substack{\\text{codim $1$ }\\Gamma\\\\ ik\\Gamma j\\ell}} \\tilde{\\gamma} \\text{ if neither \n }ij\\Delta k\\ell\\text{ nor }ik\\Delta j\\ell,\\\\\n \\sum_{\\substack{\\text{codim $1$ }\\Gamma\\\\ ij\\Gamma k\\ell}} \\tilde{\\gamma}&=\\sum_{\\substack{\\text{codim $1$ }\\Gamma\\\\ ab\\Gamma cd}} \\tilde{\\gamma} \\text{ if }ij\\Delta k\\ell \\text{ and }ab\\Delta cd.\n\\end{align*}\n\nBy Proposition~\\ref{higher}, we have an exact sequence\n \\[0\\to\\frac{\\CH(\\partial\\bar\\M_{0,\\mathcal K}\n )}{\\text{WDVV}}\\to \\CH(\\bar\\M_{0,\\mathcal K})\\to \\CH(\\M_{0,S})\\to 0.\\]\n This gives an $R_{\\mathcal K}-$module presentation for $\\CH(\\bar\\M_{0,\\mathcal K})$ as the free $R_{\\mathcal K}$-module on $1$ and $\\delta$ for codimension $1$ graphs $\\Delta$, modulo the relations\n \\begin{align*}\n \\tilde{\\gamma}\\delta, &\\text{ for codimension $1$ graphs $\\Gamma,\\Delta$ such that }\\Delta\\wedge \\Gamma=\\emptyset\\\\\n \\tilde{\\delta'}\\delta-\\tilde{\\delta}\\delta', &\\text{ for codimension $1$ graphs $\\Delta,\\Delta'$}\\\\\n \\sum_{\\substack{\\text{codim $1$ }\\Delta\\\\ ij\\Delta k\\ell}} \\delta-\\sum_{\\substack{\\text{codim $1$ }\\Delta\\\\ ik\\Delta j\\ell}} \\delta&\\text{ for distinct }i,j,k,\\ell\\in S\\\\\n \\tilde{\\delta}\\cdot 1-\\delta&\\text{ for codimension $1$ graphs $\\Delta$.}\n \\end{align*}\n We see that we only need to include $1$ as a generator. Writing all relations in terms of $1$, they all vanish. Therefore, $\\CH(\\bar\\M_{0,\\mathcal K})$ is a free $R_{\\mathcal K}$-module on the generator $1$. Therefore $h_{\\mathcal K}$ is an isomorphism.\n\\end{proof}\n\n\\begin{thm}\\label{main3}\n The Chow ring of $\\bar\\M_{0,\\mathcal K}$ is generated by the classes $[D_I]$ and $[E_{st}]$ modulo only the relations\n \\begin{align*}\n [D_I]=[D_{I^c}]&,\\\\\n [D_I][D_J]=0 &\\text{ if }I\\not\\subseteq J,J^c\\text{ and }J,J^c\\not\\subseteq I, \\\\\n [D_I][E_{st}]=0 & \\text{ if } s\\in I, t\\in I^c\\text{ or } t\\in I, s\\in I^c,\\\\\n [E_{st}][E_{tu}]=0 & \\text{ if } \\{s,t,u\\}\\notin \\mathcal K, \\text{ and}\\\\\n \\text{WDVV}(i,j,k,\\ell)& \\text{ for }i,j,k,\\ell\\in S\\text{ distinct,}\n \\end{align*}\n where the WDVV relation associated to $i,j,k,\\ell$ is \n \\[[E_{ij}]+[E_{k\\ell}]+\\sum_{\\substack{i,j\\in I\\\\ k,\\ell \\not\\in I}} [D_I]=[E_{ik}]+[E_{j\\ell}]+\\sum_{\\substack{i,k\\in I\\\\ j,\\ell \\not\\in I}} [D_I].\\]\n\\end{thm}", "post_theorem_intro_text_len": 3309, "post_theorem_intro_text": "Our main theorem is the following.\n\\begin{thm}\\label{main}\n The Chow ring of $\\bar\\M_{0,\\mathcal K}$ is generated by the classes of boundary divisors $D$, and the only relations are \n \\begin{align*}\n &[D]\\cdot [D']=0\\text{ if }D\\cap D'=\\emptyset \\text{ and}\\\\\n &\\text{the pushforward of the WDVV relations from } \\bar\\M_{0,S}.\n \\end{align*}\n\\end{thm}\nSee Theorem~\\ref{main2} and Theorem~\\ref{main3} for more explicit versions of this theorem. By letting $\\mathcal K$ be the discrete simplicial complex on $[n]$, this recovers Keel's presentation. Our proof does not generalize Keel's, so we obtain a new, short proof of Keel's result based just on the general theory of the stratification of the boundary, a Chow-Kunneth property, and some use of higher Chow groups. Moreover, Keel's original approach cannot work in the setting of an arbitrary $\\bar\\M_{0,\\mathcal K}$: Keel's proof works by describing $\\bar\\M_{0,n}$ as an iterated blow up of $(\\mathbb{P}^1)^n$, and the spaces $\\bar\\M_{0,\\mathcal K}$ are not necessarily projective \\cite[Example 11.1]{MSvAX}.\n\nIn \\cite{Me1}, the author gave new proofs of presentations of the (integer coefficient) Chow rings $\\CH(\\bar\\M_{1,n})$ for $n\\leq 4$ using higher Chow groups and the motivic Kunneth property. The same techniques are used in this paper, giving a much shorter display of their usefulness.\n\nThere is another description of $\\CH(\\bar\\M_{0,n})$ given by Kontsevich and Manin in \\cite{KM94,KM96}. This is description is additive; it describes all linear relations between the classes of the strata. The main theorem in the paper \\cite{BS2}, which greatly inspired the author's work in \\cite{Me1} and hence in the present paper, was an analogue/generalization of this description to $\\CH(\\mathfrak M_{0,n}),$ where $\\mathfrak M_{0,n}$ is the stack of prestable curves. Such a description of $\\CH(\\bar\\M_{0,\\mathcal K})$ should be true and provable following the proof given in \\cite{BS2}.\n\nThe paper is organized as follows: in section $2$, we give background on simplicially stable spaces and the techniques developed in \\cite{Me1}; in section $3$, we discuss the stratification of $\\bar\\M_{g,\\mathcal K}$ by dual graph; and in section $4$, the main theorem is proven. \n\n\\subsection*{Acknowledgments} \nThe author thanks Sebastian Bozlee for help with understanding simplicially stable spaces. The author also thanks Siddarth Kannan for a helpful discussion about previous results.\n\nThis material is based upon work supported by the National Science Foundation under Grant No. DMS-2231565.\n\n\\subsection*{Notation} \n\\begin{itemize}\n \\item We work over a field of arbitrary characteristic. All varieties are defined over this field.\n \\item For a variety $X$, $\\CH(X)$ denotes the total Chow group, i.e. it is the direct sum of cycles of all dimensions modulo rational equivalence. We will not need to refer to the grading. Similarly, $\\CH(X,j)$ is the total $j$-th higher Chow group.\n \\item The set $\\{1,2,\\dots,n\\}$ is denoted $[n]$.\n \\item For a ring $R$, we use the notation $R\\langle a_1,\\dots,a_n\\rangle$ to denote the free $R$ module on $a_1,\\dots,a_n$. Given an $R$-module with elements $f_1,\\dots,f_r$, $\\langle f_1,\\dots,f_r\\rangle$ denotes the $R$-submodule of $M$ generated by $f_1,\\dots,f_r$. \n\\end{itemize}", "sketch": "The post-theorem introduction does not give step-by-step proof details, but it indicates that the proof is \"a new, short proof\" (recovering Keel's presentation when $\\mathcal K$ is discrete) and that it is \"based just on the general theory of the stratification of the boundary, a Chow-Kunneth property, and some use of higher Chow groups.\" It also states that \"The same techniques are used in this paper\" as in \\cite{Me1} (\"using higher Chow groups and the motivic Kunneth property\"), and that \"in section $4$, the main theorem is proven.\"", "expanded_sketch": "The post-theorem introduction does not give step-by-step proof details, but it indicates that the proof is \"a new, short proof\" (recovering Keel's presentation when $\\mathcal K$ is discrete) and that it is \"based just on the general theory of the stratification of the boundary, a Chow-Kunneth property, and some use of higher Chow groups.\" It also states that \"The same techniques are used in this paper\" as in \\cite{Me1} (\"using higher Chow groups and the motivic Kunneth property\"), and that later the main theorem is proven.", "expanded_theorem": "The Chow ring of $\\bar\\M_{0,n}$ is generated by the classes $[D_I]$ modulo only the relations\n \\begin{align*}\n [D_I]=[D_{I^c}]&,\\\\\n [D_I][D_J]=0 &\\text{ if }I\\not\\subseteq J,J^c\\text{ and }J,J^c\\not\\subseteq I, \\text{ and }\\\\\n \\sum_{\\substack{i,j\\in I\\\\ k,\\ell \\not\\in I}} [D_I]=\\sum_{\\substack{i,k\\in I\\\\ j,\\ell \\not\\in I}} [D_I]& \\text{ for }i,j,k,\\ell\\in [n]\\text{ distinct (WDVV Relations).}\n \\end{align*},", "theorem_type": ["Classification or Bijection", "Universal"], "mcq": {"question": "Let \\(\\bar M_{0,n}\\) be the moduli space of stable genus-\\(0\\) curves with \\(n\\) marked points. For each subset \\(I\\subseteq [n]\\) with \\(2\\le |I|\\le n-2\\), let \\(D_I\\) denote the boundary divisor whose generic point parametrizes a 2-component curve with the markings in \\(I\\) on one component and the markings in \\(I^c\\) on the other. Which presentation describes the Chow ring \\(\\mathrm{CH}(\\bar M_{0,n})\\)?", "correct_choice": {"label": "A", "text": "\\(\\mathrm{CH}(\\bar M_{0,n})\\) is generated by the divisor classes \\([D_I]\\) and is the quotient by only the relations\n\\[\n[D_I]=[D_{I^c}],\n\\]\n\\[\n[D_I][D_J]=0 \\quad\\text{if } I\\not\\subseteq J,\\ I\\not\\subseteq J^c,\\ J\\not\\subseteq I,\\text{ and } J\\not\\subseteq I^c,\n\\]\nand, for every distinct \\(i,j,k,\\ell\\in [n]\\), the WDVV relations\n\\[\n\\sum_{\\substack{i,j\\in I\\\\ k,\\ell\\notin I}} [D_I]\n=\n\\sum_{\\substack{i,k\\in I\\\\ j,\\ell\\notin I}} [D_I].\n\\]"}, "choices": [{"label": "B", "text": "\\(\\mathrm{CH}(\\bar M_{0,n})\\) is generated by the divisor classes \\([D_I]\\) and is the quotient by only the relations\n\\[\n[D_I]=[D_{I^c}],\n\\]\n\\[\n[D_I][D_J]=0 \\quad\\text{if } I\\cap J\\neq \\varnothing,\\ I\\cap J^c\\neq \\varnothing,\\ J\\cap I^c\\neq \\varnothing,\\text{ and } J^c\\cap I^c\\neq \\varnothing,\n\\]\nand, for every distinct \\(i,j,k,\\ell\\in [n]\\), the WDVV relations\n\\[\n\\sum_{\\substack{i,j\\in I\\\\ k,\\ell\\notin I}} [D_I]\n=\n\\sum_{\\substack{i,k\\in I\\\\ j,\\ell\\notin I}} [D_I].\n\\]"}, {"label": "C", "text": "\\(\\mathrm{CH}(\\bar M_{0,n})\\) is generated by the divisor classes \\([D_I]\\) and satisfies the relations\n\\[\n[D_I]=[D_{I^c}],\n\\]\n\\[\n[D_I][D_J]=0 \\quad\\text{if } I\\not\\subseteq J,\\ I\\not\\subseteq J^c,\\ J\\not\\subseteq I,\\text{ and } J\\not\\subseteq I^c,\n\\]\nand, for every distinct \\(i,j,k,\\ell\\in [n]\\), the WDVV relations\n\\[\n\\sum_{\\substack{i,j\\in I\\\\ k,\\ell\\notin I}} [D_I]\n=\n\\sum_{\\substack{i,k\\in I\\\\ j,\\ell\\notin I}} [D_I].\n\\]"}, {"label": "D", "text": "\\(\\mathrm{CH}(\\bar M_{0,n})\\) is generated by the divisor classes \\([D_I]\\) and is the quotient by only the relations\n\\[\n[D_I]=[D_{I^c}],\n\\]\n\\[\n[D_I][D_J]=0 \\quad\\text{if } I\\not\\subseteq J,\\ I\\not\\subseteq J^c,\\ J\\not\\subseteq I,\\text{ and } J\\not\\subseteq I^c,\n\\]\nand, for every distinct \\(i,j,k,\\ell\\in [n]\\), the relations\n\\[\n\\sum_{\\substack{i,j\\in I\\\\ k,\\ell\\notin I}} [D_I]\n+\n\\sum_{\\substack{i,\\ell\\in I\\\\ j,k\\notin I}} [D_I]\n=\n\\sum_{\\substack{i,k\\in I\\\\ j,\\ell\\notin I}} [D_I].\n\\]"}, {"label": "E", "text": "\\(\\mathrm{CH}(\\bar M_{0,n})\\) is generated by the divisor classes \\([D_I]\\) with \\(2\\le |I|\\le n-2\\) and is the quotient by only the relations\n\\[\n[D_I]=[D_{I^c}],\n\\]\n\\[\n[D_I][D_J]=0 \\quad\\text{if } I\\not\\subseteq J,\\ I\\not\\subseteq J^c,\\ J\\not\\subseteq I,\\text{ and } J\\not\\subseteq I^c,\n\\]\nand, for every distinct \\(i,j,k,\\ell\\in [n]\\), the WDVV relations\n\\[\n\\sum_{\\substack{i,j\\in I\\\\ k,\\ell\\notin I}} [D_I]\n=\n\\sum_{\\substack{i,k\\in I\\\\ j,\\ell\\notin I}} [D_I],\n\\]\ntogether with the linear relations\n\\[\n\\sum_{i\\in I,\\, j\\notin I} [D_I]=0 \\qquad \\text{for every distinct } i,j\\in [n].\n\\]"}], "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": "compatibility criterion for boundary divisors", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the claim that these are the only relations / quotient by only these relations", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "exact WDVV linear relation among boundary classes", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "minimality of the presentation by only symmetry, vanishing, and WDVV relations", "template_used": "wildcard"}]}} +{"id": "2601.16972v1", "paper_link": "http://arxiv.org/abs/2601.16972v1", "theorems_cnt": 1, "theorem": {"env_name": "ErdPom", "content": "\\label{main}\nWe have the lower bound $$\\max_m f(n, m) - f(n, n) > \\frac{0.36n \\log n}{\\log \\log n}$$ for all large enough $n \\in \\mathbb{N}$. In particular, if $n$ is sufficiently large, then an interval of length $\\frac{0.36n \\log n}{\\log \\log n}$ exists that does not contain distinct multiples of $1, 2, \\ldots, n$.", "start_pos": 1319, "end_pos": 1666, "label": "main"}, "ref_dict": {"main": "\\begin{ErdPom} \\label{main}\nWe have the lower bound $$\\max_m f(n, m) - f(n, n) > \\frac{0.36n \\log n}{\\log \\log n}$$ for all large enough $n \\in \\mathbb{N}$. In particular, if $n$ is sufficiently large, then an interval of length $\\frac{0.36n \\log n}{\\log \\log n}$ exists that does not contain distinct multiples of $1, 2, \\ldots, n$. \n\\end{ErdPom}"}, "pre_theorem_intro_text_len": 541, "pre_theorem_intro_text": "Define $f(n, m)$ to be the least integer so that the interval $(m, m + f(n, m)]$ contains $n$ distinct integers $a_1, a_2, \\ldots, a_n$ such that $i$ divides $a_i$ for all $i$. Erd\\H os and Pomerance conjectured in \\mbox{\\cite{3}} that $\\max_m f(n, m) - f(n, n)$ goes to infinity with $n$. In \\mbox{\\cite{2}} Erd\\H os even offered $1000$ rupees for a solution, and it is now listed as (part of) problem \\#711 at Bloom's website \\cite{1}. In this short note we will settle their conjecture in the affirmative by proving the following theorem.", "context": "Define $f(n, m)$ to be the least integer so that the interval $(m, m + f(n, m)]$ contains $n$ distinct integers $a_1, a_2, \\ldots, a_n$ such that $i$ divides $a_i$ for all $i$. Erd\\H os and Pomerance conjectured in \\mbox{\\cite{3}} that $\\max_m f(n, m) - f(n, n)$ goes to infinity with $n$. In \\mbox{\\cite{2}} Erd\\H os even offered $1000$ rupees for a solution, and it is now listed as (part of) problem \\#711 at Bloom's website \\cite{1}. In this short note we will settle their conjecture in the affirmative by proving the following theorem.", "full_context": "Define $f(n, m)$ to be the least integer so that the interval $(m, m + f(n, m)]$ contains $n$ distinct integers $a_1, a_2, \\ldots, a_n$ such that $i$ divides $a_i$ for all $i$. Erd\\H os and Pomerance conjectured in \\mbox{\\cite{3}} that $\\max_m f(n, m) - f(n, n)$ goes to infinity with $n$. In \\mbox{\\cite{2}} Erd\\H os even offered $1000$ rupees for a solution, and it is now listed as (part of) problem \\#711 at Bloom's website \\cite{1}. In this short note we will settle their conjecture in the affirmative by proving the following theorem.\n\n\\Large\n \\begin{center}\nOn the length of an interval that contains distinct multiples of the first $n$ positive integers \\\\\n\n\\centerline{\\bf Abstract}\nConfirming a conjecture by Erd\\H os and Pomerance, we prove that there exist intervals of length $\\frac{cn\\log n}{\\log \\log n}$ that do not contain distinct multiples of $1, 2, \\ldots, n$.\n\n\\section{Introduction}\nDefine $f(n, m)$ to be the least integer so that the interval $(m, m + f(n, m)]$ contains $n$ distinct integers $a_1, a_2, \\ldots, a_n$ such that $i$ divides $a_i$ for all $i$. Erd\\H os and Pomerance conjectured in \\mbox{\\cite{3}} that $\\max_m f(n, m) - f(n, n)$ goes to infinity with $n$. In \\mbox{\\cite{2}} Erd\\H os even offered $1000$ rupees for a solution, and it is now listed as (part of) problem \\#711 at Bloom's website \\cite{1}. In this short note we will settle their conjecture in the affirmative by proving the following theorem.\n\nWe note that the second sentence of Theorem \\ref{main} immediately follows from the first, as we trivially have $f(n, n) \\ge 0$.\n\n\\begin{proof}\nReplacing both $n$ and $m$ in the definition of $f(n, m)$ by $kn$, we need to show that for every $1 \\le i \\le kn$ there is a multiple $a_i$ of $i$ with $a_i \\in (kn, k^2n + f(n, k^2n)]$, where all $a_i$ are distinct. Now, for every $i \\in (n, kn]$ we simply choose $a_i = ki \\in (kn, k^2n]$, which is certainly divisible by $i$, while all $a_i$ are distinct as $k$ is non-zero. On the other hand, by the definition of $f(n, m)$ with $m = k^2n$, it is for all $i \\in [1, n]$ possible to choose distinct multiples $a_i \\in (k^2n, k^2n + f(n, k^2n)]$. By combining the disjoint intervals we conclude that all $a_i$ are indeed contained in $(kn, k^2n + f(n, k^2n)]$.\n\\end{proof}\n\n\\begin{ErdPombounds} \\label{erdpombounds}\nFor all sufficiently large $n \\in \\mathbb{N}$ we have the following inequalities:\n\n\\begin{proof}[Proof of Theorem \\ref{main}]\nWith $n$ a sufficiently large integer, define $k := \\left \\lceil 0.6\\sqrt{\\frac{\\log n}{\\log \\log n}} \\hspace{3pt} \\right \\rceil$ and choose $\\epsilon := \\frac{1}{100}$. Using the inequality $\\frac{2}{\\sqrt{e}} > 1.21$ and the fact that $n$ is sufficiently large, the bounds from Lemma \\ref{erdpombounds} then imply, in particular, that\n\nCombining Equations (\\ref{geninq}), (\\ref{eqlower}), and (\\ref{equpper}) now finishes the proof. Indeed,\n\\begin{align*}\n\\max_m f(n, m) &\\ge f(n, k^2n) \\\\\n&\\ge kn + f(kn, kn) - k^2n \\\\\n&> (2+\\epsilon)k^2n - k^2n \\\\\n&= \\epsilon k^2n + k^2n \\\\\n&> f(n, n) + \\frac{0.36n \\log n}{\\log \\log n}. \\qedhere\n\\end{align*}\n\\end{proof}\n\n\\begin{ErdPom} \\label{main}\nWe have the lower bound $$\\max_m f(n, m) - f(n, n) > \\frac{0.36n \\log n}{\\log \\log n}$$ for all large enough $n \\in \\mathbb{N}$. In particular, if $n$ is sufficiently large, then an interval of length $\\frac{0.36n \\log n}{\\log \\log n}$ exists that does not contain distinct multiples of $1, 2, \\ldots, n$. \n\\end{ErdPom}", "post_theorem_intro_text_len": 128, "post_theorem_intro_text": "We note that the second sentence of Theorem \\ref{main} immediately follows from the first, as we trivially have $f(n, n) \\ge 0$.", "sketch": "The post-theorem text only notes that “the second sentence of Theorem \\ref{main} immediately follows from the first, as we trivially have $f(n, n) \\ge 0$.”", "expanded_sketch": "The post-theorem text only notes that “the second sentence of the main theorem immediately follows from the first, as we trivially have $f(n, n) \\ge 0$.”", "expanded_theorem": "\\label{main}\nWe have the lower bound $$\\max_m f(n, m) - f(n, n) > \\frac{0.36n \\log n}{\\log \\log n}$$ for all large enough $n \\in \\mathbb{N}$. In particular, if $n$ is sufficiently large, then an interval of length $\\frac{0.36n \\log n}{\\log \\log n}$ exists that does not contain distinct multiples of $1, 2, \\ldots, n$.", "theorem_type": ["Inequality or Bound", "Existential–Universal"], "mcq": {"question": "For integers $n\\ge 1$ and $m$, define $f(n,m)$ to be the least integer $L$ such that the interval $(m,m+L]$ contains $n$ distinct integers $a_1,a_2,\\ldots,a_n$ with $i\\mid a_i$ for every $i=1,2,\\ldots,n$. Under this definition, which quantitative estimate holds for all sufficiently large $n\\in\\mathbb N$?", "correct_choice": {"label": "A", "text": "For all sufficiently large $n\\in\\mathbb N$, one has\n$$\\max_m f(n,m)-f(n,n)>\\frac{0.36\\,n\\log n}{\\log\\log n}.$$ \nIn particular, for every sufficiently large $n$, there exists an interval of length $\\frac{0.36\\,n\\log n}{\\log\\log n}$ that does not contain $n$ distinct integers $a_1,\\ldots,a_n$ with $i\\mid a_i$ for all $i=1,\\ldots,n$; equivalently, it does not contain distinct multiples of $1,2,\\ldots,n$."}, "choices": [{"label": "B", "text": "For all sufficiently large $n\\in\\mathbb N$, one has\n$$\\max_m f(n,m)-f(n,n)\\ge \\frac{0.36\\,n\\log n}{\\log\\log n}.$$ \nIn particular, for every sufficiently large $n$, every interval of length $\\frac{0.36\\,n\\log n}{\\log\\log n}$ fails to contain $n$ distinct integers $a_1,\\ldots,a_n$ with $i\\mid a_i$ for all $i=1,\\ldots,n$."}, {"label": "C", "text": "For all sufficiently large $n\\in\\mathbb N$, there exists an interval of length $\\frac{0.36\\,n\\log n}{\\log\\log n}$ that does not contain $n$ distinct integers $a_1,\\ldots,a_n$ with $i\\mid a_i$ for all $i=1,\\ldots,n$; equivalently, it does not contain distinct multiples of $1,2,\\ldots,n$."}, {"label": "D", "text": "For all sufficiently large $n\\in\\mathbb N$, one has\n$$\\max_m f(n,m)-f(n,n)>0.36\\,n\\log n.$$ \nIn particular, for every sufficiently large $n$, there exists an interval of length $0.36\\,n\\log n$ that does not contain $n$ distinct integers $a_1,\\ldots,a_n$ with $i\\mid a_i$ for all $i=1,\\ldots,n$; equivalently, it does not contain distinct multiples of $1,2,\\ldots,n$."}, {"label": "E", "text": "For all sufficiently large $n\\in\\mathbb N$, one has\n$$f(n,n)-\\min_m f(n,m)>\\frac{0.36\\,n\\log n}{\\log\\log n}.$$ \nIn particular, for every sufficiently large $n$, there exists an interval of length $\\frac{0.36\\,n\\log n}{\\log\\log n}$ centered at $n$ that does not contain $n$ distinct integers $a_1,\\ldots,a_n$ with $i\\mid a_i$ for all $i=1,\\ldots,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": "existential consequence replaced by universal interval claim", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the explicit lower bound on \\max_m f(n,m)-f(n,n)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "growth scale denominator \\log\\log n removed", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "maximal-gap quantity replaced by unrelated minimum/centered statement", "template_used": "wildcard"}]}} +{"id": "2601.17766v1", "paper_link": "http://arxiv.org/abs/2601.17766v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "[Corollary \\ref{cor:rationalfunction}, Corollary \\ref{cor:functionalequation}, Corollary \\ref{cor:computeepsilon}]\n\\label{thm:mainresults}\nLet $ \\rho_\\lambda $ be an almost everywhere unramified $ \\lambda $-adic representation of a global function field $ K $.\n\\begin{itemize}\n\\item[(1)] There is an algorithm to compute $ \\mathcal{L}(\\rho_\\lambda, T) $ with a running time that is roughly exponential in $ n(\\rho_\\lambda)$.\n\\end{itemize}\nAssume further that $ \\rho_\\lambda $ is self-dual of weight $ w(\\rho_\\lambda) $ and sign $ c(\\rho_\\lambda) \\in \\{\\operatorname{id}, \\operatorname{cc}\\} $.\n\\begin{itemize}\n\\item[(2)] If $ \\epsilon(\\rho_\\lambda) $ is known, then there is an algorithm to compute $ \\mathcal{L}(\\rho_\\lambda, T) $ with a running time that is roughly exponential in $ \\lfloor n(\\rho_\\lambda) / 2\\rfloor $.\n\\item[(3)] Otherwise, there is an algorithm to compute $ \\epsilon(\\rho_\\lambda) $ with a running time that is typically exponential in $ \\lfloor n(\\rho_\\lambda) / 2\\rfloor $, and at worst exponential in $ n(\\rho_\\lambda) $.\n\\end{itemize}", "start_pos": 7437, "end_pos": 8482, "label": "thm:mainresults"}, "ref_dict": {"thm:mainresults": "\\begin{theorem}[Corollary \\ref{cor:rationalfunction}, Corollary \\ref{cor:functionalequation}, Corollary \\ref{cor:computeepsilon}]\n\\label{thm:mainresults}\nLet $ \\rho_\\lambda $ be an almost everywhere unramified $ \\lambda $-adic representation of a global function field $ K $.\n\\begin{itemize}\n\\item[(1)] There is an algorithm to compute $ \\LL(\\rho_\\lambda, T) $ with a running time that is roughly exponential in $ n(\\rho_\\lambda)$.\n\\end{itemize}\nAssume further that $ \\rho_\\lambda $ is self-dual of weight $ w(\\rho_\\lambda) $ and sign $ c(\\rho_\\lambda) \\in \\{\\id, \\cc\\} $.\n\\begin{itemize}\n\\item[(2)] If $ \\epsilon(\\rho_\\lambda) $ is known, then there is an algorithm to compute $ \\LL(\\rho_\\lambda, T) $ with a running time that is roughly exponential in $ \\lfloor n(\\rho_\\lambda) / 2\\rfloor $.\n\\item[(3)] Otherwise, there is an algorithm to compute $ \\epsilon(\\rho_\\lambda) $ with a running time that is typically exponential in $ \\lfloor n(\\rho_\\lambda) / 2\\rfloor $, and at worst exponential in $ n(\\rho_\\lambda) $.\n\\end{itemize}\n\\end{theorem}", "cor:computeepsilon": "\\begin{corollary}\n\\label{cor:computeepsilon}\nLet $ \\rho_\\lambda $ be an almost everywhere unramified $ \\lambda $-adic representation of a global function field $ K = \\F_q(C) $ that is self-dual of weight $ w(\\rho_\\lambda) $ and sign $ c(\\rho_\\lambda) $. Then Algorithm \\ref{alg:computeepsilon} outputs $ \\epsilon(\\rho_\\lambda) \\in F_\\lambda^\\times $ with inputs\n$$ (\\LL_\\bullet(\\rho_\\lambda, T), \\ D(\\rho_\\lambda, T), \\ n(\\rho_\\lambda), \\ 1 / q^{w(\\rho_\\lambda) + 1}, \\ c(\\rho_\\lambda)). $$\n\\end{corollary}", "cor:rationalfunction": "\\begin{corollary}\n\\label{cor:rationalfunction}\nLet $ \\rho_\\lambda $ be an almost everywhere unramified $ \\lambda $-adic representation of a global function field $ K $. Then Algorithm \\ref{alg:rationalfunction} outputs $ N(\\rho_\\lambda, T) \\in 1 + T \\cdot F_\\lambda[T] $ with inputs\n$$ (\\LL_\\bullet(\\rho_\\lambda, T), \\ D(\\rho_\\lambda, T), \\ 1, \\ n(\\rho_\\lambda), \\ n(\\rho_\\lambda)). $$\n\\end{corollary}", "cor:functionalequation": "\\begin{corollary}\n\\label{cor:functionalequation}\nLet $ \\rho_\\lambda $ be an almost everywhere unramified $ \\lambda $-adic representation of a global function field $ K = \\F_q(C) $ that is self-dual of weight $ w(\\rho_\\lambda) $ and sign $ c(\\rho_\\lambda) $. Then Algorithm \\ref{alg:functionalequation} outputs $ N(\\rho_\\lambda, T) \\in 1 + T \\cdot F_\\lambda[T] $ with inputs\n$$ (\\LL_\\bullet(\\rho_\\lambda, T), \\ D(\\rho_\\lambda, T), \\ n(\\rho_\\lambda), \\ \\epsilon(\\rho_\\lambda), \\ 1 / q^{w(\\rho_\\lambda) + 1}, \\ c(\\rho_\\lambda)). $$\n\\end{corollary}"}, "pre_theorem_intro_text_len": 5254, "pre_theorem_intro_text": "Given a global field, there are L-functions $ L(\\rho, s) $ in the complex variable $ s \\in \\mathbb{C} $ associated to suitable continuous representations $ \\rho $ of its absolute Galois group, which are conjecturally meromophic over $ \\mathbb{C} $ and conjecturally satisfy functional equations. These L-functions play the role of half of the modern Langlands philosophy, and there are many crucial algebraic and analytic questions that remain unanswered. For instance, when $ \\rho = \\chi $ is a Dirichlet character, the generalised Riemann hypothesis describes where its $ \\zeta $-function $ \\zeta(s) = L(\\chi, s) $ vanishes, and when $ \\rho = \\rho_{A, \\ell} $ arises from an abelian variety $ A $, the Birch and Swinnerton-Dyer conjecture describes the arithmetic of $ A $ in terms of its L-function $ L(A, s) = L(\\rho_{A, \\ell}, s) $. The importance of L-functions in arithmetic algebraic geometry is so much so that thousands of CPU years have been spent on computing their data for various representations in the massive L-functions and modular forms database (LMFDB) \\cite{LMFDB}, which has proven to be an immensely useful resource for number theorists.\n\nAt a cursory glance, most of the data in the LMFDB are of objects over various number fields. In this setting, their L-functions are transcendental functions of $ s $ that can be written as Dirichlet series, with the aforementioned conjectures still open for most objects. Much of their data were computed from Platt's code for degree one L-functions \\cite{Pla11} based on Booker's algorithm \\cite{Boo06}, from the \\texttt{lcalc} library in C++ based on Rubinstein's algorithm \\cite{Rub05} for Maass forms, and from the work of Booker--Sijsling--Sutherland--Voight--Yasaki \\cite{BSSVY16} for genus two curves. In a much larger generality, T Dokchitser wrote an efficient algorithm to compute the special value of a motivic L-function, assuming a conjectural functional equation that can be numerically verified \\cite{Dok04}. This was originally implemented as the PARI \\cite{PARI} package \\texttt{ComputeL}, but has since been ported to the core libraries of Magma \\cite{BCP97} as the \\texttt{LSeries} function and SageMath \\cite{SageMath} as the \\texttt{dokchitser} function.\n\nIn a stark contrast, the arithmetic of analogous objects over $ K \\coloneqq \\F_q(C) $ is very well-understood \\cite{Laf02}. In this setting, their L-functions are rational functions of $ q^{-s} $, satisfying a known functional equation and even the Riemann hypothesis. Yet, the computational ecosystem over $ K $ is severely limited, with little to no support for explicit computations in the core libraries of most computer algebra systems, including PARI and SageMath. Magma is better in this regard, including multiple constructors for $ K $, explicit class field theory computations with \\texttt{WittRing} and \\texttt{CarlitzModule}, and even an \\texttt{LFunction} for non-constant elliptic curves over $ \\F_q(t) $. Using this interface, Comeau--Lapointe--David--Lal\\'in--Li computed $ L(E, \\chi, s) = L(\\rho_{E, \\ell} \\otimes \\chi, s) $ for a fixed Legendre curve $ E $ over $ K = \\F_q(t) $ twisted by Dirichlet characters $ \\chi $ to investigate their vanishing \\cite{CLDLL22}. In private communication, Maistret--Wiersema also computed $ L(E, \\chi, s) $ for many constant elliptic curves $ E $ by hand to formulate an analogue over $ K $ of the twisted BSD-type formula of V Dokchitser--Evans--Wiersema \\cite{DEW21}. This seems to be the extent to which L-functions of $ K $ have been computed explicitly in the literature, which renders the creation of an analogue of the LMFDB over $ K $ difficult.\n\nThis paper presents an analogue of T Dokchitser's algorithm for $ K $, in the language of \\emph{$ \\lambda $-adic representations $ \\rho_\\lambda $} for some prime $ \\lambda \\nmid q $ of a fixed number field $ F $. These are the natural Galois representations associated to L-functions of motivic origin, including Riemann/Dedekind $ \\zeta $-functions, Dirichlet/Weber/Hecke/Artin L-functions, and Hasse--Weil L-functions of varieties. Then there is a rational function $ \\mathcal{L}(\\rho_\\lambda, T) \\in F_\\lambda(T) $, called the \\emph{formal L-function} of $ \\rho_\\lambda $, such that $ \\mathcal{L}(\\rho_\\lambda, q^{-s}) = L(\\rho_\\lambda, s) $. Under the condition that $ \\rho_\\lambda $ is \\emph{self-dual of weight $ w(\\rho_\\lambda) \\in \\mathbb{N} $ and sign $ c(\\rho_\\lambda) \\in \\{\\operatorname{id}, \\operatorname{cc}\\} $}, where $ \\operatorname{id} $ is the identity and $ \\operatorname{cc} $ is complex conjugation, there is a field element $ \\epsilon(\\rho_\\lambda) \\in F_\\lambda^\\times $ such that $ \\mathcal{L}(\\rho_\\lambda, T) $ satisfies a functional equation\n$$ \\mathcal{L}(\\rho_\\lambda, T) = \\epsilon(\\rho_\\lambda) \\cdot T^{n(\\rho_\\lambda) - d(\\rho_\\lambda)} \\cdot \\mathcal{L}(\\rho_\\lambda, (q^{w(\\rho_\\lambda) + 1}T)^{-1})^{c(\\rho_\\lambda)}, $$\nwhere $ n(\\rho_\\lambda), d(\\rho_\\lambda) \\in \\mathbb{N} $ are the degrees of the numerator and denominator of $ \\mathcal{L}(\\rho_\\lambda, T) $ respectively. If the genus of $ C $ and the dimension of $ \\rho_\\lambda $ are small, then $ n(\\rho_\\lambda) - d(\\rho_\\lambda) $ is essentially the degree of the \\emph{Artin conductor} of $ \\rho_\\lambda $.", "context": "Given a global field, there are L-functions $ L(\\rho, s) $ in the complex variable $ s \\in \\mathbb{C} $ associated to suitable continuous representations $ \\rho $ of its absolute Galois group, which are conjecturally meromophic over $ \\mathbb{C} $ and conjecturally satisfy functional equations. These L-functions play the role of half of the modern Langlands philosophy, and there are many crucial algebraic and analytic questions that remain unanswered. For instance, when $ \\rho = \\chi $ is a Dirichlet character, the generalised Riemann hypothesis describes where its $ \\zeta $-function $ \\zeta(s) = L(\\chi, s) $ vanishes, and when $ \\rho = \\rho_{A, \\ell} $ arises from an abelian variety $ A $, the Birch and Swinnerton-Dyer conjecture describes the arithmetic of $ A $ in terms of its L-function $ L(A, s) = L(\\rho_{A, \\ell}, s) $. The importance of L-functions in arithmetic algebraic geometry is so much so that thousands of CPU years have been spent on computing their data for various representations in the massive L-functions and modular forms database (LMFDB) \\cite{LMFDB}, which has proven to be an immensely useful resource for number theorists.\n\nAt a cursory glance, most of the data in the LMFDB are of objects over various number fields. In this setting, their L-functions are transcendental functions of $ s $ that can be written as Dirichlet series, with the aforementioned conjectures still open for most objects. Much of their data were computed from Platt's code for degree one L-functions \\cite{Pla11} based on Booker's algorithm \\cite{Boo06}, from the \\texttt{lcalc} library in C++ based on Rubinstein's algorithm \\cite{Rub05} for Maass forms, and from the work of Booker--Sijsling--Sutherland--Voight--Yasaki \\cite{BSSVY16} for genus two curves. In a much larger generality, T Dokchitser wrote an efficient algorithm to compute the special value of a motivic L-function, assuming a conjectural functional equation that can be numerically verified \\cite{Dok04}. This was originally implemented as the PARI \\cite{PARI} package \\texttt{ComputeL}, but has since been ported to the core libraries of Magma \\cite{BCP97} as the \\texttt{LSeries} function and SageMath \\cite{SageMath} as the \\texttt{dokchitser} function.\n\nIn a stark contrast, the arithmetic of analogous objects over $ K \\coloneqq \\F_q(C) $ is very well-understood \\cite{Laf02}. In this setting, their L-functions are rational functions of $ q^{-s} $, satisfying a known functional equation and even the Riemann hypothesis. Yet, the computational ecosystem over $ K $ is severely limited, with little to no support for explicit computations in the core libraries of most computer algebra systems, including PARI and SageMath. Magma is better in this regard, including multiple constructors for $ K $, explicit class field theory computations with \\texttt{WittRing} and \\texttt{CarlitzModule}, and even an \\texttt{LFunction} for non-constant elliptic curves over $ \\F_q(t) $. Using this interface, Comeau--Lapointe--David--Lal\\'in--Li computed $ L(E, \\chi, s) = L(\\rho_{E, \\ell} \\otimes \\chi, s) $ for a fixed Legendre curve $ E $ over $ K = \\F_q(t) $ twisted by Dirichlet characters $ \\chi $ to investigate their vanishing \\cite{CLDLL22}. In private communication, Maistret--Wiersema also computed $ L(E, \\chi, s) $ for many constant elliptic curves $ E $ by hand to formulate an analogue over $ K $ of the twisted BSD-type formula of V Dokchitser--Evans--Wiersema \\cite{DEW21}. This seems to be the extent to which L-functions of $ K $ have been computed explicitly in the literature, which renders the creation of an analogue of the LMFDB over $ K $ difficult.\n\nThis paper presents an analogue of T Dokchitser's algorithm for $ K $, in the language of \\emph{$ \\lambda $-adic representations $ \\rho_\\lambda $} for some prime $ \\lambda \\nmid q $ of a fixed number field $ F $. These are the natural Galois representations associated to L-functions of motivic origin, including Riemann/Dedekind $ \\zeta $-functions, Dirichlet/Weber/Hecke/Artin L-functions, and Hasse--Weil L-functions of varieties. Then there is a rational function $ \\mathcal{L}(\\rho_\\lambda, T) \\in F_\\lambda(T) $, called the \\emph{formal L-function} of $ \\rho_\\lambda $, such that $ \\mathcal{L}(\\rho_\\lambda, q^{-s}) = L(\\rho_\\lambda, s) $. Under the condition that $ \\rho_\\lambda $ is \\emph{self-dual of weight $ w(\\rho_\\lambda) \\in \\mathbb{N} $ and sign $ c(\\rho_\\lambda) \\in \\{\\operatorname{id}, \\operatorname{cc}\\} $}, where $ \\operatorname{id} $ is the identity and $ \\operatorname{cc} $ is complex conjugation, there is a field element $ \\epsilon(\\rho_\\lambda) \\in F_\\lambda^\\times $ such that $ \\mathcal{L}(\\rho_\\lambda, T) $ satisfies a functional equation\n$$ \\mathcal{L}(\\rho_\\lambda, T) = \\epsilon(\\rho_\\lambda) \\cdot T^{n(\\rho_\\lambda) - d(\\rho_\\lambda)} \\cdot \\mathcal{L}(\\rho_\\lambda, (q^{w(\\rho_\\lambda) + 1}T)^{-1})^{c(\\rho_\\lambda)}, $$\nwhere $ n(\\rho_\\lambda), d(\\rho_\\lambda) \\in \\mathbb{N} $ are the degrees of the numerator and denominator of $ \\mathcal{L}(\\rho_\\lambda, T) $ respectively. If the genus of $ C $ and the dimension of $ \\rho_\\lambda $ are small, then $ n(\\rho_\\lambda) - d(\\rho_\\lambda) $ is essentially the degree of the \\emph{Artin conductor} of $ \\rho_\\lambda $.", "full_context": "Given a global field, there are L-functions $ L(\\rho, s) $ in the complex variable $ s \\in \\mathbb{C} $ associated to suitable continuous representations $ \\rho $ of its absolute Galois group, which are conjecturally meromophic over $ \\mathbb{C} $ and conjecturally satisfy functional equations. These L-functions play the role of half of the modern Langlands philosophy, and there are many crucial algebraic and analytic questions that remain unanswered. For instance, when $ \\rho = \\chi $ is a Dirichlet character, the generalised Riemann hypothesis describes where its $ \\zeta $-function $ \\zeta(s) = L(\\chi, s) $ vanishes, and when $ \\rho = \\rho_{A, \\ell} $ arises from an abelian variety $ A $, the Birch and Swinnerton-Dyer conjecture describes the arithmetic of $ A $ in terms of its L-function $ L(A, s) = L(\\rho_{A, \\ell}, s) $. The importance of L-functions in arithmetic algebraic geometry is so much so that thousands of CPU years have been spent on computing their data for various representations in the massive L-functions and modular forms database (LMFDB) \\cite{LMFDB}, which has proven to be an immensely useful resource for number theorists.\n\nAt a cursory glance, most of the data in the LMFDB are of objects over various number fields. In this setting, their L-functions are transcendental functions of $ s $ that can be written as Dirichlet series, with the aforementioned conjectures still open for most objects. Much of their data were computed from Platt's code for degree one L-functions \\cite{Pla11} based on Booker's algorithm \\cite{Boo06}, from the \\texttt{lcalc} library in C++ based on Rubinstein's algorithm \\cite{Rub05} for Maass forms, and from the work of Booker--Sijsling--Sutherland--Voight--Yasaki \\cite{BSSVY16} for genus two curves. In a much larger generality, T Dokchitser wrote an efficient algorithm to compute the special value of a motivic L-function, assuming a conjectural functional equation that can be numerically verified \\cite{Dok04}. This was originally implemented as the PARI \\cite{PARI} package \\texttt{ComputeL}, but has since been ported to the core libraries of Magma \\cite{BCP97} as the \\texttt{LSeries} function and SageMath \\cite{SageMath} as the \\texttt{dokchitser} function.\n\nIn a stark contrast, the arithmetic of analogous objects over $ K \\coloneqq \\F_q(C) $ is very well-understood \\cite{Laf02}. In this setting, their L-functions are rational functions of $ q^{-s} $, satisfying a known functional equation and even the Riemann hypothesis. Yet, the computational ecosystem over $ K $ is severely limited, with little to no support for explicit computations in the core libraries of most computer algebra systems, including PARI and SageMath. Magma is better in this regard, including multiple constructors for $ K $, explicit class field theory computations with \\texttt{WittRing} and \\texttt{CarlitzModule}, and even an \\texttt{LFunction} for non-constant elliptic curves over $ \\F_q(t) $. Using this interface, Comeau--Lapointe--David--Lal\\'in--Li computed $ L(E, \\chi, s) = L(\\rho_{E, \\ell} \\otimes \\chi, s) $ for a fixed Legendre curve $ E $ over $ K = \\F_q(t) $ twisted by Dirichlet characters $ \\chi $ to investigate their vanishing \\cite{CLDLL22}. In private communication, Maistret--Wiersema also computed $ L(E, \\chi, s) $ for many constant elliptic curves $ E $ by hand to formulate an analogue over $ K $ of the twisted BSD-type formula of V Dokchitser--Evans--Wiersema \\cite{DEW21}. This seems to be the extent to which L-functions of $ K $ have been computed explicitly in the literature, which renders the creation of an analogue of the LMFDB over $ K $ difficult.\n\nThis paper presents an analogue of T Dokchitser's algorithm for $ K $, in the language of \\emph{$ \\lambda $-adic representations $ \\rho_\\lambda $} for some prime $ \\lambda \\nmid q $ of a fixed number field $ F $. These are the natural Galois representations associated to L-functions of motivic origin, including Riemann/Dedekind $ \\zeta $-functions, Dirichlet/Weber/Hecke/Artin L-functions, and Hasse--Weil L-functions of varieties. Then there is a rational function $ \\mathcal{L}(\\rho_\\lambda, T) \\in F_\\lambda(T) $, called the \\emph{formal L-function} of $ \\rho_\\lambda $, such that $ \\mathcal{L}(\\rho_\\lambda, q^{-s}) = L(\\rho_\\lambda, s) $. Under the condition that $ \\rho_\\lambda $ is \\emph{self-dual of weight $ w(\\rho_\\lambda) \\in \\mathbb{N} $ and sign $ c(\\rho_\\lambda) \\in \\{\\operatorname{id}, \\operatorname{cc}\\} $}, where $ \\operatorname{id} $ is the identity and $ \\operatorname{cc} $ is complex conjugation, there is a field element $ \\epsilon(\\rho_\\lambda) \\in F_\\lambda^\\times $ such that $ \\mathcal{L}(\\rho_\\lambda, T) $ satisfies a functional equation\n$$ \\mathcal{L}(\\rho_\\lambda, T) = \\epsilon(\\rho_\\lambda) \\cdot T^{n(\\rho_\\lambda) - d(\\rho_\\lambda)} \\cdot \\mathcal{L}(\\rho_\\lambda, (q^{w(\\rho_\\lambda) + 1}T)^{-1})^{c(\\rho_\\lambda)}, $$\nwhere $ n(\\rho_\\lambda), d(\\rho_\\lambda) \\in \\mathbb{N} $ are the degrees of the numerator and denominator of $ \\mathcal{L}(\\rho_\\lambda, T) $ respectively. If the genus of $ C $ and the dimension of $ \\rho_\\lambda $ are small, then $ n(\\rho_\\lambda) - d(\\rho_\\lambda) $ is essentially the degree of the \\emph{Artin conductor} of $ \\rho_\\lambda $.\n\nThis paper presents an analogue of T Dokchitser's algorithm for $ K $, in the language of \\emph{$ \\lambda $-adic representations $ \\rho_\\lambda $} for some prime $ \\lambda \\nmid q $ of a fixed number field $ F $. These are the natural Galois representations associated to L-functions of motivic origin, including Riemann/Dedekind $ \\zeta $-functions, Dirichlet/Weber/Hecke/Artin L-functions, and Hasse--Weil L-functions of varieties. Then there is a rational function $ \\LL(\\rho_\\lambda, T) \\in F_\\lambda(T) $, called the \\emph{formal L-function} of $ \\rho_\\lambda $, such that $ \\LL(\\rho_\\lambda, q^{-s}) = L(\\rho_\\lambda, s) $. Under the condition that $ \\rho_\\lambda $ is \\emph{self-dual of weight $ w(\\rho_\\lambda) \\in \\N $ and sign $ c(\\rho_\\lambda) \\in \\{\\id, \\cc\\} $}, where $ \\id $ is the identity and $ \\cc $ is complex conjugation, there is a field element $ \\epsilon(\\rho_\\lambda) \\in F_\\lambda^\\times $ such that $ \\LL(\\rho_\\lambda, T) $ satisfies a functional equation\n$$ \\LL(\\rho_\\lambda, T) = \\epsilon(\\rho_\\lambda) \\cdot T^{n(\\rho_\\lambda) - d(\\rho_\\lambda)} \\cdot \\LL(\\rho_\\lambda, (q^{w(\\rho_\\lambda) + 1}T)^{-1})^{c(\\rho_\\lambda)}, $$\nwhere $ n(\\rho_\\lambda), d(\\rho_\\lambda) \\in \\N $ are the degrees of the numerator and denominator of $ \\LL(\\rho_\\lambda, T) $ respectively. If the genus of $ C $ and the dimension of $ \\rho_\\lambda $ are small, then $ n(\\rho_\\lambda) - d(\\rho_\\lambda) $ is essentially the degree of the \\emph{Artin conductor} of $ \\rho_\\lambda $.\n\nThe algorithms in Theorem \\ref{thm:mainresults} have been implemented in Magma \\cite{Ang25} and bug-tested against existing implementations of $ L(E, s) $ and $ L(E, \\chi, s) $. The proof of (1) is elementary and only uses properties of formal power series. The proofs of (2) and (3) use a generalisation of (1), and are heavily inspired by the algorithm of Comeau-Lapointe--David--Lal\\'in--Li for $ L(E, \\chi, s) $ \\cite[Section 5.1]{CLDLL22}, although their analogue of (1) is an ad-hoc computation using Legendre symbols.\n\nNow let $ \\psi $ be a non-trivial additive character of $ K $, let $ \\mu $ be an additive Haar measure on $ K $, and let $ \\gamma(\\psi) $ be the smallest integer $ r \\in \\Z $ such that $ \\psi(\\varpi_K^r\\OO_K) = 1 $. Langlands \\cite{Lan70} and Deligne \\cite{Del73b} independently gave non-constructive proofs on the existence of a constant $ \\epsilon(\\rho, \\psi, \\mu) \\in \\C^\\times $ unique under the following properties.\n\\begin{itemize}\n\\item Additivity: if $ \\sigma $ and $ \\tau $ are Weil representations of $ K $ in a short exact sequence $ 0 \\to \\rho \\to \\sigma \\to \\tau \\to 0 $, then\n$$ \\epsilon(\\sigma, \\psi, \\mu) = \\epsilon(\\rho, \\psi, \\mu) \\cdot \\epsilon(\\tau, \\psi, \\mu). $$\n\\item Inductivity in degree zero: if $ \\rho' $ is a virtual Weil representation of degree zero over a finite separable extension $ K' $ of $ K $, and $ \\mu' $ is an additive Haar measure on $ K' $, then\n$$ \\epsilon(\\Ind_K^{K'}\\rho, \\psi, \\mu) = \\epsilon(\\rho', \\psi \\circ \\tr_{K' / K}, \\mu'). $$\n\\item Quasi-character formula: if $ \\rho $ is one-dimensional and corresponds to a quasi-character $ \\chi : W_K \\to \\C^\\times $ via local class field theory, then\n$$ \\epsilon(\\rho, \\psi, \\mu) =\n\\begin{cases}\n\\chi(\\phi_K)^{\\gamma(\\psi)} \\cdot q_K^{\\gamma(\\psi)} \\cdot \\mu(\\OO_K) & \\text{if} \\ \\rho \\ \\text{is unramified}, \\\\\n\\sum_{r \\in \\Z} \\int_{\\varpi_K^r\\OO_K} \\chi^{-1}(x)\\psi(x)\\d\\mu(x) & \\text{otherwise}.\n\\end{cases}\n$$\n\\end{itemize}\nThis depends on $ \\psi $ and $ \\mu $ in general, but Langlands and Deligne made specific choices whose differences in convention are summarised by Tate \\cite[Section 3.6]{Tat79}. The \\textbf{local $ \\epsilon $-factor} of $ \\rho_\\lambda $ is then $ \\epsilon(\\rho_\\lambda, \\psi, \\mu) \\coloneqq \\epsilon(\\rho, \\psi, \\mu) \\cdot \\delta(\\rho_\\lambda) $, where\n$$ \\delta(\\rho_\\lambda) \\coloneqq \\dfrac{\\det(-\\rho^{I_K}(\\phi_K))}{\\det(-\\ker(\\nu)^{I_K}(\\phi_K))}. $$\nThe following generalises a well-known formula for tensor products of Weil representations with one unramified factor to that of $ \\lambda $-adic representations.\n\n\\begin{proposition}\n\\label{prop:modulusepsilon}\nLet $ \\rho_\\lambda $ be a $ \\lambda $-adic representation of a global function field $ K $ that is self-dual of weight $ w(\\rho_\\lambda) $ and sign $ c(\\rho_\\lambda) $. Then\n$$ |\\epsilon(\\rho_\\lambda)| = q^{(\\deg\\ff(\\rho_\\lambda) + (2g(C) - 2)\\dim\\rho_\\lambda)(w(\\rho_\\lambda) + 1) / 2}. $$\n\\end{proposition}\n\n\\begin{theorem}\n\\label{thm:weilconjectures}\nLet $ \\rho_\\lambda $ be an almost everywhere unramified $ \\lambda $-adic representation of a global function field $ K = \\F_q(C) $.\n\\begin{enumerate}\n\\item Rationality: there are unique coprime polynomials $ N(\\rho_\\lambda, T), D(\\rho_\\lambda, T) \\in 1 + T \\cdot F_\\lambda[T] $ of degrees $ n(\\rho_\\lambda), d(\\rho_\\lambda) \\in \\N $ respectively such that\n$$ \\LL(\\rho_\\lambda, T) = \\dfrac{N(\\rho_\\lambda, T)}{D(\\rho_\\lambda, T)}. $$\n\\item Integrality: if $ \\rho_\\lambda $ has no $ \\overline{G_K} $-invariants, then $ d(\\rho_\\lambda) = 0 $.\n\\item Degree formula:\n$$ n(\\rho_\\lambda) - d(\\rho_\\lambda) = \\deg\\ff(\\rho_\\lambda) + (2g(C) - 2)\\dim\\rho_\\lambda. $$\n\\item Functional equation:\n$$ \\LL(\\rho_\\lambda, T) = \\epsilon(\\rho_\\lambda) \\cdot T^{n(\\rho_\\lambda) - d(\\rho_\\lambda)} \\cdot \\LL(\\rho_\\lambda^\\vee, (qT)^{-1}). $$\n\\item Riemann hypothesis: if $ \\rho_\\lambda $ is $ \\iota $-pure of weight $ w(\\rho_\\lambda) $, then the roots of $ N(\\rho_\\lambda, T) $ have modulus $ q^{-(w(\\rho_\\lambda) + 1) / 2} $.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{corollary}\n\\label{cor:rationalfunction}\nLet $ \\rho_\\lambda $ be an almost everywhere unramified $ \\lambda $-adic representation of a global function field $ K $. Then Algorithm \\ref{alg:rationalfunction} outputs $ N(\\rho_\\lambda, T) \\in 1 + T \\cdot F_\\lambda[T] $ with inputs\n$$ (\\LL_\\bullet(\\rho_\\lambda, T), \\ D(\\rho_\\lambda, T), \\ 1, \\ n(\\rho_\\lambda), \\ n(\\rho_\\lambda)). $$\n\\end{corollary}\n\n\\begin{corollary}\n\\label{cor:functionalequation}\nLet $ \\rho_\\lambda $ be an almost everywhere unramified $ \\lambda $-adic representation of a global function field $ K = \\F_q(C) $ that is self-dual of weight $ w(\\rho_\\lambda) $ and sign $ c(\\rho_\\lambda) $. Then Algorithm \\ref{alg:functionalequation} outputs $ N(\\rho_\\lambda, T) \\in 1 + T \\cdot F_\\lambda[T] $ with inputs\n$$ (\\LL_\\bullet(\\rho_\\lambda, T), \\ D(\\rho_\\lambda, T), \\ n(\\rho_\\lambda), \\ \\epsilon(\\rho_\\lambda), \\ 1 / q^{w(\\rho_\\lambda) + 1}, \\ c(\\rho_\\lambda)). $$\n\\end{corollary}\n\n\\begin{corollary}\n\\label{cor:computeepsilon}\nLet $ \\rho_\\lambda $ be an almost everywhere unramified $ \\lambda $-adic representation of a global function field $ K = \\F_q(C) $ that is self-dual of weight $ w(\\rho_\\lambda) $ and sign $ c(\\rho_\\lambda) $. Then Algorithm \\ref{alg:computeepsilon} outputs $ \\epsilon(\\rho_\\lambda) \\in F_\\lambda^\\times $ with inputs\n$$ (\\LL_\\bullet(\\rho_\\lambda, T), \\ D(\\rho_\\lambda, T), \\ n(\\rho_\\lambda), \\ 1 / q^{w(\\rho_\\lambda) + 1}, \\ c(\\rho_\\lambda)). $$\n\\end{corollary}\n\n\\begin{theorem}[Corollary \\ref{cor:rationalfunction}, Corollary \\ref{cor:functionalequation}, Corollary \\ref{cor:computeepsilon}]\n\\label{thm:mainresults}\nLet $ \\rho_\\lambda $ be an almost everywhere unramified $ \\lambda $-adic representation of a global function field $ K $.\n\\begin{itemize}\n\\item[(1)] There is an algorithm to compute $ \\LL(\\rho_\\lambda, T) $ with a running time that is roughly exponential in $ n(\\rho_\\lambda)$.\n\\end{itemize}\nAssume further that $ \\rho_\\lambda $ is self-dual of weight $ w(\\rho_\\lambda) $ and sign $ c(\\rho_\\lambda) \\in \\{\\id, \\cc\\} $.\n\\begin{itemize}\n\\item[(2)] If $ \\epsilon(\\rho_\\lambda) $ is known, then there is an algorithm to compute $ \\LL(\\rho_\\lambda, T) $ with a running time that is roughly exponential in $ \\lfloor n(\\rho_\\lambda) / 2\\rfloor $.\n\\item[(3)] Otherwise, there is an algorithm to compute $ \\epsilon(\\rho_\\lambda) $ with a running time that is typically exponential in $ \\lfloor n(\\rho_\\lambda) / 2\\rfloor $, and at worst exponential in $ n(\\rho_\\lambda) $.\n\\end{itemize}\n\\end{theorem}", "post_theorem_intro_text_len": 1081, "post_theorem_intro_text": "The algorithms in Theorem \\ref{thm:mainresults} have been implemented in Magma \\cite{Ang25} and bug-tested against existing implementations of $ L(E, s) $ and $ L(E, \\chi, s) $. The proof of (1) is elementary and only uses properties of formal power series. The proofs of (2) and (3) use a generalisation of (1), and are heavily inspired by the algorithm of Comeau-Lapointe--David--Lal\\'in--Li for $ L(E, \\chi, s) $ \\cite[Section 5.1]{CLDLL22}, although their analogue of (1) is an ad-hoc computation using Legendre symbols.\n\nSection \\ref{sec:adicrepresentations} reviews some useful background on invariants associated to $ \\rho_\\lambda $, including $ \\mathcal{L}(\\rho_\\lambda, T) $ and $ \\epsilon(\\rho_\\lambda) $, which are sometimes implicit in the literature without good references. Section \\ref{sec:computingfunctions} details Theorem \\ref{thm:mainresults} and briefly justifies their running time complexities. Section \\ref{sec:examplecomputations} works them out by hand for explicit examples of elliptic curves, Dirichlet characters, and other motivic objects.\n\n\\pagebreak", "sketch": "The post-theorem introduction gives the following proof sketch for Theorem~\\ref{thm:mainresults}:\n\n- For (1): “The proof of (1) is elementary and only uses properties of formal power series.”\n\n- For (2) and (3): “The proofs of (2) and (3) use a generalisation of (1),” and are “heavily inspired by the algorithm of Comeau-Lapointe--David--Lal\\'in--Li for $L(E,\\chi,s)$” (\\cite[Section 5.1]{CLDLL22}), contrasting with their “ad-hoc computation using Legendre symbols.”\n\n- The later sections “detail Theorem~\\ref{thm:mainresults} and briefly justif[y] their running time complexities.”", "expanded_sketch": "The post-theorem introduction gives the following proof sketch for the main theorem:\n\n- For (1): “The proof of (1) is elementary and only uses properties of formal power series.”\n\n- For (2) and (3): “The proofs of (2) and (3) use a generalisation of (1),” and are “heavily inspired by the algorithm of Comeau-Lapointe--David--Lal\\'in--Li for $L(E,\\chi,s)$” (\\cite[Section 5.1]{CLDLL22}), contrasting with their “ad-hoc computation using Legendre symbols.”\n\n- The later sections “detail the main theorem and briefly justif[y] their running time complexities.”", "expanded_theorem": "\\begin{corollary}\n\\label{cor:rationalfunction}\nLet $ \\rho_\\lambda $ be an almost everywhere unramified $ \\lambda $-adic representation of a global function field $ K $. Then Algorithm \\ref{alg:rationalfunction} outputs $ N(\\rho_\\lambda, T) \\in 1 + T \\cdot F_\\lambda[T] $ with inputs\n$$ (\\LL_\\bullet(\\rho_\\lambda, T), \\ D(\\rho_\\lambda, T), \\ 1, \\ n(\\rho_\\lambda), \\ n(\\rho_\\lambda)). $$\n\\end{corollary}\n\n\\begin{corollary}\n\\label{cor:functionalequation}\nLet $ \\rho_\\lambda $ be an almost everywhere unramified $ \\lambda $-adic representation of a global function field $ K = \\F_q(C) $ that is self-dual of weight $ w(\\rho_\\lambda) $ and sign $ c(\\rho_\\lambda) $. Then Algorithm \\ref{alg:functionalequation} outputs $ N(\\rho_\\lambda, T) \\in 1 + T \\cdot F_\\lambda[T] $ with inputs\n$$ (\\LL_\\bullet(\\rho_\\lambda, T), \\ D(\\rho_\\lambda, T), \\ n(\\rho_\\lambda), \\ \\epsilon(\\rho_\\lambda), \\ 1 / q^{w(\\rho_\\lambda) + 1}, \\ c(\\rho_\\lambda)). $$\n\\end{corollary}\n\n\\begin{corollary}\n\\label{cor:computeepsilon}\nLet $ \\rho_\\lambda $ be an almost everywhere unramified $ \\lambda $-adic representation of a global function field $ K = \\F_q(C) $ that is self-dual of weight $ w(\\rho_\\lambda) $ and sign $ c(\\rho_\\lambda) $. Then Algorithm \\ref{alg:computeepsilon} outputs $ \\epsilon(\\rho_\\lambda) \\in F_\\lambda^\\times $ with inputs\n$$ (\\LL_\\bullet(\\rho_\\lambda, T), \\ D(\\rho_\\lambda, T), \\ n(\\rho_\\lambda), \\ 1 / q^{w(\\rho_\\lambda) + 1}, \\ c(\\rho_\\lambda)). $$\n\\end{corollary}\n\n\\label{thm:mainresults}\nLet $ \\rho_\\lambda $ be an almost everywhere unramified $ \\lambda $-adic representation of a global function field $ K $.\n\\begin{itemize}\n\\item[(1)] There is an algorithm to compute $ \\mathcal{L}(\\rho_\\lambda, T) $ with a running time that is roughly exponential in $ n(\\rho_\\lambda)$.\n\\end{itemize}\nAssume further that $ \\rho_\\lambda $ is self-dual of weight $ w(\\rho_\\lambda) $ and sign $ c(\\rho_\\lambda) \\in \\{\\operatorname{id}, \\operatorname{cc}\\} $.\n\\begin{itemize}\n\\item[(2)] If $ \\epsilon(\\rho_\\lambda) $ is known, then there is an algorithm to compute $ \\mathcal{L}(\\rho_\\lambda, T) $ with a running time that is roughly exponential in $ \\lfloor n(\\rho_\\lambda) / 2\\rfloor $.\n\\item[(3)] Otherwise, there is an algorithm to compute $ \\epsilon(\\rho_\\lambda) $ with a running time that is typically exponential in $ \\lfloor n(\\rho_\\lambda) / 2\\rfloor $, and at worst exponential in $ n(\\rho_\\lambda) $.\n\\end{itemize}", "theorem_type": ["Algorithmic or Constructive", "Implication"], "mcq": {"question": "Let \\(\\rho_\\lambda\\) be an almost everywhere unramified \\(\\lambda\\)-adic representation of a global function field \\(K\\). Write \\(\\mathcal{L}(\\rho_\\lambda,T)\\) for its formal \\(L\\)-function, and write \\(N(\\rho_\\lambda,T) \\in 1+T\\,F_\\lambda[T]\\) for the numerator polynomial of \\(\\mathcal{L}(\\rho_\\lambda,T)\\); let \\(n(\\rho_\\lambda)\\) denote the degree of that numerator. Also let \\(D(\\rho_\\lambda,T)\\) and \\(\\mathcal{L}_\\bullet(\\rho_\\lambda,T)\\) denote the associated input data used by the algorithms. If moreover \\(K=\\mathbb F_q(C)\\) and \\(\\rho_\\lambda\\) is self-dual of weight \\(w(\\rho_\\lambda)\\) and sign \\(c(\\rho_\\lambda)\\in\\{\\operatorname{id},\\operatorname{cc}\\}\\), with functional-equation constant \\(\\epsilon(\\rho_\\lambda)\\in F_\\lambda^\\times\\), which statement about the algorithmic computation of \\(\\mathcal{L}(\\rho_\\lambda,T)\\), \\(N(\\rho_\\lambda,T)\\), and \\(\\epsilon(\\rho_\\lambda)\\) holds?", "correct_choice": {"label": "A", "text": "For every almost everywhere unramified \\(\\lambda\\)-adic representation \\(\\rho_\\lambda\\) of a global function field \\(K\\), Algorithm \\(\\mathrm{rationalfunction}\\) outputs \\(N(\\rho_\\lambda,T)\\in 1+T\\,F_\\lambda[T]\\) from the inputs \\((\\mathcal{L}_\\bullet(\\rho_\\lambda,T),\\ D(\\rho_\\lambda,T),\\ 1,\\ n(\\rho_\\lambda),\\ n(\\rho_\\lambda))\\), and consequently there is an algorithm to compute \\(\\mathcal{L}(\\rho_\\lambda,T)\\) with running time roughly exponential in \\(n(\\rho_\\lambda)\\). If, in addition, \\(\\rho_\\lambda\\) is self-dual of weight \\(w(\\rho_\\lambda)\\) and sign \\(c(\\rho_\\lambda)\\in\\{\\operatorname{id},\\operatorname{cc}\\}\\), then: (i) if \\(\\epsilon(\\rho_\\lambda)\\) is known, Algorithm \\(\\mathrm{functionalequation}\\) outputs \\(N(\\rho_\\lambda,T)\\in 1+T\\,F_\\lambda[T]\\) from the inputs \\((\\mathcal{L}_\\bullet(\\rho_\\lambda,T),\\ D(\\rho_\\lambda,T),\\ n(\\rho_\\lambda),\\ \\epsilon(\\rho_\\lambda),\\ 1/q^{w(\\rho_\\lambda)+1},\\ c(\\rho_\\lambda))\\), so there is an algorithm to compute \\(\\mathcal{L}(\\rho_\\lambda,T)\\) with running time roughly exponential in \\(\\lfloor n(\\rho_\\lambda)/2\\rfloor\\); and (ii) if \\(\\epsilon(\\rho_\\lambda)\\) is not known, Algorithm \\(\\mathrm{computeepsilon}\\) outputs \\(\\epsilon(\\rho_\\lambda)\\in F_\\lambda^\\times\\) from the inputs \\((\\mathcal{L}_\\bullet(\\rho_\\lambda,T),\\ D(\\rho_\\lambda,T),\\ n(\\rho_\\lambda),\\ 1/q^{w(\\rho_\\lambda)+1},\\ c(\\rho_\\lambda))\\), and there is an algorithm to compute \\(\\epsilon(\\rho_\\lambda)\\) with running time typically exponential in \\(\\lfloor n(\\rho_\\lambda)/2\\rfloor\\), and at worst exponential in \\(n(\\rho_\\lambda)\\)."}, "choices": [{"label": "B", "text": "For every almost everywhere unramified \\(\\lambda\\)-adic representation \\(\\rho_\\lambda\\) of a global function field \\(K\\), Algorithm \\(\\mathrm{rationalfunction}\\) outputs \\(N(\\rho_\\lambda,T)\\in 1+T\\,F_\\lambda[T]\\) from the inputs \\((\\mathcal{L}_\\bullet(\\rho_\\lambda,T),\\ D(\\rho_\\lambda,T),\\ 1,\\ n(\\rho_\\lambda),\\ n(\\rho_\\lambda))\\), and consequently there is an algorithm to compute \\(\\mathcal{L}(\\rho_\\lambda,T)\\) with running time roughly exponential in \\(n(\\rho_\\lambda)\\). If, in addition, \\(\\rho_\\lambda\\) is self-dual of weight \\(w(\\rho_\\lambda)\\) and sign \\(c(\\rho_\\lambda)\\in\\{\\operatorname{id},\\operatorname{cc}\\}\\), then: (i) Algorithm \\(\\mathrm{functionalequation}\\) outputs \\(N(\\rho_\\lambda,T)\\in 1+T\\,F_\\lambda[T]\\) from the inputs \\((\\mathcal{L}_\\bullet(\\rho_\\lambda,T),\\ D(\\rho_\\lambda,T),\\ n(\\rho_\\lambda),\\ \\epsilon(\\rho_\\lambda),\\ 1/q^{w(\\rho_\\lambda)},\\ c(\\rho_\\lambda))\\), so there is an algorithm to compute \\(\\mathcal{L}(\\rho_\\lambda,T)\\) with running time roughly exponential in \\(\\lfloor n(\\rho_\\lambda)/2\\rfloor\\); and (ii) if \\(\\epsilon(\\rho_\\lambda)\\) is not known, Algorithm \\(\\mathrm{computeepsilon}\\) outputs \\(\\epsilon(\\rho_\\lambda)\\in F_\\lambda^\\times\\) from the inputs \\((\\mathcal{L}_\\bullet(\\rho_\\lambda,T),\\ D(\\rho_\\lambda,T),\\ n(\\rho_\\lambda),\\ 1/q^{w(\\rho_\\lambda)},\\ c(\\rho_\\lambda))\\), and there is an algorithm to compute \\(\\epsilon(\\rho_\\lambda)\\) with running time typically exponential in \\(\\lfloor n(\\rho_\\lambda)/2\\rfloor\\), and at worst exponential in \\(n(\\rho_\\lambda)\\)."}, {"label": "C", "text": "For every almost everywhere unramified \\(\\lambda\\)-adic representation \\(\\rho_\\lambda\\) of a global function field \\(K\\), Algorithm \\(\\mathrm{rationalfunction}\\) outputs \\(N(\\rho_\\lambda,T)\\in 1+T\\,F_\\lambda[T]\\) from the inputs \\((\\mathcal{L}_\\bullet(\\rho_\\lambda,T),\\ D(\\rho_\\lambda,T),\\ 1,\\ n(\\rho_\\lambda),\\ n(\\rho_\\lambda))\\). If, in addition, \\(\\rho_\\lambda\\) is self-dual of weight \\(w(\\rho_\\lambda)\\) and sign \\(c(\\rho_\\lambda)\\in\\{\\operatorname{id},\\operatorname{cc}\\}\\), then: (i) if \\(\\epsilon(\\rho_\\lambda)\\) is known, Algorithm \\(\\mathrm{functionalequation}\\) outputs \\(N(\\rho_\\lambda,T)\\in 1+T\\,F_\\lambda[T]\\) from the inputs \\((\\mathcal{L}_\\bullet(\\rho_\\lambda,T),\\ D(\\rho_\\lambda,T),\\ n(\\rho_\\lambda),\\ \\epsilon(\\rho_\\lambda),\\ 1/q^{w(\\rho_\\lambda)+1},\\ c(\\rho_\\lambda))\\); and (ii) if \\(\\epsilon(\\rho_\\lambda)\\) is not known, Algorithm \\(\\mathrm{computeepsilon}\\) outputs \\(\\epsilon(\\rho_\\lambda)\\in F_\\lambda^\\times\\) from the inputs \\((\\mathcal{L}_\\bullet(\\rho_\\lambda,T),\\ D(\\rho_\\lambda,T),\\ n(\\rho_\\lambda),\\ 1/q^{w(\\rho_\\lambda)+1},\\ c(\\rho_\\lambda))\\)."}, {"label": "D", "text": "For every almost everywhere unramified \\(\\lambda\\)-adic representation \\(\\rho_\\lambda\\) of a global function field \\(K\\), Algorithm \\(\\mathrm{rationalfunction}\\) outputs \\(N(\\rho_\\lambda,T)\\in 1+T\\,F_\\lambda[T]\\) from the inputs \\((\\mathcal{L}_\\bullet(\\rho_\\lambda,T),\\ D(\\rho_\\lambda,T),\\ 1,\\ n(\\rho_\\lambda),\\ n(\\rho_\\lambda))\\), and consequently there is an algorithm to compute \\(\\mathcal{L}(\\rho_\\lambda,T)\\) with running time roughly exponential in \\(n(\\rho_\\lambda)\\). If, in addition, \\(\\rho_\\lambda\\) is self-dual of weight \\(w(\\rho_\\lambda)\\) and sign \\(c(\\rho_\\lambda)\\in\\{\\operatorname{id},\\operatorname{cc}\\}\\), then: (i) if \\(\\epsilon(\\rho_\\lambda)\\) is known, Algorithm \\(\\mathrm{functionalequation}\\) outputs \\(N(\\rho_\\lambda,T)\\in 1+T\\,F_\\lambda[T]\\) from the inputs \\((\\mathcal{L}_\\bullet(\\rho_\\lambda,T),\\ D(\\rho_\\lambda,T),\\ n(\\rho_\\lambda),\\ \\epsilon(\\rho_\\lambda),\\ 1/q^{w(\\rho_\\lambda)+1},\\ c(\\rho_\\lambda))\\), so there is an algorithm to compute \\(\\mathcal{L}(\\rho_\\lambda,T)\\) with running time roughly exponential in \\(\\lfloor n(\\rho_\\lambda)/2\\rfloor\\); and (ii) if \\(\\epsilon(\\rho_\\lambda)\\) is not known, there is an algorithm to compute \\(\\epsilon(\\rho_\\lambda)\\) with running time typically exponential in \\(\\lfloor n(\\rho_\\lambda)/2\\rfloor\\), and at worst exponential in \\(n(\\rho_\\lambda)\\), uniformly from the same input data \\((\\mathcal{L}_\\bullet(\\rho_\\lambda,T),\\ D(\\rho_\\lambda,T),\\ n(\\rho_\\lambda),\\ \\epsilon(\\rho_\\lambda),\\ 1/q^{w(\\rho_\\lambda)+1},\\ c(\\rho_\\lambda))\\)."}, {"label": "E", "text": "For every almost everywhere unramified \\(\\lambda\\)-adic representation \\(\\rho_\\lambda\\) of a global function field \\(K\\), Algorithm \\(\\mathrm{rationalfunction}\\) outputs \\(N(\\rho_\\lambda,T)\\in 1+T\\,F_\\lambda[T]\\) from the inputs \\((\\mathcal{L}_\\bullet(\\rho_\\lambda,T),\\ D(\\rho_\\lambda,T),\\ 1,\\ n(\\rho_\\lambda),\\ n(\\rho_\\lambda))\\), and consequently there is an algorithm to compute \\(\\mathcal{L}(\\rho_\\lambda,T)\\) with running time roughly exponential in \\(n(\\rho_\\lambda)\\). If, in addition, \\(\\rho_\\lambda\\) is self-dual of weight \\(w(\\rho_\\lambda)\\) and sign \\(c(\\rho_\\lambda)\\in\\{\\operatorname{id},\\operatorname{cc}\\}\\), then: (i) whether or not \\(\\epsilon(\\rho_\\lambda)\\) is known, Algorithm \\(\\mathrm{functionalequation}\\) outputs \\(N(\\rho_\\lambda,T)\\in 1+T\\,F_\\lambda[T]\\) from the inputs \\((\\mathcal{L}_\\bullet(\\rho_\\lambda,T),\\ D(\\rho_\\lambda,T),\\ n(\\rho_\\lambda),\\ 1/q^{w(\\rho_\\lambda)+1},\\ c(\\rho_\\lambda))\\), so there is an algorithm to compute \\(\\mathcal{L}(\\rho_\\lambda,T)\\) with running time roughly exponential in \\(\\lfloor n(\\rho_\\lambda)/2\\rfloor\\); and (ii) Algorithm \\(\\mathrm{computeepsilon}\\) outputs \\(\\epsilon(\\rho_\\lambda)\\in F_\\lambda^\\times\\) from the same inputs and therefore always has running time roughly exponential in \\(\\lfloor n(\\rho_\\lambda)/2\\rfloor\\)."}], "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": "functional-equation scaling factor \\(1/q^{w+1}\\)", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "running-time conclusions", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence on knowing \\(\\epsilon(\\rho_\\lambda)\\) versus computing it", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "requirement that \\(\\epsilon(\\rho_\\lambda)\\) be supplied to \\(\\mathrm{functionalequation}\\) and worst-case complexity in part (3)", "template_used": "wildcard"}]}} +{"id": "2601.17949v1", "paper_link": "http://arxiv.org/abs/2601.17949v1", "theorems_cnt": 1, "theorem": {"env_name": "thm", "content": "[{\\cite[Conjecture~1.4]{kvec-depth}}] \\label{thm:qt-sym-lodestar}\n We define $\\tC_{a,M,b}(q, t)$ as a sum over Łukasiewicz{} paths $P$ with the first (resp. last) up-step of degree $a$ (resp. $b$) and $M$ the multiset of the degrees of other up-steps:\n \\[\n \\tC_{a,M,b}(q, t) = \\sum_P q^{\\mathtt{area}(P)} t^{\\mathtt{depth}(P)}.\n \\]\n Then $\\tC_{a,M,b}(q, t)$ is symmetric in $q, t$, \\emph{i.e.}, $\\tC_{a, M, b}(q, t) = \\tC_{a, M, b}(t, q)$.", "start_pos": 6397, "end_pos": 6842, "label": "thm:qt-sym-lodestar"}, "ref_dict": {"constr:luka-tree": "\\begin{constr} \\label{constr:luka-tree}\n Given a plane tree $T$, we perform a contour walk from left to right, starting from the root, and transcribe the nodes into steps upon their first encounter: $D$ if it is a leaf, and $U_k$ if it is an internal node of degree $k + 1$. The lattice path thus obtained, denoted by $\\treetoluka(T)$, is a \\luka{} path.\n\n Conversely, given a \\luka{} path $P$, starting with an empty positioning called \\emph{bud}, we may construct a tree by reading $P$ from left to right: when reading an up-step $U_k$ (resp. down-step $D$), we replace the leftmost empty bud by an internal node with $k + 1$ empty buds below (resp. a leaf). The plane tree is denoted by $\\lukatotree(P)$. It is clear that $\\treetoluka$ and $\\lukatotree$ are bijections between $\\lukaset$ and $\\trees$, and are inverse of each other.\n\\end{constr}", "fig:luka": "\\begin{figure}\n \\centering\n \\insertfig{1}{1}\n \\caption{Example of a \\luka{} path, with its statistics $\\area$ and $\\depth$.}\n \\label{fig:luka}\n\\end{figure}", "prop:thorns-sym": "\\begin{prop} \\label{prop:thorns-sym}\n For a plane tree $T$, let $T' = \\mirror(T)$, then $\\lthorn(T) = \\rthorn(T')$ and $\\rthorn(T) = \\lthorn(T')$.\n\\end{prop}"}, "pre_theorem_intro_text_len": 2959, "pre_theorem_intro_text": "\\label{sec:intro}\n\nThe study of $q,t$-Catalan polynomials $C(q, t)$ started from the two combinatorial formulas of the bi-graded Hilbert series of the subspace of alternants in the diagonal coinvariant space with two sets of variables, both describing Dyck paths of a given size with two jointly equi-distributed statistics: with $\\mathtt{area}$ and $\\mathtt{bounce}$ by Haglund \\cite{area-bounce}, and with $\\mathtt{dinv}$ and $\\mathtt{area}$ by Haiman. The two pairs of statistics are related by the famous zeta map \\cite{zeta-map} (see also \\cite{cataland} for a simpler definition). For more details, readers are referred to \\cite{haglund-book}.\n\nFrom the algebraic definition, it is trivial that the $q,t$-Catalan polynomials are symmetric in $q$ and $t$. However, on the combinatorial side, it is totally mysterious, with no bijective explanation known. In an effort to better understand combinatorially the related statistics, Pappe, Paul and Schilling proposed in \\cite{dyck-depth} another statistics $\\mathtt{depth}$ on Dyck paths and found that it is also jointly equi-distributed with $\\mathtt{area}$ by exhibiting an involution exchanging $\\mathtt{area}$ and $\\mathtt{depth}$. The $q,t$-polynomial $\\widetilde{C}(q, t)$ defined with $\\mathtt{area}$ and $\\mathtt{depth}$, which is different from the classical $C(q, t)$, is thus also $q,t$-symmetric.\n\nThere are many generalizations of the $q,t$-Catalan polynomials, usually defined on generalizations of Dyck paths, sometimes with an algebraic motivation, but not always. For instance, Xin and Zhang considered in \\cite{kvec-stats} the so-called \\emph{$\\vec{k}$-Dyck paths}, on which they introduced the three classical statistics $\\mathtt{dinv}$, $\\mathtt{area}$ and $\\mathtt{bounce}$, even if without direct algebraic supporting background. They then defined a generalized $q,t$-Catalan polynomial $C_\\lambda(q, t)$ on $\\vec{k}$-Dyck paths using these generalized statistics and studied their properties. Unfortunately, $C_\\lambda(q, t)$ is not $q,t$-symmetric in general.\n\nIn search for generalizations of the $q,t$-Catalan polynomials defined on $\\vec{k}$-Dyck paths with better $q,t$-symmetric properties, in a recent preprint \\cite{kvec-depth}, Qu and Zhang generalized the $\\mathtt{depth}$ statistics in \\cite{dyck-depth} to $\\vec{k}$-Dyck paths. They defined a generalization of the $\\mathtt{area}$-$\\mathtt{depth}$ $q, t$-polynomial $\\widetilde{C}(q, t)$ to $\\vec{k}$-Dyck paths with a given multiset $M$ of up-steps, denoted here by $\\tC_M(q, t)$. They also defined variants of $\\tC_M(q, t)$ with extra conditions of the first and/or the last up-step. They then showed bijectively that the variant of $\\tC_M(q, t)$ fixing the first up-step is $q, t$-symmetric, and they conjectured that it remains $q, t$-symmetric even when also fixing the last up-step. The main result of this short note is a proof of this conjecture, with precise definitions of notations postponed to later sections.", "context": "\\label{sec:intro}\n\nThe study of $q,t$-Catalan polynomials $C(q, t)$ started from the two combinatorial formulas of the bi-graded Hilbert series of the subspace of alternants in the diagonal coinvariant space with two sets of variables, both describing Dyck paths of a given size with two jointly equi-distributed statistics: with $\\mathtt{area}$ and $\\mathtt{bounce}$ by Haglund \\cite{area-bounce}, and with $\\mathtt{dinv}$ and $\\mathtt{area}$ by Haiman. The two pairs of statistics are related by the famous zeta map \\cite{zeta-map} (see also \\cite{cataland} for a simpler definition). For more details, readers are referred to \\cite{haglund-book}.\n\nFrom the algebraic definition, it is trivial that the $q,t$-Catalan polynomials are symmetric in $q$ and $t$. However, on the combinatorial side, it is totally mysterious, with no bijective explanation known. In an effort to better understand combinatorially the related statistics, Pappe, Paul and Schilling proposed in \\cite{dyck-depth} another statistics $\\mathtt{depth}$ on Dyck paths and found that it is also jointly equi-distributed with $\\mathtt{area}$ by exhibiting an involution exchanging $\\mathtt{area}$ and $\\mathtt{depth}$. The $q,t$-polynomial $\\widetilde{C}(q, t)$ defined with $\\mathtt{area}$ and $\\mathtt{depth}$, which is different from the classical $C(q, t)$, is thus also $q,t$-symmetric.\n\nThere are many generalizations of the $q,t$-Catalan polynomials, usually defined on generalizations of Dyck paths, sometimes with an algebraic motivation, but not always. For instance, Xin and Zhang considered in \\cite{kvec-stats} the so-called \\emph{$\\vec{k}$-Dyck paths}, on which they introduced the three classical statistics $\\mathtt{dinv}$, $\\mathtt{area}$ and $\\mathtt{bounce}$, even if without direct algebraic supporting background. They then defined a generalized $q,t$-Catalan polynomial $C_\\lambda(q, t)$ on $\\vec{k}$-Dyck paths using these generalized statistics and studied their properties. Unfortunately, $C_\\lambda(q, t)$ is not $q,t$-symmetric in general.\n\nIn search for generalizations of the $q,t$-Catalan polynomials defined on $\\vec{k}$-Dyck paths with better $q,t$-symmetric properties, in a recent preprint \\cite{kvec-depth}, Qu and Zhang generalized the $\\mathtt{depth}$ statistics in \\cite{dyck-depth} to $\\vec{k}$-Dyck paths. They defined a generalization of the $\\mathtt{area}$-$\\mathtt{depth}$ $q, t$-polynomial $\\widetilde{C}(q, t)$ to $\\vec{k}$-Dyck paths with a given multiset $M$ of up-steps, denoted here by $\\tC_M(q, t)$. They also defined variants of $\\tC_M(q, t)$ with extra conditions of the first and/or the last up-step. They then showed bijectively that the variant of $\\tC_M(q, t)$ fixing the first up-step is $q, t$-symmetric, and they conjectured that it remains $q, t$-symmetric even when also fixing the last up-step. The main result of this short note is a proof of this conjecture, with precise definitions of notations postponed to later sections.", "full_context": "\\label{sec:intro}\n\nThe study of $q,t$-Catalan polynomials $C(q, t)$ started from the two combinatorial formulas of the bi-graded Hilbert series of the subspace of alternants in the diagonal coinvariant space with two sets of variables, both describing Dyck paths of a given size with two jointly equi-distributed statistics: with $\\mathtt{area}$ and $\\mathtt{bounce}$ by Haglund \\cite{area-bounce}, and with $\\mathtt{dinv}$ and $\\mathtt{area}$ by Haiman. The two pairs of statistics are related by the famous zeta map \\cite{zeta-map} (see also \\cite{cataland} for a simpler definition). For more details, readers are referred to \\cite{haglund-book}.\n\nFrom the algebraic definition, it is trivial that the $q,t$-Catalan polynomials are symmetric in $q$ and $t$. However, on the combinatorial side, it is totally mysterious, with no bijective explanation known. In an effort to better understand combinatorially the related statistics, Pappe, Paul and Schilling proposed in \\cite{dyck-depth} another statistics $\\mathtt{depth}$ on Dyck paths and found that it is also jointly equi-distributed with $\\mathtt{area}$ by exhibiting an involution exchanging $\\mathtt{area}$ and $\\mathtt{depth}$. The $q,t$-polynomial $\\widetilde{C}(q, t)$ defined with $\\mathtt{area}$ and $\\mathtt{depth}$, which is different from the classical $C(q, t)$, is thus also $q,t$-symmetric.\n\nThere are many generalizations of the $q,t$-Catalan polynomials, usually defined on generalizations of Dyck paths, sometimes with an algebraic motivation, but not always. For instance, Xin and Zhang considered in \\cite{kvec-stats} the so-called \\emph{$\\vec{k}$-Dyck paths}, on which they introduced the three classical statistics $\\mathtt{dinv}$, $\\mathtt{area}$ and $\\mathtt{bounce}$, even if without direct algebraic supporting background. They then defined a generalized $q,t$-Catalan polynomial $C_\\lambda(q, t)$ on $\\vec{k}$-Dyck paths using these generalized statistics and studied their properties. Unfortunately, $C_\\lambda(q, t)$ is not $q,t$-symmetric in general.\n\nIn search for generalizations of the $q,t$-Catalan polynomials defined on $\\vec{k}$-Dyck paths with better $q,t$-symmetric properties, in a recent preprint \\cite{kvec-depth}, Qu and Zhang generalized the $\\mathtt{depth}$ statistics in \\cite{dyck-depth} to $\\vec{k}$-Dyck paths. They defined a generalization of the $\\mathtt{area}$-$\\mathtt{depth}$ $q, t$-polynomial $\\widetilde{C}(q, t)$ to $\\vec{k}$-Dyck paths with a given multiset $M$ of up-steps, denoted here by $\\tC_M(q, t)$. They also defined variants of $\\tC_M(q, t)$ with extra conditions of the first and/or the last up-step. They then showed bijectively that the variant of $\\tC_M(q, t)$ fixing the first up-step is $q, t$-symmetric, and they conjectured that it remains $q, t$-symmetric even when also fixing the last up-step. The main result of this short note is a proof of this conjecture, with precise definitions of notations postponed to later sections.\n\n\\abstract{ In an effort to further understanding $q,t$-Catalan statistics, a new statistic on Dyck paths called $\\depth$ was proposed in Pappe, Paul and Schilling (2022) and was shown to be jointly equi-distributed with the well-known $\\area$ statistics. In a recent preprint, Qu and Zhang (2025) generalized $\\depth$ to so-called ``$\\vec{k}$-Dyck paths''. They showed that $\\area$ and $\\depth$ are also jointly equi-distributed over such paths with a fixed multiset of up-steps and a given first up-step, and they conjectured that the same holds when also fixing the last up-step. In this short note, we settle this conjecture on the more general context of \\luka{} paths by interpreting $\\area$ and $\\depth$ under the classical bijection between \\luka{} paths and plane trees, through which the symmetry is transparent.\n}\n\nIn search for generalizations of the $q,t$-Catalan polynomials defined on $\\vec{k}$-Dyck paths with better $q,t$-symmetric properties, in a recent preprint \\cite{kvec-depth}, Qu and Zhang generalized the $\\depth$ statistics in \\cite{dyck-depth} to $\\vec{k}$-Dyck paths. They defined a generalization of the $\\area$-$\\depth$ $q, t$-polynomial $\\tC(q, t)$ to $\\vec{k}$-Dyck paths with a given multiset $M$ of up-steps, denoted here by $\\tC_M(q, t)$. They also defined variants of $\\tC_M(q, t)$ with extra conditions of the first and/or the last up-step. They then showed bijectively that the variant of $\\tC_M(q, t)$ fixing the first up-step is $q, t$-symmetric, and they conjectured that it remains $q, t$-symmetric even when also fixing the last up-step. The main result of this short note is a proof of this conjecture, with precise definitions of notations postponed to later sections.\n\nOur proof relies essentially on the observation that both statistics $\\area$ and $\\bounce$ on \\luka{} paths can be interpreted naturally over the classical bijection between \\luka{} paths and plane trees.\n\n\\tdef{\\luka{} paths} are lattices paths in $\\mathbb{Z}^2$ with steps of the form $(1, k)$ with $k \\geq -1$ that go under the $x$-axis only at the last step, which is always $(1, -1)$. We denote by $\\lukaset$ the set of \\luka{} paths. \\Cref{fig:luka} shows an example of \\luka{} paths. For the ease of notation, we denote by $D$ the only down-step $(1, -1)$, and $U_k$ the up-step $(1, k)$ for $k \\geq 0$, and we say that $U_k$ is of \\tdef{degree} $k$. The \\tdef{degree profile} (or simply \\tdef{profile}) of a \\luka{} path $P$, denoted by $\\profile(P)$, is the sequence of degrees of up-steps in $P$ from left to right. In this case, the \\tdef{profile multiset} of $P$, denoted by $\\profmset(P)$ is the multiset of entries in $\\profile(P)$. For a finite multiset $M$ of natural numbers, we denote by $M_k$ the multiplicity of $k$ in $M$, and we write $M = [0^{M_0}, 1^{M_1}, 2^{M_2}, \\dots]$ while only listing elements in $M$. We note that $\\vec{k}$-Dyck paths, the family of paths studied in \\cite{kvec-depth}, first introduced in \\cite{kvec-sweep}, are simply \\luka{} paths with $\\profile(P) = \\vec{k}$ without any $0$, with the last down-step removed.\n\nNow we define the $q,t$-polynomial associated to $\\area$ and $\\depth$. We denote the concatenation of two vectors $P, Q$ by $P \\cdot Q$, and the union of two multisets $L, M$ by $L \\uplus M$. We further give four notations of subsets of \\luka{} paths:\n\\begin{itemize}\n\\item $\\lukaset_M$: the set of \\luka{} paths $P$ with $\\profmset(P) = M$;\n\\item $\\lukaset_{a, M}$: the set of \\luka{} paths $P$ with $U_a$ as the first up-step, and $\\profmset(P) = M \\uplus \\{a\\}$;\n\\item $\\lukaset_{M, b}$: the set of \\luka{} paths $P$ with $U_b$ as the last up-step, and $\\profmset(P) = M \\uplus \\{b\\}$;\n\\item $\\lukaset_{a, M, b}$: the set of \\luka{} paths $P$ with $U_a$ (resp. $U_b$) as the first (resp. last) up-step, and $\\profmset(P) = M \\uplus \\{a, b\\}$, with $\\{a, b\\}$ understood as a multiset.\n\\end{itemize}\nWe then define\n\\begin{align}\n \\begin{split}\n \\label{eq:qt-poly-def}\n \\tC_{M}(q, t) = \\sum_{P \\in \\lukaset_{M}} q^{\\area(P)} t^{\\depth(P)}, &\\quad \\tC_{a, M}(q, t) = \\sum_{P \\in \\lukaset_{a, M}} q^{\\area(P)} t^{\\depth(P)}, \\\\\n \\tC_{M, b}(q, t) = \\sum_{P \\in \\lukaset_{M, b}} q^{\\area(P)} t^{\\depth(P)}, &\\quad \\tC_{a, M, b}(q, t) = \\sum_{P \\in \\lukaset_{a, M, b}} q^{\\area(P)} t^{\\depth(P)}.\n \\end{split}\n\\end{align}\n\n\\begin{thm}[{\\cite[Theorem~1.1]{kvec-depth}}] \\label{prop:qt-sym-plain}\n For any $a \\in \\mathbb{N}$ and multiset $M$ with elements in $\\mathbb{N}$, we have $\\tC_{a, M}(q, t) = \\tC_{a, M}(t, q)$.\n\\end{thm}\n\\begin{proof}\n Let $P$ be a \\luka{} path, and $T = \\lukatotree(P)$. From \\Cref{constr:luka-tree}, $P \\in \\lukaset_{a, M}$ if and only if $T$ has a root of degree $a + 1$, and the multiset of the degrees of non-root internal nodes is $M' = [1^{M_0}, 2^{M_1},\\dots]$. Thus $P' = \\treetoluka(T')$ with $T' = \\mirror(T)$ is also in $\\lukaset_{a, M}$ if and only if $P \\in \\lukaset_{a, M}$, meaning that $\\treetoluka \\circ \\mirror \\circ \\lukatotree$ is an involution on $\\lukaset_{a, M}$. We then conclude by \\Cref{eq:qt-poly-def,prop:area-rthorn,prop:depth-lthorn,prop:thorns-sym}.\n\\end{proof}\n\n\\begin{coro}[{\\cite[Conjecture~1.4]{kvec-depth}}] \\label{coro:qt-sym-lodestar}\n For any $b \\in \\mathbb{N}$ and multiset $M$ with elements in $\\mathbb{N}$, we have $\\tC_{M, b}(q, t) = \\tC_{M, b}(t, q)$.\n\\end{coro}\n\\begin{proof}\n It is clear that we may write $\\tC_{M, b}$ as a sum of $\\tC_{a, M' ,b}$ where $a$ runs over elements of $M$ and $M' = M \\setminus \\{a\\}$. We then conclude by \\Cref{thm:qt-sym-lodestar}.\n\\end{proof}\n\n\\begin{rmk}\n We consider the following generating function for the two statistics $\\area$ and $\\depth$ on \\luka{} paths refined by their profile multiset. Let $F(z, q, t; p_0, p_1, \\ldots) \\equiv F(z, q, t)$ be the generating function defined as\n \\[\n F(z, q, t) = \\sum_{M \\text{ multiset}} z^{|M|} \\sum_{P \\in \\lukaset_{M}} q^{\\area(P)} t^{\\depth(P)} \\prod_{k \\geq 0} p_k^{M_k}.\n \\]\n Here, $|M|$ denotes the number of elements in the multiset $M$ counted with multiplicity. Through the lens of \\Cref{constr:luka-tree}, using \\Cref{prop:area-rthorn,prop:depth-lthorn}, by decomposing plane trees at the root and using the standard symbolic method (\\emph{cf.} \\cite{flajolet}), we have\n \\begin{equation}\n \\label{eq:fct-eq}\n F(z, q, t) = 1 + \\sum_{k \\geq 0} p_k \\prod_{\\ell = 1}^{k} F(z q^{k-\\ell} t^{\\ell - 1}, q, t).\n \\end{equation}\n Here, the first term $1$ stands for the empty tree, and the second term for trees with a root. The degree of the root is given by $k + 1$. For a node in the sub-tree induced by the $\\ell$-th child of the root, its $\\lthorn$ (resp. $\\rthorn$) increments by $\\ell - 1$ (resp. $k - \\ell$). This explains the substitution of $z$, which counts internal nodes, by $z q^{k - \\ell} t^{\\ell - 1}$.\n\\end{rmk}\n\n\\begin{constr} \\label{constr:luka-tree}\n Given a plane tree $T$, we perform a contour walk from left to right, starting from the root, and transcribe the nodes into steps upon their first encounter: $D$ if it is a leaf, and $U_k$ if it is an internal node of degree $k + 1$. The lattice path thus obtained, denoted by $\\treetoluka(T)$, is a \\luka{} path.\n\n Conversely, given a \\luka{} path $P$, starting with an empty positioning called \\emph{bud}, we may construct a tree by reading $P$ from left to right: when reading an up-step $U_k$ (resp. down-step $D$), we replace the leftmost empty bud by an internal node with $k + 1$ empty buds below (resp. a leaf). The plane tree is denoted by $\\lukatotree(P)$. It is clear that $\\treetoluka$ and $\\lukatotree$ are bijections between $\\lukaset$ and $\\trees$, and are inverse of each other.\n\\end{constr}\n\n\\begin{figure}\n \\centering\n \\insertfig{1}{1}\n \\caption{Example of a \\luka{} path, with its statistics $\\area$ and $\\depth$.}\n \\label{fig:luka}\n\\end{figure}", "post_theorem_intro_text_len": 757, "post_theorem_intro_text": "Our proof relies essentially on the observation that both statistics $\\mathtt{area}$ and $\\mathtt{bounce}$ on Łukasiewicz{} paths can be interpreted naturally over the classical bijection between Łukasiewicz{} paths and plane trees.\n\nThe rest of this article is organized as follows. We lay down the definitions of Łukasiewicz{} paths, the statistics $\\mathtt{area}$ and $\\mathtt{depth}$ and the related polynomials in \\Cref{fig:luka}. Then we revisit the classical bijection between Łukasiewicz{} paths and plane trees in \\Cref{constr:luka-tree}, along with how $\\mathtt{area}$ and $\\mathtt{depth}$ are transferred to plane trees. Finally, in \\Cref{prop:thorns-sym} we show how to obtain $q,t$-symmetry results using simple bijections in a transparent way.", "sketch": "The proof of Theorem~\\ref{thm:qt-sym-lodestar} \\\"relies essentially on the observation that both statistics $\\mathtt{area}$ and $\\mathtt{bounce}$ on \\L{}ukasiewicz{} paths can be interpreted naturally over the classical bijection between \\L{}ukasiewicz{} paths and plane trees.\\\" The argument proceeds by (i) setting up the definitions of \\L{}ukasiewicz{} paths and the statistics $\\mathtt{area}$ and $\\mathtt{depth}$ (\\Cref{fig:luka}), (ii) revisiting the classical bijection to plane trees and explaining \\\"how $\\mathtt{area}$ and $\\mathtt{depth}$ are transferred to plane trees\\\" (\\Cref{constr:luka-tree}), and (iii) using \\\"simple bijections\\\" on the tree side to \\\"obtain $q,t$-symmetry results\\\" (\\Cref{prop:thorns-sym}).", "expanded_sketch": "The proof of Theorem~\\ref{thm:qt-sym-lodestar} relies essentially on the observation that both statistics $\\mathtt{area}$ and $\\mathtt{bounce}$ on \\L{}ukasiewicz{} paths can be interpreted naturally over the classical bijection between \\L{}ukasiewicz{} paths and plane trees. The argument proceeds by (i) setting up the definitions of \\L{}ukasiewicz{} paths and the statistics $\\mathtt{area}$ and $\\mathtt{depth}$:\n\n\\begin{figure}\n \\centering\n \\insertfig{1}{1}\n \\caption{Example of a \\luka{} path, with its statistics $\\area$ and $\\depth$.}\n \\label{fig:luka}\n\\end{figure}\n\n(ii) revisiting the classical bijection to plane trees and explaining how $\\mathtt{area}$ and $\\mathtt{depth}$ are transferred to plane trees:\n\n\\begin{constr} \\label{constr:luka-tree}\n Given a plane tree $T$, we perform a contour walk from left to right, starting from the root, and transcribe the nodes into steps upon their first encounter: $D$ if it is a leaf, and $U_k$ if it is an internal node of degree $k + 1$. The lattice path thus obtained, denoted by $\\treetoluka(T)$, is a \\luka{} path.\n\n Conversely, given a \\luka{} path $P$, starting with an empty positioning called \\emph{bud}, we may construct a tree by reading $P$ from left to right: when reading an up-step $U_k$ (resp. down-step $D$), we replace the leftmost empty bud by an internal node with $k + 1$ empty buds below (resp. a leaf). The plane tree is denoted by $\\lukatotree(P)$. It is clear that $\\treetoluka$ and $\\lukatotree$ are bijections between $\\lukaset$ and $\\trees$, and are inverse of each other.\n\\end{constr}\n\nand (iii) using simple bijections on the tree side to obtain $q,t$-symmetry results, based on the following property: for a plane tree $T$, let $T' = \\mirror(T)$, then $\\lthorn(T) = \\rthorn(T')$ and $\\rthorn(T) = \\lthorn(T')$.", "expanded_theorem": "[{\\cite[Conjecture~1.4]{kvec-depth}}] \\label{thm:qt-sym-lodestar}\n We define $\\tC_{a,M,b}(q, t)$ as a sum over Łukasiewicz{} paths $P$ with the first (resp. last) up-step of degree $a$ (resp. $b$) and $M$ the multiset of the degrees of other up-steps:\n \\[\n \\tC_{a,M,b}(q, t) = \\sum_P q^{\\mathtt{area}(P)} t^{\\mathtt{depth}(P)}.\n \\]\n Then $\\tC_{a,M,b}(q, t)$ is symmetric in $q, t$, \\emph{i.e.}, $\\tC_{a, M, b}(q, t) = \\tC_{a, M, b}(t, q)$.,", "theorem_type": ["Universal", "Biconditional or Equivalence"], "mcq": {"question": "For fixed nonnegative integers \\(a\\) and \\(b\\) and a multiset \\(M\\) of nonnegative integers, define\n\\[\n\\widetilde C_{a,M,b}(q,t)=\\sum_P q^{\\operatorname{area}(P)}t^{\\operatorname{depth}(P)},\n\\]\nwhere the sum runs over Łukasiewicz paths \\(P\\) whose first up-step has degree \\(a\\), whose last up-step has degree \\(b\\), and whose other up-step degrees form the multiset \\(M\\). Which statement is equivalent to saying that \\(\\widetilde C_{a,M,b}(q,t)\\) is symmetric in \\(q\\) and \\(t\\)?", "correct_choice": {"label": "A", "text": "\\(\\widetilde C_{a,M,b}(q,t)=\\widetilde C_{a,M,b}(t,q)\\)."}, "choices": [{"label": "B", "text": "\\(\\widetilde C_{a,M,b}(q,t)=\\widetilde C_{b,M,a}(t,q)\\)."}, {"label": "C", "text": "\\(\\widetilde C_{a,M,b}(q,q)=\\widetilde C_{a,M,b}(q,q)\\)."}, {"label": "D", "text": "\\(\\widetilde C_{a,M,b}(q,t)=\\widetilde C_{a,M,b}(t,q)\\) only after summing over all possible last up-step degrees \\(b\\)."}, {"label": "E", "text": "\\(\\widetilde C_{a,M,b}(q,t)=\\widetilde C_{a,M,b}(q^{-1},t^{-1})\\)."}], "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": "fixed-endpoint degrees preserved under symmetry", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "exchange of the two variables", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "geometric_construction", "tampered_component": "symmetry holds with both first and last up-step fixed", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "symmetry means swapping variables, not inversion", "template_used": "wildcard"}]}} +{"id": "2601.18153v1", "paper_link": "http://arxiv.org/abs/2601.18153v1", "theorems_cnt": 1, "theorem": {"env_name": "thm", "content": "\\label{intro-main-thm}\n Let $\\mathbb{k}$ be an infinite field and $X=(x_{ij})_{m \\times n}$ be a matrix of indeterminates. Suppose that $S$ denotes the polynomial ring $\\mathbb{k}[x_{ij} \\mid 1 \\leq i \\leq m, 1\\leq j \\leq n]$, and $I$ is a $2$-subdeterminantal ideal of $X$. Then the following are equivalent:\n \\begin{enumerate}\n \\item[{\\rm (i)}] The ring $S/I$ is Golod.\n \\item[{\\rm (ii)}] $I$ has a linear resolution over $S$.\n \\item[{\\rm (iii)}] $I=I_2(Y)$ for some $2 \\times \\ell$ or $\\ell \\times 2$ submatrix $Y$ of $X$.\n \\item[{\\rm (iv)}] The product on the Koszul homology $\\mathcal H(\\mathcal K (x_{11}, \\ldots,x_{mn}; S/I))$ is trivial.\n \\end{enumerate}", "start_pos": 10012, "end_pos": 10733, "label": "intro-main-thm"}, "ref_dict": {"intro-main-thm": "\\begin{thm}\\label{intro-main-thm}\n Let $\\mathbb{k}$ be an infinite field and $X=(x_{ij})_{m \\times n}$ be a matrix of indeterminates. Suppose that $S$ denotes the polynomial ring $\\mathbb{k}[x_{ij} \\mid 1 \\leq i \\leq m, 1\\leq j \\leq n]$, and $I$ is a $2$-subdeterminantal ideal of $X$. Then the following are equivalent:\n \\begin{enumerate}\n \\item[{\\rm (i)}] The ring $S/I$ is Golod.\n \\item[{\\rm (ii)}] $I$ has a linear resolution over $S$.\n \\item[{\\rm (iii)}] $I=I_2(Y)$ for some $2 \\times \\ell$ or $\\ell \\times 2$ submatrix $Y$ of $X$.\n \\item[{\\rm (iv)}] The product on the Koszul homology $\\mathcal H(\\mathcal K (x_{11}, \\ldots,x_{mn}; S/I))$ is trivial.\n \\end{enumerate} \n\\end{thm}", "def:subdet-ideal": "\\begin{defn}\\label{def:subdet-ideal}\n Given any $t\\geq 1$, an ideal $I$ of $\\mathbb{k}[X]$ is called a \\emph{$t$-subdeterminantal ideal of $X$}, if $I$ is generated by a subset of the set of all $t \\times t$ minors of $X$. \n Also, we use $I_t(X)$ to denote the ideal generated by all $t \\times t$ minors of $X$. \n\\end{defn}"}, "pre_theorem_intro_text_len": 3316, "pre_theorem_intro_text": "Let $S=\\mathbb{k}[x_1, \\ldots, x_n]$ be a standard graded polynomial ring over a field $\\mathbb{k}$, $I\\subseteq (x_1, \\ldots, x_n)^2$ a homogeneous ideal, and $R=S/I$. The power series $\\mathcal{P}_{\\mathbb{k}}^R(z)= \\sum_{i \\geq 0} \\dim_{\\mathbb{k}} \\left(\\Tor_i^R(\\mathbb{k},\\mathbb{k})\\,\\right)z^i$ is called the \\textit{Poincar\\'e series of $\\mathbb{k}$ over $R$}. Due to a result of Serre \\cite{Se65}, there is a term-by-term inequality\n\\begin{align*}\n \\mathcal{P}_{\\mathbb{k}}^R(z) \\preceq \\dfrac{(1+z)^n}{1-\\sum\\limits_{i \\geq 1}\\dim_{\\mathbb{k}}\\left(\\Tor_i^S(R,\\mathbb{k})\\right)\\, z^{i+1}}. \n\\end{align*}\nIn \\cite{Go62}, Golod proved that that the equality holds if and only if all Massey operations on the homology of the Koszul complex $\\mathcal K(x_1, \\ldots, x_n; R)$ are trivial; in this case, the ring $R$ is called \\textit{Golod}. \n\nDetecting Golodness is a rather hard problem. For instance, we do not have a complete characterization of Golodness of $S/I$, even when $I$ is a monomial ideal. Attempts have been made to characterize Golodness of monomial quotients, especially those arising from combinatorial objects, such as Stanley--Reisner rings of simplicial complexes. Golod ideals over polynomial rings in at most 4 variables have also been characterized. For a non-exhaustive list of works on detecting Golodness and related criteria, we refer the interested reader to \n\\cite{Ah19,ANFY17,BJ07,DS22,DS16,Fr18,GTSW16,HH13,HRW99,IK18,IK23,IK24,Ka16,Ka17,FW14,Va22,Zo24}\n and the references therein.\n\nWhen $R$ is Golod, the triviality of all Massey operations forces the Koszul homology\n$\\mathcal H(\\mathcal K(x_1, \\ldots, x_n; R))$ to have trivial product. However, in general, \nthis condition is not sufficient for Golodness, even for monomial quotients, as the examples due to De Stefani \\cite{DS16} and Katth\\\"an \\cite{Ka17} show. \nIn some special cases, the triviality of the product on Koszul homology is indeed equivalent to Golodness. \nFor instance, if $I$ is a monomial ideal generated in degree two, then combining the results proved in \\cite{BJ07,HerzogHibiZheng, HRW99}, \nit follows that $S/I$ is Golod if and only if $I$ has a linear resolution if and only if the product on the Koszul homology is trivial. \nThe main result of our paper is in the same spirit. \n\nWe consider ideals generated by subsets of the set of all $2 \\times 2$ minors of a matrix $X=(x_{ij})_{m \\times n}$ of indeterminates, which we call \\textit{$2$-subdeterminantal ideals of $X$} (see Definition \\ref{def:subdet-ideal}). \nIn the literature, these ideals have also been referred to as \\textit{generalized binomial edge ideals} or \\textit{binomial edge ideals of a pair of graphs} (see \\cite{EHHQ14}). \nDue to this identification, these ideals are of interest in combinatorial commutative algebra, and several homological properties of the associated quotient rings, such as the Castelnuovo--Mumford regularity, Koszulness, existence of quadratic Gr\\\"obner bases, etc.~have been studied in recent years (see, e.g.,~\\cite{BEI18, EHHQ14, JK25, LMMP26}). \nThe class of ideals we consider also includes the classical determinantal ideals as a special case (see \\cite{BV88}). \nOur aim in this article is to provide a complete characterization of the Golodness of these ideals. \nOur main theorem is as follows.", "context": "Let $S=\\mathbb{k}[x_1, \\ldots, x_n]$ be a standard graded polynomial ring over a field $\\mathbb{k}$, $I\\subseteq (x_1, \\ldots, x_n)^2$ a homogeneous ideal, and $R=S/I$. The power series $\\mathcal{P}_{\\mathbb{k}}^R(z)= \\sum_{i \\geq 0} \\dim_{\\mathbb{k}} \\left(\\Tor_i^R(\\mathbb{k},\\mathbb{k})\\,\\right)z^i$ is called the \\textit{Poincar\\'e series of $\\mathbb{k}$ over $R$}. Due to a result of Serre \\cite{Se65}, there is a term-by-term inequality\n\\begin{align*}\n \\mathcal{P}_{\\mathbb{k}}^R(z) \\preceq \\dfrac{(1+z)^n}{1-\\sum\\limits_{i \\geq 1}\\dim_{\\mathbb{k}}\\left(\\Tor_i^S(R,\\mathbb{k})\\right)\\, z^{i+1}}. \n\\end{align*}\nIn \\cite{Go62}, Golod proved that that the equality holds if and only if all Massey operations on the homology of the Koszul complex $\\mathcal K(x_1, \\ldots, x_n; R)$ are trivial; in this case, the ring $R$ is called \\textit{Golod}.\n\nDetecting Golodness is a rather hard problem. For instance, we do not have a complete characterization of Golodness of $S/I$, even when $I$ is a monomial ideal. Attempts have been made to characterize Golodness of monomial quotients, especially those arising from combinatorial objects, such as Stanley--Reisner rings of simplicial complexes. Golod ideals over polynomial rings in at most 4 variables have also been characterized. For a non-exhaustive list of works on detecting Golodness and related criteria, we refer the interested reader to \n\\cite{Ah19,ANFY17,BJ07,DS22,DS16,Fr18,GTSW16,HH13,HRW99,IK18,IK23,IK24,Ka16,Ka17,FW14,Va22,Zo24}\n and the references therein.\n\nWhen $R$ is Golod, the triviality of all Massey operations forces the Koszul homology\n$\\mathcal H(\\mathcal K(x_1, \\ldots, x_n; R))$ to have trivial product. However, in general, \nthis condition is not sufficient for Golodness, even for monomial quotients, as the examples due to De Stefani \\cite{DS16} and Katth\\\"an \\cite{Ka17} show. \nIn some special cases, the triviality of the product on Koszul homology is indeed equivalent to Golodness. \nFor instance, if $I$ is a monomial ideal generated in degree two, then combining the results proved in \\cite{BJ07,HerzogHibiZheng, HRW99}, \nit follows that $S/I$ is Golod if and only if $I$ has a linear resolution if and only if the product on the Koszul homology is trivial. \nThe main result of our paper is in the same spirit.\n\nWe consider ideals generated by subsets of the set of all $2 \\times 2$ minors of a matrix $X=(x_{ij})_{m \\times n}$ of indeterminates, which we call \\textit{$2$-subdeterminantal ideals of $X$} (see Definition \\ref{def:subdet-ideal}). \nIn the literature, these ideals have also been referred to as \\textit{generalized binomial edge ideals} or \\textit{binomial edge ideals of a pair of graphs} (see \\cite{EHHQ14}). \nDue to this identification, these ideals are of interest in combinatorial commutative algebra, and several homological properties of the associated quotient rings, such as the Castelnuovo--Mumford regularity, Koszulness, existence of quadratic Gr\\\"obner bases, etc.~have been studied in recent years (see, e.g.,~\\cite{BEI18, EHHQ14, JK25, LMMP26}). \nThe class of ideals we consider also includes the classical determinantal ideals as a special case (see \\cite{BV88}). \nOur aim in this article is to provide a complete characterization of the Golodness of these ideals. \nOur main theorem is as follows.\n\n\\begin{defn}\\label{def:subdet-ideal}\n Given any $t\\geq 1$, an ideal $I$ of $\\mathbb{k}[X]$ is called a \\emph{$t$-subdeterminantal ideal of $X$}, if $I$ is generated by a subset of the set of all $t \\times t$ minors of $X$. \n Also, we use $I_t(X)$ to denote the ideal generated by all $t \\times t$ minors of $X$. \n\\end{defn}", "full_context": "Let $S=\\mathbb{k}[x_1, \\ldots, x_n]$ be a standard graded polynomial ring over a field $\\mathbb{k}$, $I\\subseteq (x_1, \\ldots, x_n)^2$ a homogeneous ideal, and $R=S/I$. The power series $\\mathcal{P}_{\\mathbb{k}}^R(z)= \\sum_{i \\geq 0} \\dim_{\\mathbb{k}} \\left(\\Tor_i^R(\\mathbb{k},\\mathbb{k})\\,\\right)z^i$ is called the \\textit{Poincar\\'e series of $\\mathbb{k}$ over $R$}. Due to a result of Serre \\cite{Se65}, there is a term-by-term inequality\n\\begin{align*}\n \\mathcal{P}_{\\mathbb{k}}^R(z) \\preceq \\dfrac{(1+z)^n}{1-\\sum\\limits_{i \\geq 1}\\dim_{\\mathbb{k}}\\left(\\Tor_i^S(R,\\mathbb{k})\\right)\\, z^{i+1}}. \n\\end{align*}\nIn \\cite{Go62}, Golod proved that that the equality holds if and only if all Massey operations on the homology of the Koszul complex $\\mathcal K(x_1, \\ldots, x_n; R)$ are trivial; in this case, the ring $R$ is called \\textit{Golod}.\n\nDetecting Golodness is a rather hard problem. For instance, we do not have a complete characterization of Golodness of $S/I$, even when $I$ is a monomial ideal. Attempts have been made to characterize Golodness of monomial quotients, especially those arising from combinatorial objects, such as Stanley--Reisner rings of simplicial complexes. Golod ideals over polynomial rings in at most 4 variables have also been characterized. For a non-exhaustive list of works on detecting Golodness and related criteria, we refer the interested reader to \n\\cite{Ah19,ANFY17,BJ07,DS22,DS16,Fr18,GTSW16,HH13,HRW99,IK18,IK23,IK24,Ka16,Ka17,FW14,Va22,Zo24}\n and the references therein.\n\nWhen $R$ is Golod, the triviality of all Massey operations forces the Koszul homology\n$\\mathcal H(\\mathcal K(x_1, \\ldots, x_n; R))$ to have trivial product. However, in general, \nthis condition is not sufficient for Golodness, even for monomial quotients, as the examples due to De Stefani \\cite{DS16} and Katth\\\"an \\cite{Ka17} show. \nIn some special cases, the triviality of the product on Koszul homology is indeed equivalent to Golodness. \nFor instance, if $I$ is a monomial ideal generated in degree two, then combining the results proved in \\cite{BJ07,HerzogHibiZheng, HRW99}, \nit follows that $S/I$ is Golod if and only if $I$ has a linear resolution if and only if the product on the Koszul homology is trivial. \nThe main result of our paper is in the same spirit.\n\nWe consider ideals generated by subsets of the set of all $2 \\times 2$ minors of a matrix $X=(x_{ij})_{m \\times n}$ of indeterminates, which we call \\textit{$2$-subdeterminantal ideals of $X$} (see Definition \\ref{def:subdet-ideal}). \nIn the literature, these ideals have also been referred to as \\textit{generalized binomial edge ideals} or \\textit{binomial edge ideals of a pair of graphs} (see \\cite{EHHQ14}). \nDue to this identification, these ideals are of interest in combinatorial commutative algebra, and several homological properties of the associated quotient rings, such as the Castelnuovo--Mumford regularity, Koszulness, existence of quadratic Gr\\\"obner bases, etc.~have been studied in recent years (see, e.g.,~\\cite{BEI18, EHHQ14, JK25, LMMP26}). \nThe class of ideals we consider also includes the classical determinantal ideals as a special case (see \\cite{BV88}). \nOur aim in this article is to provide a complete characterization of the Golodness of these ideals. \nOur main theorem is as follows.\n\n\\begin{defn}\\label{def:subdet-ideal}\n Given any $t\\geq 1$, an ideal $I$ of $\\mathbb{k}[X]$ is called a \\emph{$t$-subdeterminantal ideal of $X$}, if $I$ is generated by a subset of the set of all $t \\times t$ minors of $X$. \n Also, we use $I_t(X)$ to denote the ideal generated by all $t \\times t$ minors of $X$. \n\\end{defn}\n\n\\begin{abstract}\nLet $X=(x_{ij})_{m\\times n}$ be a matrix of indeterminates and let \n$S=\\mathbb{k}[x_{ij} \\mid 1\\le i\\le m,\\ 1\\le j\\le n]$ be a polynomial ring over an infinite field $\\mathbb{k}$. \nLet $I$ be an ideal generated by a subset of the set of all $2\\times2$ minors of $X$. \nWe show that the quotient ring $S/I$ is Golod if and only if $I=I_2(Y)$ for some $2\\times \\ell$ or $\\ell\\times2$ submatrix $Y$ of $X$. \nIn fact, we prove that Golodness of $S/I$ is equivalent to the triviality of the product on the Koszul homology of $S/I$ and to $I$ having a linear resolution.\nAlong the way, we also prove a result on the non-Golodness of tensor products of rings under certain conditions.\n\\end{abstract}\n\nWe consider ideals generated by subsets of the set of all $2 \\times 2$ minors of a matrix $X=(x_{ij})_{m \\times n}$ of indeterminates, which we call \\textit{$2$-subdeterminantal ideals of $X$} (see Definition \\ref{def:subdet-ideal}). \nIn the literature, these ideals have also been referred to as \\textit{generalized binomial edge ideals} or \\textit{binomial edge ideals of a pair of graphs} (see \\cite{EHHQ14}). \nDue to this identification, these ideals are of interest in combinatorial commutative algebra, and several homological properties of the associated quotient rings, such as the Castelnuovo--Mumford regularity, Koszulness, existence of quadratic Gr\\\"obner bases, etc.~have been studied in recent years (see, e.g.,~\\cite{BEI18, EHHQ14, JK25, LMMP26}). \nThe class of ideals we consider also includes the classical determinantal ideals as a special case (see \\cite{BV88}). \nOur aim in this article is to provide a complete characterization of the Golodness of these ideals. \nOur main theorem is as follows.\n\nOur proof of Theorem \\ref{intro-main-thm} proceeds in several steps. We begin with the case where $X$ is a $2 \\times n$ matrix, and then analyze the $3 \\times 3$ and $3 \\times n$ cases. These preliminary cases provide the foundation for the main argument. Along the way, we prove a general result on the non-Golodness of certain tensor products, which allows us to reduce the problem to a simpler setting. This result is in the same spirit as the classical statement that a Golod complete intersection must be a hypersurface. A key ingredient in the proof of our main theorem is the fact that for a Gorenstein local ring, the homology algebra of its Koszul complex is a Poincar\\'e algebra, as shown by Avramov and Golod \\cite{AG71}. Assuming that $\\mathbb{k}$ is infinite, the same result holds for standard graded Gorenstein $\\mathbb{k}$-algebras, which is the version we shall use.\n\nGiven a homogeneous ideal $I$ in a polynomial ring $S$, we let $\\mu(I)$ denote the cardinality of a minimal generating set of $I$. In other words, $\\mu(I)=\\beta_0^S(I)=\\beta_1^S(S/I)$.\n\\begin{rmk}\\label{rmk:Golod-def} {\\rm In this remark, we recall some known facts about Golodness which we shall use later. \n Let $S=\\mathbb{k}[x_1,\\ldots, x_n]$, $I$ be a homogeneous ideal of $S$, and $R=S/I$.\n\\begin{enumerate}[label={\\rm (\\alph*)}]\n \\item Since $\\beta_i^S(R)=\\dim_{\\mathbb{k}}\\left(\\Tor ^S_i(R, \\mathbb{k})\\right)$ and the Koszul complex $\\mathcal K(x_1, \\ldots, x_n; S)$ gives a minimal free resolution of $\\mathbb{k}$ over $S$, the term-by-term inequality due to Serre mentioned in Section~1 the can be rewritten as \n $$\\mathcal P^R_{\\mathbb{k}}(z) \\preceq \\dfrac{(1+z)^n}{1-z(\\mathcal P^S_R(z)-1)}.$$\n The ring $R$ is Golod if and only if the above inequality becomes an equality.\n \\item If $R$ is Golod, then the Koszul homology $\\mathcal H(\\mathcal K(x_1, \\ldots, x_n; R))$ has trivial product (see \\cite[Remark 5.2.1]{Av98}).\n\n\\begin{lemma}\\label{lem:2byn}\n Let $X = \\begin{bmatrix}\n x_1 & x_2 & \\cdots & x_n \\\\\n y_1 & y_2 & \\cdots & y_n\n\\end{bmatrix}$ be a $2 \\times n$ matrix of indeterminates, $S=\\mathbb{k}[X]$, and $I$ be a nonzero $2$-subdeterminantal ideal of $X$. Then the following are equivalent:\n\\begin{enumerate}\n \\item[{\\rm (i)}] The product on $\\mathcal H(\\mathcal K(x_1, \\ldots,x_n, y_1, \\ldots, y_n; S/I))$ is trivial.\n \\item[{\\rm (ii)}] $I=I_2(Y)$ for some $2 \\times \\ell$ submatrix $Y$ of $X$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n (ii) $\\Longrightarrow$ (i).\\ When $I=I_2(Y)$ for some $2 \\times \\ell$ matrix $Y$, the Eagon--Northcott complex (see \\cite{EN62}) gives a linear minimal free resolution of $I$ over $S$. From \\cite[Theorem 4]{HRW99}, we know that if $I$ is componentwise linear and contains no linear forms, then $S/I$ is Golod. Thus, by Remark \\ref{rmk:Golod-def}(b), we get that the product on $\\mathcal H(\\mathcal K(x_1, \\ldots,x_n, y_1, \\ldots, y_n; S/I))$ is trivial.\n\n\\begin{cor}\\label{cor:two-rows}\n Let $X=(x_{ij})_{m \\times n}$ be a matrix of indeterminates, $S=\\mathbb{k}[X]$, and let $I$ be a $2$-subdeterminantal ideal of $X$. Given any $1\\leq j_1< j_2\\leq n$, let $I^{j_1, j_2}$ be the ideal generated by the generators of $I$ of the form $$\\begin{vmatrix}\nx_{i_1j_1} & x_{i_1j_2} \\\\\nx_{i_2j_1} & x_{i_2j_2}\n\\end{vmatrix}.$$\nIf $\\mathcal{H}(\\mathcal K(x_{11},\\ldots, x_{mn}; S/I))$ has trivial product, then $I^{j_1, j_2} = I_2(Y)$ for some $\\ell \\times 2$ submatrix $Y$ of $$\\begin{bmatrix}\n x_{1j_1} & \\cdots & x_{mj_1} \\\\\n x_{1j_2} & \\cdots & x_{mj_2}\n\\end{bmatrix}^T.$$\n\nWe next focus on the $3 \\times 3$ case. \n\\begin{lemma}\\label{lem:3by3}\n Let $X=(x_{ij})_{3\\times 3}$ be a matrix of indeterminates, $S=\\mathbb{k}[X]$, and $I$ be a nonzero $2$-subdeterminantal ideal $X$.\nThen the following are equivalent:\n\\begin{enumerate}\n \\item[{\\rm (i)}] The product on $\\mathcal H(\\mathcal K(x_{11},\\ldots,x_{33}; S/I))$ is trivial.\n \\item[{\\rm (ii)}] $I=I_2(Y)$ for some $2\\times \\ell$ or $\\ell\\times 2$ submatrix $Y$ of $X$, where $\\ell=2$ or $3$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n As noted previously, if (ii) holds, then $S/I$ is Golod, and hence (i) holds.\n\n\\begin{cor}\\label{cor:3byn}\n Let $X=(x_{ij})_{3\\times n}$ be a matrix of indeterminates, $S=\\mathbb{k}[X]$, and $I$ be a nonzero $2$-subdeterminantal ideal of\n$X$.\nThen the following are equivalent:\n\\begin{enumerate}\n \\item[{\\rm (i)}] The product on $\\mathcal H(\\mathcal K(x_{11},\\ldots,x_{3n}; S/I))$ is trivial.\n \\item[{\\rm (ii)}] $I=I_2(Y)$ for some $2\\times \\ell$ or $\\ell\\times 2$ submatrix $Y$ of $X$ with $\\ell \\geq 2$.\n\\end{enumerate}\nReplacing $X$ by $X^T$, we see that an analogous statemtent holds for $n \\times 3$ matrix of indeterminates.\n\\end{cor} \n\\begin{proof}\n As noted before, (ii) $\\Longrightarrow$ (i) is true. Conversely, suppose that (i) holds. Recall that the Koszul complex $\\mathcal K(x_{11},\\ldots,x_{3n}; S/I)$ is graded with respect to the grading defined by $\\deg(x_{ij})=\\epsilon_j$. Thus, if $X^{j_1,j_2,j_3}$ is the submatrix of $X$ consisting of columns $j_1, j_2, j_3$, and if $I^{j_1,j_2,j_3}$ is the ideal of $S$ generated by the generating minors of $I$ coming from $X^{j_1,j_2,j_3}$, then by Lemma \\ref{lem:3by3}, we see that $I^{j_1, j_2, j_3}=I_2(Y)$ for some $2\\times \\ell$ or $\\ell \\times 2$ submatrix $Y$ of $X^{j_1,j_2,j_3}$.", "post_theorem_intro_text_len": 1716, "post_theorem_intro_text": "In particular, Theorem \\ref{intro-main-thm} says that the binomial edge ideal of a pair of graphs $(G_1,G_2)$ is Golod if and only if both $G_1$ and $G_2$ are complete and one of them is equal to $K_2$.\n\nOur proof of Theorem \\ref{intro-main-thm} proceeds in several steps. We begin with the case where $X$ is a $2 \\times n$ matrix, and then analyze the $3 \\times 3$ and $3 \\times n$ cases. These preliminary cases provide the foundation for the main argument. Along the way, we prove a general result on the non-Golodness of certain tensor products, which allows us to reduce the problem to a simpler setting. This result is in the same spirit as the classical statement that a Golod complete intersection must be a hypersurface. A key ingredient in the proof of our main theorem is the fact that for a Gorenstein local ring, the homology algebra of its Koszul complex is a Poincar\\'e algebra, as shown by Avramov and Golod \\cite{AG71}. Assuming that $\\mathbb{k}$ is infinite, the same result holds for standard graded Gorenstein $\\mathbb{k}$-algebras, which is the version we shall use.\n\nThe article is organized as follows. In Section~2, we recall the necessary definitions and set up the notation. We also collect relevant known results and extract some immediate consequences. \nSection~3 studies the Golod property of certain tensor products and proves a more general result for $2$-subdeterminantal ideals, which will be used in subsequent reductions. \nIn Section~4, we prove several preparatory results and establish the main theorem. We conclude with remarks and questions aimed at possible generalizations of our results, including higher-order minors, other classes of matrices, and related types of ideals.", "sketch": "Our proof of Theorem \\ref{intro-main-thm} proceeds in several steps. We begin with the case where $X$ is a $2\\times n$ matrix, and then analyze the $3\\times 3$ and $3\\times n$ cases; these preliminary cases provide the foundation for the main argument. Along the way, we prove a general result on the non-Golodness of certain tensor products, which allows us to reduce the problem to a simpler setting; this is in the same spirit as the classical statement that a Golod complete intersection must be a hypersurface. A key ingredient is that for a Gorenstein local ring, the homology algebra of its Koszul complex is a Poincar\\'e algebra (Avramov and Golod \\cite{AG71}); assuming that $\\mathbb{k}$ is infinite, the same result holds for standard graded Gorenstein $\\mathbb{k}$-algebras, which is the version used.", "expanded_sketch": "Our proof of the main theorem proceeds in several steps. We begin with the case where $X$ is a $2\\times n$ matrix, and then analyze the $3\\times 3$ and $3\\times n$ cases; these preliminary cases provide the foundation for the main argument. Along the way, we prove a general result on the non-Golodness of certain tensor products, which allows us to reduce the problem to a simpler setting; this is in the same spirit as the classical statement that a Golod complete intersection must be a hypersurface. A key ingredient is that for a Gorenstein local ring, the homology algebra of its Koszul complex is a Poincar\\'e algebra (Avramov and Golod \\cite{AG71}); assuming that $\\mathbb{k}$ is infinite, the same result holds for standard graded Gorenstein $\\mathbb{k}$-algebras, which is the version used.,", "expanded_theorem": "\\label{intro-main-thm}\n Let $\\mathbb{k}$ be an infinite field and $X=(x_{ij})_{m \\times n}$ be a matrix of indeterminates. Suppose that $S$ denotes the polynomial ring $\\mathbb{k}[x_{ij} \\mid 1 \\leq i \\leq m, 1\\leq j \\leq n]$, and $I$ is a $2$-subdeterminantal ideal of $X$. Then the following are equivalent:\n \\begin{enumerate}\n \\item[{\\rm (i)}] The ring $S/I$ is Golod.\n \\item[{\\rm (ii)}] $I$ has a linear resolution over $S$.\n \\item[{\\rm (iii)}] $I=I_2(Y)$ for some $2 \\times \\ell$ or $\\ell \\times 2$ submatrix $Y$ of $X$.\n \\item[{\\rm (iv)}] The product on the Koszul homology $\\mathcal H(\\mathcal K (x_{11}, \\ldots,x_{mn}; S/I))$ is trivial.\n \\end{enumerate}", "theorem_type": ["Biconditional or Equivalence", "Classification or Bijection"], "mcq": {"question": "Let k be an infinite field, let X = (x_ij) be an m × n matrix of indeterminates, and let S = k[x_ij | 1 ≤ i ≤ m, 1 ≤ j ≤ n]. Suppose I is a 2-subdeterminantal ideal of X, meaning that I is generated by a subset of the 2 × 2 minors of X. Which statement gives the complete classification of when the quotient ring S/I is Golod?", "correct_choice": {"label": "A", "text": "S/I is Golod exactly when I = I_2(Y) for some 2 × ℓ or ℓ × 2 submatrix Y of X, where I_2(Y) denotes the ideal generated by all 2 × 2 minors of Y; equivalently, this is also exactly when I has a linear resolution over S, and exactly when the product on the Koszul homology H(K(x_11, ..., x_mn; S/I)) is trivial."}, "choices": [{"label": "B", "text": "S/I is Golod exactly when I = I_2(Y) for some submatrix Y of X with at least two rows and at least two columns; equivalently, this is also exactly when I has a linear resolution over S, and exactly when the product on the Koszul homology H(K(x_11, ..., x_mn; S/I)) is trivial."}, {"label": "C", "text": "If S/I is Golod, then the product on the Koszul homology H(K(x_11, ..., x_mn; S/I)) is trivial."}, {"label": "D", "text": "S/I is Golod exactly when, after possibly restricting to a submatrix Y of X, the ideal I is generated by some subset of the 2 × 2 minors of a 2 × ℓ or ℓ × 2 submatrix Y; equivalently, this is also exactly when the product on the Koszul homology H(K(x_11, ..., x_mn; S/I)) is trivial."}, {"label": "E", "text": "S/I is Golod exactly when I = I_2(Y) for some 2 × ℓ or ℓ × 2 submatrix Y of X; equivalently, this is also exactly when the product on the Koszul homology H(K(x_11, ..., x_mn; S/I)) is trivial, and in this case S/I is a complete intersection."}], "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": "two-row-or-two-column classification narrowed to arbitrary larger submatrices", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped converse and classification/linear-resolution equivalences, keeping only Golod implies trivial product", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "equality I=I_2(Y) weakened to being generated by a subset of minors of such a submatrix", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "confuses the tensor-product/non-Golod reduction with a complete-intersection conclusion", "template_used": "wildcard"}]}} +{"id": "2601.18153v1", "paper_link": "http://arxiv.org/abs/2601.18153v1", "theorems_cnt": 1, "theorem": {"env_name": "thm", "content": "\\label{intro-main-thm}\n Let $\\mathbb{k}$ be an infinite field and $X=(x_{ij})_{m \\times n}$ be a matrix of indeterminates. Suppose that $S$ denotes the polynomial ring $\\mathbb{k}[x_{ij} \\mid 1 \\leq i \\leq m, 1\\leq j \\leq n]$, and $I$ is a $2$-subdeterminantal ideal of $X$. Then the following are equivalent:\n \\begin{enumerate}\n \\item[{\\rm (i)}] The ring $S/I$ is Golod.\n \\item[{\\rm (ii)}] $I$ has a linear resolution over $S$.\n \\item[{\\rm (iii)}] $I=I_2(Y)$ for some $2 \\times \\ell$ or $\\ell \\times 2$ submatrix $Y$ of $X$.\n \\item[{\\rm (iv)}] The product on the Koszul homology $\\mathcal H(\\mathcal K (x_{11}, \\ldots,x_{mn}; S/I))$ is trivial.\n \\end{enumerate}", "start_pos": 10012, "end_pos": 10733, "label": "intro-main-thm"}, "ref_dict": {"intro-main-thm": "\\begin{thm}\\label{intro-main-thm}\n Let $\\mathbb{k}$ be an infinite field and $X=(x_{ij})_{m \\times n}$ be a matrix of indeterminates. Suppose that $S$ denotes the polynomial ring $\\mathbb{k}[x_{ij} \\mid 1 \\leq i \\leq m, 1\\leq j \\leq n]$, and $I$ is a $2$-subdeterminantal ideal of $X$. Then the following are equivalent:\n \\begin{enumerate}\n \\item[{\\rm (i)}] The ring $S/I$ is Golod.\n \\item[{\\rm (ii)}] $I$ has a linear resolution over $S$.\n \\item[{\\rm (iii)}] $I=I_2(Y)$ for some $2 \\times \\ell$ or $\\ell \\times 2$ submatrix $Y$ of $X$.\n \\item[{\\rm (iv)}] The product on the Koszul homology $\\mathcal H(\\mathcal K (x_{11}, \\ldots,x_{mn}; S/I))$ is trivial.\n \\end{enumerate} \n\\end{thm}", "def:subdet-ideal": "\\begin{defn}\\label{def:subdet-ideal}\n Given any $t\\geq 1$, an ideal $I$ of $\\mathbb{k}[X]$ is called a \\emph{$t$-subdeterminantal ideal of $X$}, if $I$ is generated by a subset of the set of all $t \\times t$ minors of $X$. \n Also, we use $I_t(X)$ to denote the ideal generated by all $t \\times t$ minors of $X$. \n\\end{defn}"}, "pre_theorem_intro_text_len": 3316, "pre_theorem_intro_text": "Let $S=\\mathbb{k}[x_1, \\ldots, x_n]$ be a standard graded polynomial ring over a field $\\mathbb{k}$, $I\\subseteq (x_1, \\ldots, x_n)^2$ a homogeneous ideal, and $R=S/I$. The power series $\\mathcal{P}_{\\mathbb{k}}^R(z)= \\sum_{i \\geq 0} \\dim_{\\mathbb{k}} \\left(\\Tor_i^R(\\mathbb{k},\\mathbb{k})\\,\\right)z^i$ is called the \\textit{Poincar\\'e series of $\\mathbb{k}$ over $R$}. Due to a result of Serre \\cite{Se65}, there is a term-by-term inequality\n\\begin{align*}\n \\mathcal{P}_{\\mathbb{k}}^R(z) \\preceq \\dfrac{(1+z)^n}{1-\\sum\\limits_{i \\geq 1}\\dim_{\\mathbb{k}}\\left(\\Tor_i^S(R,\\mathbb{k})\\right)\\, z^{i+1}}. \n\\end{align*}\nIn \\cite{Go62}, Golod proved that that the equality holds if and only if all Massey operations on the homology of the Koszul complex $\\mathcal K(x_1, \\ldots, x_n; R)$ are trivial; in this case, the ring $R$ is called \\textit{Golod}. \n\nDetecting Golodness is a rather hard problem. For instance, we do not have a complete characterization of Golodness of $S/I$, even when $I$ is a monomial ideal. Attempts have been made to characterize Golodness of monomial quotients, especially those arising from combinatorial objects, such as Stanley--Reisner rings of simplicial complexes. Golod ideals over polynomial rings in at most 4 variables have also been characterized. For a non-exhaustive list of works on detecting Golodness and related criteria, we refer the interested reader to \n\\cite{Ah19,ANFY17,BJ07,DS22,DS16,Fr18,GTSW16,HH13,HRW99,IK18,IK23,IK24,Ka16,Ka17,FW14,Va22,Zo24}\n and the references therein.\n\nWhen $R$ is Golod, the triviality of all Massey operations forces the Koszul homology\n$\\mathcal H(\\mathcal K(x_1, \\ldots, x_n; R))$ to have trivial product. However, in general, \nthis condition is not sufficient for Golodness, even for monomial quotients, as the examples due to De Stefani \\cite{DS16} and Katth\\\"an \\cite{Ka17} show. \nIn some special cases, the triviality of the product on Koszul homology is indeed equivalent to Golodness. \nFor instance, if $I$ is a monomial ideal generated in degree two, then combining the results proved in \\cite{BJ07,HerzogHibiZheng, HRW99}, \nit follows that $S/I$ is Golod if and only if $I$ has a linear resolution if and only if the product on the Koszul homology is trivial. \nThe main result of our paper is in the same spirit. \n\nWe consider ideals generated by subsets of the set of all $2 \\times 2$ minors of a matrix $X=(x_{ij})_{m \\times n}$ of indeterminates, which we call \\textit{$2$-subdeterminantal ideals of $X$} (see Definition \\ref{def:subdet-ideal}). \nIn the literature, these ideals have also been referred to as \\textit{generalized binomial edge ideals} or \\textit{binomial edge ideals of a pair of graphs} (see \\cite{EHHQ14}). \nDue to this identification, these ideals are of interest in combinatorial commutative algebra, and several homological properties of the associated quotient rings, such as the Castelnuovo--Mumford regularity, Koszulness, existence of quadratic Gr\\\"obner bases, etc.~have been studied in recent years (see, e.g.,~\\cite{BEI18, EHHQ14, JK25, LMMP26}). \nThe class of ideals we consider also includes the classical determinantal ideals as a special case (see \\cite{BV88}). \nOur aim in this article is to provide a complete characterization of the Golodness of these ideals. \nOur main theorem is as follows.", "context": "Let $S=\\mathbb{k}[x_1, \\ldots, x_n]$ be a standard graded polynomial ring over a field $\\mathbb{k}$, $I\\subseteq (x_1, \\ldots, x_n)^2$ a homogeneous ideal, and $R=S/I$. The power series $\\mathcal{P}_{\\mathbb{k}}^R(z)= \\sum_{i \\geq 0} \\dim_{\\mathbb{k}} \\left(\\Tor_i^R(\\mathbb{k},\\mathbb{k})\\,\\right)z^i$ is called the \\textit{Poincar\\'e series of $\\mathbb{k}$ over $R$}. Due to a result of Serre \\cite{Se65}, there is a term-by-term inequality\n\\begin{align*}\n \\mathcal{P}_{\\mathbb{k}}^R(z) \\preceq \\dfrac{(1+z)^n}{1-\\sum\\limits_{i \\geq 1}\\dim_{\\mathbb{k}}\\left(\\Tor_i^S(R,\\mathbb{k})\\right)\\, z^{i+1}}. \n\\end{align*}\nIn \\cite{Go62}, Golod proved that that the equality holds if and only if all Massey operations on the homology of the Koszul complex $\\mathcal K(x_1, \\ldots, x_n; R)$ are trivial; in this case, the ring $R$ is called \\textit{Golod}.\n\nDetecting Golodness is a rather hard problem. For instance, we do not have a complete characterization of Golodness of $S/I$, even when $I$ is a monomial ideal. Attempts have been made to characterize Golodness of monomial quotients, especially those arising from combinatorial objects, such as Stanley--Reisner rings of simplicial complexes. Golod ideals over polynomial rings in at most 4 variables have also been characterized. For a non-exhaustive list of works on detecting Golodness and related criteria, we refer the interested reader to \n\\cite{Ah19,ANFY17,BJ07,DS22,DS16,Fr18,GTSW16,HH13,HRW99,IK18,IK23,IK24,Ka16,Ka17,FW14,Va22,Zo24}\n and the references therein.\n\nWhen $R$ is Golod, the triviality of all Massey operations forces the Koszul homology\n$\\mathcal H(\\mathcal K(x_1, \\ldots, x_n; R))$ to have trivial product. However, in general, \nthis condition is not sufficient for Golodness, even for monomial quotients, as the examples due to De Stefani \\cite{DS16} and Katth\\\"an \\cite{Ka17} show. \nIn some special cases, the triviality of the product on Koszul homology is indeed equivalent to Golodness. \nFor instance, if $I$ is a monomial ideal generated in degree two, then combining the results proved in \\cite{BJ07,HerzogHibiZheng, HRW99}, \nit follows that $S/I$ is Golod if and only if $I$ has a linear resolution if and only if the product on the Koszul homology is trivial. \nThe main result of our paper is in the same spirit.\n\nWe consider ideals generated by subsets of the set of all $2 \\times 2$ minors of a matrix $X=(x_{ij})_{m \\times n}$ of indeterminates, which we call \\textit{$2$-subdeterminantal ideals of $X$} (see Definition \\ref{def:subdet-ideal}). \nIn the literature, these ideals have also been referred to as \\textit{generalized binomial edge ideals} or \\textit{binomial edge ideals of a pair of graphs} (see \\cite{EHHQ14}). \nDue to this identification, these ideals are of interest in combinatorial commutative algebra, and several homological properties of the associated quotient rings, such as the Castelnuovo--Mumford regularity, Koszulness, existence of quadratic Gr\\\"obner bases, etc.~have been studied in recent years (see, e.g.,~\\cite{BEI18, EHHQ14, JK25, LMMP26}). \nThe class of ideals we consider also includes the classical determinantal ideals as a special case (see \\cite{BV88}). \nOur aim in this article is to provide a complete characterization of the Golodness of these ideals. \nOur main theorem is as follows.\n\n\\begin{defn}\\label{def:subdet-ideal}\n Given any $t\\geq 1$, an ideal $I$ of $\\mathbb{k}[X]$ is called a \\emph{$t$-subdeterminantal ideal of $X$}, if $I$ is generated by a subset of the set of all $t \\times t$ minors of $X$. \n Also, we use $I_t(X)$ to denote the ideal generated by all $t \\times t$ minors of $X$. \n\\end{defn}", "full_context": "Let $S=\\mathbb{k}[x_1, \\ldots, x_n]$ be a standard graded polynomial ring over a field $\\mathbb{k}$, $I\\subseteq (x_1, \\ldots, x_n)^2$ a homogeneous ideal, and $R=S/I$. The power series $\\mathcal{P}_{\\mathbb{k}}^R(z)= \\sum_{i \\geq 0} \\dim_{\\mathbb{k}} \\left(\\Tor_i^R(\\mathbb{k},\\mathbb{k})\\,\\right)z^i$ is called the \\textit{Poincar\\'e series of $\\mathbb{k}$ over $R$}. Due to a result of Serre \\cite{Se65}, there is a term-by-term inequality\n\\begin{align*}\n \\mathcal{P}_{\\mathbb{k}}^R(z) \\preceq \\dfrac{(1+z)^n}{1-\\sum\\limits_{i \\geq 1}\\dim_{\\mathbb{k}}\\left(\\Tor_i^S(R,\\mathbb{k})\\right)\\, z^{i+1}}. \n\\end{align*}\nIn \\cite{Go62}, Golod proved that that the equality holds if and only if all Massey operations on the homology of the Koszul complex $\\mathcal K(x_1, \\ldots, x_n; R)$ are trivial; in this case, the ring $R$ is called \\textit{Golod}.\n\nDetecting Golodness is a rather hard problem. For instance, we do not have a complete characterization of Golodness of $S/I$, even when $I$ is a monomial ideal. Attempts have been made to characterize Golodness of monomial quotients, especially those arising from combinatorial objects, such as Stanley--Reisner rings of simplicial complexes. Golod ideals over polynomial rings in at most 4 variables have also been characterized. For a non-exhaustive list of works on detecting Golodness and related criteria, we refer the interested reader to \n\\cite{Ah19,ANFY17,BJ07,DS22,DS16,Fr18,GTSW16,HH13,HRW99,IK18,IK23,IK24,Ka16,Ka17,FW14,Va22,Zo24}\n and the references therein.\n\nWhen $R$ is Golod, the triviality of all Massey operations forces the Koszul homology\n$\\mathcal H(\\mathcal K(x_1, \\ldots, x_n; R))$ to have trivial product. However, in general, \nthis condition is not sufficient for Golodness, even for monomial quotients, as the examples due to De Stefani \\cite{DS16} and Katth\\\"an \\cite{Ka17} show. \nIn some special cases, the triviality of the product on Koszul homology is indeed equivalent to Golodness. \nFor instance, if $I$ is a monomial ideal generated in degree two, then combining the results proved in \\cite{BJ07,HerzogHibiZheng, HRW99}, \nit follows that $S/I$ is Golod if and only if $I$ has a linear resolution if and only if the product on the Koszul homology is trivial. \nThe main result of our paper is in the same spirit.\n\nWe consider ideals generated by subsets of the set of all $2 \\times 2$ minors of a matrix $X=(x_{ij})_{m \\times n}$ of indeterminates, which we call \\textit{$2$-subdeterminantal ideals of $X$} (see Definition \\ref{def:subdet-ideal}). \nIn the literature, these ideals have also been referred to as \\textit{generalized binomial edge ideals} or \\textit{binomial edge ideals of a pair of graphs} (see \\cite{EHHQ14}). \nDue to this identification, these ideals are of interest in combinatorial commutative algebra, and several homological properties of the associated quotient rings, such as the Castelnuovo--Mumford regularity, Koszulness, existence of quadratic Gr\\\"obner bases, etc.~have been studied in recent years (see, e.g.,~\\cite{BEI18, EHHQ14, JK25, LMMP26}). \nThe class of ideals we consider also includes the classical determinantal ideals as a special case (see \\cite{BV88}). \nOur aim in this article is to provide a complete characterization of the Golodness of these ideals. \nOur main theorem is as follows.\n\n\\begin{defn}\\label{def:subdet-ideal}\n Given any $t\\geq 1$, an ideal $I$ of $\\mathbb{k}[X]$ is called a \\emph{$t$-subdeterminantal ideal of $X$}, if $I$ is generated by a subset of the set of all $t \\times t$ minors of $X$. \n Also, we use $I_t(X)$ to denote the ideal generated by all $t \\times t$ minors of $X$. \n\\end{defn}\n\n\\begin{abstract}\nLet $X=(x_{ij})_{m\\times n}$ be a matrix of indeterminates and let \n$S=\\mathbb{k}[x_{ij} \\mid 1\\le i\\le m,\\ 1\\le j\\le n]$ be a polynomial ring over an infinite field $\\mathbb{k}$. \nLet $I$ be an ideal generated by a subset of the set of all $2\\times2$ minors of $X$. \nWe show that the quotient ring $S/I$ is Golod if and only if $I=I_2(Y)$ for some $2\\times \\ell$ or $\\ell\\times2$ submatrix $Y$ of $X$. \nIn fact, we prove that Golodness of $S/I$ is equivalent to the triviality of the product on the Koszul homology of $S/I$ and to $I$ having a linear resolution.\nAlong the way, we also prove a result on the non-Golodness of tensor products of rings under certain conditions.\n\\end{abstract}\n\nWe consider ideals generated by subsets of the set of all $2 \\times 2$ minors of a matrix $X=(x_{ij})_{m \\times n}$ of indeterminates, which we call \\textit{$2$-subdeterminantal ideals of $X$} (see Definition \\ref{def:subdet-ideal}). \nIn the literature, these ideals have also been referred to as \\textit{generalized binomial edge ideals} or \\textit{binomial edge ideals of a pair of graphs} (see \\cite{EHHQ14}). \nDue to this identification, these ideals are of interest in combinatorial commutative algebra, and several homological properties of the associated quotient rings, such as the Castelnuovo--Mumford regularity, Koszulness, existence of quadratic Gr\\\"obner bases, etc.~have been studied in recent years (see, e.g.,~\\cite{BEI18, EHHQ14, JK25, LMMP26}). \nThe class of ideals we consider also includes the classical determinantal ideals as a special case (see \\cite{BV88}). \nOur aim in this article is to provide a complete characterization of the Golodness of these ideals. \nOur main theorem is as follows.\n\nOur proof of Theorem \\ref{intro-main-thm} proceeds in several steps. We begin with the case where $X$ is a $2 \\times n$ matrix, and then analyze the $3 \\times 3$ and $3 \\times n$ cases. These preliminary cases provide the foundation for the main argument. Along the way, we prove a general result on the non-Golodness of certain tensor products, which allows us to reduce the problem to a simpler setting. This result is in the same spirit as the classical statement that a Golod complete intersection must be a hypersurface. A key ingredient in the proof of our main theorem is the fact that for a Gorenstein local ring, the homology algebra of its Koszul complex is a Poincar\\'e algebra, as shown by Avramov and Golod \\cite{AG71}. Assuming that $\\mathbb{k}$ is infinite, the same result holds for standard graded Gorenstein $\\mathbb{k}$-algebras, which is the version we shall use.\n\nGiven a homogeneous ideal $I$ in a polynomial ring $S$, we let $\\mu(I)$ denote the cardinality of a minimal generating set of $I$. In other words, $\\mu(I)=\\beta_0^S(I)=\\beta_1^S(S/I)$.\n\\begin{rmk}\\label{rmk:Golod-def} {\\rm In this remark, we recall some known facts about Golodness which we shall use later. \n Let $S=\\mathbb{k}[x_1,\\ldots, x_n]$, $I$ be a homogeneous ideal of $S$, and $R=S/I$.\n\\begin{enumerate}[label={\\rm (\\alph*)}]\n \\item Since $\\beta_i^S(R)=\\dim_{\\mathbb{k}}\\left(\\Tor ^S_i(R, \\mathbb{k})\\right)$ and the Koszul complex $\\mathcal K(x_1, \\ldots, x_n; S)$ gives a minimal free resolution of $\\mathbb{k}$ over $S$, the term-by-term inequality due to Serre mentioned in Section~1 the can be rewritten as \n $$\\mathcal P^R_{\\mathbb{k}}(z) \\preceq \\dfrac{(1+z)^n}{1-z(\\mathcal P^S_R(z)-1)}.$$\n The ring $R$ is Golod if and only if the above inequality becomes an equality.\n \\item If $R$ is Golod, then the Koszul homology $\\mathcal H(\\mathcal K(x_1, \\ldots, x_n; R))$ has trivial product (see \\cite[Remark 5.2.1]{Av98}).\n\n\\begin{lemma}\\label{lem:2byn}\n Let $X = \\begin{bmatrix}\n x_1 & x_2 & \\cdots & x_n \\\\\n y_1 & y_2 & \\cdots & y_n\n\\end{bmatrix}$ be a $2 \\times n$ matrix of indeterminates, $S=\\mathbb{k}[X]$, and $I$ be a nonzero $2$-subdeterminantal ideal of $X$. Then the following are equivalent:\n\\begin{enumerate}\n \\item[{\\rm (i)}] The product on $\\mathcal H(\\mathcal K(x_1, \\ldots,x_n, y_1, \\ldots, y_n; S/I))$ is trivial.\n \\item[{\\rm (ii)}] $I=I_2(Y)$ for some $2 \\times \\ell$ submatrix $Y$ of $X$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n (ii) $\\Longrightarrow$ (i).\\ When $I=I_2(Y)$ for some $2 \\times \\ell$ matrix $Y$, the Eagon--Northcott complex (see \\cite{EN62}) gives a linear minimal free resolution of $I$ over $S$. From \\cite[Theorem 4]{HRW99}, we know that if $I$ is componentwise linear and contains no linear forms, then $S/I$ is Golod. Thus, by Remark \\ref{rmk:Golod-def}(b), we get that the product on $\\mathcal H(\\mathcal K(x_1, \\ldots,x_n, y_1, \\ldots, y_n; S/I))$ is trivial.\n\n\\begin{cor}\\label{cor:two-rows}\n Let $X=(x_{ij})_{m \\times n}$ be a matrix of indeterminates, $S=\\mathbb{k}[X]$, and let $I$ be a $2$-subdeterminantal ideal of $X$. Given any $1\\leq j_1< j_2\\leq n$, let $I^{j_1, j_2}$ be the ideal generated by the generators of $I$ of the form $$\\begin{vmatrix}\nx_{i_1j_1} & x_{i_1j_2} \\\\\nx_{i_2j_1} & x_{i_2j_2}\n\\end{vmatrix}.$$\nIf $\\mathcal{H}(\\mathcal K(x_{11},\\ldots, x_{mn}; S/I))$ has trivial product, then $I^{j_1, j_2} = I_2(Y)$ for some $\\ell \\times 2$ submatrix $Y$ of $$\\begin{bmatrix}\n x_{1j_1} & \\cdots & x_{mj_1} \\\\\n x_{1j_2} & \\cdots & x_{mj_2}\n\\end{bmatrix}^T.$$\n\nWe next focus on the $3 \\times 3$ case. \n\\begin{lemma}\\label{lem:3by3}\n Let $X=(x_{ij})_{3\\times 3}$ be a matrix of indeterminates, $S=\\mathbb{k}[X]$, and $I$ be a nonzero $2$-subdeterminantal ideal $X$.\nThen the following are equivalent:\n\\begin{enumerate}\n \\item[{\\rm (i)}] The product on $\\mathcal H(\\mathcal K(x_{11},\\ldots,x_{33}; S/I))$ is trivial.\n \\item[{\\rm (ii)}] $I=I_2(Y)$ for some $2\\times \\ell$ or $\\ell\\times 2$ submatrix $Y$ of $X$, where $\\ell=2$ or $3$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n As noted previously, if (ii) holds, then $S/I$ is Golod, and hence (i) holds.\n\n\\begin{cor}\\label{cor:3byn}\n Let $X=(x_{ij})_{3\\times n}$ be a matrix of indeterminates, $S=\\mathbb{k}[X]$, and $I$ be a nonzero $2$-subdeterminantal ideal of\n$X$.\nThen the following are equivalent:\n\\begin{enumerate}\n \\item[{\\rm (i)}] The product on $\\mathcal H(\\mathcal K(x_{11},\\ldots,x_{3n}; S/I))$ is trivial.\n \\item[{\\rm (ii)}] $I=I_2(Y)$ for some $2\\times \\ell$ or $\\ell\\times 2$ submatrix $Y$ of $X$ with $\\ell \\geq 2$.\n\\end{enumerate}\nReplacing $X$ by $X^T$, we see that an analogous statemtent holds for $n \\times 3$ matrix of indeterminates.\n\\end{cor} \n\\begin{proof}\n As noted before, (ii) $\\Longrightarrow$ (i) is true. Conversely, suppose that (i) holds. Recall that the Koszul complex $\\mathcal K(x_{11},\\ldots,x_{3n}; S/I)$ is graded with respect to the grading defined by $\\deg(x_{ij})=\\epsilon_j$. Thus, if $X^{j_1,j_2,j_3}$ is the submatrix of $X$ consisting of columns $j_1, j_2, j_3$, and if $I^{j_1,j_2,j_3}$ is the ideal of $S$ generated by the generating minors of $I$ coming from $X^{j_1,j_2,j_3}$, then by Lemma \\ref{lem:3by3}, we see that $I^{j_1, j_2, j_3}=I_2(Y)$ for some $2\\times \\ell$ or $\\ell \\times 2$ submatrix $Y$ of $X^{j_1,j_2,j_3}$.", "post_theorem_intro_text_len": 1716, "post_theorem_intro_text": "In particular, Theorem \\ref{intro-main-thm} says that the binomial edge ideal of a pair of graphs $(G_1,G_2)$ is Golod if and only if both $G_1$ and $G_2$ are complete and one of them is equal to $K_2$.\n\nOur proof of Theorem \\ref{intro-main-thm} proceeds in several steps. We begin with the case where $X$ is a $2 \\times n$ matrix, and then analyze the $3 \\times 3$ and $3 \\times n$ cases. These preliminary cases provide the foundation for the main argument. Along the way, we prove a general result on the non-Golodness of certain tensor products, which allows us to reduce the problem to a simpler setting. This result is in the same spirit as the classical statement that a Golod complete intersection must be a hypersurface. A key ingredient in the proof of our main theorem is the fact that for a Gorenstein local ring, the homology algebra of its Koszul complex is a Poincar\\'e algebra, as shown by Avramov and Golod \\cite{AG71}. Assuming that $\\mathbb{k}$ is infinite, the same result holds for standard graded Gorenstein $\\mathbb{k}$-algebras, which is the version we shall use.\n\nThe article is organized as follows. In Section~2, we recall the necessary definitions and set up the notation. We also collect relevant known results and extract some immediate consequences. \nSection~3 studies the Golod property of certain tensor products and proves a more general result for $2$-subdeterminantal ideals, which will be used in subsequent reductions. \nIn Section~4, we prove several preparatory results and establish the main theorem. We conclude with remarks and questions aimed at possible generalizations of our results, including higher-order minors, other classes of matrices, and related types of ideals.", "sketch": "Our proof of Theorem \\ref{intro-main-thm} proceeds in several steps. We begin with the case where $X$ is a $2\\times n$ matrix, and then analyze the $3\\times 3$ and $3\\times n$ cases; these preliminary cases provide the foundation for the main argument. Along the way, we prove a general result on the non-Golodness of certain tensor products, which allows us to reduce the problem to a simpler setting; this is in the same spirit as the classical statement that a Golod complete intersection must be a hypersurface. A key ingredient is that for a Gorenstein local ring, the homology algebra of its Koszul complex is a Poincar\\'e algebra (Avramov and Golod \\cite{AG71}); assuming that $\\mathbb{k}$ is infinite, the same result holds for standard graded Gorenstein $\\mathbb{k}$-algebras, which is the version used.", "expanded_sketch": "Our proof of the main theorem proceeds in several steps. We begin with the case where $X$ is a $2\\times n$ matrix, and then analyze the $3\\times 3$ and $3\\times n$ cases; these preliminary cases provide the foundation for the main argument. Along the way, we prove a general result on the non-Golodness of certain tensor products, which allows us to reduce the problem to a simpler setting; this is in the same spirit as the classical statement that a Golod complete intersection must be a hypersurface. A key ingredient is that for a Gorenstein local ring, the homology algebra of its Koszul complex is a Poincar\\'e algebra (Avramov and Golod \\cite{AG71}); assuming that $\\mathbb{k}$ is infinite, the same result holds for standard graded Gorenstein $\\mathbb{k}$-algebras, which is the version used.,", "expanded_theorem": "\\label{intro-main-thm}\n Let $\\mathbb{k}$ be an infinite field and $X=(x_{ij})_{m \\times n}$ be a matrix of indeterminates. Suppose that $S$ denotes the polynomial ring $\\mathbb{k}[x_{ij} \\mid 1 \\leq i \\leq m, 1\\leq j \\leq n]$, and $I$ is a $2$-subdeterminantal ideal of $X$. Then the following are equivalent:\n \\begin{enumerate}\n \\item[{\\rm (i)}] The ring $S/I$ is Golod.\n \\item[{\\rm (ii)}] $I$ has a linear resolution over $S$.\n \\item[{\\rm (iii)}] $I=I_2(Y)$ for some $2 \\times \\ell$ or $\\ell \\times 2$ submatrix $Y$ of $X$.\n \\item[{\\rm (iv)}] The product on the Koszul homology $\\mathcal H(\\mathcal K (x_{11}, \\ldots,x_{mn}; S/I))$ is trivial.\n \\end{enumerate}", "theorem_type": ["Biconditional or Equivalence", "Classification or Bijection"], "mcq": {"question": "Let \\(\\mathbb{k}\\) be an infinite field, let \\(X=(x_{ij})_{m\\times n}\\) be a matrix of indeterminates, and let \\(S=\\mathbb{k}[x_{ij}\\mid 1\\le i\\le m,\\ 1\\le j\\le n]\\). Suppose that \\(I\\) is a \\(2\\)-subdeterminantal ideal of \\(X\\), meaning that \\(I\\) is generated by a subset of the set of all \\(2\\times 2\\) minors of \\(X\\). According to the main theorem, which of the following gives the explicit structural characterization of \\(I\\) that is equivalent to both the assertion that \\(S/I\\) is Golod (equivalently, that the Poincar\\'e series of \\(\\mathbb{k}\\) over \\(S/I\\) attains equality in Serre's upper bound) and the assertion that \\(I\\) has a linear resolution over \\(S\\)?", "correct_choice": {"label": "A", "text": "There exists a submatrix \\(Y\\) of \\(X\\) of size \\(2\\times \\ell\\) or \\(\\ell\\times 2\\) such that \\(I=I_2(Y)\\), where \\(I_2(Y)\\) denotes the ideal generated by all \\(2\\times 2\\) minors of \\(Y\\)."}, "choices": [{"label": "B", "text": "There exists a submatrix \\(Y\\) of \\(X\\) of size \\(2\\times \\ell\\) or \\(\\ell\\times 2\\) such that \\(I\\subseteq I_2(Y)\\), where \\(I_2(Y)\\) denotes the ideal generated by all \\(2\\times 2\\) minors of \\(Y\\)."}, {"label": "C", "text": "The product on the Koszul homology \\(\\mathcal H(\\mathcal K(x_{11},\\ldots,x_{mn}; S/I))\\) is trivial."}, {"label": "D", "text": "There exists a submatrix \\(Y\\) of \\(X\\) of size \\(2\\times \\ell\\) or \\(\\ell\\times 2\\) such that \\(I\\) is generated by a subset of the set of all \\(2\\times 2\\) minors of \\(Y\\)."}, {"label": "E", "text": "For every field extension \\(\\mathbb{K}/\\mathbb{k}\\), if \\(S_{\\mathbb{K}}=\\mathbb{K}[x_{ij}\\mid 1\\le i\\le m,\\ 1\\le j\\le n]\\) and \\(I_{\\mathbb{K}}=I S_{\\mathbb{K}}\\), then there exists a submatrix \\(Y\\) of \\(X\\) of size \\(2\\times \\ell\\) or \\(\\ell\\times 2\\) such that \\(I_{\\mathbb{K}}=I_2(Y)\\)."}], "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": "equality_with_full_minor_ideal", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped structural classification by submatrix", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "case_split", "tampered_component": "all_minors_of_the_submatrix_vs_arbitrary_subset", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "characteristic", "tampered_component": "existence over the given infinite field replaced by uniform base-change statement", "template_used": "uniformity_effectivity"}]}} +{"id": "2601.18460v1", "paper_link": "http://arxiv.org/abs/2601.18460v1", "theorems_cnt": 6, "theorem": {"env_name": "thm", "content": "\\label{thm:main}\n Let $\\rho(x,t)$ be the global-in-time solution to an unstable initial datum $\\rho_0 \\not\\equiv \\rho_s$, where \n $$\n \\rho_s(x_2) = \\begin{cases}\n 1, & x_2 > 0,\\\\\n -1, & x_2 < 0,\n \\end{cases}\n $$\n is the unstable stratified state. Moreover, suppose that $\\Gamma_0$ is centrally symmetric and evenly symmetric with respect to $x_1 = \\pm\\frac{\\pi}{2}$. Then for any $\\epsilon > 0$, we have\n \\begin{equation}\n \\label{est:main}\n \\limsup_{t \\to \\infty} t^{-{\\frac{1}{17}} + \\epsilon} (L(t) + \\mathcal{K}(t))= \\infty.\n \\end{equation}", "start_pos": 11647, "end_pos": 12249, "label": "thm:main"}, "ref_dict": {"est:main": "\\begin{equation}\n \\label{est:main}\n \\limsup_{t \\to \\infty} t^{-{\\frac{1}{17}} + \\eps} (L(t) + \\calK(t))= \\infty.\n \\end{equation}", "eq:Stokes": "\\begin{equation}\\label{eq:Stokes}\n \\begin{cases}\n -\\Delta u^{\\pm} = -\\nabla p^\\pm - (0,\\rho^\\pm)^T,&\\text{in }\\Omega^\\pm(t),\\\\\n \\nabla\\cdot u^\\pm = 0,&\\text{in }\\Omega^\\pm(t),\\\\\n \\llbracket(\\nabla u + \\nabla u^T - p \\, \\text{Id})n\\rrbracket = 0,&\\text{on }\\Gamma(t),\\\\\n \\llbracket u \\rrbracket = 0,&\\text{on }\\Gamma(t),\\\\\n z_t(\\alpha,t) = u(z(\\alpha,t),t),&\\text{on }\\Gamma(t),\n \\end{cases}\n\\end{equation}", "thm:turning": "\\begin{thm} \\label{thm:turning}\n Let $\\bar{z}(\\alpha) = (\\bar{z}_1(\\alpha),\\bar{z}_2(\\alpha))$ a curve such that \n \\begin{enumerate}\n \\item $\\bar{z}_i$ are smooth and have odd symmetry.\n \\item $\\partial_\\alpha \\bar{z}_1(\\alpha) >0$ for $\\alpha \\neq 0$, $\\partial_\\alpha \\bar{z}_1(0) =0$ and $\\partial_\\alpha \\bar z_2(0) >0$.\n \\item $\\partial_\\alpha \\bar u_1(0) = \\partial_\\alpha \\partial_t \\bar{z}_1(0) <0$.\n \\end{enumerate}\n Then, if we set $\\bar{z}(\\alpha)$ as the initial datum for the Contour Dynamics Equation \\eqref{eq:cde} in the Rayleigh-Taylor stable case of the densities, the evolution of $\\bar{z}(\\alpha)$ exhibits a turning instability at $\\alpha =0$ in finite time.\n Moreover, we prove that there exists a family of initial data $\\bar{z}$ fulfilling the conditions 1-3 and we construct another family of curves that satisfy the extra symmetry assumption:\n \\begin{enumerate}\n \\setcounter{enumi}{3}\n \\item $\\bar{z}_i$ are symmetric with respect to $\\frac{\\pi}{2}$ in the sense of \\eqref{evensym}.\n \\end{enumerate}\n For the latter family of curves fulfilling conditions 1-4, we prove that the turning instability occurs at $\\alpha = 0$ and the endpoints $\\alpha = \\pm \\pi$ due to the even symmetry of $\\bar{z}_2$.\n\\end{thm}", "thm:2": "\\begin{thm}\\label{thm:2}\nFix $T>0$ and $\\mu>0$. There exists a family of initial interfaces $(\\alpha,g_0(\\alpha))$ such that their corresponding solutions to the Stokes system $(\\alpha,g(\\alpha,t))$ satisfy that they are analytic in the strip $\\{\\alpha+i\\beta\\text{ with } |\\beta|\\leq \\nu^*\\}$. Then $[-\\pi, \\pi]=I_\\mu\\cup R_\\mu$, where $I_\\mu$ is an union of at most $[\\frac{4\\pi}{\\nu^*}]$ intervals open in $[-\\pi, \\pi]$, and\n\\begin{itemize}\n\\item $|\\partial_\\alpha g(\\alpha,t)| \\leq \\mu, \\text{ for all }\\alpha\\in I_\\mu,$\n\\item $\\text{Card}\\{\\alpha \\in R_\\mu : \\partial_\\alpha g(\\alpha,t)=0\\}\\leq\nF(t),$\n\\end{itemize}\nwhere the growth of $F(t)$ is bounded from below as follows\n$$\nC(\\nu^*,\\mu,g_0)(1+\\sqrt{t})\\leq F(t).\n$$\n\\end{thm}", "eq:cde": "\\begin{equation}\n \\label{eq:cde}\n \\p_t z(\\alpha, t) = (\\rho^- - \\rho^+)\\int_\\T \\calS(z(\\alpha, t) - z(\\beta, t)) \\, \\dotz^\\perp(\\beta, t)z_2(\\beta,t) d\\beta,\\quad \\alpha \\in \\T,\n\\end{equation}", "evensym": "\\begin{equation}\\label{evensym}\n\\begin{split}\n z_{0,1}(\\alpha) &= -\\pi - z_{0,1}(-\\pi - \\alpha),\\quad z_{0,2}(\\alpha) = z_{0,2}(-\\pi - \\alpha),\\quad \\alpha \\in (-\\pi,0],\\\\\n z_{0,1}(\\alpha) &= \\pi - z_{0,1}(\\pi - \\alpha),\\quad z_{0,2}(\\alpha) = z_{0,2}(\\pi - \\alpha),\\quad \\alpha \\in (0,\\pi],\n\\end{split}\n\\end{equation}", "thm:main": "\\begin{thm}\\label{thm:main}\n Let $\\rho(x,t)$ be the global-in-time solution to an unstable initial datum $\\rho_0 \\not\\equiv \\rho_s$, where \n $$\n \\rho_s(x_2) = \\begin{cases}\n 1, & x_2 > 0,\\\\\n -1, & x_2 < 0,\n \\end{cases}\n $$\n is the unstable stratified state. Moreover, suppose that $\\Gamma_0$ is centrally symmetric and evenly symmetric with respect to $x_1 = \\pm\\frac{\\pi}{2}$. Then for any $\\eps > 0$, we have\n \\begin{equation}\n \\label{est:main}\n \\limsup_{t \\to \\infty} t^{-{\\frac{1}{17}} + \\eps} (L(t) + \\calK(t))= \\infty.\n \\end{equation}\n\\end{thm}"}, "pre_theorem_intro_text_len": 8991, "pre_theorem_intro_text": "In this paper, we study various unstable behavior of the sharp interface problem for the two-dimensional gravity Stokes flow. Indeed, we focus on the study of the two-phase Stokes system for incompressible, homogeneous fluids: \n\\begin{equation}\\label{eq:Stokes}\n \\begin{cases}\n -\\Delta u^{\\pm} = -\\nabla p^\\pm - (0,\\rho^\\pm)^T,&\\text{in }\\Omega^\\pm(t),\\\\\n \\nabla\\cdot u^\\pm = 0,&\\text{in }\\Omega^\\pm(t),\\\\\n \\llbracket(\\nabla u + \\nabla u^T - p \\, \\text{Id})n\\rrbracket = 0,&\\text{on }\\Gamma(t),\\\\\n \\llbracket u \\rrbracket = 0,&\\text{on }\\Gamma(t),\\\\\n z_t(\\alpha,t) = u(z(\\alpha,t),t),&\\text{on }\\Gamma(t),\n \\end{cases}\n\\end{equation}\nwhere $\\rho^\\pm$ are constant but different densities of the fluids located in the upper region \n$\\Omega^+$ and the lower region $\\Omega^-$, respectively. We consider a horizontally periodic flow, namely, we restrict ourselves to the case of\n$$\n\\Omega^+(t)\\cup\\Omega^-(t)\\cup\\Gamma(t)=\\mathbb{T}\\times \\mathbb{R}\n$$\nwhere $\\mathbb{T} = [-\\pi, \\pi)$ is the standard torus and $\\partial\\Omega^{\\pm}(t)=\\Gamma(t)$ is the moving free boundary. Here, we also use the notation $\\llbracket f \\rrbracket = f^+ - f^-$ on $\\Gamma(t)$. The free boundary is also denoted by the curve $z(\\alpha,t)$ for some parametrization, with $\\alpha \\in \\mathbb{T}$. The main motivation to study such a free boundary problem comes from the fact that at low Reynolds numbers, the viscosity forces dominate the inertial forces acting on the fluid. Thus, in a certain limiting regime, the fluids flow in the Stokes regime and their dynamics is captured by \\eqref{eq:Stokes}. For more studies on steady viscous flow we refer to the pioneer work Ladyzhenskaya \\& Solonnikov \\cite{ladyzhenskaya1983determination} and the references therein.\n\nThe previous multiphase system can also be viewed as a sharp-interface case of the following, so called Stokes-Transport system \\cite{hofer2025sedimentation}\n\n\\begin{equation}\\label{eq:StokesTr}\n \\begin{cases}\n -\\Delta u= -\\nabla p - (0,\\rho)^T,&\\text{in }\\mathbb{T}\\times \\mathbb{R},\\\\\n \\nabla\\cdot u = 0,&\\text{in }\\mathbb{T}\\times \\mathbb{R},\\\\\n\\partial_t \\rho+\\nabla\\cdot(u\\rho)=0, &\\text{in }\\mathbb{T}\\times \\mathbb{R}. \\end{cases}\n\\end{equation}\n\nThis active scalar equation has received significant attention in recent years. In particular, this system was derived by Höfer \\cite{hofer2018sedimentation} and Mecherbet \\cite{mecherbet2019sedimentation} as a model for sedimentation of particles in a viscous fluid, in the case of negligible inertia. The existence and uniqueness of solutions under certain integrability and regularity assumptions for the density have been proved in several recent works (e.g. \\cite{cobb2023wellposednessfractionalstokestransport,dalibard2025long,grayer2023dynamics,lazar2025global,leblond2022wellposedness,mecherbet2021onthesedimentation}). For instance, Grayer II \\cite{grayer2023dynamics} showed the global well-posedness for patch-like and $L^1 \\cap L^\\infty$ initial density, as well as regularity persistence of the boundary of the patch. More recently, it was proved in \\cite{lazar2025global} that higher Hölder regularities are also propagated. Moreover, Dalibard, Guillod \\& Leblond \\cite{dalibard2025long} proved global well-posedness for bounded initial density in the case of bounded domains and the strip $\\Omega = \\mathbb{T} \\times (0,1)$. We refer the reader to \\cite{mecherbet2024afewremarks} for a detailed review of the literature and well-posedness results for densities in $L^1 \\cap L^p$. Our setting differs to the previous frameworks in the sense that the density field is merely a piecewise constant function, so it lacks integrability in the chosen unbounded region.\n\nIn \\cite{GGS25_SIMA} it was proved that one can equivalently write \\eqref{eq:Stokes} in the form of the following Contour Dynamics Equation (CDE):\n\\begin{equation}\n \\label{eq:cde}\n \\p_t z(\\alpha, t) = (\\rho^- - \\rho^+)\\int_\\mathbb{T} \\mathcal{S}(z(\\alpha, t) - z(\\beta, t)) \\, \\dot{z}^\\perp(\\beta, t)z_2(\\beta,t) d\\beta,\\quad \\alpha \\in \\mathbb{T},\n\\end{equation}\nwhere $\\dot{z}(\\beta) := \\p_\\beta z(\\beta)$. Moreover, the kernel $\\mathcal{S}(x)$ is the $x_1$--periodic Stokeslet, which can be explicitly written as \n$$\n\\mathcal{S}(x_1,x_2) = \\frac{1}{8\\pi}\\log(2(\\cosh(x_2) - \\cos(x_1))) \\, \\text{Id} - \\frac{x_2}{8\\pi(\\cosh(x_2) - \\cos(x_1))}\\begin{bmatrix}\n -\\sinh(x_2) & \\sin(x_1)\\\\\n \\sin(x_1) & \\sinh(x_2)\n\\end{bmatrix}.\n$$\nAlthough for gravity $x_1-$periodic flows such contour dynamics approach was new, an integral approach has been used before for the case of capillarity-driven Stokes interfaces in the whole 2D plane. In this regard, the interested reader is referred for instance to the papers by Badea \\& Duchon and Matioc \\& Prokert \\cite{badea1998capillary,matioc2021two} (see also \\cite{matioc2022two,matioc2023capillarity}). More recently, the combined case of gravity and capillarity forces was studied in \\cite{bohme2025wellposedness,böhme2025wellposednessrayleightaylorinstabilitytwophase} in the horizontally-periodic case. See also \\cite{JaeHoChoi} for the capillarity-driven stability of nearly circular closed curves in the absence of gravity.\n\nA remarkable fact about the problem \\eqref{eq:cde} is that the equation is globally well-posed, as long as we impose the initial regularity assumption $z_0 \\in C^{k,\\gamma}(\\mathbb{T})$ for arbitrary $k\\in\\mathbb{N}\\cup\\{0\\}$ and $0<\\gamma< 1$, as shown in \\cite{GGS25}. In particular, this global well-posedness result holds regardless of the size of the initial data or the initial stratification. Namely, the solution exists globally whether the denser fluid lies above or below the lighter one. Consequently, this global existence result is independent of the Rayleigh–Taylor condition. This behavior stands in sharp contrast with that of the two-phase Muskat problem, a closely related active scalar equation, which can in fact be viewed as a more singular counterpart of \\eqref{eq:cde}. \n\nMoreover, if one restricts to the behavior of \\eqref{eq:cde} near stably stratified states, \\cite{GGS25_SIMA} shows that perturbations of such states are asymptotically stable in both Sobolev spaces and in certain spaces of analytic functions. These stability results are based on the fact that, on the linear level, \\eqref{eq:cde} exhibits a weak damping effect where the linear operator is given by the Neumann-to-Dirichlet operator (instead of the more common Dirichlet-to-Neumann operator appearing in the linearized version of both the Muskat, the water waves and even the capillary Stokes problems). On the other hand, in \\cite{GGS25_SIMA} it is shown that unstable solutions grow. This instability result roughly reads as follows: fix an arbitrary $T$, then there exists an initial interface that grows exponentially at least for such $T$. Due to the way these solutions are constructed, both, the initial interface and its corresponding solution at time $T$ are in fact small in certain sense.\n\nHowever, the fact that the solution exists globally regardless of the size of the initial data and the sign of the density jump raises the following rather big question: \\emph{what is the dynamics of these globally defined solutions?} Some possible \\emph{a priori} answer, for instance, would be that the solutions stabilize around particular stable, steady, non-flat interfaces. A different possible answer would be that every solution keeps growing (in a certain sense) for arbitrary large times. It is in this question that we focus on. Indeed, with the discussion above, the exploration of both instabilities and finite-time singular behavior of the system \\eqref{eq:Stokes} is rather limited, both when the initial configuration is stable ($\\rho^+ > \\rho^-$) and unstable ($\\rho^+ > \\rho^-$). Hence, in this paper, we concern ourselves with the study of possible instability scenarios in both stable and unstable regimes. \n\nOur first main result aims to capture long-time instabilities for initial data belonging to the unstable regime. To the best of the authors' knowledge, there are no rigorous results discussing the long-time behavior of the solutions to such initial data, except for the preservation of boundary regularity as shown in \\cite{GGS25}. In the first main result of this paper, we show that the infinite-in-time growth of some key geometric quantities of the free interface is rather generic given initial data in the unstable regime. More precisely, we denote the the perimeter of the interface $\\Gamma_t$ as\n\\[\nL(t) := |\\Gamma_t| = \\int_\\mathbb{T} |\\dot{z}(\\beta,t)| d\\beta\n\\]\nand the maximal curvature of the interface $\\Gamma_t$ as\n\\[\n\\mathcal{K}(t) := \\max_{x \\in \\Gamma_t}|\\kappa(t)|.\n\\]\nThis result can be stated as follows: for an initial interface $\\Gamma_0$ under certain symmetry conditions, either the perimeter or the maximal curvature of the evolved interface $\\Gamma_t$ must grow infinitely in time with some explicit power-law rates.", "context": "\\begin{equation}\\label{eq:StokesTr}\n \\begin{cases}\n -\\Delta u= -\\nabla p - (0,\\rho)^T,&\\text{in }\\mathbb{T}\\times \\mathbb{R},\\\\\n \\nabla\\cdot u = 0,&\\text{in }\\mathbb{T}\\times \\mathbb{R},\\\\\n\\partial_t \\rho+\\nabla\\cdot(u\\rho)=0, &\\text{in }\\mathbb{T}\\times \\mathbb{R}. \\end{cases}\n\\end{equation}\n\nIn \\cite{GGS25_SIMA} it was proved that one can equivalently write \\eqref{eq:Stokes} in the form of the following Contour Dynamics Equation (CDE):\n\\begin{equation}\n \\label{eq:cde}\n \\p_t z(\\alpha, t) = (\\rho^- - \\rho^+)\\int_\\mathbb{T} \\mathcal{S}(z(\\alpha, t) - z(\\beta, t)) \\, \\dot{z}^\\perp(\\beta, t)z_2(\\beta,t) d\\beta,\\quad \\alpha \\in \\mathbb{T},\n\\end{equation}\nwhere $\\dot{z}(\\beta) := \\p_\\beta z(\\beta)$. Moreover, the kernel $\\mathcal{S}(x)$ is the $x_1$--periodic Stokeslet, which can be explicitly written as \n$$\n\\mathcal{S}(x_1,x_2) = \\frac{1}{8\\pi}\\log(2(\\cosh(x_2) - \\cos(x_1))) \\, \\text{Id} - \\frac{x_2}{8\\pi(\\cosh(x_2) - \\cos(x_1))}\\begin{bmatrix}\n -\\sinh(x_2) & \\sin(x_1)\\\\\n \\sin(x_1) & \\sinh(x_2)\n\\end{bmatrix}.\n$$\nAlthough for gravity $x_1-$periodic flows such contour dynamics approach was new, an integral approach has been used before for the case of capillarity-driven Stokes interfaces in the whole 2D plane. In this regard, the interested reader is referred for instance to the papers by Badea \\& Duchon and Matioc \\& Prokert \\cite{badea1998capillary,matioc2021two} (see also \\cite{matioc2022two,matioc2023capillarity}). More recently, the combined case of gravity and capillarity forces was studied in \\cite{bohme2025wellposedness,böhme2025wellposednessrayleightaylorinstabilitytwophase} in the horizontally-periodic case. See also \\cite{JaeHoChoi} for the capillarity-driven stability of nearly circular closed curves in the absence of gravity.\n\nMoreover, if one restricts to the behavior of \\eqref{eq:cde} near stably stratified states, \\cite{GGS25_SIMA} shows that perturbations of such states are asymptotically stable in both Sobolev spaces and in certain spaces of analytic functions. These stability results are based on the fact that, on the linear level, \\eqref{eq:cde} exhibits a weak damping effect where the linear operator is given by the Neumann-to-Dirichlet operator (instead of the more common Dirichlet-to-Neumann operator appearing in the linearized version of both the Muskat, the water waves and even the capillary Stokes problems). On the other hand, in \\cite{GGS25_SIMA} it is shown that unstable solutions grow. This instability result roughly reads as follows: fix an arbitrary $T$, then there exists an initial interface that grows exponentially at least for such $T$. Due to the way these solutions are constructed, both, the initial interface and its corresponding solution at time $T$ are in fact small in certain sense.\n\nHowever, the fact that the solution exists globally regardless of the size of the initial data and the sign of the density jump raises the following rather big question: \\emph{what is the dynamics of these globally defined solutions?} Some possible \\emph{a priori} answer, for instance, would be that the solutions stabilize around particular stable, steady, non-flat interfaces. A different possible answer would be that every solution keeps growing (in a certain sense) for arbitrary large times. It is in this question that we focus on. Indeed, with the discussion above, the exploration of both instabilities and finite-time singular behavior of the system \\eqref{eq:Stokes} is rather limited, both when the initial configuration is stable ($\\rho^+ > \\rho^-$) and unstable ($\\rho^+ > \\rho^-$). Hence, in this paper, we concern ourselves with the study of possible instability scenarios in both stable and unstable regimes.\n\nOur first main result aims to capture long-time instabilities for initial data belonging to the unstable regime. To the best of the authors' knowledge, there are no rigorous results discussing the long-time behavior of the solutions to such initial data, except for the preservation of boundary regularity as shown in \\cite{GGS25}. In the first main result of this paper, we show that the infinite-in-time growth of some key geometric quantities of the free interface is rather generic given initial data in the unstable regime. More precisely, we denote the the perimeter of the interface $\\Gamma_t$ as\n\\[\nL(t) := |\\Gamma_t| = \\int_\\mathbb{T} |\\dot{z}(\\beta,t)| d\\beta\n\\]\nand the maximal curvature of the interface $\\Gamma_t$ as\n\\[\n\\mathcal{K}(t) := \\max_{x \\in \\Gamma_t}|\\kappa(t)|.\n\\]\nThis result can be stated as follows: for an initial interface $\\Gamma_0$ under certain symmetry conditions, either the perimeter or the maximal curvature of the evolved interface $\\Gamma_t$ must grow infinitely in time with some explicit power-law rates.", "full_context": "\\begin{equation}\\label{eq:StokesTr}\n \\begin{cases}\n -\\Delta u= -\\nabla p - (0,\\rho)^T,&\\text{in }\\mathbb{T}\\times \\mathbb{R},\\\\\n \\nabla\\cdot u = 0,&\\text{in }\\mathbb{T}\\times \\mathbb{R},\\\\\n\\partial_t \\rho+\\nabla\\cdot(u\\rho)=0, &\\text{in }\\mathbb{T}\\times \\mathbb{R}. \\end{cases}\n\\end{equation}\n\nIn \\cite{GGS25_SIMA} it was proved that one can equivalently write \\eqref{eq:Stokes} in the form of the following Contour Dynamics Equation (CDE):\n\\begin{equation}\n \\label{eq:cde}\n \\p_t z(\\alpha, t) = (\\rho^- - \\rho^+)\\int_\\mathbb{T} \\mathcal{S}(z(\\alpha, t) - z(\\beta, t)) \\, \\dot{z}^\\perp(\\beta, t)z_2(\\beta,t) d\\beta,\\quad \\alpha \\in \\mathbb{T},\n\\end{equation}\nwhere $\\dot{z}(\\beta) := \\p_\\beta z(\\beta)$. Moreover, the kernel $\\mathcal{S}(x)$ is the $x_1$--periodic Stokeslet, which can be explicitly written as \n$$\n\\mathcal{S}(x_1,x_2) = \\frac{1}{8\\pi}\\log(2(\\cosh(x_2) - \\cos(x_1))) \\, \\text{Id} - \\frac{x_2}{8\\pi(\\cosh(x_2) - \\cos(x_1))}\\begin{bmatrix}\n -\\sinh(x_2) & \\sin(x_1)\\\\\n \\sin(x_1) & \\sinh(x_2)\n\\end{bmatrix}.\n$$\nAlthough for gravity $x_1-$periodic flows such contour dynamics approach was new, an integral approach has been used before for the case of capillarity-driven Stokes interfaces in the whole 2D plane. In this regard, the interested reader is referred for instance to the papers by Badea \\& Duchon and Matioc \\& Prokert \\cite{badea1998capillary,matioc2021two} (see also \\cite{matioc2022two,matioc2023capillarity}). More recently, the combined case of gravity and capillarity forces was studied in \\cite{bohme2025wellposedness,böhme2025wellposednessrayleightaylorinstabilitytwophase} in the horizontally-periodic case. See also \\cite{JaeHoChoi} for the capillarity-driven stability of nearly circular closed curves in the absence of gravity.\n\nMoreover, if one restricts to the behavior of \\eqref{eq:cde} near stably stratified states, \\cite{GGS25_SIMA} shows that perturbations of such states are asymptotically stable in both Sobolev spaces and in certain spaces of analytic functions. These stability results are based on the fact that, on the linear level, \\eqref{eq:cde} exhibits a weak damping effect where the linear operator is given by the Neumann-to-Dirichlet operator (instead of the more common Dirichlet-to-Neumann operator appearing in the linearized version of both the Muskat, the water waves and even the capillary Stokes problems). On the other hand, in \\cite{GGS25_SIMA} it is shown that unstable solutions grow. This instability result roughly reads as follows: fix an arbitrary $T$, then there exists an initial interface that grows exponentially at least for such $T$. Due to the way these solutions are constructed, both, the initial interface and its corresponding solution at time $T$ are in fact small in certain sense.\n\nHowever, the fact that the solution exists globally regardless of the size of the initial data and the sign of the density jump raises the following rather big question: \\emph{what is the dynamics of these globally defined solutions?} Some possible \\emph{a priori} answer, for instance, would be that the solutions stabilize around particular stable, steady, non-flat interfaces. A different possible answer would be that every solution keeps growing (in a certain sense) for arbitrary large times. It is in this question that we focus on. Indeed, with the discussion above, the exploration of both instabilities and finite-time singular behavior of the system \\eqref{eq:Stokes} is rather limited, both when the initial configuration is stable ($\\rho^+ > \\rho^-$) and unstable ($\\rho^+ > \\rho^-$). Hence, in this paper, we concern ourselves with the study of possible instability scenarios in both stable and unstable regimes.\n\nOur first main result aims to capture long-time instabilities for initial data belonging to the unstable regime. To the best of the authors' knowledge, there are no rigorous results discussing the long-time behavior of the solutions to such initial data, except for the preservation of boundary regularity as shown in \\cite{GGS25}. In the first main result of this paper, we show that the infinite-in-time growth of some key geometric quantities of the free interface is rather generic given initial data in the unstable regime. More precisely, we denote the the perimeter of the interface $\\Gamma_t$ as\n\\[\nL(t) := |\\Gamma_t| = \\int_\\mathbb{T} |\\dot{z}(\\beta,t)| d\\beta\n\\]\nand the maximal curvature of the interface $\\Gamma_t$ as\n\\[\n\\mathcal{K}(t) := \\max_{x \\in \\Gamma_t}|\\kappa(t)|.\n\\]\nThis result can be stated as follows: for an initial interface $\\Gamma_0$ under certain symmetry conditions, either the perimeter or the maximal curvature of the evolved interface $\\Gamma_t$ must grow infinitely in time with some explicit power-law rates.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics{fig_initial.pdf}\n \\label{fig:init}\n \\caption{An example of an unstable, un-stratified configuration satisfying our symmetry assumptions in Theorem~\\ref{thm:main}. Note that $\\Gamma_0$ is centrally symmetric about the point $(0,0)$, and evenly symmetric with respect to the orange dashed lines $x_1 = \\pm\\frac{\\pi}{2}$.}\n\\end{figure}\n\nIn this case, we would have $L(t) \\le 2\\beta t^{\\alpha}$ for all $t$ sufficiently large. From Lemma \\ref{lem:perimeterlb} and the fact that $L(t) \\ge M(t) + m(t)$, we immediately have $E(t) \\le C(\\beta)t^{2\\alpha}$ for $t$ sufficiently large. Using \\eqref{eq:potential}, one can extract an increasing sequence of times $t_n \\nearrow \\infty$ such that\n$$\n\\delta(t_n) \\le C(\\alpha,\\beta)t_n^{2\\alpha - 1}.\n$$\nHence in case where $\\alpha < 1/2$ and $t \\gg 1$, the heuristics above behooves us to consider the geometry of $\\Gamma_t$ when $\\delta(t) = \\|(-\\Delta)^{-1}\\p_1 \\rho\\|_{L^2}^2 \\ll 1$. As we will see in Section \\ref{sec:finiteE}, the maximal curvature $\\calK(t)$ will be bounded from below by some negative powers of $\\delta(t)$. In particular, we will show that for sufficiently large $n$:\n\\begin{equation}\\label{est:case2main}\n\\calK(t_n) \\ge C(\\alpha,\\beta)\\delta(t_n)^{-1/12} \\ge C(\\alpha,\\beta) t_n^{\\frac{1-2\\alpha}{12}},\n\\end{equation}\nafter a careful choice of $\\alpha$ and $\\beta$. But \\eqref{est:case2main} implies that\n$$\n\\limsup_{t\\to\\infty} t^{\\frac{2\\alpha-1}{12} + \\eps}\\calK(t) =\\infty\n$$\nfor any $\\eps > 0.$ The main result (Theorem \\ref{thm:main}) follows from summarizing the above two cases after optimizing over the choices for $\\alpha$ and $\\beta$.\n\nWith the preparations above, we may start with deriving a quantitative estimate of $\\delta(t)$.\n\\begin{lem}\\label{lem:h-2}\n Suppose there exist an $\\eps > 0$ and two disks $B_- = B_\\epsilon (x^-) \\subset D^-$, $B_+ = B_\\eps (x^+) \\subset D^+$, where $x_2^+ = x_2^-$. Then there exists a universal constant $C_0 > 0$ such that\n \\begin{equation}\n \\label{est:radiusbd}\n \\eps \\le C_0\\delta(t)^{\\frac15}.\n \\end{equation}\n\\end{lem}\n\\begin{proof}\n Without loss of generality, assume $x_1^+ < x_1^-$. To begin with, we note that for an arbitrary test function $\\varphi \\in C^\\infty_c (\\T \\times \\R)$, the following computation holds:\n \\begin{equation}\\label{radbdaux1}\n \\begin{split}\n \\int_{\\T \\times \\R} \\p_1 \\varphi \\rho dx &= -\\int_{\\T \\times \\R} \\varphi \\p_1\\rho dx = -\\int_{\\T \\times \\R} \\varphi (-\\Delta)(-\\Delta)^{-1}\\p_1 \\rho dx\\\\\n &= -\\int_{\\T \\times \\R} (-\\Delta \\varphi)(-\\Delta)^{-1}\\p_1 \\rho dx \\le \\|\\Delta \\varphi\\|_{L^2}\\delta(t)^{1/2}.\n \\end{split}\n \\end{equation}\n On the other hand, let us consider the test function $\\varphi(x) = g(x_1)h(x_2)$, where\n \\begin{align*}\n g(x_1) :=\\begin{cases}\n 1+ \\sin\\left(\\frac{\\pi}{2\\eps}(x_1 - x_1^+)\\right),& |x_1 - x_1^+| \\le \\eps,\\\\\n 2,& x_1^+ + \\eps < x_1 < x_1^- - \\eps,\\\\\n 1 - \\sin\\left(\\frac{\\pi}{2\\eps}(x_1 - x_1^-)\\right),& |x_1 - x_1^-| \\le \\eps,\\\\\n 0,& \\text{otherwise},\n \\end{cases}\n \\end{align*}\n and\n \\begin{align*}\n h(x_2) = \\begin{cases}\n 1 + \\cos\\left(\\frac{\\pi}{2\\eps}(x_2 - x_2^+)\\right),& |x_2 - x_2^+| \\le \\eps,\\\\\n 0,& \\text{otherwise}.\n \\end{cases}\n \\end{align*}\n From the construction, we know that $$\\supp(\\nabla^2\\varphi) \\subset \\{(x_1,x_2 )\\in \\T\\times \\R\\;:\\; x_1^+ - \\eps\\le x_1 \\le x_1^- + \\eps, x_2^+ - \\eps \\le x_2 \\le x_2^+ +\\eps\\},$$ and $\\|\\Delta \\varphi\\|_{L^\\infty} \\le \\eps^{-2}$. Thus an elementary computation yields\n \\begin{equation}\\label{radbdaux2}\n \\|\\Delta\\varphi\\|_{L^2} \\le \\|\\Delta \\varphi\\|_{L^\\infty}|\\supp(\\nabla^2\\varphi)|^{1/2} \\lesssim \\eps^{-2}\\cdot \\eps^{\\frac12} = \\eps^{-\\frac32}.\n \\end{equation}\n Moreover, using the fact that $\\rho \\equiv \\pm1$ in $B_\\pm$ and $\\supp(\\p_1 \\varphi) \\subset B_+ \\cup B_-$, we have\n \\begin{equation}\\label{radbdaux3}\n \\int_{\\T \\times \\R}\\p_1\\varphi \\rho dx = \\int_{B_+}\\p_1 \\varphi dx - \\int_{B_-}\\p_1\\varphi dx.\n \\end{equation}\n By the construction of $\\varphi$, we note that\n \\begin{equation}\\label{h-2aux1}\n \\begin{split}\n \\int_{B_+} \\p_1\\varphi dx &= \\frac{\\pi}{2\\eps}\\int_{B_+}\\cos\\left(\\frac{\\pi}{2\\eps}(x_1 - x_1^+)\\right)\\left(1 + \\cos\\left(\\frac{\\pi}{2\\eps}(x_2 - x_2^+)\\right)\\right) dx\\\\\n &= \\frac{2\\eps}{\\pi}\\int_{B_{\\pi/2}(0)}\\cos(u_1)(1+\\cos(u_2)) du\\\\\n &= c_0\\eps,\n \\end{split}\n \\end{equation}\n where $c_0$ is a universal constant. A similar computation also yields\n \\begin{equation}\\label{h-2aux2}\n \\int_{B_-}\\p_1\\varphi = -c_0\\eps.\n \\end{equation}\n Thus combining \\eqref{h-2aux1}, \\eqref{h-2aux2} with \\eqref{radbdaux2}, we obtain\n $$\n \\int_{\\T\\times\\R}\\p_1\\varphi \\rho dx = 2c_0\\eps.\n $$\n Further combining with \\eqref{radbdaux1} and \\eqref{radbdaux2} yields the desired bound \\eqref{est:radiusbd}.\n\\end{proof}\n\n\\section{Estimate on the Evolution of the Number of Fingers}\\label{sec:relaxation}\nIn this section we give an estimate on the number of fingers that could be developed by the flow. In order to do that we recall the following result (see \\cite{grujic2000spatial} and the references therein)\n\\begin{lem}\\label{grujic}\nLet $L$, $\\tau>0$, and let $u$ be analytic in the neighborhood of $\\{\\alpha+i\\beta: |\\beta|\\leq \\tau\\}$ and $L$-periodic in the $\\alpha$-direction. Then, for any $\\mu>0$, $[0, L]=I_\\mu\\cup R_\\mu$, where $I_\\mu$ is an union of at most $[\\frac{2L}{\\tau}]$ intervals open in $[0, L]$, and\n\\begin{itemize}\n\\item $|\\partial_\\alpha u(\\alpha)| \\leq \\mu, \\text{ for all }\\alpha\\in I_\\mu,$\n\\item $\\text{Card}\\{\\alpha \\in R_\\mu : \\partial_\\alpha u(\\alpha)=0\\}\\leq\n\\frac{2}{\\log 2}\\frac{L}{\\tau}\\log\\left(\\frac{\\max_{|\\beta|\\leq \\tau}|\\partial_\\alpha u(\\alpha+i\\beta)|}{\\mu}\\right).$\n\\end{itemize}\n\\end{lem}\n\\begin{proof}[Proof of Theorem \\ref{thm:2}]\nLet us introduce the following space of analytic functions\n\\begin{equation}\\label{wiener}\nA^{s}_\\nu=\\{u\\in L^2(\\mathbb{T}): \\sum_{k=-\\infty}^\\infty e^{\\nu |k|}|k|^{s}|\\hat{u}(k)|<\\infty\\},\\quad\\mbox{where}\\quad \\hat{u}(k)=\\frac1{2\\pi}\\int_\\T u(\\alpha)e^{-ik\\alpha}d\\alpha.\n\\end{equation}\nIn our case, fixing $\\nu_0$ and invoking Theorems 2 and 3 in \\cite{GGS25_SIMA}, we can ensure the existence of initial data $h_0(\\alpha)$ such that\n\\begin{enumerate}\n\\item the corresponding solutions are analytic in a strip of width $\\nu^*\\leq \\nu_0/24$,\n\\item the corresponding solutions satisfy the bound\n$$\n\\max_{|\\beta|\\leq \\nu^*}|\\partial_\\alpha h(\\alpha+i\\beta,t)|\\leq \\|h_0\\|_{A^1_{\\nu_0}}.\n$$\n\\end{enumerate}\nIf we now fix $T$ and define the functions\n$$\ng(\\alpha,t)=h(\\alpha,T-t),\n$$\nwe have that $g(\\alpha,t)$ satisfy the RT unstable Stokes-Transport system. Recalling the ideas of Theorem 4 in \\cite{GGS25_SIMA}, we have that\n$$\n\\|g(T)\\|_{A^1_{\\nu^*}}\\geq \\|g(T)\\|_{A^0_{\\nu^*}}\\geq C_1e^{C_2 \\sqrt{T}}\\|g_0\\|_{A^0},\n$$\nwhere $C_1, C_2$ are harmless constants only depending on the parameters $\\nu^\\ast, \\rho^+, \\rho^-$ and the initial data. Invoking the previous lemma we can ensure the desired result: the spatial domain can be split as $[-\\pi, \\pi]=I_\\mu\\cup R_\\mu$, where $I_\\mu$ is an union of at most $[\\frac{4\\pi}{\\nu^*}]$ open intervals and\n\\begin{itemize}\n\\item $|\\partial_\\alpha g(\\alpha,t)| \\leq \\mu, \\text{ for all }\\alpha\\in I_\\mu,$\n\\item $\\text{Card}\\{\\alpha \\in R_\\mu : \\partial_\\alpha g(\\alpha,T)=0\\}\\leq\nF(T),$\n\\end{itemize}\nwhere the growth of $F(t)$ is bounded below as follows\n$$\nC(1+\\sqrt{T})\\leq \\frac{2}{\\log 2}\\frac{2\\pi}{\\nu^*}\\log\\left(\\frac{C_1e^{C_2 \\sqrt{T}}\\|g_0\\|_{A^0}}{\\mu}\\right)\\leq F(T).\n$$\n\\end{proof}", "post_theorem_intro_text_len": 7546, "post_theorem_intro_text": "\\begin{figure}[h]\n \\centering\n \\includegraphics{fig_initial.pdf}\n \\label{fig:init}\n \\caption{An example of an unstable, un-stratified configuration satisfying our symmetry assumptions in Theorem~\\ref{thm:main}. Note that $\\Gamma_0$ is centrally symmetric about the point $(0,0)$, and evenly symmetric with respect to the orange dashed lines $x_1 = \\pm\\frac{\\pi}{2}$.}\n\\end{figure}\n\n\\begin{rmk}\n \\begin{enumerate}\n \\item The growth phenomenon described in Theorem \\ref{thm:main} qualitatively agrees with the numerical simulations reported in Section \\ref{sec:numerics}, which contain evolutions displaying large growth in the length of the interface and boundedness in interface's curvature. These numerical evidences indeed suggest the validity of the dichotomy in Theorem \\ref{thm:main}. \n\n \\item We do not expect the growth rate in \\eqref{est:main} to be sharp. In fact, the numerical examples in Section \\ref{sec:numerics} seem to suggests a linear rate of growth, which is much faster than the sublinear rate obtained in our main theorem.\n \\end{enumerate}\n\\end{rmk}\n\nThe proof of Theorem \\ref{thm:main} crucially exploits a natural potential energy possessed by the Stokes-Transport system, which is monotone as the solution evolves in time (See Section~\\ref{sec_energy} for derivation). The majority of the analysis will focus on the analysis of this potential energy, where we obtain inequalities that connect its time derivative with the perimeter and maximal curvature of the evolved surface $\\Gamma(t)$. We remark that the analysis of monotone quantities is also important in proving large-time growth phenomena and creation of small scales for other fundamental incompressible fluid models. We refer the readers to \\cite{drivas2024twisting,kiselev2024small,kiselev2023small,park2024growth} and references therein.\n\nWe also highlight a connection between the growth phenomena found in Theorem \\ref{thm:main} and the formation of viscous fingering, which occurs at the interface separating two immiscible fluids in a porous medium driven by gravity and is highly related to the well-known Rayleigh-Taylor instability. We refer the readers to \\cite{menon2005dynamic,otto1997viscous} for a more thorough discussion about growth rate of fingers in this regime. In this regard, we give the second main result of this work, where we obtain an estimate on the number of {fingers} for some class of solutions given that certain {relaxation} in the number of {fingers} is observed in the numerics (see Section \\ref{sec:numerics}).\n\nA precise statement concerning the number of fingers is as follows:\n\\begin{thm}\\label{thm:2}\nFix $T>0$ and $\\mu>0$. There exists a family of initial interfaces $(\\alpha,g_0(\\alpha))$ such that their corresponding solutions to the Stokes system $(\\alpha,g(\\alpha,t))$ satisfy that they are analytic in the strip $\\{\\alpha+i\\beta\\text{ with } |\\beta|\\leq \\nu^*\\}$. Then $[-\\pi, \\pi]=I_\\mu\\cup R_\\mu$, where $I_\\mu$ is an union of at most $[\\frac{4\\pi}{\\nu^*}]$ intervals open in $[-\\pi, \\pi]$, and\n\\begin{itemize}\n\\item $|\\partial_\\alpha g(\\alpha,t)| \\leq \\mu, \\text{ for all }\\alpha\\in I_\\mu,$\n\\item $\\text{Card}\\{\\alpha \\in R_\\mu : \\partial_\\alpha g(\\alpha,t)=0\\}\\leq\nF(t),$\n\\end{itemize}\nwhere the growth of $F(t)$ is bounded from below as follows\n$$\nC(\\nu^*,\\mu,g_0)(1+\\sqrt{t})\\leq F(t).\n$$\n\\end{thm}\n\nIt should be noted that we are not able to say anything on the possible relaxation on the number of fingers. Instead we can say that, in the case of maximal growth on the number of fingers, such maximal growth is at least as $\\sqrt{t}$.\n\nOur third main result concerns finite-time singular behavior in the stable regime. In particular, we prove that there exist initial smooth curves in the stable stratification of the densities that exhibit a turning instability, that is, the interface fails to be a graph in finite time, and the Rayleigh-Taylor condition breaks down. The proof follows \\cite{castro2012rayleigh} but the operators involved in our proof are more regular. This turning phenomenon is described in the following result: \n\\begin{thm} \\label{thm:turning}\n Let $\\bar{z}(\\alpha) = (\\bar{z}_1(\\alpha),\\bar{z}_2(\\alpha))$ a curve such that \n \\begin{enumerate}\n \\item $\\bar{z}_i$ are smooth and have odd symmetry.\n \\item $\\partial_\\alpha \\bar{z}_1(\\alpha) >0$ for $\\alpha \\neq 0$, $\\partial_\\alpha \\bar{z}_1(0) =0$ and $\\partial_\\alpha \\bar z_2(0) >0$.\n \\item $\\partial_\\alpha \\bar u_1(0) = \\partial_\\alpha \\partial_t \\bar{z}_1(0) <0$.\n \\end{enumerate}\n Then, if we set $\\bar{z}(\\alpha)$ as the initial datum for the Contour Dynamics Equation \\eqref{eq:cde} in the Rayleigh-Taylor stable case of the densities, the evolution of $\\bar{z}(\\alpha)$ exhibits a turning instability at $\\alpha =0$ in finite time.\n Moreover, we prove that there exists a family of initial data $\\bar{z}$ fulfilling the conditions 1-3 and we construct another family of curves that satisfy the extra symmetry assumption:\n \\begin{enumerate}\n \\setcounter{enumi}{3}\n \\item $\\bar{z}_i$ are symmetric with respect to $\\frac{\\pi}{2}$ in the sense of \\eqref{evensym}.\n \\end{enumerate}\n For the latter family of curves fulfilling conditions 1-4, we prove that the turning instability occurs at $\\alpha = 0$ and the endpoints $\\alpha = \\pm \\pi$ due to the even symmetry of $\\bar{z}_2$.\n\\end{thm}\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=0.5\\textwidth]{turning.png}\n \\caption{Sketch of the turning instability at $\\alpha = 0$. The blue curve represents the initial interface and the red curve represents its evolution into a partially unstable regime.}\n\\end{figure}\n\\begin{rmk}\nIn the above result, we construct a curve that fulfills the symmetry assumptions needed in Theorem \\ref{thm:main}. However, we cannot conclude that after the Rayleigh-Taylor condition breaks down, the dynamics turns to the scenario in Theorem \\ref{thm:main}. In the latter, we need an infinite mass of the denser fluid on top, while in the turning instability, there is just a finite mass that enters in the unstable regime. It is not clear what the behavior of the fluid will be after the turning instability: it might become stable again (see \\cite{CGZ17}) or develop infinite growth similar to the one shown in Theorem \\ref{thm:main}.\n\\end{rmk}\n\\begin{rmk}\nTheorem \\ref{thm:turning} also holds for the Rayleigh-Taylor unstable case of the densities. In fact, in the unstable case, the conditions for the initial data to develop a turning instability are expected to be weaker. It is direct to check that the same choice of $\\bar{z}_1$ as in the proof of Theorem \\ref{thm:turning} and the simple choice $\\bar{z}_2(\\alpha) = \\sin(\\alpha)$ is enough to ensure turning in the unstable scenario.\n\\end{rmk}\n\nWe conclude the introduction by laying out the organization of this article. In Section \\ref{sec:prelim}, we discuss several important properties concerning the Stokes-Transport system, including a class of symmetries and a potential energy. Section \\ref{sec:finiteE} is dedicated to proving Theorem \\ref{thm:main} on infinite-in-time growth in the unstable regime. In Section \\ref{sec:relaxation}, we show Theorem \\ref{thm:2} concerning the fingering phenomena. In Section \\ref{sec:RTbreakdown}, we prove Theorem \\ref{thm:turning} on Rayleigh-Taylor breakdown in the stable regime. Finally in Section \\ref{sec:numerics}, we provide some numerical examples regarding the growth phenomena proved in Theorem \\ref{thm:main} and \\ref{thm:2}.", "sketch": "The proof of Theorem~\\ref{thm:main} “crucially exploits a natural potential energy possessed by the Stokes-Transport system, which is monotone as the solution evolves in time.” The analysis “will focus on the analysis of this potential energy,” and in particular on deriving “inequalities that connect its time derivative with the perimeter and maximal curvature of the evolved surface $\\Gamma(t)$.”", "expanded_sketch": "The proof of the main theorem “crucially exploits a natural potential energy possessed by the Stokes-Transport system, which is monotone as the solution evolves in time.” The analysis “will focus on the analysis of this potential energy,” and in particular on deriving “inequalities that connect its time derivative with the perimeter and maximal curvature of the evolved surface $\\Gamma(t)$.”", "expanded_theorem": "\\label{thm:main}\n Let $\\rho(x,t)$ be the global-in-time solution to an unstable initial datum $\\rho_0 \\not\\equiv \\rho_s$, where \n $$\n \\rho_s(x_2) = \\begin{cases}\n 1, & x_2 > 0,\\\\\n -1, & x_2 < 0,\n \\end{cases}\n $$\n is the unstable stratified state. Moreover, suppose that $\\Gamma_0$ is centrally symmetric and evenly symmetric with respect to $x_1 = \\pm\\frac{\\pi}{2}$. Then for any $\\epsilon > 0$, we have\n \\begin{equation}\n \\label{est:main}\n \\limsup_{t \\to \\infty} t^{-\\frac{1}{17} + \\epsilon} (L(t) + \\mathcal{K}(t))= \\infty.\n \\end{equation}", "theorem_type": ["Asymptotic or Limit", "Universal"], "mcq": {"question": "Consider a global-in-time solution \\(\\rho(x,t)\\) of the free-boundary Stokes-transport problem in \\(\\mathbb T\\times\\mathbb R\\), with evolving interface \\(\\Gamma_t\\), arising from an unstable initial datum \\(\\rho_0\\not\\equiv \\rho_s\\), where the unstable stratified state is\n\\[\n\\rho_s(x_2)=\\begin{cases}\n1,& x_2>0,\\\\\n-1,& x_2<0.\n\\end{cases}\n\\]\nAssume that the initial interface \\(\\Gamma_0\\) is centrally symmetric and evenly symmetric with respect to the vertical lines \\(x_1=\\pm \\frac{\\pi}{2}\\). Let\n\\[\nL(t)=|\\Gamma_t|=\\int_{\\mathbb T}|\\partial_\\beta z(\\beta,t)|\\,d\\beta\n\\]\ndenote the perimeter of \\(\\Gamma_t\\), where \\(z(\\beta,t)\\) is a parametrization of the interface, and let\n\\[\n\\mathcal K(t)=\\max_{x\\in \\Gamma_t}|\\kappa(t)|\n\\]\ndenote the maximal curvature of \\(\\Gamma_t\\). Which statement holds for every such solution?", "correct_choice": {"label": "A", "text": "For every \\(\\epsilon>0\\),\n\\[\n\\limsup_{t\\to\\infty} t^{-\\frac{1}{17}+\\epsilon}\\bigl(L(t)+\\mathcal K(t)\\bigr)=\\infty.\n\\]"}, "choices": [{"label": "B", "text": "There exists \\(\\epsilon>0\\) such that\n\\[\n\\limsup_{t\\to\\infty} t^{-\\frac{1}{17}+\\epsilon}\\bigl(L(t)+\\mathcal K(t)\\bigr)=\\infty.\n\\]"}, {"label": "C", "text": "For every \\(\\epsilon>0\\),\n\\[\n\\limsup_{t\\to\\infty} t^{-\\frac{1}{17}+\\epsilon}L(t)=\\infty\n\\quad\\text{or}\\quad\n\\limsup_{t\\to\\infty} t^{-\\frac{1}{17}+\\epsilon}\\mathcal K(t)=\\infty.\n\\]"}, {"label": "D", "text": "For every \\(\\epsilon>0\\),\n\\[\n\\liminf_{t\\to\\infty} t^{-\\frac{1}{17}+\\epsilon}\\bigl(L(t)+\\mathcal K(t)\\bigr)=\\infty.\n\\]"}, {"label": "E", "text": "For every \\(\\epsilon>0\\), there exists \\(C_\\epsilon>0\\) such that for all sufficiently large \\(t\\),\n\\[\nL(t)+\\mathcal K(t)\\ge C_\\epsilon \\, t^{\\frac{1}{17}-\\epsilon}.\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": "universal-for-all-epsilon quantifier", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "sum-growth conclusion weakened to growth of at least one summand", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "limsup along subsequences replaced by uniform eventual lower growth", "template_used": "stronger_trap"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "energy-based intermittent growth upgraded to effective eventual polynomial lower bound", "template_used": "wildcard"}]}} +{"id": "2601.18507v1", "paper_link": "http://arxiv.org/abs/2601.18507v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{main}\nThere exists $\\varepsilon>0$ and a set $A\\subset \\mathbb N$ such that:\n\\begin{enumerate}\n\\item For all sufficiently large $m$, $r_A(m)>\\varepsilon\\log m$,\n\\item $A$ contains no minimal additive subbasis of order $2$.\n\\end{enumerate}", "start_pos": 1939, "end_pos": 2214, "label": "main"}, "ref_dict": {}, "pre_theorem_intro_text_len": 896, "pre_theorem_intro_text": "Let $\\mathbb{N}$ denote the positive integers and $A\\subset \\mathbb{N}$ be any subset. We write $r_A(m)$ for the number of pairs $(a,b)\\in A^2$ such that $a+b=m$ and $a\\le b$.\nWe say such pairs give \\emph{representations} of $m$.\nWe say $A$ is an \\emph{additive basis} (of order $2$, always in this paper) if $r_A(m)>0$ for all $m$ sufficiently large.\nAn additive basis is \\emph{minimal} if it contains no proper additive subbasis, i.e. no proper subset which is an additive basis in its own right.\nGiven $A$, a \\emph{summand} of $m$ is an element $a\\in A$ such that $m-a\\in A$ as well. (A set other than $A$ may be specified.)\n\nErd\\H{o}s and Nathanson \\cite{EN79} proved that if $c>\\log\\bigl(\\frac43\\bigr)^{-1}$ and $r_A(m) > c\\log m$ for all sufficiently large $m$, then $A$ must contain a minimal subbasis.\nThey conjectured that this is not true for all positive $c$. We prove this conjecture.", "context": "Let $\\mathbb{N}$ denote the positive integers and $A\\subset \\mathbb{N}$ be any subset. We write $r_A(m)$ for the number of pairs $(a,b)\\in A^2$ such that $a+b=m$ and $a\\le b$.\nWe say such pairs give \\emph{representations} of $m$.\nWe say $A$ is an \\emph{additive basis} (of order $2$, always in this paper) if $r_A(m)>0$ for all $m$ sufficiently large.\nAn additive basis is \\emph{minimal} if it contains no proper additive subbasis, i.e. no proper subset which is an additive basis in its own right.\nGiven $A$, a \\emph{summand} of $m$ is an element $a\\in A$ such that $m-a\\in A$ as well. (A set other than $A$ may be specified.)\n\nErd\\H{o}s and Nathanson \\cite{EN79} proved that if $c>\\log\\bigl(\\frac43\\bigr)^{-1}$ and $r_A(m) > c\\log m$ for all sufficiently large $m$, then $A$ must contain a minimal subbasis.\nThey conjectured that this is not true for all positive $c$. We prove this conjecture.", "full_context": "Let $\\mathbb{N}$ denote the positive integers and $A\\subset \\mathbb{N}$ be any subset. We write $r_A(m)$ for the number of pairs $(a,b)\\in A^2$ such that $a+b=m$ and $a\\le b$.\nWe say such pairs give \\emph{representations} of $m$.\nWe say $A$ is an \\emph{additive basis} (of order $2$, always in this paper) if $r_A(m)>0$ for all $m$ sufficiently large.\nAn additive basis is \\emph{minimal} if it contains no proper additive subbasis, i.e. no proper subset which is an additive basis in its own right.\nGiven $A$, a \\emph{summand} of $m$ is an element $a\\in A$ such that $m-a\\in A$ as well. (A set other than $A$ may be specified.)\n\nErd\\H{o}s and Nathanson \\cite{EN79} proved that if $c>\\log\\bigl(\\frac43\\bigr)^{-1}$ and $r_A(m) > c\\log m$ for all sufficiently large $m$, then $A$ must contain a minimal subbasis.\nThey conjectured that this is not true for all positive $c$. We prove this conjecture.\n\n\\begin{abstract}\nThere exists a set $A$ of positive integers such that the number of representations of a large positive integer $m$ as a sum of two elements of $A$ grows \nwith a lower bound of order $\\log m$, but for which there is no subset $D$ of $A$ minimal for the property that $D+D$ contains all sufficiently large positive integers.\n\\end{abstract}\n\nLet $\\N$ denote the positive integers and $A\\subset \\N$ be any subset. We write $r_A(m)$ for the number of pairs $(a,b)\\in A^2$ such that $a+b=m$ and $a\\le b$.\nWe say such pairs give \\emph{representations} of $m$.\nWe say $A$ is an \\emph{additive basis} (of order $2$, always in this paper) if $r_A(m)>0$ for all $m$ sufficiently large.\nAn additive basis is \\emph{minimal} if it contains no proper additive subbasis, i.e. no proper subset which is an additive basis in its own right.\nGiven $A$, a \\emph{summand} of $m$ is an element $a\\in A$ such that $m-a\\in A$ as well. (A set other than $A$ may be specified.)\n\nErd\\H{o}s and Nathanson \\cite{EN79} proved that if $c>\\log\\bigl(\\frac43\\bigr)^{-1}$ and $r_A(m) > c\\log m$ for all sufficiently large $m$, then $A$ must contain a minimal subbasis.\nThey conjectured that this is not true for all positive $c$. We prove this conjecture.\n\nLet $X_n=2^{2^n}$, and let\n$$I_n=[X_n,X_{n+1})\\cap\\mathbb N.$$\nHenceforth, we will not bother writing $\\N$ and will understand intervals to consist only of integers.\nWe will construct random subsets $A_n\\subset I_n$ and then modify $A_n$, \nfirst by removing a small set of elements to obtain $A'_n$, and then by adding in another small set of elements to obtain $A''_n$.\nLet $A(n)$ (resp. $A''(n)$) be the union of $A_i$ (resp. $A''_i$) as $i$ goes from $1$ to $n$. \nOur sequence of approximations to the final set therefore looks like\n$$\\cdots \\to A''(n-1)\\to A''(n-1)\\cup A_n\\to A''(n-1)\\cup A'_n\\to A''(n-1)\\cup A''_n=A''(n)\\to \\cdots$$\n\nThe idea for constructing $A'_n$ from $A_n$ is first to choose a small subset $B_n\\subset I_n$ \nand then to remove from $A_n$ all elements which are summands of elements of $B_n$ with respect to \n$A''(n-1)\\cup A_n$, in order to reduce the number of representations of each element of $B_n$ as a sum of two elements of $A''(n-1)\\cup A'_n$ to zero. \nWe then construct $A''_n$ (which then determines $A''(n)$) by adding some new elements to $A'_n$ in such a way that every element in\n$b\\in B_n$ has at least $\\varepsilon\\log b$ representations over $A''(n)$. Furthermore, this modification is carried out in such a way that, for large $n$ and\nany subset $D\\subset A''(n)$ such that $B_n\\subset D+D$, we have almost surely that every element in $I_{n-10}\\setminus B_{n-10}$ has at least two different representations as a sum in $D$,\nand, moreover, the smallest summand of any element of $B_n$ in $D$ goes to infinity as $n\\to \\infty$. We will see that this implies that $D$ cannot be a minimal basis.\n\nLet \n$$p(x) = \\begin{cases}\n\\min\\biggl(1,40\\sqrt{\\frac{\\log x}x}\\biggr) & \\text{ if $x\\ge 1$}\\\\\n0&\\text{ otherwise.}\n\\end{cases}$$\nThis is a log-convex function for $x$ sufficiently large.\nLet $\\Y_2,\\Y_3,\\Y_4 \\ldots$ denote independent Bernoulli random variables such that \n$\\Pr[\\Y_m=1]=p(m)$\nfor $m\\ge 2$.\nWe define $A_n = \\{m\\in I_n\\mid \\Y_m=1\\}$ and $A(n) = A_1\\cup\\cdots\\cup A_n$.\nThe $p(m)$ are chosen to ensure that\n$r_A(m)$ has upper and lower bounds which are constant multiples of $\\log m$ for all sufficiently large $m$.\n\n\\def\\ext{\\mathrm{ext}}\n\\begin{lemma}\n\\label{bounds}\nLet $A_n^{\\ext} = [X_n/4,X_{n+1})\\cap A(n)$. It is asymptotically true that for all $m\\in I_n$ we have \n$$r_{A_n^{\\ext} }(m) > 160 \\log m$$\nand\n$$r_{A(n)}(m) \\ll \\log m.$$\n\\end{lemma}\n\n\\begin{lemma}\n\\label{almost disjoint}\nLet $S\\subset [X_n/4,X_{n+1})$ be a set of positive integers, and let $k$ and $\\ell$ be positive integers. If a subset $Z$ of $[X_{n+1}/6,X_{n+1}/4)$ of\ncardinality $k$ is chosen uniformly, \nthen the probability that $|S\\cap Z| \\ge \\ell$ is $\\ll_\\ell \\Bigl(\\frac{k|S|}{X_n^2}\\Bigr)^\\ell$.\nIf instead, $Z\\subset \\N$ whose elements are each chosen independently, with probability at most $\\epsilon$, then the probability that $|S\\cap Z| \\ge \\ell$ is $\\le (\\epsilon |S|)^\\ell$.\n\\end{lemma}", "post_theorem_intro_text_len": 2862, "post_theorem_intro_text": "Let $X_n=2^{2^n}$, and let\n$$I_n=[X_n,X_{n+1})\\cap\\mathbb N.$$\nHenceforth, we will not bother writing $\\mathbb{N}$ and will understand intervals to consist only of integers.\nWe will construct random subsets $A_n\\subset I_n$ and then modify $A_n$, \nfirst by removing a small set of elements to obtain $A'_n$, and then by adding in another small set of elements to obtain $A''_n$.\nLet $A(n)$ (resp. $A''(n)$) be the union of $A_i$ (resp. $A''_i$) as $i$ goes from $1$ to $n$. \nOur sequence of approximations to the final set therefore looks like\n$$\\cdots \\to A''(n-1)\\to A''(n-1)\\cup A_n\\to A''(n-1)\\cup A'_n\\to A''(n-1)\\cup A''_n=A''(n)\\to \\cdots$$\n\nLet $A''$ be the union of all $A''_i$. \nOnce we have determined $A''_n$, we no longer make any changes within $I_n$, so $A''(n)$ is the correct initial part of our final set $A''$.\nWith probability $1$, the set $A''$ will satisfy the conditions\n(1) and (2). \n\nThe idea for constructing $A'_n$ from $A_n$ is first to choose a small subset $B_n\\subset I_n$ \nand then to remove from $A_n$ all elements which are summands of elements of $B_n$ with respect to \n$A''(n-1)\\cup A_n$, in order to reduce the number of representations of each element of $B_n$ as a sum of two elements of $A''(n-1)\\cup A'_n$ to zero. \nWe then construct $A''_n$ (which then determines $A''(n)$) by adding some new elements to $A'_n$ in such a way that every element in\n$b\\in B_n$ has at least $\\varepsilon\\log b$ representations over $A''(n)$. Furthermore, this modification is carried out in such a way that, for large $n$ and\nany subset $D\\subset A''(n)$ such that $B_n\\subset D+D$, we have almost surely that every element in $I_{n-10}\\setminus B_{n-10}$ has at least two different representations as a sum in $D$,\nand, moreover, the smallest summand of any element of $B_n$ in $D$ goes to infinity as $n\\to \\infty$. We will see that this implies that $D$ cannot be a minimal basis.\n\nThis strategy is based on \\cite{EN89}, which proves that for any fixed $t$, there exists a set $A$ with $r_A(m)\\ge t$ for all sufficiently large $m$ which nevertheless does not contain any minimal subbasis. One can think of our sets $B_n$ as being generalizations of their singletons $\\{N_n\\}$. The key trick of Erd\\H{o}s and Nathanson is to arrange matters so that \n$\\{N_1,N_2,\\ldots\\}$ serves as a ``canary in the coal mine,'' whose elements cease to be representable when the set $A$ is thinned, long before elements in its complement.\nThis makes the analysis of which subsets are subbases much more tractable. The cost of making the $N_n$ ``fragile'' is that $r_A(N_n)$ must be small (in their case bounded).\nBecause we are able to create many more fragile elements per generation, we are able to spread the load and make the representation count of elements in $B_n$\ngrow logarithmically (only smaller by a bounded factor than regular elements).", "sketch": "To prove Theorem~\\ref{main}, the construction proceeds in \"generations\" on intervals $I_n=[X_n,X_{n+1})$ where $X_n=2^{2^n}$. One constructs random subsets $A_n\\subset I_n$ and then modifies them in two small steps $A_n\\to A'_n\\to A''_n$, forming partial unions $A''(n)=\\bigcup_{i\\le n}A''_i$ and finally $A''=\\bigcup_n A''_n$. Once $A''_n$ is fixed, no further changes are made inside $I_n$, so $A''(n)$ is the correct initial segment of the final set; \"With probability $1$, the set $A''$ will satisfy\" conditions (1) and (2).\n\nThe modification $A_n\\to A'_n$ is designed to destroy representations of a chosen small set $B_n\\subset I_n$: one \"remove[s] from $A_n$ all elements which are summands of elements of $B_n$ with respect to $A''(n-1)\\cup A_n$, in order to reduce the number of representations of each element of $B_n$\" over $A''(n-1)\\cup A'_n$ \"to zero.\"\n\nThen one forms $A''_n$ by \"adding some new elements to $A'_n$\" so that every $b\\in B_n$ has at least $\\varepsilon\\log b$ representations over $A''(n)$. Simultaneously, the modification is arranged so that for large $n$ and any subset $D\\subset A''(n)$ with $B_n\\subset D+D$, almost surely (i) every element of $I_{n-10}\\setminus B_{n-10}$ has at least two different representations as a sum in $D$, and (ii) \"the smallest summand of any element of $B_n$ in $D$ goes to infinity as $n\\to\\infty$.\" The paper then indicates: \"We will see that this implies that $D$ cannot be a minimal basis,\" yielding condition (2) (no minimal additive subbasis of order $2$) while (1) follows from ensuring $r_{A''}(m)\\gtrsim \\log m$.\n\nThe overall strategy is described as based on \\cite{EN89}: the sets $B_n$ play the role of Erd\\H{o}s--Nathanson's special elements $\\{N_n\\}$, acting as a \"canary in the coal mine\" whose elements become non-representable when the set is thinned, \"long before\" elements in the complement, making the subbasis/minimality analysis tractable; by having \"many more fragile elements per generation\" one can \"spread the load\" so that $r_A(b)$ for $b\\in B_n$ can still \"grow logarithmically.\"", "expanded_sketch": "To prove the main theorem, the construction proceeds in \"generations\" on intervals $I_n=[X_n,X_{n+1})$ where $X_n=2^{2^n}$. One constructs random subsets $A_n\\subset I_n$ and then modifies them in two small steps $A_n\\to A'_n\\to A''_n$, forming partial unions $A''(n)=\\bigcup_{i\\le n}A''_i$ and finally $A''=\\bigcup_n A''_n$. Once $A''_n$ is fixed, no further changes are made inside $I_n$, so $A''(n)$ is the correct initial segment of the final set; \"With probability $1$, the set $A''$ will satisfy\" conditions (1) and (2).\n\nThe modification $A_n\\to A'_n$ is designed to destroy representations of a chosen small set $B_n\\subset I_n$: one \"remove[s] from $A_n$ all elements which are summands of elements of $B_n$ with respect to $A''(n-1)\\cup A_n$, in order to reduce the number of representations of each element of $B_n$\" over $A''(n-1)\\cup A'_n$ \"to zero.\"\n\nThen one forms $A''_n$ by \"adding some new elements to $A'_n$\" so that every $b\\in B_n$ has at least $\\varepsilon\\log b$ representations over $A''(n)$. Simultaneously, the modification is arranged so that for large $n$ and any subset $D\\subset A''(n)$ with $B_n\\subset D+D$, almost surely (i) every element of $I_{n-10}\\setminus B_{n-10}$ has at least two different representations as a sum in $D$, and (ii) \"the smallest summand of any element of $B_n$ in $D$ goes to infinity as $n\\to\\infty$.\" The paper then indicates: \"We will see that this implies that $D$ cannot be a minimal basis,\" yielding condition (2) (no minimal additive subbasis of order $2$) while (1) follows from ensuring $r_{A''}(m)\\gtrsim \\log m$.\n\nThe overall strategy is described as based on \\cite{EN89}: the sets $B_n$ play the role of Erd\\H{o}s--Nathanson's special elements $\\{N_n\\}$, acting as a \"canary in the coal mine\" whose elements become non-representable when the set is thinned, \"long before\" elements in the complement, making the subbasis/minimality analysis tractable; by having \"many more fragile elements per generation\" one can \"spread the load\" so that $r_A(b)$ for $b\\in B_n$ can still \"grow logarithmically.\"", "expanded_theorem": "\\label{main}\nThere exists $\\varepsilon>0$ and a set $A\\subset \\mathbb N$ such that:\n\\begin{enumerate}\n\\item For all sufficiently large $m$, $r_A(m)>\\varepsilon\\log m$,\n\\item $A$ contains no minimal additive subbasis of order $2$.\n\\end{enumerate}", "theorem_type": ["Existential–Universal", "Existence"], "mcq": {"question": "Let \\(\\mathbb N\\) be the positive integers. For a set \\(A\\subset \\mathbb N\\), define \\(r_A(m)\\) to be the number of pairs \\((a,b)\\in A^2\\) with \\(a+b=m\\) and \\(a\\le b\\). A set \\(D\\subset \\mathbb N\\) is called an additive basis of order \\(2\\) if \\(r_D(m)>0\\) for all sufficiently large integers \\(m\\). Such a basis is minimal if it contains no proper subset that is also an additive basis of order \\(2\\). Under these definitions, which existence statement holds?", "correct_choice": {"label": "A", "text": "There exist a real number \\(\\varepsilon>0\\) and a set \\(A\\subset \\mathbb N\\) such that \\(r_A(m)>\\varepsilon\\log m\\) for all sufficiently large integers \\(m\\), and \\(A\\) contains no subset \\(D\\subseteq A\\) that is a minimal additive basis of order \\(2\\)."}, "choices": [{"label": "B", "text": "There exist a real number \\(\\varepsilon>0\\) and a set \\(A\\subset \\mathbb N\\) such that \\(r_A(m)\\ge \\varepsilon\\log m\\) for all sufficiently large integers \\(m\\), and every subset \\(D\\subseteq A\\) that is an additive basis of order \\(2\\) is minimal."}, {"label": "C", "text": "There exist a real number \\(\\varepsilon>0\\) and a set \\(A\\subset \\mathbb N\\) such that \\(r_A(m)>\\varepsilon\\log m\\) for all sufficiently large integers \\(m\\), and \\(A\\) is not itself a minimal additive basis of order \\(2\\)."}, {"label": "D", "text": "For every real number \\(\\varepsilon>0\\), there exists a set \\(A\\subset \\mathbb N\\) such that \\(r_A(m)>\\varepsilon\\log m\\) for all sufficiently large integers \\(m\\), and \\(A\\) contains no subset \\(D\\subseteq A\\) that is a minimal additive basis of order \\(2\\)."}, {"label": "E", "text": "There exist a real number \\(\\varepsilon>0\\) and a set \\(A\\subset \\mathbb N\\) such that \\(r_A(m)>\\varepsilon\\log m\\) for all sufficiently large integers \\(m\\), and no subset \\(D\\subseteq A\\) is an additive basis of order \\(2\\)."}], "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": "direction of the minimality conclusion for subbases", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the stronger claim excluding all minimal additive subbases inside A", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "existential epsilon replaced by uniform-for-all epsilon", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "confuses absence of minimal additive subbasis with absence of any additive subbasis", "template_used": "property_confusion"}]}} +{"id": "2601.18519v1", "paper_link": "http://arxiv.org/abs/2601.18519v1", "theorems_cnt": 2, "theorem": {"env_name": "thm", "content": "\\label{thm:main}\nConsider an algebraic variety $X\\subset\\mathbb{K}^n\\setminus\\{0\\}$, given by an ideal\n$I\\subset\\mathbb{K}[x_1,\\ldots,x_n]$.\n\\begin{enumerate}\n\\item The set of initial parts can be described as a fibration of algebraic sets\n\\[\\Init_\\nu(X)=\\bigsqcup_{\\alpha\\in\\mathbb{R}}\\mathbb{V}(\\IN_\\alpha(I))\\setminus\\{0\\}.\\]\n\\item There exists a set of critical values $\\beta_0<\\ldots<\\beta_r$ such that, for\n$i=0,\\ldots,r+1$ the function $\\alpha\\in(\\beta_{i-1},\\beta_i)\\mapsto\\IN_\\alpha(I)$ is\nconstant and gives a homogeneous ideal. The ideals $\\IN_{\\beta_i}(I)$ are not homogeneous\n(we set $\\beta_{-1}=-\\infty$ and $\\beta_{r+1}=+\\infty$).\n\\end{enumerate}", "start_pos": 7986, "end_pos": 8644, "label": "thm:main"}, "ref_dict": {"prop:crit_values": "\\begin{prop}\\label{prop:crit_values}\nFix an ideal $I$ of $\\K[x_1,\\ldots,x_n]$. There is a finite set of values\n$\\beta_0<\\ldots<\\beta_r$ in $\\Gamma$ such that $\\forall\\alpha\\in(\\beta_{i},\\beta_{i+1})$\nthe initial ideal $\\IN_{\\underline{\\alpha}}(I)$ is constant and homogeneous.\n\\end{prop}", "eq:diffeo": "\\begin{equation}\\label{eq:diffeo}\n\\lim_{t\\to+\\infty}R_{\\log(t)^{-1}}(A_t)=[\\![e^\\alpha B+e^{-\\alpha}(B^*)^{adj}]\\!],\n\\end{equation}", "app:alt_proof": "\\label{app:alt_proof}\n\nIn this appendix, we briefly sketch out an alternative proof of \\cref{thm:phase_trop}, which\nis independent of \\cite[Prop. 3.2.11, p. 108]{MS}. We recall the statement: for any", "app:diffeo": "\\label{app:diffeo}\n\nLet $\\tilde R_h\\colon\\mathrm{SL}_2(\\C)\\to\\mathrm{SL}_2(\\C)$ be a diffeomorphism defined in terms\nof the polar decomposition by $PU\\mapsto P^hU$. Consider an $\\mathrm{SL}_2(\\C)$-val", "thm:phase_trop": "\\begin{thm}\\label{thm:phase_trop}\nFor any affine algebraic set $X=\\mathbb{V}(I)\\subset\\K^n$\n\\[\\forall\\alpha\\in\\Gamma_\\nu,\\quad \\Init_\\nu(X)_{\\alpha}=\n\\{\\alpha\\}\\times\\mathbb{V}_\\alpha(\\IN_{\\underline{\\alpha}}(I))),\n\\quad\\mathrm{where}\\quad\\underline{\\alpha}=(\\alpha,\\ldots,\\alpha).\\]\n\\end{thm}", "ex:counter-ex-dim3": "\\begin{ex}\\label{ex:counter-ex-dim3}\nThese results do not immediately generalize to $\\mathrm{SL}_\\ell$ for $\\ell\\geqslant3$.\nFix a value $\\alpha\\in(0,1)$ and consider, for instance, the diagonal matrix\n\\[A_t=\\begin{pmatrix} t & & \\\\ & t^\\alpha & \\\\ & & t^{-1-\\alpha}\\end{pmatrix}\n=t\\begin{pmatrix} 1 && \\\\ &0& \\\\ &&0\\end{pmatrix}+o(t).\\]\nClearly\n\\[\\widetilde{\\mathrm{VAL}}(A)=\\begin{pmatrix} e && \\\\ & e^\\alpha & \\\\ && e^{-1-\\alpha}\n\\end{pmatrix}\\]\ndepends on $\\alpha$, however the first asymptotic of $A_t$ does not recover $\\alpha$.\nOne needs to account for the lower order asymptotics.\n\\end{ex}", "thm:main": "\\begin{thm}\\label{thm:main}\nConsider an algebraic variety $X\\subset\\K^n\\setminus\\{0\\}$, given by an ideal\n$I\\subset\\K[x_1,\\ldots,x_n]$.\n\\begin{enumerate}\n\\item The set of initial parts can be described as a fibration of algebraic sets\n\\[\\Init_\\nu(X)=\\bigsqcup_{\\alpha\\in\\R}\\mathbb{V}(\\IN_\\alpha(I))\\setminus\\{0\\}.\\]\n\\item There exists a set of critical values $\\beta_0<\\ldots<\\beta_r$ such that, for\n$i=0,\\ldots,r+1$ the function $\\alpha\\in(\\beta_{i-1},\\beta_i)\\mapsto\\IN_\\alpha(I)$ is\nconstant and gives a homogeneous ideal. The ideals $\\IN_{\\beta_i}(I)$ are not homogeneous\n(we set $\\beta_{-1}=-\\infty$ and $\\beta_{r+1}=+\\infty$).\n\\end{enumerate}\n\\end{thm}"}, "pre_theorem_intro_text_len": 1702, "pre_theorem_intro_text": "\\label{sec:intro}\n\nKapranov's tropical geometric theorem is a foundational result relating tropicalizations\nof toric varieties to their defining ideals (see for instance \\cite[3.1.3 and 3.2.3]{MS}).\nIn fact the theorem is stronger and gives the defining ideal of phase tropicalizations.\nIn this paper we extend these results to affine varieties in the following way. Consider an\nalgebraic variety $X$ defined over the field of (reversed) Hahn series $(\\mathbb{K},\\nu)$, where $\\nu$\nis its associated valuation. Elements in this field can be written as $ct^\\alpha+o(t^\\alpha)$,\nwhere $c\\in\\mathbb{C},\\:\\alpha\\in\\mathbb{R}$ and the term $o(t^\\alpha)$ designates lower order terms. Any vector\n$A\\in\\mathbb{K}^n\\setminus\\{0\\}$ can be written as $t^\\alpha B+o(t^\\alpha),\\:B\\in\\mathbb{C}^n\\setminus\\{0\\}$ by\ntaking the entries of highest order. Define $\\Init_\\nu$ as the map that associates to this vector\nits leading part $(\\alpha, B)$. Our main result states that these leading terms form explicit\nalgebraic sets.\n\nFor any $f=\\sum_uc_ux^u\\in\\mathbb{K}[x_1,\\ldots,x_n]$, set $c_u=\\lambda_ut^{\\alpha_u}+o(t^{\\alpha_u})$ and\nwrite $\\nu_\\alpha(f):=\\max\\{\\nu(c_u)+\\alpha|u|\\::\\:c_u\\neq0\\}$, where\n$|(u_1,\\ldots,u_n)|=u_1+\\cdots+u_n$. Define then\n\\[\\IN_\\alpha(f):=\\sum_{\\nu(c_u)+\\alpha|u|=\\nu_\\alpha(f)}\\lambda_uX^u\\in\\mathbb{C}[X_1,\\ldots,X_n].\\]\nFor any ideal $I$ in $\\mathbb{K}[x_1,\\ldots,x_n]$, let $\\IN_\\alpha(I)$ be the ideal in\n$\\mathbb{C}[X_1,\\ldots,X_n]$ generated by the $\\IN_\\alpha(f),\\,f\\in I$. Recall that the disjoint union\nof a family of sets $(S_j)_{j\\in J}$ is constructed as $\\bigsqcup_{j\\in J}S_j:=\n\\bigcup_{j\\in J}\\{j\\}\\times S_j$. The following result is the main contribution of this paper.", "context": "\\label{sec:intro}\n\nKapranov's tropical geometric theorem is a foundational result relating tropicalizations\nof toric varieties to their defining ideals (see for instance \\cite[3.1.3 and 3.2.3]{MS}).\nIn fact the theorem is stronger and gives the defining ideal of phase tropicalizations.\nIn this paper we extend these results to affine varieties in the following way. Consider an\nalgebraic variety $X$ defined over the field of (reversed) Hahn series $(\\mathbb{K},\\nu)$, where $\\nu$\nis its associated valuation. Elements in this field can be written as $ct^\\alpha+o(t^\\alpha)$,\nwhere $c\\in\\mathbb{C},\\:\\alpha\\in\\mathbb{R}$ and the term $o(t^\\alpha)$ designates lower order terms. Any vector\n$A\\in\\mathbb{K}^n\\setminus\\{0\\}$ can be written as $t^\\alpha B+o(t^\\alpha),\\:B\\in\\mathbb{C}^n\\setminus\\{0\\}$ by\ntaking the entries of highest order. Define $\\Init_\\nu$ as the map that associates to this vector\nits leading part $(\\alpha, B)$. Our main result states that these leading terms form explicit\nalgebraic sets.\n\nFor any $f=\\sum_uc_ux^u\\in\\mathbb{K}[x_1,\\ldots,x_n]$, set $c_u=\\lambda_ut^{\\alpha_u}+o(t^{\\alpha_u})$ and\nwrite $\\nu_\\alpha(f):=\\max\\{\\nu(c_u)+\\alpha|u|\\::\\:c_u\\neq0\\}$, where\n$|(u_1,\\ldots,u_n)|=u_1+\\cdots+u_n$. Define then\n\\[\\IN_\\alpha(f):=\\sum_{\\nu(c_u)+\\alpha|u|=\\nu_\\alpha(f)}\\lambda_uX^u\\in\\mathbb{C}[X_1,\\ldots,X_n].\\]\nFor any ideal $I$ in $\\mathbb{K}[x_1,\\ldots,x_n]$, let $\\IN_\\alpha(I)$ be the ideal in\n$\\mathbb{C}[X_1,\\ldots,X_n]$ generated by the $\\IN_\\alpha(f),\\,f\\in I$. Recall that the disjoint union\nof a family of sets $(S_j)_{j\\in J}$ is constructed as $\\bigsqcup_{j\\in J}S_j:=\n\\bigcup_{j\\in J}\\{j\\}\\times S_j$. The following result is the main contribution of this paper.", "full_context": "\\label{sec:intro}\n\nKapranov's tropical geometric theorem is a foundational result relating tropicalizations\nof toric varieties to their defining ideals (see for instance \\cite[3.1.3 and 3.2.3]{MS}).\nIn fact the theorem is stronger and gives the defining ideal of phase tropicalizations.\nIn this paper we extend these results to affine varieties in the following way. Consider an\nalgebraic variety $X$ defined over the field of (reversed) Hahn series $(\\mathbb{K},\\nu)$, where $\\nu$\nis its associated valuation. Elements in this field can be written as $ct^\\alpha+o(t^\\alpha)$,\nwhere $c\\in\\mathbb{C},\\:\\alpha\\in\\mathbb{R}$ and the term $o(t^\\alpha)$ designates lower order terms. Any vector\n$A\\in\\mathbb{K}^n\\setminus\\{0\\}$ can be written as $t^\\alpha B+o(t^\\alpha),\\:B\\in\\mathbb{C}^n\\setminus\\{0\\}$ by\ntaking the entries of highest order. Define $\\Init_\\nu$ as the map that associates to this vector\nits leading part $(\\alpha, B)$. Our main result states that these leading terms form explicit\nalgebraic sets.\n\nFor any $f=\\sum_uc_ux^u\\in\\mathbb{K}[x_1,\\ldots,x_n]$, set $c_u=\\lambda_ut^{\\alpha_u}+o(t^{\\alpha_u})$ and\nwrite $\\nu_\\alpha(f):=\\max\\{\\nu(c_u)+\\alpha|u|\\::\\:c_u\\neq0\\}$, where\n$|(u_1,\\ldots,u_n)|=u_1+\\cdots+u_n$. Define then\n\\[\\IN_\\alpha(f):=\\sum_{\\nu(c_u)+\\alpha|u|=\\nu_\\alpha(f)}\\lambda_uX^u\\in\\mathbb{C}[X_1,\\ldots,X_n].\\]\nFor any ideal $I$ in $\\mathbb{K}[x_1,\\ldots,x_n]$, let $\\IN_\\alpha(I)$ be the ideal in\n$\\mathbb{C}[X_1,\\ldots,X_n]$ generated by the $\\IN_\\alpha(f),\\,f\\in I$. Recall that the disjoint union\nof a family of sets $(S_j)_{j\\in J}$ is constructed as $\\bigsqcup_{j\\in J}S_j:=\n\\bigcup_{j\\in J}\\{j\\}\\times S_j$. The following result is the main contribution of this paper.\n\nFor any $f=\\sum_uc_ux^u\\in\\K[x_1,\\ldots,x_n]$, set $c_u=\\lambda_ut^{\\alpha_u}+o(t^{\\alpha_u})$ and\nwrite $\\nu_\\alpha(f):=\\max\\{\\nu(c_u)+\\alpha|u|\\::\\:c_u\\neq0\\}$, where\n$|(u_1,\\ldots,u_n)|=u_1+\\cdots+u_n$. Define then\n\\[\\IN_\\alpha(f):=\\sum_{\\nu(c_u)+\\alpha|u|=\\nu_\\alpha(f)}\\lambda_uX^u\\in\\C[X_1,\\ldots,X_n].\\]\nFor any ideal $I$ in $\\K[x_1,\\ldots,x_n]$, let $\\IN_\\alpha(I)$ be the ideal in\n$\\C[X_1,\\ldots,X_n]$ generated by the $\\IN_\\alpha(f),\\,f\\in I$. Recall that the disjoint union\nof a family of sets $(S_j)_{j\\in J}$ is constructed as $\\bigsqcup_{j\\in J}S_j:=\n\\bigcup_{j\\in J}\\{j\\}\\times S_j$. The following result is the main contribution of this paper.\n\nAs an application, we retrieve the general picture of double-hyperbolic tropicalizations of\nsurfaces \\cref{thm:surfaces}. We henceforth work with the field $\\K$ of Hahn series and thus\nidentify $\\Gr_\\nu(\\K^n)$ with $\\R\\times\\C^n$. We recall briefly that for a variety\n$X\\subset\\mathrm{SL}_2(\\K)$, its double-hyperbolic tropicalization is\n$\\widehat{\\varkappa}(\\widetilde{\\mathrm{VAL}}(X))$, where\n$\\widehat{\\varkappa}(C)=(CC^*,C^*C)$ for a matrix $C\\in\\mathrm{SL}_2(\\C)$.\nThe layered structure that is stated in \\cref{thm:surfaces} stems from the general picture\nof the valuation tropicalization $\\Init_\\nu(X)$ since it dominates the\ndouble-hyperbolic tropicalization. First of all, $\\mathrm{SL}_2(\\K)$ is cut out by a single\nequation $\\det-1$, thus by \\cref{ex:principal}, $\\Init_\\nu(\\mathrm{SL}_2(\\K))=\\bigsqcup_\\alpha\n\\mathbb{V}(\\IN_{\\underline{\\alpha}}(\\det-1))$. We clearly have\n\\[\\IN_{\\underline{\\alpha}}(\\det-1)=\\left\\{\\begin{array}{ll}\n 1 & \\text{if } \\alpha<0\\\\\n \\det-1 & \\text{if } \\alpha=0\\\\\n \\det & \\text{if } \\alpha>0.\n\\end{array}\\right.\\]\nBy directly using the method in the proof of \\cref{prop:crit_values}, we see that\n$\\mathrm{SL}_2(\\K)$ has only one critical level, $\\beta=0$ and\n\\[\\Init_\\nu(\\mathrm{SL}_2(\\K))=\\Big(\\{0\\}\\times\\mathrm{SL}_2(\\C)\\Big)\n\\cup\\Big((0,+\\infty)\\times\\{\\det=0\\}\\Big).\\]\nSince $X\\subset\\mathrm{SL}_2(\\K)$, the critical levels $0=\\beta_0<\\ldots<\\beta_r$ of $X$\nare non-negative. Let $I$ be the defining ideal of $X$. We know that the ideal\n$\\IN_{\\underline{\\alpha}}(I)$ is constant and homogeneous for\n$\\alpha\\in(\\beta_{i},\\beta_{i+1})$. By \\cref{rke:dimension}, this amounts to saying that its\ndouble-hyperbolic tropicalization will be the union of cylinders $(\\beta_{i},\\beta_{i+1})\\times\nC_i$, for some complex projective curve $C_i$ and the fibers over the critical levels. By the\nsame remark, $\\IN_{\\underline{\\beta_i}}(I)$ is an inhomogeneous ideal of $\\C[x_1,x_2,x_3,x_4]$\nof height $2$. Thus when projectivizing $\\{\\det=0\\}$ to $Q_2(\\C)$, the image of\n$\\mathbb{V}(\\IN_{\\underline{\\beta_i}}(I))$ dominates $Q_2(\\C)$. This amounts to saying that\nabove the positive critical levels, the double-hyperbolic tropicalizations are of the form\n$\\{\\beta_i\\}\\times Q_2(\\C)$. Finally, the fiber over the vertex at the level $\\beta=0$\ncollapses to a point in the double-hyperbolic tropicalization.\n\nOne can further simplify the expression of $f$ to a reduced function $\\widetilde{f}$ such\nthat $\\mathrm{Trop}(f)=\\mathrm{Trop}(\\widetilde{f})$. Consider $f=\\sum_{u\\in U}c_ux^u$, with\n$c_u\\neq0\\iff u\\in U$ and set\n\\[\\widetilde{U}:=\\{u\\in U:\\:\\exists\\alpha\\in\\R,\\:\n\\mathrm{Trop}(f)\\cdot\\alpha=\\mathrm{Trop}(c_ux^u)=\\nu(c_u)+\\alpha|u|\\}.\\]\nWe define $\\widetilde{f}=\\sum_{u\\in\\widetilde{U}}c_ux^u$ and furthermore decompose it in\nhomogeneous components $\\widetilde{f}=\\widetilde{f}_0+\\cdots+\\widetilde{f}_d$.\nSince $f$ and $\\widetilde{f}$ have the same tropical polynomial, they have the same tropical\nroots. Furthermore, it is easy to check that they also have the same initial forms:\n$\\forall\\alpha,\\:\\IN_{\\underline{\\alpha}}(f)=\\IN_{\\underline{\\alpha}}(\\widetilde{f})$.\nIt may happen that $\\widetilde{f}_i=0$, for which we set\n$\\IN_{\\underline{\\alpha}}(\\widetilde{f}_i)=0$ for all $\\alpha$.\nIt is clear to see how the different $\\IN_{\\underline{\\alpha}}(\\widetilde{f})$ arise.\nCall $\\beta_1<\\ldots<\\beta_r$ the tropical roots of $\\mathrm{Trop}(\\widetilde{f})$ and set\n$\\beta_0=0$. There are degrees $0\\leqslant d_0\\leqslant\\ldots\\leqslant d_r$ such that\n\\begin{align*}\n\\IN_{\\underline{\\alpha}}(\\widetilde{f}) &=\n \\IN_{\\underline{\\alpha}}(\\widetilde{f}_{d_i}),\\: \\forall\\alpha\\in(\\beta_i,\\beta_{i+1})\n \\text{ and} \\\\\n\\IN_{\\underline{\\beta_i}}(\\widetilde{f}) &=\n \\IN_{\\underline{\\alpha}}(\\widetilde{f}_{d_{i-1}})+\\cdots+\n \\IN_{\\underline{\\beta_i}}(\\widetilde{f}_{d_i}),\\:i=1,\\ldots,r.\n\\end{align*}\nOne can simplify the expression of $\\widetilde{f}$ even further. If we are working over the\nfield of Hahn series $\\K$ one can replace every $c_ux^u,\\:u\\in\\widetilde{U}$ with $\\widehat{c_u}t^{\\nu(c_u)}$ with $\\widehat{c_u}\\in\\C$ and $c_u=\\widehat{c_u}t^{\\nu(c_u)}+\no(t^{\\nu(c_u)})$. We denote the ensuing polynomial $\\widehat f$. Since we have reduced $f$ to\n$\\widetilde{f}$, it is clear that $\\widehat{f}=t^{\\gamma_0}\\widehat{f}_0+\\cdots+\nt^{\\gamma_d}\\widehat{f}_d$ with $\\widehat{f}_i\\in\\C[x_1,\\ldots,x_n]$. The following result\nshows that valuative tropicalizations of general irreducible surfaces inside\n$\\mathrm{SL}_2(\\K)$ are the same as those given by such simplified expressions.\n\n\\begin{itemize}\n\\item \\textit{Step 0:} We show the inclusion $\\Init_\\nu(X)_\\alpha\\subset\\{\\alpha\\}\\times\n\\mathbb{V}_\\alpha(\\IN_{\\underline{\\alpha}}(I)))$. This has been done already in the proof of\n\\cref{thm:phase_trop}. Thus we now focus on the reverse inclusion. Fix\n$(\\init_\\nu(z_1),\\ldots,\\init_\\nu(z_n))\\in\\mathbb{V}_\\alpha(\\IN_{\\underline{\\alpha}}(I)))$\n\\item \\textit{Step 1:} We show \\cref{thm:phase_trop} when $X$ is hypersurface case.\nThe basic strategy consists in looking for solutions of the form $x_i=z_i+y_i$ where\n$\\nu(y_i)<\\alpha$. We leave out the details of this process.\n\\item \\textit{Step 2:} We reduce to the irreducible or prime ideal case (just like in the\nfirst part of the proof of \\cref{thm:phase_trop}).\n\\item \\textit{Step 3:} We prove a very subtle Noether normalization theorem. We assume our\nbase field $\\K$ to be infinite. Set a variety $X_0\\hookrightarrow\\A^n_\\K$ of dimension $d$,\ncodimension $c=n-d$ and fix a point $x\\in X_0$. One can find a general linear projection\n$\\pi:\\A^n_\\K\\to\\A^d_\\K$, such that the induced morphism $X_0\\to\\pi(X_0)=X_d\\subset\\A^d_\\K$\nis finite. We can refine this process and show that $\\pi$ can be factored into two linear\nprojections $\\pi=p\\circ q$ with $p:\\A^n_\\K\\to\\A^{d+1}_\\K$ and $q:\\A^{d+1}_\\K\\to\\A^d_\\K$, such\nthat $q^{-1}(q(x))=\\{x\\}$. In this situation $p(X_0)=X_{d-1}\\subset\\A^{d+1}_\\K$ is a\nhypersurface. Both $X_{d-1}$ and $X_d$ are irreducible varieties. This is an affine version\nof the refinement of the projective version of Noether's normalization that can be found in\n\\cite[Proposition (2.32), p. 38]{Mu}.\n\\item \\textit{Step 4:} We apply the previous step to $X_0=X$. Set $I_0,I_{d-1},I_d$ the\nrespective ideals of $X_0,X_{d-1}$ and $X_d$. By functoriality we obtain the following\n\\[\\begin{tikzcd}\n\\A^{n}_{\\gr_\\nu(\\K)} \\ar[r,\"\\Init_\\nu(p)\"] & \\A^{d-1}_{\\gr_\\nu(\\K)}\n \\ar[r,\"\\Init_\\nu(q)\"] & \\A^{d}_{\\gr_\\nu(\\K)} \\\\\n\\mathbb{V}_\\alpha(\\IN_{\\underline{\\alpha}}(I_0))\n \\ar[u,closed] \\ar[r,\"\\widetilde{\\Init_\\nu(p)}\"] &\n \\mathbb{V}_\\alpha(\\IN_{\\underline{\\alpha}}(I_{d-1}))\n \\ar[u,closed] \\ar[r,\"\\widetilde{\\Init_\\nu(q)}\"] &\n \\mathbb{V}_\\alpha(\\IN_{\\underline{\\alpha}}(I_d)) \\ar[u,equal] \\\\\n\\Init_\\nu(X_0) \\ar[u,closed] \\ar[r,\"\\widehat{\\Init_\\nu(p)}\"] &\n \\Init_\\nu(X_{d-1}) \\ar[u,equal] \\ar[r,\"\\widehat{\\Init_\\nu(q)}\"] &\n \\Init_\\nu(X_d) \\ar[u,equal]\n\\end{tikzcd}\\]\nIt is important to maintain the genericity conditions on the projections $\\Init_\\nu(p)$ and\n$\\Init_\\nu(q)$ by translating them into genericity conditions on $p$ and $q$.\n\\item \\textit{Step 5:} From the hypersurface case, we can conclude that\n$\\mathbb{V}_\\alpha(\\IN_{\\underline{\\alpha}}(I_{d-1}))=\\Init_\\nu(X_{d-1})$. We conclude via a\nsimple diagram chase, considering that $\\widehat{\\Init_\\nu(p)}$ is surjective and step 4.\n\\end{itemize}", "post_theorem_intro_text_len": 6773, "post_theorem_intro_text": "Our main application of this result concerns $\\mathrm{SL}_2$ (or $\\mathrm{PSL}_2$)\ntropicalizations as they were introduced in \\cite{MS22}. Simplifying, they associate\nto a family of varieties $X=(X_t)_{t>0}$ with $X_t\\subset\\mathrm{PSL}_2(\\mathbb{C})$ a limit degeneration\nthat we denote $\\mathrm{VAL}(X)\\subset\\mathrm{PSL}_2(\\mathbb{C})$. Consider the polar decomposition of\n$\\mathrm{PSL}_2(\\mathbb{C})\\simeq\\mathbb{H}^3\\times\\mathrm{PSU}(2)$. The set $\\mathbb{H}^3$ of positive\ndefinite, Hermitian and unimodular matrices is a (Hermitian) model of the hyperbolic space.\nWe write $O$ for its center, which is the identity matrix. The natural metric gives rise to a\ndistance function that can be calculated explicitly: $\\mathrm{dist}(O,B)=|\\ln(\\lambda(B))|$,\nwhere $\\lambda(B)$ is one of the eigenvalues of $B$. Focusing on the Hermitian parts\nof $\\mathrm{VAL}(X)$, we obtain the hyperbolic tropicalization of $X$. The authors of \\cite{MS22}\nobtain that the hyperbolic tropicalizations of families of complex algebraic curves are unions of\nconcentric spheres around $O$ and segments extending to geodesics passing through $O$. In\n\\cite{PS25}, it is shown that the hyperbolic tropicalization of a family of surfaces $(X_t)_{t>0}$\nin $\\mathrm{PSL}_2(\\mathbb{C})$ is a complement to an open ball in $\\mathbb{H}^3$ centered at the image of\nthe identity matrix denoted by $O$.\n\nIn this work, \\cref{thm:main} completes \\cite[Thm. 3, 4]{SP25}, thus proving an\n$\\mathrm{SL}_2$ version of Kapranov's theorem at the algebraic level for complex surfaces, and\nreprove and refine the analogous theorem for curves. Write $\\widetilde{\\mathrm{VAL}}(X)$ for the\n$\\mathrm{SL}_2$ tropicalization of $X$. When restricting our varieties to convergent Hahn series,\none can still see $X$ as a family of varieties\n\\footnote{More specifically, suppose $X$ is given by equations $X=\\mathbb{V}(f_1,\\ldots,f_s)$.\nEach polynomial $f_j$ has coefficients in $\\mathbb{K}$, thus they are functions in $t$, so we\ncan write $f_j=f_j(t)$. We can thus define the family $(X_t)_{t>0}$ by\n$X_t=\\mathbb{V}(f_1(t),\\ldots,f_s(t))$.}. In this situation, the critical levels\ncorrespond to the radii of the concentric spheres in $\\mathbb{H}^3$ of the hyperbolic\ntropicalization.\n\nTo help in describing the hyperbolic degenerations for surfaces we upgrade our hyperbolic picture\nto a double hyperbolic one. More precisely, consider the two polar decompositions of a matrix\n$C\\in\\mathrm{SL}_2(\\mathbb{C})$, $C=PU$ and its dual $B=U'P'$ with $P^*=P$ positive definite, unimodular,\n$UU^*=U^*U=1$ and likewise for $P',U'$. Then associate to $C$, its two Hermitian parts $(P,P')$\nand denote it $\\widehat{\\varkappa}(C)$. We prove the following in \\cref{sec:layered}.\n\n\\begin{thm}\\label{thm:surfaces}\nConsider a surface $X\\subset\\mathrm{SL}_2(\\mathbb{K})$ and let $\\beta_0<\\beta_1<\\dots<\\beta_r$ be its\ncritical levels. Then the double-hyperbolic tropicalization is\n\\[\\widehat{\\varkappa}(\\widetilde{\\mathrm{VAL}}(X))=\\bigcup_{i=0}^r\\{\\beta_i\\}\\times Q_2(\\mathbb{C})\n\\cup\\bigcup_{k=0}^r(\\beta_i,\\beta_{i+1})\\times C_k,\\]\nwhere $\\beta_{r+1}=+\\infty$ and $C_0,\\dots,C_r$ are complex algebraic curves on\n$Q_2(\\mathbb{C})$ of symmetric bi-degree increasing with $i$.\n\\end{thm}\n\nTo summarize, this work settles the question of how phase tropical limits of surfaces in\n$\\mathrm{SL}_2(\\mathbb{C})$ may look like in general. In particular we resolve the previous issue\nof showing that inclusions in Theorems 3 and 4 of \\cite{SP25} are actually equalities.\nThis opens the door to extending the general principle that phase tropicalizations restore\nthe topology of the initial variety. The first instance of this idea was Viro's patchworking\n\\cite{V83}, which, in essence, is a real phase toric tropicalization. The second instance\nwas accomplished in theorems for complex phases, of Kerr and Zharkov \\cite{KZ18},\nand of Kim and Nisse \\cite{KN21}. The $\\mathrm{SL}_2$ versions of these facts are subjects\nof current investigation.\n\nWe reiterate that what follows is building foundations of a more general framework\nwhich may be referred to as {\\it affine initial forms}, inspired by and applied to\n$\\mathrm{SL}_2$ phase tropicalization, but not limited to it. The closest set of examples\nare affine quadrics as ambient spaces with the first relevant instance being akin to\nBrugallé's conic floor diagrams \\cite{B15}. Nevertheless in order to generalize the above\nstatements to other groups, one would be required to go beyond the first-order term in the\nasymptotic expansion (see \\cref{ex:counter-ex-dim3}). This necessitates an adequate analytic,\ngeometric, representation, and valuation theoretic sophistication that has yet to be developed.\n\nThe paper is organized as follows. In \\cref{sec:fibers} we detail different types of\ntropicalizations that help clarify the general picture of the hyperbolic degenerations.\nIn \\cref{sec:prelim} and \\cref{sec:composing_init} we introduce the elements of valuation\nthat are needed for the proof of \\cref{thm:main}. Notably we make extensive use of\nthe graded algebra associated to a valuative pair. Additionally, we introduce a similar\nconstruction, that we name phase space. This is the structure inside which the initial terms\nof vectors live. It is more convenient for our proofs and future applications to establish\nthe formalism in full generality. One important property of graded algebras is functoriality.\nRoughly speaking, functoriality allows one to make a change of variables. It is the key\ningredient in our arguments towards the Kapranov-like theorem. Our proof is divided between\n\\cref{sec:lifting} and \\cref{sec:layered}. More precisely, the first point of \\cref{thm:main}\nis shown in \\cref{thm:phase_trop} and the second is contained in \\cref{prop:crit_values}.\nWe sketch an alternative proof to \\cref{thm:phase_trop} in \\cref{app:alt_proof} and in\n\\cref{app:diffeo} we detail the proof of \\eqref{eq:diffeo} and show how it defines a diffeomorphism\nbetween $\\mathrm{SL}_2(\\mathbb{C})$ and its non-abelian tropicalization.\n\n\\textbf{Acknowledgments.} We would like to extend our gratitude to Peter Petrov, Grigory Mikhalkin\nand Ilia Zharkov for proposing to us this project, their ample support and fruitful discussions.\nThe first author was supported by a \"Peter Beron i NIE\"\nfellowship [KP-06-DB-5] from the Bulgarian Science Fund.\nThe second author gratefully acknowledges the hospitality of IMPA (Rio de Janeiro), where this work was completed during his visit, and is supported by the Simons Foundation, grant SFI-MPS-T-Institutes-00007697,\nand the Ministry of Education and Science of the Republic of Bulgaria,\ngrant DO1-239/10.12.2024, as well as by the National Science Fund,\nThe Ministry of Education and Science of the Republic of Bulgaria, under contract KP-06-N92/2.", "sketch": "The post-theorem text does not give a detailed argument, but it does outline where the two parts of \\cref{thm:main} are proved and what tools are used. The authors say that, for the proof of \\cref{thm:main}, they “introduce the elements of valuation” and “make extensive use of the graded algebra associated to a valuative pair,” and also introduce a “phase space” as “the structure inside which the initial terms of vectors live.” They emphasize that “one important property of graded algebras is functoriality,” and that “functoriality allows one to make a change of variables” and is “the key ingredient in our arguments towards the Kapranov-like theorem.” They then specify the division of the proof by results/sections: “the first point of \\cref{thm:main} is shown in \\cref{thm:phase_trop} and the second is contained in \\cref{prop:crit_values},” with an “alternative proof to \\cref{thm:phase_trop}” sketched in \\cref{app:alt_proof}.", "expanded_sketch": "The post-theorem text does not give a detailed argument, but it does outline where the two parts of the main theorem are proved and what tools are used. The authors say that, in establishing the main theorem, they “introduce the elements of valuation” and “make extensive use of the graded algebra associated to a valuative pair,” and also introduce a “phase space” as “the structure inside which the initial terms of vectors live.” They emphasize that “one important property of graded algebras is functoriality,” and that “functoriality allows one to make a change of variables” and is “the key ingredient in our arguments towards the Kapranov-like theorem.”\n\nThey then specify the division of the proof by results/sections: the first point of the main theorem is shown in the following theorem.\n\n\\begin{thm}\\label{thm:phase_trop}\nFor any affine algebraic set $X=\\mathbb{V}(I)\\subset\\K^n$\n\\[\\forall\\alpha\\in\\Gamma_\\nu,\\quad \\Init_\\nu(X)_{\\alpha}=\n\\{\\alpha\\}\\times\\mathbb{V}_\\alpha(\\IN_{\\underline{\\alpha}}(I))),\n\\quad\\mathrm{where}\\quad\\underline{\\alpha}=(\\alpha,\\ldots,\\alpha).\\]\n\\end{thm}\n\nThe second point is contained in the following proposition: \\begin{prop}\\label{prop:crit_values}\nFix an ideal $I$ of $\\K[x_1,\\ldots,x_n]$. There is a finite set of values\n$\\beta_0<\\ldots<\\beta_r$ in $\\Gamma$ such that $\\forall\\alpha\\in(\\beta_{i},\\beta_{i+1})$\nthe initial ideal $\\IN_{\\underline{\\alpha}}(I)$ is constant and homogeneous.\n\\end{prop}\n\nThey also mention that an alternative proof to the theorem above is sketched in the following appendix text:\n\n\\label{app:alt_proof}\n\nIn this appendix, we briefly sketch out an alternative proof of \\cref{thm:phase_trop}, which\nis independent of \\cite[Prop. 3.2.11, p. 108]{MS}. We recall the statement: for any", "expanded_theorem": "\\label{thm:main}\nConsider an algebraic variety $X\\subset\\mathbb{K}^n\\setminus\\{0\\}$, given by an ideal\n$I\\subset\\mathbb{K}[x_1,\\ldots,x_n]$.\n\\begin{enumerate}\n\\item The set of initial parts can be described as a fibration of algebraic sets\n\\[\\Init_\\nu(X)=\\bigsqcup_{\\alpha\\in\\mathbb{R}}\\mathbb{V}(\\IN_\\alpha(I))\\setminus\\{0\\}.\\]\n\\item There exists a set of critical values $\\beta_0<\\ldots<\\beta_r$ such that, for\n$i=0,\\ldots,r+1$ the function $\\alpha\\in(\\beta_{i-1},\\beta_i)\\mapsto\\IN_\\alpha(I)$ is\nconstant and gives a homogeneous ideal. The ideals $\\IN_{\\beta_i}(I)$ are not homogeneous\n(we set $\\beta_{-1}=-\\infty$ and $\\beta_{r+1}=+\\infty$).\n\\end{enumerate},", "theorem_type": ["Existence", "Universal"], "mcq": {"question": "Let \\(\\mathbb{K}\\) be the field of Hahn series with valuation \\(\\nu\\). For any nonzero vector \\(A\\in\\mathbb{K}^n\\setminus\\{0\\}\\), write \\(A=t^\\alpha B+o(t^\\alpha)\\) with \\(\\alpha\\in\\mathbb{R}\\) and \\(B\\in\\mathbb{C}^n\\setminus\\{0\\}\\), and define its initial part by \\(\\Init_\\nu(A)=(\\alpha,B)\\). For an algebraic variety \\(X\\subset \\mathbb{K}^n\\setminus\\{0\\}\\), let \\(\\Init_\\nu(X)\\) be the set of initial parts of points of \\(X\\). If \\(I\\subset \\mathbb{K}[x_1,\\ldots,x_n]\\) is the defining ideal of \\(X\\), and for \\(f=\\sum_u c_u x^u\\) with \\(c_u=\\lambda_u t^{\\alpha_u}+o(t^{\\alpha_u})\\) one sets\n\\[\n\\nu_\\alpha(f):=\\max\\{\\nu(c_u)+\\alpha|u|:c_u\\neq 0\\},\\qquad\n\\IN_\\alpha(f):=\\sum_{\\nu(c_u)+\\alpha|u|=\\nu_\\alpha(f)}\\lambda_u X^u\\in \\mathbb{C}[X_1,\\ldots,X_n],\n\\]\nwhere \\(|u|=u_1+\\cdots+u_n\\), and \\(\\IN_\\alpha(I)\\) is the ideal generated by all \\(\\IN_\\alpha(f)\\) with \\(f\\in I\\), while \\(\\mathbb{V}(J)\\) denotes the zero locus of an ideal \\(J\\subset\\mathbb{C}[X_1,\\ldots,X_n]\\), which statement holds? Here \\(\\bigsqcup_{\\alpha\\in\\mathbb{R}} S_\\alpha\\) denotes the disjoint union \\(\\bigcup_{\\alpha\\in\\mathbb{R}} \\{\\alpha\\}\\times S_\\alpha\\).", "correct_choice": {"label": "A", "text": "The set of initial parts is exactly\n\\[\n\\Init_\\nu(X)=\\bigsqcup_{\\alpha\\in\\mathbb{R}} \\mathbb{V}(\\IN_\\alpha(I))\\setminus\\{0\\},\n\\]\nand there exist real numbers \\(\\beta_0<\\cdots<\\beta_r\\) such that, with the conventions \\(\\beta_{-1}=-\\infty\\) and \\(\\beta_{r+1}=+\\infty\\), for every \\(i=0,\\ldots,r+1\\) the map \\(\\alpha\\in(\\beta_{i-1},\\beta_i)\\mapsto \\IN_\\alpha(I)\\) is constant and its common value is a homogeneous ideal, whereas each ideal \\(\\IN_{\\beta_i}(I)\\) is not homogeneous."}, "choices": [{"label": "B", "text": "The set of initial parts is exactly\n\\[\n\\Init_\\nu(X)=\\bigsqcup_{\\alpha\\in\\mathbb{R}} \\mathbb{V}(\\IN_\\alpha(I))\\setminus\\{0\\},\n\\]\nand there exist real numbers \\(\\beta_0<\\cdots<\\beta_r\\) such that, with the conventions \\(\\beta_{-1}=-\\infty\\) and \\(\\beta_{r+1}=+\\infty\\), for every \\(i=0,\\ldots,r+1\\) the map \\(\\alpha\\in[\\beta_{i-1},\\beta_i]\\mapsto \\IN_\\alpha(I)\\) is constant and its common value is a homogeneous ideal."}, {"label": "C", "text": "The set of initial parts satisfies\n\\[\n\\Init_\\nu(X)=\\bigsqcup_{\\alpha\\in\\mathbb{R}} \\mathbb{V}(\\IN_\\alpha(I))\\setminus\\{0\\},\n\\]\nand there exist real numbers \\(\\beta_0<\\cdots<\\beta_r\\) such that, with the conventions \\(\\beta_{-1}=-\\infty\\) and \\(\\beta_{r+1}=+\\infty\\), for every \\(i=0,\\ldots,r+1\\) the map \\(\\alpha\\in(\\beta_{i-1},\\beta_i)\\mapsto \\IN_\\alpha(I)\\) is constant."}, {"label": "D", "text": "The set of initial parts is exactly\n\\[\n\\Init_\\nu(X)=\\bigsqcup_{\\alpha\\in\\mathbb{R}} \\mathbb{V}(\\IN_\\alpha(I))\\setminus\\{0\\},\n\\]\nand there exists a finite set of real numbers \\(\\beta_0<\\cdots<\\beta_r\\) such that, with the conventions \\(\\beta_{-1}=-\\infty\\) and \\(\\beta_{r+1}=+\\infty\\), for every \\(i=0,\\ldots,r+1\\) and every \\(\\alpha\\in(\\beta_{i-1},\\beta_i)\\), the ideal \\(\\IN_\\alpha(I)\\) is homogeneous, and moreover each critical ideal \\(\\IN_{\\beta_i}(I)\\) is also homogeneous."}, {"label": "E", "text": "The set of initial parts is exactly\n\\[\n\\Init_\\nu(X)=\\bigsqcup_{\\alpha\\in\\mathbb{R}} \\mathbb{V}(\\IN_\\alpha(I))\\setminus\\{0\\},\n\\]\nand for every \\(\\alpha\\in\\mathbb{R}\\) the ideal \\(\\IN_\\alpha(I)\\) is homogeneous; in particular one may take no critical values at all, i.e. the map \\(\\alpha\\mapsto \\IN_\\alpha(I)\\) is constant on all of \\(\\mathbb{R}\\)."}], "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": "open_interval_vs_closed_interval_for_constancy", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped_homogeneous_on_open_intervals_and_nonhomogeneous_at_critical_values", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "critical_levels_are_exceptional_nonhomogeneous", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "finiteness", "tampered_component": "finite_piecewise_constant_stratification_replaced_by_global_constancy", "template_used": "wildcard"}]}} +{"id": "2601.18738v1", "paper_link": "http://arxiv.org/abs/2601.18738v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{JPX} \nLet $k\\geq 5$, and $a_1,\\dots,a_k\\in\\mathbb{Z}\\setminus\\{0\\}$ with $\\sum_{i=1}^k a_i=0$. \nLet $A\\subseteq [N]$ be $K_{s,t}$-free and assume that $A$ lacks nontrivial solutions of $a_1x_1+\\cdots +a_kx_k=0$. Then\n\\[\n|A|\\ll N^{1-1/s} \\exp(-O_{a_i,s,t}((\\log\\log N)^{\\frac17})).\n\\]", "start_pos": 5596, "end_pos": 5915, "label": "JPX"}, "ref_dict": {"KST": "\\begin{equation} \\label{KST}\n\\operatorname{ex}(n,K_{s,t}) \\ll_{s,t} n^{2-1/s}.\n\\end{equation}", "JPX2": "\\begin{theorem}\\label{JPX2}\nLet $q$ be a fixed odd prime power. Let $k\\ge 5$ and $a_1,\\dots,a_k\\in\\F_q^\\times$ satisfy $\\sum_{i=1}^k a_i=0$.\nLet $A\\subseteq \\F_q^n$ be $K_{s,t}$-free and assume that $A$ contains no nontrivial solution to\n$a_1x_1+\\cdots+a_kx_k=0$. Then, by writing $N=q^n$, there exists $c=c(q,k,s,t)>0$ such that\n\\[\n|A|\\ll N^{1-1/s}(\\log N)^{-c}.\n\\]\n\\end{theorem}", "JPX": "\\begin{theorem}\\label{JPX} \nLet $k\\geq 5$, and $a_1,\\dots,a_k\\in\\mathbb{Z}\\setminus\\{0\\}$ with $\\sum_{i=1}^k a_i=0$. \nLet $A\\subseteq [N]$ be $K_{s,t}$-free and assume that $A$ lacks nontrivial solutions of $a_1x_1+\\cdots +a_kx_k=0$. Then\n\\[\n|A|\\ll N^{1-1/s} \\exp(-O_{a_i,s,t}((\\log\\log N)^{\\frac17})).\n\\]\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2264, "pre_theorem_intro_text": "Fix integers $2\\le s\\le t$ and write $K_{s,t}$ for the complete bipartite graph with parts of sizes\n$s$ and $t$. Let $\\operatorname{ex}(n,K_{s,t})$ denote the maximum number of edges in an $n$-vertex graph containing no copy of $K_{s,t}$. A well-known theorem of K\\H{o}v\\'ari--S\\'os--Tur\\'an~\\cite{KST54} gives\n\\begin{equation} \\label{KST}\n\\operatorname{ex}(n,K_{s,t}) \\ll_{s,t} n^{2-1/s}.\n\\end{equation}\nThere are several remarkable algebraic constructions showing that \\eqref{KST} is optimal (up to constant factors) for various values of $s$ and $t$, in particular when $t$ is sufficiently large compared to $s$. See for example the norm graph constructions from ~\\cite{KRSz96,ARSz99}, and the more recent\nrandom-algebraic construction from \\cite{BukhCrazy}. There are also many other values of $s$ and $t$ for which the asymptotic value of $\\operatorname{ex}(n,K_{s,t})$ is poorly understood. For example, even for $s=t=4$, the problem is wide open. \n\nIn this paper, we consider the following arithmetic analogue, first introduced by Erd\\H{o}s and Harzheim in \\cite{ErdosHarzheim1986} (see also \\cite{CT18}). For integers $2\\le s\\le t$, we say that a set $A\\subseteq [N]$ is \\emph{$K_{s,t}$-free} if it contains no configuration of the form\n\\[\n\\{x_i+y_j : 1\\le i\\le s,\\ 1\\le j\\le t\\} \\subseteq A\n\\]\nwith integers $x_1,\\dots,x_s$ distinct and integers $y_1,\\dots,y_t$ distinct integers. Equivalently, $A$ contains no subset of\nthe form $B+C$ with $|B|=s$ and $|C|=t$. A standard sum-graph reduction shows that the K\\H{o}v\\'ari--S\\'os--Tur\\'an theorem immediately implies\nthe trivial density bound\n\\[\n|A| \\ll_{s,t} N^{1-1/s}.\n\\]\nWhen $s=t=2$, such sets are precisely \\emph{Sidon sets} (also called $B_2$-sets), and the above\ncorrespondence recovers the classical link between Sidon sets and $C_4$-free graphs. See \\cite{CT18} for more details and further generalizations.\n\n\\medskip\n\\subsection*{Our results}\nBuilding upon recent work of Conlon--Fox--Sudakov--Zhao~\\cite{CFSZ} and Prendiville~\\cite{Prendi}, we show that all dense $K_{s,t}$-free subsets of $\\{1,\\dots,N\\}$ must possess arithmetic structure, in the sense that they must always contain a solution to any translation-invariant linear equation\nin at least five variables, with all variables distinct.", "context": "Fix integers $2\\le s\\le t$ and write $K_{s,t}$ for the complete bipartite graph with parts of sizes\n$s$ and $t$. Let $\\operatorname{ex}(n,K_{s,t})$ denote the maximum number of edges in an $n$-vertex graph containing no copy of $K_{s,t}$. A well-known theorem of K\\H{o}v\\'ari--S\\'os--Tur\\'an~\\cite{KST54} gives\n\\begin{equation} \\label{KST}\n\\operatorname{ex}(n,K_{s,t}) \\ll_{s,t} n^{2-1/s}.\n\\end{equation}\nThere are several remarkable algebraic constructions showing that \\eqref{KST} is optimal (up to constant factors) for various values of $s$ and $t$, in particular when $t$ is sufficiently large compared to $s$. See for example the norm graph constructions from ~\\cite{KRSz96,ARSz99}, and the more recent\nrandom-algebraic construction from \\cite{BukhCrazy}. There are also many other values of $s$ and $t$ for which the asymptotic value of $\\operatorname{ex}(n,K_{s,t})$ is poorly understood. For example, even for $s=t=4$, the problem is wide open.\n\nIn this paper, we consider the following arithmetic analogue, first introduced by Erd\\H{o}s and Harzheim in \\cite{ErdosHarzheim1986} (see also \\cite{CT18}). For integers $2\\le s\\le t$, we say that a set $A\\subseteq [N]$ is \\emph{$K_{s,t}$-free} if it contains no configuration of the form\n\\[\n\\{x_i+y_j : 1\\le i\\le s,\\ 1\\le j\\le t\\} \\subseteq A\n\\]\nwith integers $x_1,\\dots,x_s$ distinct and integers $y_1,\\dots,y_t$ distinct integers. Equivalently, $A$ contains no subset of\nthe form $B+C$ with $|B|=s$ and $|C|=t$. A standard sum-graph reduction shows that the K\\H{o}v\\'ari--S\\'os--Tur\\'an theorem immediately implies\nthe trivial density bound\n\\[\n|A| \\ll_{s,t} N^{1-1/s}.\n\\]\nWhen $s=t=2$, such sets are precisely \\emph{Sidon sets} (also called $B_2$-sets), and the above\ncorrespondence recovers the classical link between Sidon sets and $C_4$-free graphs. See \\cite{CT18} for more details and further generalizations.\n\n\\medskip\n\\subsection*{Our results}\nBuilding upon recent work of Conlon--Fox--Sudakov--Zhao~\\cite{CFSZ} and Prendiville~\\cite{Prendi}, we show that all dense $K_{s,t}$-free subsets of $\\{1,\\dots,N\\}$ must possess arithmetic structure, in the sense that they must always contain a solution to any translation-invariant linear equation\nin at least five variables, with all variables distinct.", "full_context": "Fix integers $2\\le s\\le t$ and write $K_{s,t}$ for the complete bipartite graph with parts of sizes\n$s$ and $t$. Let $\\operatorname{ex}(n,K_{s,t})$ denote the maximum number of edges in an $n$-vertex graph containing no copy of $K_{s,t}$. A well-known theorem of K\\H{o}v\\'ari--S\\'os--Tur\\'an~\\cite{KST54} gives\n\\begin{equation} \\label{KST}\n\\operatorname{ex}(n,K_{s,t}) \\ll_{s,t} n^{2-1/s}.\n\\end{equation}\nThere are several remarkable algebraic constructions showing that \\eqref{KST} is optimal (up to constant factors) for various values of $s$ and $t$, in particular when $t$ is sufficiently large compared to $s$. See for example the norm graph constructions from ~\\cite{KRSz96,ARSz99}, and the more recent\nrandom-algebraic construction from \\cite{BukhCrazy}. There are also many other values of $s$ and $t$ for which the asymptotic value of $\\operatorname{ex}(n,K_{s,t})$ is poorly understood. For example, even for $s=t=4$, the problem is wide open.\n\nIn this paper, we consider the following arithmetic analogue, first introduced by Erd\\H{o}s and Harzheim in \\cite{ErdosHarzheim1986} (see also \\cite{CT18}). For integers $2\\le s\\le t$, we say that a set $A\\subseteq [N]$ is \\emph{$K_{s,t}$-free} if it contains no configuration of the form\n\\[\n\\{x_i+y_j : 1\\le i\\le s,\\ 1\\le j\\le t\\} \\subseteq A\n\\]\nwith integers $x_1,\\dots,x_s$ distinct and integers $y_1,\\dots,y_t$ distinct integers. Equivalently, $A$ contains no subset of\nthe form $B+C$ with $|B|=s$ and $|C|=t$. A standard sum-graph reduction shows that the K\\H{o}v\\'ari--S\\'os--Tur\\'an theorem immediately implies\nthe trivial density bound\n\\[\n|A| \\ll_{s,t} N^{1-1/s}.\n\\]\nWhen $s=t=2$, such sets are precisely \\emph{Sidon sets} (also called $B_2$-sets), and the above\ncorrespondence recovers the classical link between Sidon sets and $C_4$-free graphs. See \\cite{CT18} for more details and further generalizations.\n\n\\medskip\n\\subsection*{Our results}\nBuilding upon recent work of Conlon--Fox--Sudakov--Zhao~\\cite{CFSZ} and Prendiville~\\cite{Prendi}, we show that all dense $K_{s,t}$-free subsets of $\\{1,\\dots,N\\}$ must possess arithmetic structure, in the sense that they must always contain a solution to any translation-invariant linear equation\nin at least five variables, with all variables distinct.\n\n\\medskip\n\\subsection*{Our results}\nBuilding upon recent work of Conlon--Fox--Sudakov--Zhao~\\cite{CFSZ} and Prendiville~\\cite{Prendi}, we show that all dense $K_{s,t}$-free subsets of $\\{1,\\dots,N\\}$ must possess arithmetic structure, in the sense that they must always contain a solution to any translation-invariant linear equation\nin at least five variables, with all variables distinct.\n\nThe qualitative version of this result in the case of Sidon sets (when $s=t=2$, and with $|A| = o(N^{1/2})$) was first established by Conlon, Fox, Sudakov, and Zhao~\\cite{CFSZ}, who proved it using a regularity lemma for graphs with few cycles of length four. Not too long after, Prendiville developed a Fourier-analytic transference principle to show the improved quantitative bound $|A| = O\\!\\left(N^{1/2}(\\log\\log N)^{-1}\\right)$. This argument used an $L^{2}$-dense model theorem in the style of Helfgott and de Roton~\\cite{HeRo} for Roth's theorem in the primes (originally established by Green~\\cite{Gre05}). To prove Theorem~\\ref{JPX}, we take a similar Fourier-analytic transference path, relying instead on an $L^{s}$-dense model theorem, in the spirit of Naslund's work~\\cite{Naslund}.\n\n\\begin{theorem}\\label{JPX2}\nLet $q$ be a fixed odd prime power. Let $k\\ge 5$ and $a_1,\\dots,a_k\\in\\F_q^\\times$ satisfy $\\sum_{i=1}^k a_i=0$.\nLet $A\\subseteq \\F_q^n$ be $K_{s,t}$-free and assume that $A$ contains no nontrivial solution to\n$a_1x_1+\\cdots+a_kx_k=0$. Then, by writing $N=q^n$, there exists $c=c(q,k,s,t)>0$ such that\n\\[\n|A|\\ll N^{1-1/s}(\\log N)^{-c}.\n\\]\n\\end{theorem}\n\n\\begin{lemma}[Ko\\'sciuszko~\\cite{K25}]\\label{Kosciuszko counting}\nLet $a_1,\\dots,a_k \\in \\Z\\setminus \\{0\\}$ with $k\\ge 5$ and $a_1 +\\cdots +a_k = 0$. Then for any $A \\subseteq [N]$ of size $\\delta N$, we have the lower bound\n\\[\n\\sum_{a_1x_1+\\cdots +a_kx_k = 0} \\e_A(x_1)\\cdots \\e_A(x_k)\n\\ge \\exp(-O_{a_i} ( \\log^7(1/\\delta)))N^{k-1}.\n\\]\n\\end{lemma}\n\n\\begin{lemma}\\label{L^p Kosciuszko counting}\n Let $a_1, a_2, \\cdots, a_k \\in \\mathbb{Z}\\backslash \\{0\\}$ with $k\\ge 5$ and $a_1 +\\cdots a_k = 0$, $p\\geq2$. Let $f:I\\to[0,\\infty)$ be a function defined in an interval $I\\subseteq\\mathbb{Z}$ of size $|I|=N$. Suppose $\\sum_n f(n)\\geq \\delta N$ and $\\sum_n f(n)^p\\leq N$. \n Then we have the lower bound\n\\[ \n\\sum_{a_1x_1+\\cdots +a_kx_k = 0} f(x_1) f(x_2)\\cdots f(x_k) \\ge \\exp(-O_{a_i,p} ( \\log^7(1/\\delta)))N^{k-1}.\n\\] \n\\end{lemma}\n\\begin{proof}\n Let $Z=f(n)$ be a random variable on $I$, which is uniformly and independent distributed for $n\\in I$. Then the conditions are the same as saying $\\mathbb E(Z)\\geq\\delta$ and $\\mathbb E (Z^p)\\leq 1$. By the Paley--Zygmund level set inequality \\cite{PZineq},\n \\[\n\\mathbb{P}\\Big(Z\\geq \\frac12\\mathbb E(Z)\\Big)\\geq \\Big(\\frac{1}{2}\\Big)^{\\frac{p}{p-1}}\\frac{(\\mathbb E Z)^{\\frac{p}{p-1}}}{\\mathbb (\\mathbb E(Z^p))^{\\frac{1}{p-1}}}\\geq\\Big(\\frac{\\delta}{2}\\Big)^{\\frac{p}{p-1}}. \n \\]\n This implies that, by defining $A=\\{n: f(n)\\geq \\delta/2\\}$, we have $|A|\\geq (\\frac{\\delta}{2})^{\\frac{p}{p-1}}N$. Now by Lemma~\\ref{Kosciuszko counting}, for $k\\geq 5$,\n \\[ \n\\sum_{a_1x_1+\\cdots +a_kx_k = 0} 1_A(x_1) 1_A(x_2)\\cdots 1_A(x_k) \\ge \\exp(-O_{a_i,p} ( \\log^7(1/\\delta)))N^{k-1}.\n\\]\nAs $f(n)\\geq \\delta/2$ on $A$, we then have\n \\begin{align*}\n\\sum_{a_1x_1+\\cdots +a_kx_k = 0} f(x_1) f(x_2)\\cdots f(x_k) &\\geq (\\delta/2)^k\\exp(-O_{a_i,p} ( \\log^7(1/\\delta^2)))N^{k-1}\\\\\n&\\geq \\exp(-O_{a_i,p} ( \\log^7(1/\\delta)))N^{k-1},\n\\end{align*}\n as the $(\\delta/2)^k$ factor will be absorbed by the $\\exp(-O_{a_i,p} ( \\log^7(1/\\delta)))$ term. \n\\end{proof}\n\n\\begin{lemma}[Counting lemma for $E_2$]\\label{lem: counting}\nLet $a_1,\\dots,a_k\\in\\Z \\setminus \\{0\\}$ with $k\\geq 5$ and $a_1 +\\cdots + a_k = 0$. Let $\\nu : I \\to [0, \\infty)$ be a function defined on an interval $I \\subseteq \\Z$ of length $N$. Suppose $\\sum_n \\nu(n)\\ll N$ and\n\\[\nE_2(\\nu,\\nu)\\ll N^{3}. \n\\]\nThen for every $|f_i|\\le \\nu$ we have\n\\[\n\\left| \\sum_{a_1x_1+\\cdots +a_k x_k = 0} \\prod_{j=1}^k f_j(x_j)\\right|\\ll N^{k-2}\\min_i \\|\\widehat{f_i}\\|_\\infty. \n\\]\n\\end{lemma}\n\nNext, we transfer the counting result to $A$.\nSet $F:=N^{1/s}\\e_A$, $g:=f-F$, and $\\nu:=f+F$.\nThen $\\|\\widehat g\\|_\\infty\\ll \\varepsilon N$ by (ii) above.\nBy Lemma~\\ref{lem: upper bound on A} and Lemma~\\ref{lem: E2}, as well as H\\\"older inequality, we have\n\\begin{align*}\n E_2(\\nu)^{\\frac{1}{4}}&=\\| \\widehat{\\nu}\\|_{4}\\ll \\| \\widehat{f}\\|_{4}+N^{\\frac{1}{s}}\\| \\widehat{\\e_A}\\|_{4}\\\\\n &\\leq N^{\\frac{1}{4}}\\| f\\|_2^{\\frac{1}{2}}\\|\\widehat{f}\\|_2^{\\frac{1}{2}}+ N^{\\frac{1}{s}}E_2(\\e_A,\\e_{-A})^{\\frac{1}{4}}\\ll N^{\\frac{3}{4}}. \n\\end{align*}\nNotice that\n$$\\sum_n \\nu(n) = \\widehat{f}(0)+N^{1/s}\\widehat{\\e_A}(0) \\leq 2\\widehat{f}(0) + \\varepsilon N \\leq 2 (2N)^{1/2}\\| f\\|_2+\\varepsilon N\\ll N.$$ We use \nthe telescoping identity to get\n\\[\n\\prod_{j=1}^k f(x_j)-\\prod_{j=1}^k F(x_j)\n=\\sum_{i=1}^k g(x_i)\\Bigl(\\prod_{ji} F(x_j)\\Bigr). \n\\]\nNote that $f, F \\ll \\nu$ and an application of Lemma~\\ref{lem: counting} (viewing functions $g, F$ as $f_i$ in the lemma) gives \n\\begin{equation}\\label{eq:transfer}\n|T(f)-T(F)|\\ll_k N^{k-2}\\|\\widehat g\\|_\\infty \\ll_k \\varepsilon N^{k-1}.\n\\end{equation}\nSince $A$ has no nontrivial solutions, the only solutions counted by $T(F)$ are diagonal, \n\\[\nT(F)=N^{k/s}\\sum_{a_1x_1+\\cdots +a_kx_k=0}\\e_A(x_1)\\cdots\\e_A(x_k)\n=N^{k/s}|A|=\\delta\\,N^{1+\\frac{k-1}{s}}.\n\\]\nSince $k\\ge 5$ and $s\\ge 2$, we have $1+\\frac{k-1}{s} 0$. Recalling that $|A|=\\delta N^{1-1/s}$ completes the proof.\n\\end{proof}\n\nLet $0<\\varepsilon<1$ be a parameter to be chosen at the end.\nBy Lemma~\\ref{lem:dense-model-ff} (with $\\eta=o(1)$ supplied by Lemma~\\ref{lem: E_s}) there exists\n$f:G\\to[0,\\infty)$ such that\n\\begin{enumerate}[label=(\\roman*)]\n\\item $\\sum_x f(x)=N^{1/s}|A|=\\delta N$;\n\\item $\\|\\widehat f-N^{1/s}\\widehat{\\e_A}\\|_\\infty\\ll \\varepsilon N$;\n\\item $\\sum_x f(x)^s\\ll_{s,t} N$.\n\\end{enumerate}\nRecall the definition\n\\[\nT(h_1,\\dots,h_k):=\\sum_{a_1x_1+\\cdots+a_kx_k=0}\\prod_{j=1}^k h_j(x_j),\n\\qquad T(h):=T(h,\\dots,h).\n\\]\nApplying Lemma~\\ref{lem:FF-Roth-count} to $f$ yields\n\\begin{equation}\\label{eq:Tf-lower-FF}\nT(f)\\gg_{k,s,t,q}\\delta^{\\,k+\\frac{s}{s-1}C_{q,k}}\\,N^{k-1}.\n\\end{equation}\nSet $\\nu:=f+N^{1/s}\\e_A\\ge 0$ and $g:=f-N^{1/s}\\e_A$. Then $\\|\\widehat g\\|_\\infty\\ll \\varepsilon N$ by (ii).\nAs in the integer case, using (iii) and Lemma~\\ref{lem: E2} one has $\\sum_x\\nu(x)\\ll N$ and $E_2(\\nu,\\nu)\\ll N^3$.\nLemma~\\ref{lem: counting} therefore applies in $G$ and, via the same telescoping identity as before, gives\n\\begin{equation}\\label{eq:transfer-FF}\n|T(f)-T(N^{1/s}\\e_A)|\\ll_k \\varepsilon\\,N^{k-1}.\n\\end{equation}\nSince $A$ has only trivial solutions to \\eqref{eq:FF-eqn},\n\\[\nT(N^{1/s}\\e_A)=N^{k/s}\\cdot |A|=\\delta\\,N^{1+\\frac{k-1}{s}}.\n\\]\nCombining with \\eqref{eq:transfer-FF} gives\n\\[\nT(f)\\ll N^{1+\\frac{k-1}{s}}+\\varepsilon N^{k-1}.\n\\]\nComparing with \\eqref{eq:Tf-lower-FF} and dividing by $N^{k-1}$ yields, for $N$ large,\n\\[\n\\delta^{\\,k+\\frac{s}{s-1}C_{q,k}}\\ll_{k,s,t,q}\\varepsilon.\n\\]\nFinally choose $\\varepsilon:=(\\log N)^{-c_0}$ with $c_0>0$ sufficiently small. Then\n\\[\n\\delta\\ll_{k,s,t,q}(\\log N)^{-c},\n\\qquad\nc:=\\frac{c_0}{\\,k+\\frac{s}{s-1}C_{q,k}\\,}>0,\n\\]\nand hence $|A|=\\delta N^{1-1/s}\\ll N^{1-1/s}(\\log N)^{-c}$.\n\\end{proof}\n\n\\begin{theorem}\\label{JPX} \nLet $k\\geq 5$, and $a_1,\\dots,a_k\\in\\mathbb{Z}\\setminus\\{0\\}$ with $\\sum_{i=1}^k a_i=0$. \nLet $A\\subseteq [N]$ be $K_{s,t}$-free and assume that $A$ lacks nontrivial solutions of $a_1x_1+\\cdots +a_kx_k=0$. Then\n\\[\n|A|\\ll N^{1-1/s} \\exp(-O_{a_i,s,t}((\\log\\log N)^{\\frac17})).\n\\]\n\\end{theorem}", "post_theorem_intro_text_len": 6452, "post_theorem_intro_text": "The qualitative version of this result in the case of Sidon sets (when $s=t=2$, and with $|A| = o(N^{1/2})$) was first established by Conlon, Fox, Sudakov, and Zhao~\\cite{CFSZ}, who proved it using a regularity lemma for graphs with few cycles of length four. Not too long after, Prendiville developed a Fourier-analytic transference principle to show the improved quantitative bound $|A| = O\\!\\left(N^{1/2}(\\log\\log N)^{-1}\\right)$. This argument used an $L^{2}$-dense model theorem in the style of Helfgott and de Roton~\\cite{HeRo} for Roth's theorem in the primes (originally established by Green~\\cite{Gre05}). To prove Theorem~\\ref{JPX}, we take a similar Fourier-analytic transference path, relying instead on an $L^{s}$-dense model theorem, in the spirit of Naslund's work~\\cite{Naslund}. \n\n\\medskip\n\n\\subsection*{Finite-field analogue.}\nIn Section~\\ref{sec:finite-field} we study the same question in the vector space $\\F_q^n$\n($q$ a fixed odd prime power), where we establish the following theorem. \n\n\\begin{theorem}\\label{JPX2}\nLet $q$ be a fixed odd prime power. Let $k\\ge 5$ and $a_1,\\dots,a_k\\in\\F_q^\\times$ satisfy $\\sum_{i=1}^k a_i=0$.\nLet $A\\subseteq \\F_q^n$ be $K_{s,t}$-free and assume that $A$ contains no nontrivial solution to\n$a_1x_1+\\cdots+a_kx_k=0$. Then, by writing $N=q^n$, there exists $c=c(q,k,s,t)>0$ such that\n\\[\n|A|\\ll N^{1-1/s}(\\log N)^{-c}.\n\\]\n\\end{theorem}\n\nTo obtain the polylogarithmic from Theorem~\\ref{JPX2}, we take advantage of the polynomial method developments around the cap set problem~\\cite{CLP17,EG}, in particular the improved bounds in the arithmetic $k$-cycle removal lemma in $\\mathbb{F}_{q}^{n}$ due to Fox, Lov\\'asz, and Sauermann~\\cite{FLS}.\n\nWe remark that $k\\ge 5$ is optimal in the sense that if we want to avoid nontrivial solutions in any given $k$ translation invariant linear equation. For example, in the $K_{2,2}$ Sidon case, both $x_1+x_2=x_3+x_4$ ($k=4$) and $x+y=2z$ ($k=3$) are excluded by the Sidon condition.\n\n\\medskip\n\\subsection*{Proof ideas}\\label{subsec:proof-ideas}\n\nOur arguments follow the same overarching Fourier analytic transference principle from \\cite{Prendi}: we build a \\emph{dense model} for a sparse set (or weight) and then transfer a \\emph{dense counting statement} back to the original object. It is conceptually helpful to split the proof into two independent components.\n\n\\medskip\n\\noindent{\\bf (1) Dense model step (Fourier approximation).}\nGiven a sparse set $A$ in the natural extremal scaling (for instance $|A|\\asymp N^{1-1/s}$ for\n$K_{s,t}$-free sets), we introduce a renormalized\nnonnegative weight $\\nu$ (e.g.\\ $\\nu=N^{1/s}\\mathbf{1}_A$) so that\n$\\sum \\nu$ is of order $N$. The goal is to construct a nonnegative function $f$ with\n\\[\n\\sum f=\\sum \\nu\n\\qquad\\text{and}\\qquad\n\\|\\widehat f-\\widehat\\nu\\|_\\infty \\ \\ \\text{small},\n\\]\nwhile keeping some $L^s$ norm of $f$ bounded ($\\sum f^s\\ll N$ in the $K_{s,t}$ setting). Over the integers, the standard way to produce $f$ is to smooth by a Bohr set built from the large\nspectrum; controlling the rank of this Bohr set is a key difficulty, and this step uses the $K_{s,t}$-freeness of $A$. Over $G=\\F_q^n$ some of these steps will be simplified substantially, because the analogue of a Bohr set is an actual subspace.\n\n\\medskip\n\\noindent{\\bf (2) Dense counting step (supersaturation).}\nFix a translation-invariant equation in $\\mathbb{Z}[x_1, \\dots, x_k]$, \n\\[\na_1x_1+\\cdots+a_kx_k=0,\n\\qquad a_i\\ne0,\\ \\sum_{i=1}^k a_i=0,\n\\]\nand write the associated counting functional\n\\[\nT(h_1,\\dots,h_k)\n:=\\sum_{a_1x_1+\\cdots+a_kx_k=0}\\ \\prod_{i=1}^k h_i(x_i),\n\\qquad\nT(h):=T(h,\\dots,h).\n\\]\nThe dense model $f$ is built so that it behaves ``like a dense set'' from the point of view of\nFourier analysis; one then applies a dense-set input to obtain a nontrivial lower bound for\n$T(f)$. In the integer case this dense input comes from quantitative results on solutions to\ntranslation-invariant equations in dense subsets of $[N]$ (Bloom-type bounds \\cite{Bloom} and their refinements).\nIn the finite-field case we can instead use the arithmetic $k$-cycle removal lemma of\nFox--Lov\\'asz--Sauermann \\cite{FLS} (which in turn builds upon the works of Croot--Lev--Pach \\cite{CLP17} and Ellenberg--Gijswijt \\cite{EG}) and it yields polynomial supersaturation bounds.\n\n\\medskip\n\\noindent{\\bf (3) Transference (telescoping + moment control).}\nThe final step is to compare $T(f)$ with $T(\\nu_A)$. A standard telescoping identity expands\n\\[\n\\prod_{i=1}^k f(x_i)-\\prod_{i=1}^k \\nu(x_i)\n\\]\nas a sum of $k$ terms, each containing one copy of $f-\\nu$ and $k-1$ copies of either $f$ or $\\nu$.\nTo bound each resulting multilinear form we use a Fourier/H\\\"older estimate whose strength depends\non an available moment bound for a suitable \\emph{majorant} $\\omega\\ge |f|+|\\nu|$.\nIn the $K_{s,t}$-free setting, the underlying graph-freeness yields strong control on the second\nmoment energy $E_2$, which provides the $L^4$ Fourier bound needed for the counting lemma.\nCombining the dense lower bound for $T(f)$ with the transference error bound\n\\[\n|T(f)-T(\\nu)|\\ \\ll\\ N^{k-2}\\,\\|\\widehat{f-\\nu}\\|_\\infty,\n\\]\none concludes that $T(\\nu)$ is also large. Finally, if $A$ is assumed to have only ``trivial''\nsolutions, then $T(\\nu)$ can be computed explicitly, which leads to an upper bound that contradicts the transferred lower bound for $N$ sufficiently large. \n\n\\medskip\n\n\\subsection*{Notation}\nFor a function $f:\\mathbb{Z}\\to\\mathbb{C}$ with finite support, we define its Fourier transform on $\\mathbb{T}$ by\n\\[\n\\widehat{f}(\\alpha) = \\sum_{n\\in\\mathbb{Z}} f(n)e(-n\\alpha),\n\\qquad \\alpha\\in\\mathbb{T}.\n\\] \nThe convolution of two functions on $\\mathbb{Z}$ is denoted as\n\\[\nf*g(x) = \\sum_{y\\in\\mathbb{Z}} f(y)g(x-y).\n\\]\nWith respect to Haar probability measure on $\\mathbb{T}$, define the $L^p$ norm of $F:\\mathbb{T}\\to\\mathbb{C}$ by \n\\[\n\\|F\\|_{p} = \\Big(\\int_{\\mathbb{T}}|F(\\alpha)|^{p} \\,\\mathrm{d}\\alpha \\Big)^{\\frac{1}{p}},\n\\qquad\n\\|F\\|_{\\infty}:= \\sup_{\\alpha \\in \\mathbb{T}} |F(\\alpha)|.\n\\]\nThroughout the paper, we write $f=O(g)$ or $f\\ll g$ if there exists a positive constant $C$ such that $|f(x)|\\le C g(x)$ for all $x$, and $f\\asymp g$ if $f=O(g)$ and $g=O(f)$. We say $f= o(g)$ if for any $\\varepsilon>0$, $|f(x)|\\le \\varepsilon g(x)$ for all sufficiently large $x$.\n\n\\medskip\n\n\\subsection*{Acknowledgments} CP was supported by NSF grant DMS-2246659. MWX is supported by a Simons Junior Fellowship from the Simons Foundation.", "sketch": "To prove Theorem~\\ref{JPX}, the authors “take a similar Fourier-analytic transference path, relying instead on an $L^{s}$-dense model theorem.” The proof is split into “two independent components” plus a final transference step:\n\n\\noindent\\textbf{(1) Dense model step (Fourier approximation).} For a sparse $K_{s,t}$-free set $A$ in the extremal scaling (e.g. $|A|\\asymp N^{1-1/s}$), introduce a renormalized weight $\\nu$ (e.g. $\\nu=N^{1/s}\\mathbf{1}_A$) so that $\\sum \\nu$ is order $N$. Construct a nonnegative function $f$ with\n\\[\n\\sum f=\\sum \\nu\n\\qquad\\text{and}\\qquad\n\\|\\widehat f-\\widehat\\nu\\|_\\infty\\ \\ \\text{small},\n\\]\nwhile keeping an $L^s$ norm bounded (in the $K_{s,t}$ setting “$\\sum f^s\\ll N$”). Over the integers, $f$ is produced by “smooth[ing] by a Bohr set built from the large spectrum”; “controlling the rank of this Bohr set is a key difficulty,” and “this step uses the $K_{s,t}$-freeness of $A$.”\n\n\\noindent\\textbf{(2) Dense counting step (supersaturation).} For the translation-invariant equation $a_1x_1+\\cdots+a_kx_k=0$ (with $\\sum a_i=0$), define the counting functional\n\\[\nT(h_1,\\dots,h_k):=\\sum_{a_1x_1+\\cdots+a_kx_k=0}\\ \\prod_{i=1}^k h_i(x_i),\\qquad T(h):=T(h,\\dots,h).\n\\]\nSince $f$ is built to behave “like a dense set” in Fourier terms, apply a dense-set input to get a “nontrivial lower bound for $T(f)$.” In the integer case, this dense input comes from “quantitative results on solutions to translation-invariant equations in dense subsets of $[N]$ (Bloom-type bounds \\cite{Bloom} and their refinements).”\n\n\\noindent\\textbf{(3) Transference (telescoping + moment control).} Compare $T(f)$ with $T(\\nu)$. A “standard telescoping identity” expands $\\prod_{i=1}^k f(x_i)-\\prod_{i=1}^k \\nu(x_i)$ into a sum of $k$ terms each containing one $f-\\nu$. Each multilinear form is bounded using a “Fourier/H\\\"older estimate” whose strength depends on a moment bound for a “majorant $\\omega\\ge |f|+|\\nu|$.” In the $K_{s,t}$-free setting, “graph-freeness yields strong control on the second moment energy $E_2$, which provides the $L^4$ Fourier bound needed for the counting lemma.” This gives the transference error bound\n\\[\n|T(f)-T(\\nu)|\\ \\ll\\ N^{k-2}\\,\\|\\widehat{f-\\nu}\\|_\\infty.\n\\]\nCombining the dense lower bound for $T(f)$ with this error shows $T(\\nu)$ is also large. Finally, assuming $A$ has only “trivial” solutions, “$T(\\nu)$ can be computed explicitly,” yielding an upper bound that “contradicts the transferred lower bound for $N$ sufficiently large,” giving the claimed bound on $|A|$.", "expanded_sketch": "To prove the main theorem, the authors “take a similar Fourier-analytic transference path, relying instead on an $L^{s}$-dense model theorem.” The proof is split into “two independent components” plus a final transference step:\n\n\\noindent\\textbf{(1) Dense model step (Fourier approximation).} For a sparse $K_{s,t}$-free set $A$ in the extremal scaling (e.g. $|A|\\asymp N^{1-1/s}$), introduce a renormalized weight $\\nu$ (e.g. $\\nu=N^{1/s}\\mathbf{1}_A$) so that $\\sum \\nu$ is order $N$. Construct a nonnegative function $f$ with\n\\[\n\\sum f=\\sum \\nu\n\\qquad\\text{and}\\qquad\n\\|\\widehat f-\\widehat\\nu\\|_\\infty\\ \\ \\text{small},\n\\]\nwhile keeping an $L^s$ norm bounded (in the $K_{s,t}$ setting “$\\sum f^s\\ll N$”). Over the integers, $f$ is produced by “smooth[ing] by a Bohr set built from the large spectrum”; “controlling the rank of this Bohr set is a key difficulty,” and “this step uses the $K_{s,t}$-freeness of $A$.”\n\n\\noindent\\textbf{(2) Dense counting step (supersaturation).} For the translation-invariant equation $a_1x_1+\\cdots+a_kx_k=0$ (with $\\sum a_i=0$), define the counting functional\n\\[\nT(h_1,\\dots,h_k):=\\sum_{a_1x_1+\\cdots+a_kx_k=0}\\ \\prod_{i=1}^k h_i(x_i),\\qquad T(h):=T(h,\\dots,h).\n\\]\nSince $f$ is built to behave “like a dense set” in Fourier terms, apply a dense-set input to get a “nontrivial lower bound for $T(f)$.” In the integer case, this dense input comes from “quantitative results on solutions to translation-invariant equations in dense subsets of $[N]$ (Bloom-type bounds \\cite{Bloom} and their refinements).”\n\n\\noindent\\textbf{(3) Transference (telescoping + moment control).} Compare $T(f)$ with $T(\\nu)$. A “standard telescoping identity” expands $\\prod_{i=1}^k f(x_i)-\\prod_{i=1}^k \\nu(x_i)$ into a sum of $k$ terms each containing one $f-\\nu$. Each multilinear form is bounded using a “Fourier/H\\\"older estimate” whose strength depends on a moment bound for a “majorant $\\omega\\ge |f|+|\\nu|$.” In the $K_{s,t}$-free setting, “graph-freeness yields strong control on the second moment energy $E_2$, which provides the $L^4$ Fourier bound needed for the counting lemma.” This gives the transference error bound\n\\[\n|T(f)-T(\\nu)|\\ \\ll\\ N^{k-2}\\,\\|\\widehat{f-\\nu}\\|_\\infty.\n\\]\nCombining the dense lower bound for $T(f)$ with this error shows $T(\\nu)$ is also large. Finally, assuming $A$ has only “trivial” solutions, “$T(\\nu)$ can be computed explicitly,” yielding an upper bound that “contradicts the transferred lower bound for $N$ sufficiently large,” giving the claimed bound on $|A|$.", "expanded_theorem": "\\label{JPX} \nLet $k\\geq 5$, and $a_1,\\dots,a_k\\in\\mathbb{Z}\\setminus\\{0\\}$ with $\\sum_{i=1}^k a_i=0$. \nLet $A\\subseteq [N]$ be $K_{s,t}$-free and assume that $A$ lacks nontrivial solutions of $a_1x_1+\\cdots +a_kx_k=0$. Then\n\\[\n|A|\\ll N^{1-1/s} \\exp(-O_{a_i,s,t}((\\log\\log N)^{\\frac17})).\n\\],", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let integers $s,t$ satisfy $2\\le s\\le t$, let $[N]=\\{1,2,\\dots,N\\}$, and let $k\\ge 5$. Suppose $a_1,\\dots,a_k\\in\\mathbb Z\\setminus\\{0\\}$ satisfy $\\sum_{i=1}^k a_i=0$. Let $A\\subseteq [N]$ be $K_{s,t}$-free in the arithmetic sense that there do not exist distinct integers $x_1,\\dots,x_s$ and distinct integers $y_1,\\dots,y_t$ with\n\\[\n\\{x_i+y_j:1\\le i\\le s,\\ 1\\le j\\le t\\}\\subseteq A.\n\\]\nAssume also that $A$ has no nontrivial solution to\n\\[\na_1x_1+\\cdots+a_kx_k=0,\n\\]\nwhere a nontrivial solution means one that is not diagonal (that is, not all $x_1,\\dots,x_k$ are equal). Under these hypotheses, which statement about $|A|$ holds?", "correct_choice": {"label": "A", "text": "There exist constants $C,c>0$, depending only on $a_1,\\dots,a_k,s,t$, such that\n\\[\n|A|\\le C\\,N^{1-1/s}\\exp\\bigl(-c(\\log\\log N)^{1/7}\\bigr).\n\\]\nEquivalently,\n\\[\n|A|\\ll N^{1-1/s}\\exp\\bigl(-O_{a_i,s,t}((\\log\\log N)^{1/7})\\bigr).\n\\]"}, "choices": [{"label": "B", "text": "There exist constants $C,c>0$, depending only on $a_1,\\dots,a_k,s,t$, such that\n\\[\n|A|\\le C\\,N^{1-1/s}\\exp\\bigl(-c(\\log N)^{1/7}\\bigr).\n\\]\nEquivalently,\n\\[\n|A|\\ll N^{1-1/s}\\exp\\bigl(-O_{a_i,s,t}((\\log N)^{1/7})\\bigr).\n\\]"}, {"label": "C", "text": "There exists a constant $C>0$, depending only on $a_1,\\dots,a_k,s,t$, such that\n\\[\n|A|\\le C\\,N^{1-1/s}.\n\\]\nEquivalently,\n\\[\n|A|\\ll_{a_i,s,t} N^{1-1/s}.\n\\]"}, {"label": "D", "text": "There exist constants $C,c>0$, depending only on $a_1,\\dots,a_k,s,t$, such that\n\\[\n|A|\\le C\\,N^{1-1/s}\\exp\\bigl(-c(\\log\\log N)^{1/7}\\bigr)\n\\]\nfor every $K_{s,t}$-free set $A\\subseteq [N]$, without any assumption that $A$ has no nontrivial solution to\n\\[\na_1x_1+\\cdots+a_kx_k=0.\n\\]"}, {"label": "E", "text": "There exist constants $C,c>0$, depending only on $a_1,\\dots,a_k,s,t$, such that\n\\[\n|A|\\le C\\,N^{1-1/t}\\exp\\bigl(-c(\\log\\log N)^{1/7}\\bigr).\n\\]\nEquivalently,\n\\[\n|A|\\ll N^{1-1/t}\\exp\\bigl(-O_{a_i,s,t}((\\log\\log N)^{1/7})\\bigr).\n\\]"}], "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": "final decay scale from the dense-counting/transference comparison", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "the logarithmic saving factor", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "trace_identity", "tampered_component": "need for the explicit diagonal-only evaluation of $T(\\nu)$ using absence of nontrivial solutions", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "extremal exponent determined by the $L^s$ dense model / K_{s,t}-free scaling", "template_used": "boundary_range"}]}} +{"id": "2601.19282v1", "paper_link": "http://arxiv.org/abs/2601.19282v1", "theorems_cnt": 3, "theorem": {"env_name": "theorem", "content": "[Well-posedness in $L^1$] \\label{th:existence weak sol}\n\tAssume that $h$ satisfies \\eqref{eq:comportement h -infty} and \\eqref{eq:comportement h +infty}. Then, for any initial data $u_0\\in L^1(\\mathbb{R})$, there exists a unique weak solution $(u,N_u)\\in L^\\infty(\\mathbb{R}^+;L^1(\\mathbb{R}))\\times L^1_{loc}(\\mathbb{R}^+)$ of Eq.~\\eqref{eq:FPSL lin}, i.e., that satisfies~\\eqref{eq:formulation faible FPS} and~\\eqref{eq:Conservation mass}.\n \t In addition, we have \n \t \t\\begin{equation}\\label{eq:L1Contract}\n \t \\text{for a.e. } t > 0, \\quad \\int_\\mathbb{R} |u(t,x)|\\,dx \\leq \\int_\\mathbb{R} |u_0(x)| \\,dx,\n \t\\end{equation}\n \t\\begin{equation}\\label{eq: U_T bound}\n \t \\forall T\\geq 0, \\quad \\sup_{x\\in \\mathbb{R}}\\int_0^T|h(x)u(t,x)|\\,dt <\\infty \\quad \\text{and}\\quad U_T = \\int_0^Tu(t)\\,dt \\in W^{1,\\infty}(\\mathbb{R}).\n \t\\end{equation}", "start_pos": 10831, "end_pos": 11646, "label": "th:existence weak sol"}, "ref_dict": {"eq:Conservation mass": "\\begin{equation}\\label{eq:Conservation mass}\n\t\t\t\\text{for } a.e. \\; t\\geq0 \\quad \n\t\t\t\\int_\\R u(t,x)\\,dx = \\int_\\R u_0(x)\\,dx .\n\t\t\t\\end{equation}", "eq:comportement h +infty": "\\begin{equation}\\label{eq:comportement h +infty}\n\t\\begin{cases}\n\t\t\\exists x_1 \\geq 0,\\; \\fa x\\geq x_1, \\quad h(x)>0, \\quad h'(x)>0, \n\t\t\\\\[2pt]\n\t\t\\displaystyle \\int_{x_1}^\\infty\\frac{dx}{h(x)}<\\infty,\n\t\t\\\\[2pt]\n\t\th(x)\\sup_{y\\geq x}\\frac{h'(y)^2}{h(y)^4}\\in L^1(x_1,\\infty).\n\t\t\\end{cases}\n\t\\end{equation}", "eq:comportement h -infty": "\\begin{equation}\\label{eq:comportement h -infty}\n\t\t\\exists x_0 \\leq 0,\\; h_0\\in \\R \\quad \\text{such that} \\quad \\fa x\\leq x_0,\\quad h(x) = -x+h_0.\n\t\\end{equation}", "eq:formulation faible FPS": "\\begin{align} \n\t\t\t\t\\int_0^\\infty \\! \\int_\\R u(t,x)\\Big(-\\partial_t\\varphi(t,x)&-h(x)\\partial_x\\varphi(t,x)-\\partial_x^2\\varphi(t,x)\\Big)\\,dx\\,dt \\notag\n\t\t\t\t\\\\\n\t\t\t\t& = \\int_0^\\infty N_u(t)\\varphi(t,0)\\,dt\n\t\t\t\t+\\int_\\R u_0(x)\\varphi(0,x)\\,dx. \\label{eq:formulation faible FPS} \n\t\t\t\\end{align}", "eq:FPSL lin": "\\begin{equation}\\label{eq:FPSL lin}\n\t\t\\left\\{\\begin{aligned}\n\t\t\t& \\frac{\\partial u}{\\partial t}(t,x) +\\frac{\\partial}{\\partial x}\\big((h(x)u(t,x)\\big) - \\frac{\\partial ^2u}{\\partial x^2}(t,x) = \\delta_{0}(x)N(t), \\quad x\\in \\mathbb{R}, t\\geq 0,\n\t\t\t\\\\\n\t\t\t& N(t) = \\lim_{x\\rightarrow \\infty}h(x)u(t,x),\n\t\t\t\\\\\n\t\t\t& u(t=0,x) = u_0(x)\\in L^1(\\R).\n\t\t\\end{aligned}\\right.\n\t\\end{equation}"}, "pre_theorem_intro_text_len": 6932, "pre_theorem_intro_text": "In recent literature, many works have focused on the {\\em integrate-and-fire} (I\\&F) models for neurons potentials, which now constitute a widely studied reference framework (see the survey \\cite{CR2025}). However, several approaches from biophysics and computational neuroscience suggest a specificity, namely that intrinsic dynamics with super-linear growth for large potentials, notably quadratic or exponential, would provide a more realistic description of certain neuronal behaviors. This motivation is notably inspired by the work of \\cite{BL_2003, BretteG2005} on the Quadratic or Exponential Integrate-and-Fire model, which showed that introducing a super-linearity in the drift better captures neuronal firing dynamics and certain oscillations, compared to classical linear models. We propose a partial differential equation (PDE) that incorporates this super-linear component in order to develop a theory and a qualitative analysis of neuronal dynamics within this framework. To date, this theoretical framework has never been subject to a rigorous mathematical study via a Fokker-Planck type PDE. The associated linear model we propose is formally as follows:\n\t\\begin{equation}\\label{eq:FPSL lin}\n\t\t\\left\\{\\begin{aligned}\n\t\t\t& \\frac{\\partial u}{\\partial t}(t,x) +\\frac{\\partial}{\\partial x}\\big((h(x)u(t,x)\\big) - \\frac{\\partial ^2u}{\\partial x^2}(t,x) = \\delta_{0}(x)N(t), \\quad x\\in \\mathbb{R}, t\\geq 0,\n\t\t\t\\\\\n\t\t\t& N(t) = \\lim_{x\\rightarrow \\infty}h(x)u(t,x),\n\t\t\t\\\\\n\t\t\t& u(t=0,x) = u_0(x)\\in L^1(\\mathbb{R}).\n\t\t\\end{aligned}\\right.\n\t\\end{equation}\nThe specificity is to work on the full line with a flux of firing neurons $h(x)u(x)$ at infinity which occurs because we assume that $\\int_{\\cdot}^{\\infty} \\frac{1}{h(x)}dx <+\\infty$. This specificity also makes the mathematical interest and new difficulty for the analysis of the model. \n\n\\subsection{Motivations}\nThe study of this equation fundamentally differs, in its structure, from the classical linear-drift I\\&F model introduced in \\cite{BrHa}, \n\t\\begin{equation*}\n\t\t\\left\\{\\begin{aligned}\n\t\t\t& \\frac{\\partial p}{\\partial t}(t,x) +\\frac{\\partial}{\\partial x}(h(x) p(t,x)) - \\frac{\\partial ^2p}{\\partial x}(t,x) = \\delta_{V_R}(x)N(t), \\qquad t\\geq 0, \\; x\\leq V_F,\\\\\n\t\t\t& p(t,V_F) = 0\\\\\n\t\t\t& N(t) = -\\frac{\\partial p}{\\partial x}(t,V_F).\n\t\t\\end{aligned}\\right.\n\t\\end{equation*}\nThis equation describes the probability density of neurons, according to their membrane potential~$x$. \tWhen a neuron reaches the potential of firing, $V_F$, then it is instantaneously re-injected with a membrane potential of reset, $V_R< V_F$ \n(we have chosen $V_R=0$ in~\\eqref{eq:FPSL lin} for simplicity). \n\nIn such a model, the mathematical theory does not take into account that the drift is quadratic or exponential. The super-linearity of the drift is a concept for $x\\to \\infty$. This leads us to consider the equation on the full line and compute the flux of firing neurons $N(t)$ through a source term located at infinity, unlike in the classical case where it is imposed by a Dirichlet boundary condition at $V_F$. \n\nNotice that the novelty is the {\\em flux condition at infinity}, which justifies the superlinear growth of $h(x)$ and impose to extend the I\\&F equation to the full line with $V_F=\\infty$. This extension to the full line has also been used, mainly to understand when the drift depends on the neural activity $N(t)$, the complex possible dynamics including periodic solutions, \\cite{CaPe, DPSZ2024,JGLiuZZ21}.\n\nWe aim to develop a theory around this new formulation \\eqref{eq:FPSL lin} with a super-linear drift, in a first step, in the linear setting, that is, in term of modeling, the neurons are independent the one with each others. \n\\\\\n\n\\subsection{Assumptions}\n\n\tIn order to catch the superlinear behaviour of the drift, we consider a generalized drift function $h\\in \\mathcal{C}^1$ that satisfies the following asymptotic behavior.\n\t\\\\\n\t\\textbf{Behavior at $-\\infty$:}\n\t\\begin{equation}\\label{eq:comportement h -infty}\n\t\t\\exists x_0 \\leq 0,\\; h_0\\in \\mathbb{R} \\quad \\text{such that} \\quad \\forall x\\leq x_0,\\quad h(x) = -x+h_0.\n\t\\end{equation}\n\tThis assumption could be generalized, this specific form allows us to perform explicit calculations which avoid some technicalities.\n\t\\\\\n\t\\textbf{Behavior at $+\\infty$:}\n\t\\begin{equation}\\label{eq:comportement h +infty}\n\t\\begin{cases}\n\t\t\\exists x_1 \\geq 0,\\; \\forall x\\geq x_1, \\quad h(x)>0, \\quad h'(x)>0, \n\t\t\\\\[2pt]\n\t\t\\displaystyle \\int_{x_1}^\\infty\\frac{dx}{h(x)}<\\infty,\n\t\t\\\\[2pt]\n\t\th(x)\\sup_{y\\geq x}\\frac{h'(y)^2}{h(y)^4}\\in L^1(x_1,\\infty).\n\t\t\\end{cases}\n\t\\end{equation}\nBoth the quadratic and exponential models used in the literature satisfy these assumption.\tThroughout this paper we use these two assumptions without necessarily mentioning them. We also use the notation, \n\\begin{equation}\\label{def:calV}\n\\mathcal{H}(x) = \\int_0^x h(y)dy, \\quad \\text{hence} \\quad \\mathcal{H}(x) \\underset{-\\infty}{\\sim} -\\frac{x^2}{2}, \\quad \\lim_{x\\to \\infty} \\mathcal{H}(x) = +\\infty .\n\\end{equation}\n\tThe third line of assumptions \\eqref{eq:comportement h +infty} ensures that \n\t\\begin{equation}\\label{eq:limh'/h^2}\n \\zeta(x):= \\sup_{y\\geq x}\\frac{h'(y)}{h(y)^2}\\in L^1(x_1,\\infty) \\qquad \\text{and} \\qquad \\zeta(x) \\to 0, \\quad \\text{as}\\quad x\\rightarrow \\infty.\n \\end{equation}\n\tIndeed, we define $f(x):= h(x)\\sup_{y\\geq x}\\frac{h'(y)^2}{h(y)^4}$ and then, for $x\\geq x_1$, $\\zeta(x) = \\big(\\frac{f(x)}{h(x)}\\big)^{1/2}$. According to assumption \\eqref{eq:comportement h +infty}, both $\\frac{1}{h}$ and $f$ belong to $L^1(x_1,\\infty)$ so does $\\zeta$. Since it is non-increasing, the result follows. \n\n\tSuch a drift $h$ has the property to send any potential that is located after $x_1$ to $+\\infty$ in finite time. Indeed, ignoring the diffusion, the characteristics, $X(t)$, are determined by\n\t\\[\n\t\t\\frac{dX}{dt} = h(X(t)).\n\t\\]\n\tWhen $X(0)>x_1$, the solution $X(t)$ blows-up in finite time, given by\n\t\\[\n\t\\lim_{t\\rightarrow t_0}X(t) = +\\infty, \\qquad \\text{with}\\qquad t_0= \\int_{X(0)}^\\infty\\frac{dy}{h(y)}.\n\t\\]\n\tThis kind of property is still true, with non-zero probability, when the membrane potential evolves according to the stochastic differential equation \n\t\\begin{equation}\\label{eq:EDS IF}\n\t dX_t = h(X_t)\\,dt +\\sqrt{2} dB_t .\n\t\\end{equation}\nIn both cases, the reset is obtained by setting $X_{t^+} = 0$ when $\\lim_{t\\to t^-} X_{t} =+\\infty$.\n The interest of the linear model lies in building a solid theoretical foundation for the existence and regularity of solutions, as well as their asymptotic behavior, on order to better address the nonlinear case.\n \\\\\n One of the main difficulty is to make sense of the flux $N(t)$ of particles reaching $+\\infty$ when we only handle a weak solution, $u(t)\\in L^1(\\mathbb{R})$ which is not defined pointwise.\n\n\\subsection{Main results}\n\n\tOur first result concerns well-posedness in $L^1$ with minimal assumptions.", "context": "In recent literature, many works have focused on the {\\em integrate-and-fire} (I\\&F) models for neurons potentials, which now constitute a widely studied reference framework (see the survey \\cite{CR2025}). However, several approaches from biophysics and computational neuroscience suggest a specificity, namely that intrinsic dynamics with super-linear growth for large potentials, notably quadratic or exponential, would provide a more realistic description of certain neuronal behaviors. This motivation is notably inspired by the work of \\cite{BL_2003, BretteG2005} on the Quadratic or Exponential Integrate-and-Fire model, which showed that introducing a super-linearity in the drift better captures neuronal firing dynamics and certain oscillations, compared to classical linear models. We propose a partial differential equation (PDE) that incorporates this super-linear component in order to develop a theory and a qualitative analysis of neuronal dynamics within this framework. To date, this theoretical framework has never been subject to a rigorous mathematical study via a Fokker-Planck type PDE. The associated linear model we propose is formally as follows:\n \\begin{equation}\\label{eq:FPSL lin}\n \\left\\{\\begin{aligned}\n & \\frac{\\partial u}{\\partial t}(t,x) +\\frac{\\partial}{\\partial x}\\big((h(x)u(t,x)\\big) - \\frac{\\partial ^2u}{\\partial x^2}(t,x) = \\delta_{0}(x)N(t), \\quad x\\in \\mathbb{R}, t\\geq 0,\n \\\\\n & N(t) = \\lim_{x\\rightarrow \\infty}h(x)u(t,x),\n \\\\\n & u(t=0,x) = u_0(x)\\in L^1(\\mathbb{R}).\n \\end{aligned}\\right.\n \\end{equation}\nThe specificity is to work on the full line with a flux of firing neurons $h(x)u(x)$ at infinity which occurs because we assume that $\\int_{\\cdot}^{\\infty} \\frac{1}{h(x)}dx <+\\infty$. This specificity also makes the mathematical interest and new difficulty for the analysis of the model.\n\n\\subsection{Motivations}\nThe study of this equation fundamentally differs, in its structure, from the classical linear-drift I\\&F model introduced in \\cite{BrHa}, \n \\begin{equation*}\n \\left\\{\\begin{aligned}\n & \\frac{\\partial p}{\\partial t}(t,x) +\\frac{\\partial}{\\partial x}(h(x) p(t,x)) - \\frac{\\partial ^2p}{\\partial x}(t,x) = \\delta_{V_R}(x)N(t), \\qquad t\\geq 0, \\; x\\leq V_F,\\\\\n & p(t,V_F) = 0\\\\\n & N(t) = -\\frac{\\partial p}{\\partial x}(t,V_F).\n \\end{aligned}\\right.\n \\end{equation*}\nThis equation describes the probability density of neurons, according to their membrane potential~$x$. When a neuron reaches the potential of firing, $V_F$, then it is instantaneously re-injected with a membrane potential of reset, $V_R< V_F$ \n(we have chosen $V_R=0$ in~\\eqref{eq:FPSL lin} for simplicity).\n\nIn order to catch the superlinear behaviour of the drift, we consider a generalized drift function $h\\in \\mathcal{C}^1$ that satisfies the following asymptotic behavior.\n \\\\\n \\textbf{Behavior at $-\\infty$:}\n \\begin{equation}\\label{eq:comportement h -infty}\n \\exists x_0 \\leq 0,\\; h_0\\in \\mathbb{R} \\quad \\text{such that} \\quad \\forall x\\leq x_0,\\quad h(x) = -x+h_0.\n \\end{equation}\n This assumption could be generalized, this specific form allows us to perform explicit calculations which avoid some technicalities.\n \\\\\n \\textbf{Behavior at $+\\infty$:}\n \\begin{equation}\\label{eq:comportement h +infty}\n \\begin{cases}\n \\exists x_1 \\geq 0,\\; \\forall x\\geq x_1, \\quad h(x)>0, \\quad h'(x)>0, \n \\\\[2pt]\n \\displaystyle \\int_{x_1}^\\infty\\frac{dx}{h(x)}<\\infty,\n \\\\[2pt]\n h(x)\\sup_{y\\geq x}\\frac{h'(y)^2}{h(y)^4}\\in L^1(x_1,\\infty).\n \\end{cases}\n \\end{equation}\nBoth the quadratic and exponential models used in the literature satisfy these assumption. Throughout this paper we use these two assumptions without necessarily mentioning them. We also use the notation, \n\\begin{equation}\\label{def:calV}\n\\mathcal{H}(x) = \\int_0^x h(y)dy, \\quad \\text{hence} \\quad \\mathcal{H}(x) \\underset{-\\infty}{\\sim} -\\frac{x^2}{2}, \\quad \\lim_{x\\to \\infty} \\mathcal{H}(x) = +\\infty .\n\\end{equation}\n The third line of assumptions \\eqref{eq:comportement h +infty} ensures that \n \\begin{equation}\\label{eq:limh'/h^2}\n \\zeta(x):= \\sup_{y\\geq x}\\frac{h'(y)}{h(y)^2}\\in L^1(x_1,\\infty) \\qquad \\text{and} \\qquad \\zeta(x) \\to 0, \\quad \\text{as}\\quad x\\rightarrow \\infty.\n \\end{equation}\n Indeed, we define $f(x):= h(x)\\sup_{y\\geq x}\\frac{h'(y)^2}{h(y)^4}$ and then, for $x\\geq x_1$, $\\zeta(x) = \\big(\\frac{f(x)}{h(x)}\\big)^{1/2}$. According to assumption \\eqref{eq:comportement h +infty}, both $\\frac{1}{h}$ and $f$ belong to $L^1(x_1,\\infty)$ so does $\\zeta$. Since it is non-increasing, the result follows.\n\nSuch a drift $h$ has the property to send any potential that is located after $x_1$ to $+\\infty$ in finite time. Indeed, ignoring the diffusion, the characteristics, $X(t)$, are determined by\n \\[\n \\frac{dX}{dt} = h(X(t)).\n \\]\n When $X(0)>x_1$, the solution $X(t)$ blows-up in finite time, given by\n \\[\n \\lim_{t\\rightarrow t_0}X(t) = +\\infty, \\qquad \\text{with}\\qquad t_0= \\int_{X(0)}^\\infty\\frac{dy}{h(y)}.\n \\]\n This kind of property is still true, with non-zero probability, when the membrane potential evolves according to the stochastic differential equation \n \\begin{equation}\\label{eq:EDS IF}\n dX_t = h(X_t)\\,dt +\\sqrt{2} dB_t .\n \\end{equation}\nIn both cases, the reset is obtained by setting $X_{t^+} = 0$ when $\\lim_{t\\to t^-} X_{t} =+\\infty$.\n The interest of the linear model lies in building a solid theoretical foundation for the existence and regularity of solutions, as well as their asymptotic behavior, on order to better address the nonlinear case.\n \\\\\n One of the main difficulty is to make sense of the flux $N(t)$ of particles reaching $+\\infty$ when we only handle a weak solution, $u(t)\\in L^1(\\mathbb{R})$ which is not defined pointwise.\n\n\\subsection{Main results}\n\nOur first result concerns well-posedness in $L^1$ with minimal assumptions.", "full_context": "In recent literature, many works have focused on the {\\em integrate-and-fire} (I\\&F) models for neurons potentials, which now constitute a widely studied reference framework (see the survey \\cite{CR2025}). However, several approaches from biophysics and computational neuroscience suggest a specificity, namely that intrinsic dynamics with super-linear growth for large potentials, notably quadratic or exponential, would provide a more realistic description of certain neuronal behaviors. This motivation is notably inspired by the work of \\cite{BL_2003, BretteG2005} on the Quadratic or Exponential Integrate-and-Fire model, which showed that introducing a super-linearity in the drift better captures neuronal firing dynamics and certain oscillations, compared to classical linear models. We propose a partial differential equation (PDE) that incorporates this super-linear component in order to develop a theory and a qualitative analysis of neuronal dynamics within this framework. To date, this theoretical framework has never been subject to a rigorous mathematical study via a Fokker-Planck type PDE. The associated linear model we propose is formally as follows:\n \\begin{equation}\\label{eq:FPSL lin}\n \\left\\{\\begin{aligned}\n & \\frac{\\partial u}{\\partial t}(t,x) +\\frac{\\partial}{\\partial x}\\big((h(x)u(t,x)\\big) - \\frac{\\partial ^2u}{\\partial x^2}(t,x) = \\delta_{0}(x)N(t), \\quad x\\in \\mathbb{R}, t\\geq 0,\n \\\\\n & N(t) = \\lim_{x\\rightarrow \\infty}h(x)u(t,x),\n \\\\\n & u(t=0,x) = u_0(x)\\in L^1(\\mathbb{R}).\n \\end{aligned}\\right.\n \\end{equation}\nThe specificity is to work on the full line with a flux of firing neurons $h(x)u(x)$ at infinity which occurs because we assume that $\\int_{\\cdot}^{\\infty} \\frac{1}{h(x)}dx <+\\infty$. This specificity also makes the mathematical interest and new difficulty for the analysis of the model.\n\n\\subsection{Motivations}\nThe study of this equation fundamentally differs, in its structure, from the classical linear-drift I\\&F model introduced in \\cite{BrHa}, \n \\begin{equation*}\n \\left\\{\\begin{aligned}\n & \\frac{\\partial p}{\\partial t}(t,x) +\\frac{\\partial}{\\partial x}(h(x) p(t,x)) - \\frac{\\partial ^2p}{\\partial x}(t,x) = \\delta_{V_R}(x)N(t), \\qquad t\\geq 0, \\; x\\leq V_F,\\\\\n & p(t,V_F) = 0\\\\\n & N(t) = -\\frac{\\partial p}{\\partial x}(t,V_F).\n \\end{aligned}\\right.\n \\end{equation*}\nThis equation describes the probability density of neurons, according to their membrane potential~$x$. When a neuron reaches the potential of firing, $V_F$, then it is instantaneously re-injected with a membrane potential of reset, $V_R< V_F$ \n(we have chosen $V_R=0$ in~\\eqref{eq:FPSL lin} for simplicity).\n\nIn order to catch the superlinear behaviour of the drift, we consider a generalized drift function $h\\in \\mathcal{C}^1$ that satisfies the following asymptotic behavior.\n \\\\\n \\textbf{Behavior at $-\\infty$:}\n \\begin{equation}\\label{eq:comportement h -infty}\n \\exists x_0 \\leq 0,\\; h_0\\in \\mathbb{R} \\quad \\text{such that} \\quad \\forall x\\leq x_0,\\quad h(x) = -x+h_0.\n \\end{equation}\n This assumption could be generalized, this specific form allows us to perform explicit calculations which avoid some technicalities.\n \\\\\n \\textbf{Behavior at $+\\infty$:}\n \\begin{equation}\\label{eq:comportement h +infty}\n \\begin{cases}\n \\exists x_1 \\geq 0,\\; \\forall x\\geq x_1, \\quad h(x)>0, \\quad h'(x)>0, \n \\\\[2pt]\n \\displaystyle \\int_{x_1}^\\infty\\frac{dx}{h(x)}<\\infty,\n \\\\[2pt]\n h(x)\\sup_{y\\geq x}\\frac{h'(y)^2}{h(y)^4}\\in L^1(x_1,\\infty).\n \\end{cases}\n \\end{equation}\nBoth the quadratic and exponential models used in the literature satisfy these assumption. Throughout this paper we use these two assumptions without necessarily mentioning them. We also use the notation, \n\\begin{equation}\\label{def:calV}\n\\mathcal{H}(x) = \\int_0^x h(y)dy, \\quad \\text{hence} \\quad \\mathcal{H}(x) \\underset{-\\infty}{\\sim} -\\frac{x^2}{2}, \\quad \\lim_{x\\to \\infty} \\mathcal{H}(x) = +\\infty .\n\\end{equation}\n The third line of assumptions \\eqref{eq:comportement h +infty} ensures that \n \\begin{equation}\\label{eq:limh'/h^2}\n \\zeta(x):= \\sup_{y\\geq x}\\frac{h'(y)}{h(y)^2}\\in L^1(x_1,\\infty) \\qquad \\text{and} \\qquad \\zeta(x) \\to 0, \\quad \\text{as}\\quad x\\rightarrow \\infty.\n \\end{equation}\n Indeed, we define $f(x):= h(x)\\sup_{y\\geq x}\\frac{h'(y)^2}{h(y)^4}$ and then, for $x\\geq x_1$, $\\zeta(x) = \\big(\\frac{f(x)}{h(x)}\\big)^{1/2}$. According to assumption \\eqref{eq:comportement h +infty}, both $\\frac{1}{h}$ and $f$ belong to $L^1(x_1,\\infty)$ so does $\\zeta$. Since it is non-increasing, the result follows.\n\nSuch a drift $h$ has the property to send any potential that is located after $x_1$ to $+\\infty$ in finite time. Indeed, ignoring the diffusion, the characteristics, $X(t)$, are determined by\n \\[\n \\frac{dX}{dt} = h(X(t)).\n \\]\n When $X(0)>x_1$, the solution $X(t)$ blows-up in finite time, given by\n \\[\n \\lim_{t\\rightarrow t_0}X(t) = +\\infty, \\qquad \\text{with}\\qquad t_0= \\int_{X(0)}^\\infty\\frac{dy}{h(y)}.\n \\]\n This kind of property is still true, with non-zero probability, when the membrane potential evolves according to the stochastic differential equation \n \\begin{equation}\\label{eq:EDS IF}\n dX_t = h(X_t)\\,dt +\\sqrt{2} dB_t .\n \\end{equation}\nIn both cases, the reset is obtained by setting $X_{t^+} = 0$ when $\\lim_{t\\to t^-} X_{t} =+\\infty$.\n The interest of the linear model lies in building a solid theoretical foundation for the existence and regularity of solutions, as well as their asymptotic behavior, on order to better address the nonlinear case.\n \\\\\n One of the main difficulty is to make sense of the flux $N(t)$ of particles reaching $+\\infty$ when we only handle a weak solution, $u(t)\\in L^1(\\mathbb{R})$ which is not defined pointwise.\n\n\\subsection{Main results}\n\nOur first result concerns well-posedness in $L^1$ with minimal assumptions.\n\n\\subsection{Main results}\n\n\\begin{proof}\n Let $u_0 \\in \\mathcal{C}^1_c(\\mathbb{R})$. By Proposition \\ref{prop:regularity of u}, $u$ is continuous on $\\mathbb{R}^+ \\times \\mathbb{R}$, which implies that the trace $t \\mapsto w(t,0)$ is well-defined. Notice in addition that according to Proposition \\ref{prop:MaxPrinc}, $w$ satisfies $w\\in L^\\infty(\\R^+\\times \\R)$.\n Using Eq. \\eqref{eq:FPSL lin}, $w$ satisfies the following equation\n \\begin{align*}\n \\partial_t w &= \\frac{1}{u_\\infty} \\left( \\partial^2_x(w u_\\infty) - \\partial_x(h w u_\\infty) + N_u \\delta_0 \\right) \\\\\n &= \\partial_x^2 w + \\partial_x w \\left( 2 \\frac{u'_\\infty}{u_\\infty} - h \\right) + \\frac{w}{u_\\infty} \\underbrace{(u''_\\infty - \\partial_x(h u_\\infty))}_{= -N_\\infty \\delta_0} + \\frac{N_u}{u_\\infty} \\delta_0 \\\\\n &= \\partial_x^2 w + \\partial_x w \\left( 2 \\frac{u'_\\infty}{u_\\infty} - h \\right) + \\left( \\frac{N_u}{u_\\infty} - N_\\infty \\frac{w}{u_\\infty} \\right) \\delta_0.\n \\end{align*}\n Let $R > 0$ and let $\\chi$ be defined as in Theorem \\ref{th:entropy}. We introduce the cut-off function $\\rho_R(x) = \\int_x^\\infty \\chi(y-R) \\, dy$.\n Define the truncated entropy functional by\n \\[\n I_R(t) = \\int_{\\mathbb{R}} \\rho_R(x) H(w(t,x)) u_\\infty(x) \\, dx.\n \\]\n Differentiating with respect to time for $t \\geq 0$, we obtain:\n \\begin{align*}\n I'_R(t) &= \\int_{\\mathbb{R}} \\rho_R(x) \\left\\{ \\partial_x^2 w + \\partial_x w \\left( 2 \\frac{u'_\\infty}{u_\\infty} - h \\right) + \\left( \\frac{N_u}{u_\\infty} - N_\\infty \\frac{w}{u_\\infty} \\right) \\delta_0 \\right\\} H'(w) u_\\infty \\, dx.\n \\end{align*}\n Handling the Dirac term at $x=0$, the expression becomes\n \\begin{equation} \\label{eq:IR_deriv_interm}\n \\begin{aligned}\n I'_R(t) &= - \\rho_R(0) N_\\infty H'(w(t,0)) \\left( w(t,0) - \\frac{N_u(t)}{N_\\infty} \\right) \\\\\n &\\quad + \\int_{\\mathbb{R}} \\rho_R(x) \\left\\{ \\partial_x^2 w u_\\infty + \\partial_x w (2 u'_\\infty - h u_\\infty) \\right\\} H'(w) \\, dx.\n \\end{aligned}\n \\end{equation}\n Next we use the identity $H'(w) \\partial_x^2 w = \\partial_x^2 H(w) - (\\partial_x w)^2 H''(w)$ and integrating by parts, the term involving $\\partial_x^2 H(w)$ becomes\n \\begin{align*}\n \\int_{\\mathbb{R}} \\rho_R u_\\infty \\partial_x^2 H(w) \\, dx\n &= - \\int_{\\mathbb{R}} \\rho'_R u_\\infty \\partial_x w H'(w) \\, dx - \\int_{\\mathbb{R}} \\rho_R u'_\\infty \\partial_x w H'(w) \\, dx.\n \\end{align*}\n Substituting this back into \\eqref{eq:IR_deriv_interm}, we get\n \\begin{align*}\n I'_R(t) &= - \\rho_R(0) N_\\infty H'(w(t,0)) \\left( w(t,0) - \\frac{N_u(t)}{N_\\infty} \\right) - \\int_{\\mathbb{R}} \\rho'_R u_\\infty \\partial_x w H'(w) \\, dx\\\\\n &\\quad - \\int_{\\mathbb{R}} \\rho_R (\\partial_x w)^2 H''(w) u_\\infty \\, dx + \\int_{\\mathbb{R}} \\rho_R \\partial_x w H'(w) \\underbrace{(u'_\\infty - h u_\\infty)}_{= -N_\\infty \\mathbf{1}_{x \\geq 0}} \\, dx.\n \\end{align*}\n Let us analyze the last integral involving the indicator function. We compute\n \\begin{align*}\n \\int_{\\mathbb{R}} \\rho_R \\partial_x w H'(w) (-N_\\infty \\mathbf{1}_{x \\geq 0}) \\, dx\n &= - N_\\infty \\int_0^\\infty \\rho_R(x) \\partial_x H(w(t,x)) \\, dx \\\\\n &= - N_\\infty \\left[ \\rho_R(x) H(w(t,x)) \\right]_{0}^\\infty + N_\\infty \\int_0^\\infty \\rho'_R(x) H(w(t,x)) \\, dx \\\\\n &= N_\\infty \\rho_R(0) H(w(t,0)) + N_\\infty \\int_{\\mathbb{R}} \\rho'_R(x) H(w(t,x)) \\, dx.\n \\end{align*}\n Now we pass to the limit as $R \\to \\infty$.\n Using the bound $u_\\infty^{1/2} \\partial_x w \\in L^2(\\mathbb{R}^+ \\times \\mathbb{R})$ from \\eqref{eq: borne d_x w } and that $H'(w) \\in L^\\infty$, the Cauchy-Schwarz inequality gives\n \\[\n \\left| \\int_{\\mathbb{R}} \\rho'_R u_\\infty \\partial_x w H'(w) \\, dx \\right| \\leq C \\left( \\int_{\\mathbb{R}} |\\rho'_R| u_\\infty \\, dx \\right)^{1/2} \\left( \\int_{\\mathbb{R}} |\\rho'_R| u_\\infty (\\partial_x w)^2 \\, dx \\right)^{1/2}.\n \\]\n Thus, for almost every $t \\geq 0$, this term vanishes as $R \\to \\infty$.\n Furthermore, $\\lim_{R \\to \\infty} \\rho_R(0) = 1$, and by the definition of $\\rho_R$ we have\n \\[\n \\lim_{R \\to \\infty} N_\\infty \\int_{\\mathbb{R}} \\rho'_R(x) H(w(t,x)) \\, dx = \\lim_{R \\to \\infty} - N_\\infty \\int_{\\mathbb{R}} \\chi(x-R) H(w(t,x)) \\, dx.\n \\]\n Recalling that in the weak sense $\\lim_{R \\to \\infty} \\int_{\\mathbb{R}} \\chi(x-R) w(t,x) \\, dx = \\frac{N_u(t)}{N_\\infty}$, and using Jensen's inequality, we have\n \\[\n \\liminf_{R \\to \\infty} \\int_{\\mathbb{R}} \\chi(x-R) H(w(t,x)) \\, dx \\geq H \\left( \\frac{N_u(t)}{N_\\infty} \\right).\n \\]\n Finally, gathering all terms and taking the limit $R \\to \\infty$, we obtain\n \\begin{align*}\n \\frac{d}{dt} \\int_{\\mathbb{R}} H(w) u_\\infty \\, dx\n &\\leq - N_\\infty \\left[ H \\left( \\frac{N_u}{N_\\infty} \\right) - H(w(t,0)) - H'(w(t,0)) \\left( \\frac{N_u}{N_\\infty} - w(t,0) \\right) \\right]\n \\\\\n &\\quad - \\int_{\\mathbb{R}} (\\partial_x w)^2 H''(w) u_\\infty \\, dx.\n \\end{align*}\n Since $H$ is convex, the term in bracket is non-negative, as well as the integral term. \\\\\n\n\\begin{proof}\n To get the desired lower bound, we first introduce the function $\\varphi$. For $x \\in \\mathbb{R}$, define\n \\[\n \\varphi(x)\n = \\int_{0}^{x} e^{-\\mathcal{H}(y)}\n \\left( \\int_{-\\infty}^{y} e^{\\mathcal{H}(z)}\\,dz \\right) dy .\n \\] \n This function satisfies the following properties\n \\begin{equation}\\label{eq:eqdif sur phi}\n \\fa x \\in \\R, \\qquad\\varphi''(x)+h(x)\\varphi'(x) = 1,\n \\end{equation}\n \\begin{equation}\\label{eq:behavior phi}\n \\varphi(x) \\sim_{x\\rightarrow -\\infty}- \\log|x|\\:\\:,\\:\\:\n \\lim_{x\\rightarrow \\infty} \\varphi(x) = c_\\varphi <\\infty \\text{ and } \\varphi \\text{ is non-decreasing on }\\R, \n \\end{equation}\n \\begin{equation}\\label{eq:signe phi}\n \\varphi(x) \\geq 0\\Longleftrightarrow x\\geq 0.\n \\end{equation}\nAlso, since $|\\varphi(x)| \\leq M \\big((-x)_++1\\big)$, for a constant $M\\geq 0$, for any $f\\in E$, we have\n \\begin{equation}\\label{eq:encadrement phi E}\n \\int_\\R|\\varphi(x)f(x)|dx\\leq M\\int_\\R\\big((-x)_++1\\big)|f(x)|dx=M\\|f\\|_E.\n \\end{equation}\n Now, let $I(t) = \\int_\\R\\varphi(x)u(t,x)\\,dx$. By \\eqref{eq:encadrement phi E} together with point~2 of Lemma~\\ref{lem:estimation u_E}, this quantity is well defined for all $t \\geq 0$. Then, differentiating $I(t)$, we obtain\n \\begin{align*}\n I'(t) &= \\int_\\R\\big(-\\partial_x\\big(h(x)u\\big)+\\partial_x^2u + N_u(t)\\delta_0\\big)\\varphi(x)\\,dx \\\\\n &= \\int_\\R u(t,x)\\big(h(x)\\varphi'(x)+\\varphi''(x)\\big)\\,dx - \\varphi(0)N_u(t) \\\\\n &= \\int_\\R u(t,x)\\,dx - c_\\varphi N_u(t) =1-c_\\varphi N_u(t).\n \\end{align*}\n Therefore, integrating on $(0,t)$ we obtain thanks to \\eqref{eq:signe phi}\n $$\n \\fa t\\geq 0,\\qquad \\int_0^t N_u(\\tau)\\,d\\tau = \\frac{1}{c_\\varphi}\\big(t+I(0)-I(t)\\big).\n $$\n Now, using the properties \\eqref{eq:encadrement phi E} and \\eqref{eq:behavior phi} for $I(t)$, we estimate \n \\begin{align*}\n I(0)-I(t) &= \\int_\\R\\varphi(x)\\big(u_0(x)-u(t,x)\\big)\\,dx \n \\\\\n &\\geq -\\int_{-\\infty}^0|\\varphi(x)|u_0(x)\\,dx - \\int_{0}^\\infty|\\varphi(x)|u(t,x)\\,dx \\\\\n &\\geq -M \\|u_0\\|_{E} - c_\\varphi \\int_0^\\infty u(t,x)\\,dx \\quad \\text{according to \\eqref{eq:encadrement phi E}} \\text{ and } \\eqref{eq:behavior phi},\\\\\n &\\geq -M C - c_\\varphi \\qquad\\text{since}\\quad \\int_0^\\infty u(t,x)dx \\leq 1. \\\\\n \\end{align*}\n Therefore, taking $t_C = MC+1+c_\\varphi$, we get, for any $t\\geq t_C$,\n $$\n \\int_0^t N_u(\\tau)\\,d\\tau \\geq \\frac{1}{c_\\varphi}>0.\n $$\n\\end{proof}\nUsing this lower bound on the source term, we construct an explicit subsolution which allows us to establish the Doeblin condition.\n\\begin{prop}\\label{prop:minoration S(t)u}\n For any $ C\\geq 0$, there exists a time $t'_C\\geq 0$, and a probability density $\\mu\\in L^1(\\R)$ such that for any probability density $u_0\\in E$ that satisfies $\\|u_0\\|_E\\leq C$, we have\n \\[\n \\fa t \\geq t_C',\\quad S(t)u_0\\geq \\eta(t) \\times \\mu.\n \\]\n where $\\eta$ is a strictly positive continuous function. \n \\end{prop}", "post_theorem_intro_text_len": 2577, "post_theorem_intro_text": "The precise definition of weak solutions, including the delicate question of the condition at infinity, that is \\eqref{eq:formulation faible FPS} and~\\eqref{eq:Conservation mass}, is treated in Section~\\ref{sec:exist} where we also prove existence relying on a truncated equation. Uniqueness uses a regularization argument and is treated in Section~\\ref{sec:unique}.\n\\\\\n\nWith stronger initial data, the solution can enjoy further regularity, however limited by the Dirac mass in the right hand side of \\eqref{eq:FPSL lin}.\n\\begin{prop}[Regularity of the solution]\\label{prop:regularity of u}.\n Let $u_0\\in \\mathcal{C}^0_c(\\mathbb{R})$, the weak solution of \\eqref{eq:FPSL lin} satisfies \n\t \\begin{equation}\\label{eq:borne du/dx}\n\t\\forall T>0, \\qquad \\int_0^T \\! \\int_\\mathbb{R}\\frac{\\big(\\partial_xu(t,x)\\big)^2}{u_\\infty(x)}\\,dx\\,dt<\\infty.\n\t \\end{equation}\nAssume additionally that $u_0\\in H^1(\\mathbb{R})$, then for any $T>0$, we have $u\\in L^\\infty((0,T);H^1_{loc}(\\mathbb{R}))$ and \n\\[\n\\forall A>0, \\qquad \\sup_{t\\in (0,T)}\\sup_{|x|\\leq A}|u(t+\\eta, x)-u(t,x)| = O(\\eta^{1/4}),\n\\] \n \\[\n \\lim_{\\eta \\rightarrow 0}\\sup_{t\\in (0,T)}\\|u(t+\\eta, \\cdot)-u(t,\\cdot)\\|_{\\infty}+\\|u(t+\\eta, \\cdot)-u(t,\\cdot)\\|_{1}=0.\n \\]\n In particular, $u$ is continuous with respect to $x$ and $t$. \n Finally, for all $ u_0\\in L^1(\\mathbb{R})$, $t \\mapsto u(t) \\in \\mathcal{C}^0\\big(\\mathbb{R}^+;L^1(\\mathbb{R})\\big)$.\n \\end{prop}\n\n This proposition, and several other regularity statements, are proved in Section~\\ref{sec:regularity}. In particular it allows us to state the flux condition at infinity in a stronger form, to prove the relative entropy equality and to establish a Poincar\\'e inequality for suitable forms of $h$.\n \\\\\n\n Although the Poincar\\'e inequality implies exponential decay at infinity, the Doeblin-Harris method provides us with much more general results.\n\\begin{theorem} [Long-term convergence] \\label{th:conv DH} \n There are constants $M, \\lambda > 0$ such that for any initial probability density $u_0$, the solution $u(t)$ of Eq.~\\eqref{eq:FPSL lin} with initial data $u_0$ satisfies\n $$\n \\forall t \\geq 0, \\qquad \\int_{\\mathbb{R}} \\big(1 + (-x)_+\\big) |u(t)-u_\\infty|\\, dx \\leq Me^{-\\lambda t} \\int_{\\mathbb{R}} \\big(1 + (-x)_+\\big) |u_0-u_\\infty | \\, dx \\leq +\\infty, \n $$\n where $(x)_+ = \\max(x, 0)$.\n\\end{theorem}\nThe proof is detailed in Section~\\ref{sec:LongTermDH}.\n\\\\\n\nThese results open other directions which are given in the conclusion, see Section~\\ref{sec:conclusion}.", "sketch": "Existence (for Theorem~\\ref{th:existence weak sol}) is proved in Section~\\ref{sec:exist} and “rel[ies] on a truncated equation.” Uniqueness “uses a regularization argument” and is treated in Section~\\ref{sec:unique}. The “precise definition of weak solutions, including the delicate question of the condition at infinity,” namely \\eqref{eq:formulation faible FPS} and \\eqref{eq:Conservation mass}, is also handled in Section~\\ref{sec:exist}.", "expanded_sketch": "Existence (for the main theorem) is proved next and “rel[ies] on a truncated equation.” Uniqueness “uses a regularization argument” and is treated later. The “precise definition of weak solutions, including the delicate question of the condition at infinity,” namely\n\\begin{align} \n\t\t\t\t\\int_0^\\infty \\! \\int_\\R u(t,x)\\Big(-\\partial_t\\varphi(t,x)&-h(x)\\partial_x\\varphi(t,x)-\\partial_x^2\\varphi(t,x)\\Big)\\,dx\\,dt \\notag\n\t\t\t\t\\\\\n\t\t\t\t& = \\int_0^\\infty N_u(t)\\varphi(t,0)\\,dt\n\t\t\t\t+\\int_\\R u_0(x)\\varphi(0,x)\\,dx. \\label{eq:formulation faible FPS} \n\t\t\t\\end{align}\nand\n\\begin{equation}\\label{eq:Conservation mass}\n\t\t\t\\text{for } a.e. \\; t\\geq0 \\quad \n\t\t\t\\int_\\R u(t,x)\\,dx = \\int_\\R u_0(x)\\,dx .\n\t\t\t\\end{equation}\nare also handled next.", "expanded_theorem": "[Well-posedness in $L^1$] \\label{th:existence weak sol}\n\tAssume that $h$ satisfies\n\t\\begin{equation}\\label{eq:comportement h -infty}\n\t\t\\exists x_0 \\leq 0,\\; h_0\\in \\R \\quad \\text{such that} \\quad \\fa x\\leq x_0,\\quad h(x) = -x+h_0.\n\t\\end{equation}\n\tand\n\t\\begin{equation}\\label{eq:comportement h +infty}\n\t\\begin{cases}\n\t\t\\exists x_1 \\geq 0,\\; \\fa x\\geq x_1, \\quad h(x)>0, \\quad h'(x)>0, \n\t\t\\\\[2pt]\n\t\t\\displaystyle \\int_{x_1}^\\infty\\frac{dx}{h(x)}<\\infty,\n\t\t\\\\[2pt]\n\t\th(x)\\sup_{y\\geq x}\\frac{h'(y)^2}{h(y)^4}\\in L^1(x_1,\\infty).\n\t\t\\end{cases}\n\t\\end{equation}\n\tThen, for any initial data $u_0\\in L^1(\\mathbb{R})$, there exists a unique weak solution $(u,N_u)\\in L^\\infty(\\mathbb{R}^+;L^1(\\mathbb{R}))\\times L^1_{loc}(\\mathbb{R}^+)$ of\n\t\\begin{equation}\\label{eq:FPSL lin}\n\t\t\\left\\{\\begin{aligned}\n\t\t\t& \\frac{\\partial u}{\\partial t}(t,x) +\\frac{\\partial}{\\partial x}\\big((h(x)u(t,x)\\big) - \\frac{\\partial ^2u}{\\partial x^2}(t,x) = \\delta_{0}(x)N(t), \\quad x\\in \\mathbb{R}, t\\geq 0,\n\t\t\t\\\\\n\t\t\t& N(t) = \\lim_{x\\rightarrow \\infty}h(x)u(t,x),\n\t\t\t\\\\\n\t\t\t& u(t=0,x) = u_0(x)\\in L^1(\\R).\n\t\t\\end{aligned}\\right.\n\t\\end{equation}\n\ti.e., that satisfies\n\t\\begin{align} \n\t\t\t\t\\int_0^\\infty \\! \\int_\\R u(t,x)\\Big(-\\partial_t\\varphi(t,x)&-h(x)\\partial_x\\varphi(t,x)-\\partial_x^2\\varphi(t,x)\\Big)\\,dx\\,dt \\notag\n\t\t\t\t\\\\\n\t\t\t\t& = \\int_0^\\infty N_u(t)\\varphi(t,0)\\,dt\n\t\t\t\t+\\int_\\R u_0(x)\\varphi(0,x)\\,dx. \\label{eq:formulation faible FPS} \n\t\t\t\\end{align}\n\tand\n\t\\begin{equation}\\label{eq:Conservation mass}\n\t\t\t\\text{for } a.e. \\; t\\geq0 \\quad \n\t\t\t\\int_\\R u(t,x)\\,dx = \\int_\\R u_0(x)\\,dx .\n\t\t\t\\end{equation}\n\tIn addition, we have \n\t \t\\begin{equation}\\label{eq:L1Contract}\n\t \\text{for a.e. } t > 0, \\quad \\int_\\mathbb{R} |u(t,x)|\\,dx \\leq \\int_\\mathbb{R} |u_0(x)| \\,dx,\n\t\\end{equation}\n\t\\begin{equation}\\label{eq: U_T bound}\n\t \\forall T\\geq 0, \\quad \\sup_{x\\in \\mathbb{R}}\\int_0^T|h(x)u(t,x)|\\,dt <\\infty \\quad \\text{and}\\quad U_T = \\int_0^Tu(t)\\,dt \\in W^{1,\\infty}(\\mathbb{R}).\n\t\\end{equation}", "theorem_type": ["Uniqueness", "Existence"], "mcq": {"question": "Let $h\\in C^1(\\mathbb{R})$ satisfy the following two assumptions:\n1. There exist $x_0\\le 0$ and $h_0\\in\\mathbb{R}$ such that $h(x)=-x+h_0$ for all $x\\le x_0$.\n2. There exists $x_1\\ge 0$ such that for all $x\\ge x_1$, one has $h(x)>0$ and $h'(x)>0$, together with\n$$\\int_{x_1}^{\\infty}\\frac{dx}{h(x)}<\\infty,$$\nand\n$$x\\mapsto h(x)\\sup_{y\\ge x}\\frac{h'(y)^2}{h(y)^4}\\in L^1(x_1,\\infty).$$\nFor an initial datum $u_0\\in L^1(\\mathbb{R})$, consider the Fokker--Planck problem on $\\mathbb{R}\\times [0,\\infty)$\n$$\\partial_t u(t,x)+\\partial_x\\big(h(x)u(t,x)\\big)-\\partial_x^2 u(t,x)=\\delta_0(x)N(t),$$\n$$N(t)=\\lim_{x\\to\\infty} h(x)u(t,x),\\qquad u(0,x)=u_0(x),$$\nwhere $\\delta_0$ is the Dirac mass at $0$. Which statement holds for every such $u_0$?", "correct_choice": {"label": "A", "text": "For every $u_0\\in L^1(\\mathbb{R})$, there exists a unique weak solution $(u,N_u)\\in L^\\infty(\\mathbb{R}^+;L^1(\\mathbb{R}))\\times L^1_{\\mathrm{loc}}(\\mathbb{R}^+)$ of this problem, with $N_u(t)=\\lim_{x\\to\\infty}h(x)u(t,x)$, such that\n$$\\int_0^\\infty\\!\\int_{\\mathbb{R}} u(t,x)\\Big(-\\partial_t\\varphi(t,x)-h(x)\\partial_x\\varphi(t,x)-\\partial_x^2\\varphi(t,x)\\Big)\\,dx\\,dt\n=\\int_0^\\infty N_u(t)\\varphi(t,0)\\,dt+\\int_{\\mathbb{R}}u_0(x)\\varphi(0,x)\\,dx,$$\nand, for a.e. $t\\ge 0$,\n$$\\int_{\\mathbb{R}}u(t,x)\\,dx=\\int_{\\mathbb{R}}u_0(x)\\,dx.$$ \nIn addition, for a.e. $t>0$,\n$$\\int_{\\mathbb{R}}|u(t,x)|\\,dx\\le \\int_{\\mathbb{R}}|u_0(x)|\\,dx,$$\nand for every $T\\ge 0$,\n$$\\sup_{x\\in\\mathbb{R}}\\int_0^T |h(x)u(t,x)|\\,dt<\\infty,\\qquad U_T(x):=\\int_0^T u(t,x)\\,dt\\in W^{1,\\infty}(\\mathbb{R}).$$"}, "choices": [{"label": "B", "text": "For every $u_0\\in L^1(\\mathbb{R})$, there exists a unique weak solution $(u,N_u)\\in L^\\infty(\\mathbb{R}^+;L^1(\\mathbb{R}))\\times L^\\infty_{\\mathrm{loc}}(\\mathbb{R}^+)$ of this problem, with $N_u(t)=\\lim_{x\\to\\infty}h(x)u(t,x)$, such that\n$$\\int_0^\\infty\\!\\int_{\\mathbb{R}} u(t,x)\\Big(-\\partial_t\\varphi(t,x)-h(x)\\partial_x\\varphi(t,x)-\\partial_x^2\\varphi(t,x)\\Big)\\,dx\\,dt\n=\\int_0^\\infty N_u(t)\\varphi(t,0)\\,dt+\\int_{\\mathbb{R}}u_0(x)\\varphi(0,x)\\,dx,$$\nand, for every $t\\ge 0$,\n$$\\int_{\\mathbb{R}}u(t,x)\\,dx=\\int_{\\mathbb{R}}u_0(x)\\,dx.$$ \nIn addition, for every $t>0$,\n$$\\int_{\\mathbb{R}}|u(t,x)|\\,dx\\le \\int_{\\mathbb{R}}|u_0(x)|\\,dx,$$\nand for every $T\\ge 0$,\n$$\\sup_{x\\in\\mathbb{R}}\\int_0^T |h(x)u(t,x)|\\,dt<\\infty,\\qquad U_T(x):=\\int_0^T u(t,x)\\,dt\\in W^{1,\\infty}(\\mathbb{R}).$$"}, {"label": "C", "text": "For every $u_0\\in L^1(\\mathbb{R})$, there exists at least one weak solution $(u,N_u)\\in L^\\infty(\\mathbb{R}^+;L^1(\\mathbb{R}))\\times L^1_{\\mathrm{loc}}(\\mathbb{R}^+)$ of this problem, with $N_u(t)=\\lim_{x\\to\\infty}h(x)u(t,x)$, such that\n$$\\int_0^\\infty\\!\\int_{\\mathbb{R}} u(t,x)\\Big(-\\partial_t\\varphi(t,x)-h(x)\\partial_x\\varphi(t,x)-\\partial_x^2\\varphi(t,x)\\Big)\\,dx\\,dt\n=\\int_0^\\infty N_u(t)\\varphi(t,0)\\,dt+\\int_{\\mathbb{R}}u_0(x)\\varphi(0,x)\\,dx,$$\nand, for a.e. $t\\ge 0$,\n$$\\int_{\\mathbb{R}}u(t,x)\\,dx=\\int_{\\mathbb{R}}u_0(x)\\,dx.$$"}, {"label": "D", "text": "For every $u_0\\in L^1(\\mathbb{R})$, there exists a unique weak solution $(u,N_u)\\in L^\\infty(\\mathbb{R}^+;L^1(\\mathbb{R}))\\times L^1_{\\mathrm{loc}}(\\mathbb{R}^+)$ of this problem, and the weak formulation is\n$$\\int_0^\\infty\\!\\int_{\\mathbb{R}} u(t,x)\\Big(-\\partial_t\\varphi(t,x)-h(x)\\partial_x\\varphi(t,x)-\\partial_x^2\\varphi(t,x)\\Big)\\,dx\\,dt\n=\\int_0^\\infty N_u(t)\\varphi(t,\\infty)\\,dt+\\int_{\\mathbb{R}}u_0(x)\\varphi(0,x)\\,dx,$$\nwhere $N_u(t)=-\\partial_x u(t,0)$, and, for a.e. $t\\ge 0$,\n$$\\int_{\\mathbb{R}}u(t,x)\\,dx=\\int_{\\mathbb{R}}u_0(x)\\,dx.$$ \nIn addition, for a.e. $t>0$,\n$$\\int_{\\mathbb{R}}|u(t,x)|\\,dx\\le \\int_{\\mathbb{R}}|u_0(x)|\\,dx,$$\nand for every $T\\ge 0$,\n$$\\sup_{x\\in\\mathbb{R}}\\int_0^T |h(x)u(t,x)|\\,dt<\\infty,\\qquad U_T(x):=\\int_0^T u(t,x)\\,dt\\in W^{1,\\infty}(\\mathbb{R}).$$"}, {"label": "E", "text": "For every $u_0\\in L^1(\\mathbb{R})$, there exists a unique classical solution $(u,N_u)$ of this problem on $[0,\\infty)\\times\\mathbb{R}$, with $u\\in C^1([0,\\infty);L^1(\\mathbb{R}))\\cap C([0,\\infty);W^{2,1}(\\mathbb{R}))$ and $N_u\\in C([0,\\infty))$, such that\n$$\\int_0^\\infty\\!\\int_{\\mathbb{R}} u(t,x)\\Big(-\\partial_t\\varphi(t,x)-h(x)\\partial_x\\varphi(t,x)-\\partial_x^2\\varphi(t,x)\\Big)\\,dx\\,dt\n=\\int_0^\\infty N_u(t)\\varphi(t,0)\\,dt+\\int_{\\mathbb{R}}u_0(x)\\varphi(0,x)\\,dx,$$\nand, for a.e. $t\\ge 0$,\n$$\\int_{\\mathbb{R}}u(t,x)\\,dx=\\int_{\\mathbb{R}}u_0(x)\\,dx.$$ \nIn addition, for a.e. $t>0$,\n$$\\int_{\\mathbb{R}}|u(t,x)|\\,dx\\le \\int_{\\mathbb{R}}|u_0(x)|\\,dx,$$\nand for every $T\\ge 0$,\n$$\\sup_{x\\in\\mathbb{R}}\\int_0^T |h(x)u(t,x)|\\,dt<\\infty,\\qquad U_T(x):=\\int_0^T u(t,x)\\,dt\\in W^{1,\\infty}(\\mathbb{R}).$$"}], "meta": {"weaker_true_label": "C", "false_labels": ["B", "D", "E"], "wildcard_false_label": "D"}, "sketch_usage_meta": [{"label": "B", "sketch_hook_type": "regularity", "tampered_component": "time-regularity/integrability of the flux and almost-everywhere statements", "template_used": "uniformity_effectivity"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "uniqueness and additional contractive/regularity conclusions", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "delicate condition at infinity encoded by the source term at $x=0$ and the boundary functional", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "weak-solution level of the theorem versus unjustified classical regularity", "template_used": "stronger_trap"}]}} +{"id": "2601.19744v1", "paper_link": "http://arxiv.org/abs/2601.19744v1", "theorems_cnt": 3, "theorem": {"env_name": "Theorem", "content": "\\label{main result}\nLet $n = 2$ or $3$ and $\\gamma > 1$. For any given $T > 0$, consider any weak solution ${\\bf u}\\in L^\\infty(0,T;L^2(\\mathbb{T}^n))$ to the incompressible Euler equation \\eqref{incom Euler} with initial data ${\\bf u}^0\\in L^2(\\mathbb{T}^n)$, as defined in Definition \\ref{definition of weak solution incom}. Then for any $T' < T$, \n there exists a small number $\\delta_0>0$ such that for any Mach number $\\delta<\\delta_0$, there exists a family of initial data \n$$\n(\\rho_{\\delta}^0,\\m_{\\delta}^0)\\in C^0(\\mathbb{T}^n)\\times L^2(\\mathbb{T}^n)\n$$\nwith the following properties.\n\\begin{itemize}\n\\item For any fixed $\\delta>0$, there exist infinitely many weak solutions $(\\rho_{\\delta},\\rho_{\\delta}\\widehat{{\\bf v}}_{\\delta})$ on $(0, T')$ to the rescaled compressible Euler equations \\eqref{scalling system 111} emanating from the above data $(\\rho_{\\delta}^0,\\m_{\\delta}^0)$, in the sense of Definition \\ref{definition of weak solutions compressible}.\n\\item The following asymptotic limits hold:\n\\begin{equation*}\n\\rho_{\\delta}\\to 1\\;\\;\\;\\text{ in } C^0([0,T']\\times \\mathbb{T}^n),\\;\\;\\;\\;\\; \\rho_{\\delta}\\widehat{{\\bf v}}_{\\delta}\\to {\\bf u} \\;\\text{ in } L^{2}(0,T';L^2(\\mathbb{T}^n)),\n\\end{equation*}\nand the initial data \n$$\n\\m_{\\delta}^0\\to {\\bf u}^0\\;\\;\\;\\text{ weakly in }\\;\\; L^2(\\mathbb{T}^n),\n$$\nas $\\delta$ tends to zero.\n\\end{itemize}", "start_pos": 12552, "end_pos": 13919, "label": "main result"}, "ref_dict": {"incom Euler": "\\begin{equation}\\label{incom Euler} \n\\left\\{\n\\begin{split}&\n\\partial_t \\u + \\u\\cdot\\nabla\\u+\\nabla \\pi=0,\n\\\\& \\Dv \\u=0.\n\\end{split}\\right.\n\\end{equation}", "eq unif": "\\begin{equation}\\label{eq unif}\n\\int \\frac{|\\widetilde{V}|^2}{\\rho_\\delta}\\,dxdt \\le C \\int \\trace R_\\delta \\,dxdt.\n\\end{equation}", "lem conv int": "\\begin{Lemma}\\label{lem conv int}\nLet $(\\rho_0, V_0, R_0) \\in C^0(\\R \\times \\R^n; \\R \\times \\R^n \\times \\mathbb{S}_0^{n\\times n})$\n be given with $\\rho_0$ satisfying \\eqref{bounds rho} and $R_0$ being positive definite in some open set $P \\subset \\R \\times \\R^n$. Let $U_0$ be given as in \\eqref{U0 condition}. There exist infinitely many $(\\widetilde{V},\\widetilde{U}) \\in L^\\infty(\\R \\times \\R^n; \\R^n \\times \\mathbb{S}_0^{n\\times n})$ which are compactly supported in $P$ and satisfy \\eqref{ci system} and \\eqref{restriction =}. In addition, there exists a positive number $C_0>0$, such that \n $$\\int_0^T\\int_{\\mathbb{T}^n}\\frac{|\\widetilde{V}|^2}{\\rho_0}\\,dx\\,dt\\leq C_0\\int_0^T\\int_{\\mathbb{T}^n} \\tr R_0\\,dx\\,dt.$$\n\\end{Lemma}", "proposition points": "\\begin{Proposition}\n\\label{proposition points}\nLet $(\\widehat{V},\\widehat{U})\\in X$ be a point of continuity of the identity map $\\id$ from $(X,d)$ to $L^2(\\R^n\\times\\R)$. Then, $(\\widehat{V},\\widehat{U})$ satisfies \\eqref{restriction =}. In addition, there exists a positive number $C_0>0$, such that \n $$\n \\int_0^T\\int_{\\mathbb{T}^n}\\frac{|\\widehat{V}|^2}{\\rho_0}\\,dx\\,dt\\leq C_0\\int_0^T\\int_{\\mathbb{T}^n} \\trace R_0\\,dx\\,dt.\n $$\n\\end{Proposition}", "L1-coercivity": "\\begin{Lemma}[$L^1$-coercivity]\n\\label{L1-coercivity}\nFor any $(\\widetilde{V}, \\widetilde{U}) \\in X_0$, where $X_0$ is defined in \\eqref{defn X_0}, there exists a sequence $\\{(\\widetilde{V}_i, \\widetilde{U}_i)\\} \\subset X_0$ that converges weakly-$*$ to $(\\widetilde{V}, \\widetilde{U})$, such that\n$$\n\\|\\widetilde{V}_i - \\widetilde{V}\\|_{L^1(P)} \\geq \\frac{c_0}{\\Lambda} \\int_{P} \\trace \\, M \\, dx \\, dt,\n$$\nfor some positive geometric constant $c_0$, where $\\Lambda$ is given in \\eqref{bounds rho}.\n\\end{Lemma}", "eq new constr": "\\begin{equation}\\label{eq new constr}\n\\left| \\int \\frac{V_\\delta \\cdot \\widetilde{V}}{\\rho_\\delta} \\,dxdt \\right| \\le \\frac18 \\int \\frac{|\\widetilde{V}|^2}{\\rho_\\delta}\\,dxdt,\n\\end{equation}", "lem subsoln": "\\begin{Lemma}\\label{lem subsoln}\n\\label{subsolution}\nFor any $\\varepsilon>0$, consider $(\\u^\\varepsilon, \\pi^\\varepsilon, R^\\varepsilon)$ as in Lemma \\ref{incom smooth}. There exists a $\\delta_0 = \\delta_0(\\eps) >0$ such that for any $\\delta\\leq \\delta_0$, there exist $\\v_\\delta \\in C^{\\infty}([0,T-\\eps]\\times \\mathbb{T}^n)$ and $\\tilde{R}_{\\delta} \\in C^\\infty([0, T - \\varepsilon] \\times \\mathbb{T}^n; \\mathbb{S}^{n \\times n})$ with $\\tilde{R}_{\\delta} > 0$, solving \n\\begin{equation}\\label{eq subsoln}\n\\left\\{\n\\begin{split}\n&\\partial_t\\rho_{\\delta}+\\Dv(\\rho_{\\delta} \\v_{\\delta})=0,\n\\\\&\\partial_t(\\rho_{\\delta} \\v_{\\delta})+\\Dv(\\rho_{\\delta} \\v_{\\delta}\\otimes \\v_{\\delta})+\\nabla\\frac{\\rho_{\\delta}^\\gamma}{\\delta^2}+\\Dv(R^{\\varepsilon}+\\tilde{R}_{\\delta})=0,\n\\end{split}\\right.\n\\end{equation}\nwhere \n\\begin{equation}\\label{eq def rho}\n\\rho_\\delta(t,x) := 1 + \\delta^2 \\varrho_{\\delta, K_\\ast}(t,x) \n\\end{equation}\nwith $\\varrho_{\\delta, K_\\ast}$ being given in \\eqref{eq varrho}. Moreover we have\n\\begin{equation}\\label{eq subsoln est}\n\\begin{split}\n\\|\\rho_{\\delta}-1\\|_{C^0([0,T-\\varepsilon] \\times\\mathbb{T}^n)} + \\|\\rho_{\\delta}\\v_{\\delta}-\\u^{\\varepsilon}\\|_{L^2([0,T-\\varepsilon] \\times \\mathbb{T}^n)} + \\|\\tilde{R_{\\delta}}\\|_{L^\\infty([0,T-\\varepsilon] \\times \\mathbb{T}^n)}\\leq \\varepsilon.\n\\end{split}\n\\end{equation}\n\\end{Lemma}", "definition of weak solution incom": "\\begin{Definition}\n\\label{definition of weak solution incom}\nWe say $\\u \\in L^{\\infty}(0,T;L^2(\\mathbb{T}^n)) $ is a weak solution of \\eqref{incom Euler} with intial data $\\u^0 \\in L^2$ if it is divergence-free in the sense of distribution and \n\\begin{itemize}\n\\item for any $\\phi \\in C^\\infty_c(\\mathbb{T}^n \\times [0, T); \\R^n)$ with $\\Dv \\, \\phi = 0$,\n\\begin{equation}\\label{weak incomp euler eqn}\n\\int^\\infty_0 \\int_{\\mathbb{T}^n} \\u \\cdot \\partial_t \\phi + \\u \\otimes \\u : \\nabla \\phi \\,dxdt = -\\int_{\\mathbb{T}^n} \\u^0 \\cdot \\phi(\\cdot, 0) \\,dx.\n\\end{equation}\n\\end{itemize}\n\\end{Definition}", "incom smooth": "\\begin{Lemma}\n\\label{incom smooth} \nFor any weak solutions $\\u$ to the incompressible Euler equations \\eqref{incom Euler} and any $\\eps > 0$,\nthere exist $\\u^{\\varepsilon}, \\pi^\\varepsilon \\in C^{\\infty}([0,T-\\eps] \\times \\mathbb{T}^n)$ and $R^\\varepsilon \\in C^\\infty([0, T - \\varepsilon] \\times \\mathbb{T}^n; \\mathbb{S}^{n \\times n})$, where $\\mathbb{S}^{n \\times n}$ is the set of symmetric matrices in dimension $n$, such that\n\\begin{itemize}\n\\item\n$(\\u^{\\varepsilon}, \\pi^{\\varepsilon}, R^\\varepsilon)$ is a weak solution to \n\\begin{equation}\n\\label{smooth euler}\n\\left\\{\n\\begin{split}\n& \\partial_t \\u^{\\varepsilon} + \\u^{\\varepsilon}\\cdot\\nabla \\u^{\\varepsilon}+\\nabla \\pi^{\\varepsilon}+\\Dv R^{\\varepsilon}=0,\n\\\\& \\Dv {\\u^{\\varepsilon}}=0.\n\\end{split}\n\\right.\n\\end{equation}\n\\item $R^{\\varepsilon}>\\frac{\\varepsilon}{4} \\id_n$ is a symmetric matrix, where $\\id_n$ is the $n \\times n$ identity matrix.\n\\item There exists a constant $C>0$ independent of $\\eps$ such that\n\\begin{equation}\\label{eq reg est}\n\\|\\u^{\\varepsilon}-\\u\\|_{L^2([0,T - \\eps] \\times \\mathbb{T}^n)}+\\|R^{\\varepsilon}\\|_{L^1([0,T - \\eps] \\times \\mathbb{T}^n)}\\leq C \\varepsilon.\n\\end{equation}\n\\end{itemize}\n\\end{Lemma}", "definition of weak solutions compressible": "\\begin{Definition}\n\\label{definition of weak solutions compressible} \nFor $\\gamma>1$, we say $(\\rho_{\\delta}, \\rho_{\\delta} \\v_{\\delta})$ is a weak solution of \\eqref{scalling system 111} with initial value $(\\rho^0_{\\delta},\\m_{\\delta}^0) \\in L^\\gamma \\times L^2$ if the following holds true.\n\\begin{itemize}\n\\item $\\rho_{\\delta}\\in L^{\\infty}(0,T;L^{\\gamma}(\\mathbb{T}^n)), \\;\\;\\sqrt{\\rho_{\\delta}} \\v_{\\delta}\\in L^{\\infty}(0,T;L^2(\\mathbb{T}^n))$;\n\\item for any $\\varphi \\in C^\\infty_c(\\mathbb{T}^3 \\times [0, T))$,\n\\[\n\\int_0^\\infty \\int_{\\mathbb{T}^n} \\left( \\rho_{\\delta} \\, \\partial_t \\varphi + \\rho_{\\delta} \\v_{\\delta} \\cdot \\nabla \\varphi \\right) \\, dx \\, dt + \\int_{\\mathbb{T}^n} \\rho_{\\delta}^0(x) \\varphi(0,x) \\, dx = 0;\n\\]\n\n \\item for any $\\phi \\in C^\\infty_c(\\mathbb{T}^n \\times [0, T); \\R^n)$,\n\\begin{equation*}\n\\begin{split}\n\\int^\\infty_0 \\int_{\\mathbb{T}^n} \\rho_{\\delta} \\v_{\\delta} \\cdot \\partial_t \\phi &+ \\rho_{\\delta} \\v_{\\delta} \\otimes \\v_{\\delta} : \\nabla \\phi \\,dxdt +\\int_0^\\infty\\int_{\\mathbb{T}^n}\\frac{\\rho_{\\delta}^{\\gamma}}{\\delta^2}\\Dv \\phi\\,dx\\,dt\n\\\\&= -\\int_{\\mathbb{T}^n}\\m_{\\delta}^0 \\cdot \\phi(\\cdot, 0) \\,dx.\n\\end{split}\n\\end{equation*}\n\\end{itemize}\n\\end{Definition}", "scalling system 111": "\\begin{equation}\n\\label{scalling system 111}\n\\left\\{\n\\begin{split}\n&\\partial_t\\rho_{\\delta}+\\Dv(\\rho_{\\delta} \\v_{\\delta})=0,\n\\\\&\\partial_t(\\rho_{\\delta} \\v_{\\delta})+\\Dv(\\rho_{\\delta} \\v_{\\delta}\\otimes \\v_{\\delta})+\\nabla\\frac{\\rho_{\\delta}^\\gamma}{\\delta^2}=0.\n\\end{split}\n\\right.\n\\end{equation}", "main result": "\\begin{Theorem}\n\\label{main result}\nLet $n = 2$ or $3$ and $\\gamma > 1$. For any given $T > 0$, consider any weak solution $\\u\\in L^\\infty(0,T;L^2(\\mathbb{T}^n))$ to the incompressible Euler equation \\eqref{incom Euler} with initial data $\\u^0\\in L^2(\\mathbb{T}^n)$, as defined in Definition \\ref{definition of weak solution incom}. Then for any $T' < T$, \n there exists a small number $\\delta_0>0$ such that for any Mach number $\\delta<\\delta_0$, there exists a family of initial data \n$$\n(\\rho_{\\delta}^0,\\m_{\\delta}^0)\\in C^0(\\mathbb{T}^n)\\times L^2(\\mathbb{T}^n)\n$$\nwith the following properties.\n\\begin{itemize}\n\\item For any fixed $\\delta>0$, there exist infinitely many weak solutions $(\\rho_{\\delta},\\rho_{\\delta}\\widehat{\\v}_{\\delta})$ on $(0, T')$ to the rescaled compressible Euler equations \\eqref{scalling system 111} emanating from the above data $(\\rho_{\\delta}^0,\\m_{\\delta}^0)$, in the sense of Definition \\ref{definition of weak solutions compressible}.\n\\item The following asymptotic limits hold:\n\\begin{equation*}\n\\rho_{\\delta}\\to 1\\;\\;\\;\\text{ in } C^0([0,T']\\times \\mathbb{T}^n),\\;\\;\\;\\;\\; \\rho_{\\delta}\\widehat{\\v}_{\\delta}\\to \\u \\;\\text{ in } L^{2}(0,T';L^2(\\mathbb{T}^n)),\n\\end{equation*}\nand the initial data \n$$\n\\m_{\\delta}^0\\to \\u^0\\;\\;\\;\\text{ weakly in }\\;\\; L^2(\\mathbb{T}^n),\n$$\nas $\\delta$ tends to zero.\n\\end{itemize}\n\\end{Theorem}", "compressible Euler": "\\begin{equation} \n\\label{compressible Euler}\n\\left\\{\n\\begin{split}\n&\\partial_t \\rho + \\Dv (\\rho \\v)= 0,\n\\\\&\\partial_t (\\rho \\v) + \\Dv (\\rho \\v \\otimes \\v) + \\nabla P = 0,\n\\end{split}\n\\right.\n\\end{equation}"}, "pre_theorem_intro_text_len": 9672, "pre_theorem_intro_text": "The incompressible Euler equations are universally regarded as the formal low Mach number limit of the compressible Euler equations. This physical premise is rooted in the observation that when fluid velocities are much smaller than the speed of sound, density variations become negligible, and the flow behaves as if it were incompressible \\cite{KM2, Majda, MaS}. The mathematical theory of this singular limit aims to rigorously justify the transition from solutions of the compressible Euler system\n\\begin{equation} \n\\label{compressible Euler}\n\\left\\{\n\\begin{split}\n&\\partial_t \\rho + {\\rm div} (\\rho {\\bf v})= 0,\n\\\\&\\partial_t (\\rho {\\bf v}) + {\\rm div} (\\rho {\\bf v} \\otimes {\\bf v}) + \\nabla P = 0,\n\\end{split}\n\\right.\n\\end{equation} \nwhere $\\rho$ is the density, ${\\bf v}$ the velocity, and $P(\\rho) = a \\rho^\\gamma$ the pressure ($a > 0$, $\\gamma > 1$), to those of the incompressible Euler system\n\\begin{equation}\\label{incom Euler} \n\\left\\{\n\\begin{split}&\n\\partial_t {\\bf u} + {\\bf u}\\cdot\\nabla{\\bf u}+\\nabla \\pi=0,\n\\\\& {\\rm div} {\\bf u}=0.\n\\end{split}\\right.\n\\end{equation}\nTo capture this low Mach number regime, one introduces a scaling where time is slowed down and velocity is rescaled by a small parameter\n\\[\n{\\rho}(x,t) =: \\rho_\\delta(x,\\delta t), \\quad {{\\bf v}}(x,t) =: \\delta \\v_\\delta(x,\\delta t).\n\\]\nThis leads from \\eqref{compressible Euler} to the rescaled compressible Euler equations\n\\begin{equation}\n\\label{scalling system 111}\n\\left\\{\n\\begin{split}\n&\\partial_t\\rho_{\\delta}+{\\rm div}(\\rho_{\\delta} \\v_{\\delta})=0,\n\\\\&\\partial_t(\\rho_{\\delta} \\v_{\\delta})+{\\rm div}(\\rho_{\\delta} \\v_{\\delta}\\otimes \\v_{\\delta})+\\nabla\\frac{\\rho_{\\delta}^\\gamma}{\\delta^2}=0.\n\\end{split}\n\\right.\n\\end{equation} \nFormally, as $\\delta \\to 0$, we expect $\\rho_\\delta$ to converge to a constant (say, $1$), and the velocity field to become divergence-free, recovering \\eqref{incom Euler}. \n\nThe study of this limit was initiated by the seminal works of Ebin \\cite{Ebin} and Klainerman--Majda \\cite{KM, KM2}, who treated the case of smooth, well-prepared initial data. Subsequent research expanded this theory to various contexts, including ill-prepared data \\cite{Ukai} , bounded domains \\cite{Scho}, non-isentropic flows \\cite{MS2,MS0}, and the framework of weak solutions for the Navier--Stokes equations \\cite{Feireisl-incom,LM, LM2, LM3}. This extensive body of work (see also \\cite{BDGL,BFH, CJ, FKMV, Fu,SEC}) typically relies on strong compactness arguments or relative entropy methods. These techniques require sufficient a priori estimates or structural assumptions on the solutions, effectively filtering out highly oscillatory or turbulent behavior.\nTo obtain global-in-time results, Feireisl--Klingenberg--Markfelder \\cite{FKM} introduced the framework of dissipative measure-valued (DMV) solutions, establishing convergence to the smooth incompressible solution for finite-energy data. Their result demonstrates that under the DMV framework, the incompressible limit acts as a selective filter for families of approximate solutions. \n\nIn contrast, our work asks if the incompressible system is a \\emph{universal attractor}: can every incompressible weak solution be targeted and approximated by a sequence of weak solutions to the compressible system?\n\n\\subsection*{A New Perspective from Convex Integration}\nThis question is motivated by the modern theory of weak solutions offered by \\emph{convex integration}. In a marked departure from prior approaches, this framework has revealed that for a dense set of initial data, the incompressible and compressible isentropic Euler system admits infinitely many weak solutions, even when imposing the energy inequality as an admissibility criterion \\cite{CVY,SW}. The main achievement in our prior work \\cite{CVY} is the development of a convex integration scheme capable of handling a general positive definite Reynolds stress tensor, a generalization beyond earlier methods.\n\nWe now re-examine the foundational low Mach number limit through this lens. The discovery of ubiquitous non-uniqueness prompts us to shift the analytical paradigm: instead of investigating whether a given family of compressible solutions converges, we constructively address whether any arbitrary incompressible weak solution ${\\bf u}$ can be realized as the limit (as $\\delta \\to 0$) of a sequence of compressible weak solutions built via convex integration. An affirmative answer would demonstrate a form of universality, thereby establishing that the incompressible system is indeed a universal attractor within this framework, independent of the small-scale turbulent microstructure permitted by the convex integration machinery.\n\n\\subsection*{Strategy and the Key Innovation}\nOur strategy builds directly on the framework of \\cite{CVY} and proceeds in three conceptual stages. First, from a given incompressible solution ${\\bf u}$, we construct a smooth incompressible subsolution $({\\bf u}^{\\varepsilon}, \\pi^{\\varepsilon}, R^{\\varepsilon})$ to a relaxed system, with a positive definite Reynolds stress $R^{\\varepsilon} > 0$. Second, we design a corresponding compressible subsolution $(\\rho_\\delta, V_\\delta, R_\\delta)$ to \\eqref{scalling system 111} that approximates the incompressible one, ensuring $\\rho_\\delta \\sim 1$ and preserving the crucial property $R_\\delta > 0$. Third, we apply the convex integration scheme of \\cite{CVY} to this subsolution to generate infinitely many true weak solutions $(\\rho_\\delta, \\widehat{V}_\\delta)$ of \\eqref{scalling system 111}. \n\nThe central new challenge, absent in \\cite{CVY}, is the requirement of uniform approximation: the constructed compressible solutions must converge strongly to ${\\bf u}$ as $\\delta \\to 0$. This demand for controlled asymptotics forces a critical refinement of the convex integration technique. Our principal innovation is the introduction of a new $L^2$-type constraint into the non-constructive $L^\\infty$ convex integration scheme. We define the set of admissible perturbations $X_0$ to include only those $\\widetilde{V}$ satisfying\n\\begin{equation}\\label{eq new constr}\n\\left| \\int \\frac{V_\\delta \\cdot \\widetilde{V}}{\\rho_\\delta} \\,dxdt \\right| \\le \\frac18 \\int \\frac{|\\widetilde{V}|^2}{\\rho_\\delta}\\,dxdt,\n\\end{equation}\nwhere $(\\rho_\\delta, V_\\delta, R_\\delta)$ is the underlying subsolution. This constraint is the key that unlocks a uniform energy estimate. It enables us to prove that the kinetic energy of the perturbations is controlled by the trace of the Reynolds stress:\n\\begin{equation}\\label{eq unif}\n\\int \\frac{|\\widetilde{V}|^2}{\\rho_\\delta}\\,dxdt \\le C \\int \\mathrm{tr} R_\\delta \\,dxdt.\n\\end{equation}\nSince our subsolution construction ensures that $\\mathrm{tr} R_\\delta$ remains $O(\\delta)$, this estimate \ntames the turbulent oscillations uniformly with respect to the Mach number, ensuring their vanishing in the limit and thereby securing the desired strong convergence.\n\nSuch a control of the {\\it turbulent} perturbations by the Reynolds tensor comes naturally when considering the $C^\\alpha$ theory of convex integration (the constructivist theory). It was already used, in this context, to obtain the inviscid limit of the incompressible Navier--Stokes equation for instance (see \\cite{Annals}). However, to our knowledge, it was not yet derived for the $L^\\infty$ theory of convex integration (using the Bair\\'e category theorem). \n\n\\subsection*{Main Result}\nEmploying this convex integration strategy, we prove that the incompressible system is indeed a universal attractor within this framework. For the precise notions of weak solutions to the incompressible Euler system \\eqref{incom Euler} and the rescaled compressible Euler system \\eqref{scalling system 111}, see the definitions below.\n\\begin{Definition}\n\\label{definition of weak solution incom}\nWe say ${\\bf u} \\in L^{\\infty}(0,T;L^2(\\mathbb{T}^n)) $ is a weak solution of \\eqref{incom Euler} with intial data ${\\bf u}^0 \\in L^2$ if it is divergence-free in the sense of distribution and \n\\begin{itemize}\n\\item for any $\\phi \\in C^\\infty_c(\\mathbb{T}^n \\times [0, T); {\\mathbb R}^n)$ with ${\\rm div} \\, \\phi = 0$,\n\\begin{equation}\\label{weak incomp euler eqn}\n\\int^\\infty_0 \\int_{\\mathbb{T}^n} {\\bf u} \\cdot \\partial_t \\phi + {\\bf u} \\otimes {\\bf u} : \\nabla \\phi \\,dxdt = -\\int_{\\mathbb{T}^n} {\\bf u}^0 \\cdot \\phi(\\cdot, 0) \\,dx.\n\\end{equation}\n\\end{itemize}\n\\end{Definition}\n\n\\begin{Definition}\n\\label{definition of weak solutions compressible} \nFor $\\gamma>1$, we say $(\\rho_{\\delta}, \\rho_{\\delta} \\v_{\\delta})$ is a weak solution of \\eqref{scalling system 111} with initial value $(\\rho^0_{\\delta},\\m_{\\delta}^0) \\in L^\\gamma \\times L^2$ if the following holds true.\n\\begin{itemize}\n\\item $\\rho_{\\delta}\\in L^{\\infty}(0,T;L^{\\gamma}(\\mathbb{T}^n)), \\;\\;\\sqrt{\\rho_{\\delta}} \\v_{\\delta}\\in L^{\\infty}(0,T;L^2(\\mathbb{T}^n))$;\n\\item for any $\\varphi \\in C^\\infty_c(\\mathbb{T}^3 \\times [0, T))$,\n\\[\n\\int_0^\\infty \\int_{\\mathbb{T}^n} \\left( \\rho_{\\delta} \\, \\partial_t \\varphi + \\rho_{\\delta} \\v_{\\delta} \\cdot \\nabla \\varphi \\right) \\, dx \\, dt + \\int_{\\mathbb{T}^n} \\rho_{\\delta}^0(x) \\varphi(0,x) \\, dx = 0;\n\\]\n\n \\item for any $\\phi \\in C^\\infty_c(\\mathbb{T}^n \\times [0, T); {\\mathbb R}^n)$,\n\\begin{equation*}\n\\begin{split}\n\\int^\\infty_0 \\int_{\\mathbb{T}^n} \\rho_{\\delta} \\v_{\\delta} \\cdot \\partial_t \\phi &+ \\rho_{\\delta} \\v_{\\delta} \\otimes \\v_{\\delta} : \\nabla \\phi \\,dxdt +\\int_0^\\infty\\int_{\\mathbb{T}^n}\\frac{\\rho_{\\delta}^{\\gamma}}{\\delta^2}{\\rm div} \\phi\\,dx\\,dt\n\\\\&= -\\int_{\\mathbb{T}^n}\\m_{\\delta}^0 \\cdot \\phi(\\cdot, 0) \\,dx.\n\\end{split}\n\\end{equation*}\n\\end{itemize}\n\\end{Definition}\n\n \\smallskip\n\nOur main result is stated as follows.", "context": "The incompressible Euler equations are universally regarded as the formal low Mach number limit of the compressible Euler equations. This physical premise is rooted in the observation that when fluid velocities are much smaller than the speed of sound, density variations become negligible, and the flow behaves as if it were incompressible \\cite{KM2, Majda, MaS}. The mathematical theory of this singular limit aims to rigorously justify the transition from solutions of the compressible Euler system\n\\begin{equation} \n\\label{compressible Euler}\n\\left\\{\n\\begin{split}\n&\\partial_t \\rho + {\\rm div} (\\rho {\\bf v})= 0,\n\\\\&\\partial_t (\\rho {\\bf v}) + {\\rm div} (\\rho {\\bf v} \\otimes {\\bf v}) + \\nabla P = 0,\n\\end{split}\n\\right.\n\\end{equation} \nwhere $\\rho$ is the density, ${\\bf v}$ the velocity, and $P(\\rho) = a \\rho^\\gamma$ the pressure ($a > 0$, $\\gamma > 1$), to those of the incompressible Euler system\n\\begin{equation}\\label{incom Euler} \n\\left\\{\n\\begin{split}&\n\\partial_t {\\bf u} + {\\bf u}\\cdot\\nabla{\\bf u}+\\nabla \\pi=0,\n\\\\& {\\rm div} {\\bf u}=0.\n\\end{split}\\right.\n\\end{equation}\nTo capture this low Mach number regime, one introduces a scaling where time is slowed down and velocity is rescaled by a small parameter\n\\[\n{\\rho}(x,t) =: \\rho_\\delta(x,\\delta t), \\quad {{\\bf v}}(x,t) =: \\delta \\v_\\delta(x,\\delta t).\n\\]\nThis leads from \\eqref{compressible Euler} to the rescaled compressible Euler equations\n\\begin{equation}\n\\label{scalling system 111}\n\\left\\{\n\\begin{split}\n&\\partial_t\\rho_{\\delta}+{\\rm div}(\\rho_{\\delta} \\v_{\\delta})=0,\n\\\\&\\partial_t(\\rho_{\\delta} \\v_{\\delta})+{\\rm div}(\\rho_{\\delta} \\v_{\\delta}\\otimes \\v_{\\delta})+\\nabla\\frac{\\rho_{\\delta}^\\gamma}{\\delta^2}=0.\n\\end{split}\n\\right.\n\\end{equation} \nFormally, as $\\delta \\to 0$, we expect $\\rho_\\delta$ to converge to a constant (say, $1$), and the velocity field to become divergence-free, recovering \\eqref{incom Euler}.\n\nThe central new challenge, absent in \\cite{CVY}, is the requirement of uniform approximation: the constructed compressible solutions must converge strongly to ${\\bf u}$ as $\\delta \\to 0$. This demand for controlled asymptotics forces a critical refinement of the convex integration technique. Our principal innovation is the introduction of a new $L^2$-type constraint into the non-constructive $L^\\infty$ convex integration scheme. We define the set of admissible perturbations $X_0$ to include only those $\\widetilde{V}$ satisfying\n\\begin{equation}\\label{eq new constr}\n\\left| \\int \\frac{V_\\delta \\cdot \\widetilde{V}}{\\rho_\\delta} \\,dxdt \\right| \\le \\frac18 \\int \\frac{|\\widetilde{V}|^2}{\\rho_\\delta}\\,dxdt,\n\\end{equation}\nwhere $(\\rho_\\delta, V_\\delta, R_\\delta)$ is the underlying subsolution. This constraint is the key that unlocks a uniform energy estimate. It enables us to prove that the kinetic energy of the perturbations is controlled by the trace of the Reynolds stress:\n\\begin{equation}\\label{eq unif}\n\\int \\frac{|\\widetilde{V}|^2}{\\rho_\\delta}\\,dxdt \\le C \\int \\mathrm{tr} R_\\delta \\,dxdt.\n\\end{equation}\nSince our subsolution construction ensures that $\\mathrm{tr} R_\\delta$ remains $O(\\delta)$, this estimate \ntames the turbulent oscillations uniformly with respect to the Mach number, ensuring their vanishing in the limit and thereby securing the desired strong convergence.\n\n\\subsection*{Main Result}\nEmploying this convex integration strategy, we prove that the incompressible system is indeed a universal attractor within this framework. For the precise notions of weak solutions to the incompressible Euler system \\eqref{incom Euler} and the rescaled compressible Euler system \\eqref{scalling system 111}, see the definitions below.\n\\begin{Definition}\n\\label{definition of weak solution incom}\nWe say ${\\bf u} \\in L^{\\infty}(0,T;L^2(\\mathbb{T}^n)) $ is a weak solution of \\eqref{incom Euler} with intial data ${\\bf u}^0 \\in L^2$ if it is divergence-free in the sense of distribution and \n\\begin{itemize}\n\\item for any $\\phi \\in C^\\infty_c(\\mathbb{T}^n \\times [0, T); {\\mathbb R}^n)$ with ${\\rm div} \\, \\phi = 0$,\n\\begin{equation}\\label{weak incomp euler eqn}\n\\int^\\infty_0 \\int_{\\mathbb{T}^n} {\\bf u} \\cdot \\partial_t \\phi + {\\bf u} \\otimes {\\bf u} : \\nabla \\phi \\,dxdt = -\\int_{\\mathbb{T}^n} {\\bf u}^0 \\cdot \\phi(\\cdot, 0) \\,dx.\n\\end{equation}\n\\end{itemize}\n\\end{Definition}\n\n\\begin{Definition}\n\\label{definition of weak solutions compressible} \nFor $\\gamma>1$, we say $(\\rho_{\\delta}, \\rho_{\\delta} \\v_{\\delta})$ is a weak solution of \\eqref{scalling system 111} with initial value $(\\rho^0_{\\delta},\\m_{\\delta}^0) \\in L^\\gamma \\times L^2$ if the following holds true.\n\\begin{itemize}\n\\item $\\rho_{\\delta}\\in L^{\\infty}(0,T;L^{\\gamma}(\\mathbb{T}^n)), \\;\\;\\sqrt{\\rho_{\\delta}} \\v_{\\delta}\\in L^{\\infty}(0,T;L^2(\\mathbb{T}^n))$;\n\\item for any $\\varphi \\in C^\\infty_c(\\mathbb{T}^3 \\times [0, T))$,\n\\[\n\\int_0^\\infty \\int_{\\mathbb{T}^n} \\left( \\rho_{\\delta} \\, \\partial_t \\varphi + \\rho_{\\delta} \\v_{\\delta} \\cdot \\nabla \\varphi \\right) \\, dx \\, dt + \\int_{\\mathbb{T}^n} \\rho_{\\delta}^0(x) \\varphi(0,x) \\, dx = 0;\n\\]\n\n\\smallskip\n\nOur main result is stated as follows.", "full_context": "The incompressible Euler equations are universally regarded as the formal low Mach number limit of the compressible Euler equations. This physical premise is rooted in the observation that when fluid velocities are much smaller than the speed of sound, density variations become negligible, and the flow behaves as if it were incompressible \\cite{KM2, Majda, MaS}. The mathematical theory of this singular limit aims to rigorously justify the transition from solutions of the compressible Euler system\n\\begin{equation} \n\\label{compressible Euler}\n\\left\\{\n\\begin{split}\n&\\partial_t \\rho + {\\rm div} (\\rho {\\bf v})= 0,\n\\\\&\\partial_t (\\rho {\\bf v}) + {\\rm div} (\\rho {\\bf v} \\otimes {\\bf v}) + \\nabla P = 0,\n\\end{split}\n\\right.\n\\end{equation} \nwhere $\\rho$ is the density, ${\\bf v}$ the velocity, and $P(\\rho) = a \\rho^\\gamma$ the pressure ($a > 0$, $\\gamma > 1$), to those of the incompressible Euler system\n\\begin{equation}\\label{incom Euler} \n\\left\\{\n\\begin{split}&\n\\partial_t {\\bf u} + {\\bf u}\\cdot\\nabla{\\bf u}+\\nabla \\pi=0,\n\\\\& {\\rm div} {\\bf u}=0.\n\\end{split}\\right.\n\\end{equation}\nTo capture this low Mach number regime, one introduces a scaling where time is slowed down and velocity is rescaled by a small parameter\n\\[\n{\\rho}(x,t) =: \\rho_\\delta(x,\\delta t), \\quad {{\\bf v}}(x,t) =: \\delta \\v_\\delta(x,\\delta t).\n\\]\nThis leads from \\eqref{compressible Euler} to the rescaled compressible Euler equations\n\\begin{equation}\n\\label{scalling system 111}\n\\left\\{\n\\begin{split}\n&\\partial_t\\rho_{\\delta}+{\\rm div}(\\rho_{\\delta} \\v_{\\delta})=0,\n\\\\&\\partial_t(\\rho_{\\delta} \\v_{\\delta})+{\\rm div}(\\rho_{\\delta} \\v_{\\delta}\\otimes \\v_{\\delta})+\\nabla\\frac{\\rho_{\\delta}^\\gamma}{\\delta^2}=0.\n\\end{split}\n\\right.\n\\end{equation} \nFormally, as $\\delta \\to 0$, we expect $\\rho_\\delta$ to converge to a constant (say, $1$), and the velocity field to become divergence-free, recovering \\eqref{incom Euler}.\n\nThe central new challenge, absent in \\cite{CVY}, is the requirement of uniform approximation: the constructed compressible solutions must converge strongly to ${\\bf u}$ as $\\delta \\to 0$. This demand for controlled asymptotics forces a critical refinement of the convex integration technique. Our principal innovation is the introduction of a new $L^2$-type constraint into the non-constructive $L^\\infty$ convex integration scheme. We define the set of admissible perturbations $X_0$ to include only those $\\widetilde{V}$ satisfying\n\\begin{equation}\\label{eq new constr}\n\\left| \\int \\frac{V_\\delta \\cdot \\widetilde{V}}{\\rho_\\delta} \\,dxdt \\right| \\le \\frac18 \\int \\frac{|\\widetilde{V}|^2}{\\rho_\\delta}\\,dxdt,\n\\end{equation}\nwhere $(\\rho_\\delta, V_\\delta, R_\\delta)$ is the underlying subsolution. This constraint is the key that unlocks a uniform energy estimate. It enables us to prove that the kinetic energy of the perturbations is controlled by the trace of the Reynolds stress:\n\\begin{equation}\\label{eq unif}\n\\int \\frac{|\\widetilde{V}|^2}{\\rho_\\delta}\\,dxdt \\le C \\int \\mathrm{tr} R_\\delta \\,dxdt.\n\\end{equation}\nSince our subsolution construction ensures that $\\mathrm{tr} R_\\delta$ remains $O(\\delta)$, this estimate \ntames the turbulent oscillations uniformly with respect to the Mach number, ensuring their vanishing in the limit and thereby securing the desired strong convergence.\n\n\\subsection*{Main Result}\nEmploying this convex integration strategy, we prove that the incompressible system is indeed a universal attractor within this framework. For the precise notions of weak solutions to the incompressible Euler system \\eqref{incom Euler} and the rescaled compressible Euler system \\eqref{scalling system 111}, see the definitions below.\n\\begin{Definition}\n\\label{definition of weak solution incom}\nWe say ${\\bf u} \\in L^{\\infty}(0,T;L^2(\\mathbb{T}^n)) $ is a weak solution of \\eqref{incom Euler} with intial data ${\\bf u}^0 \\in L^2$ if it is divergence-free in the sense of distribution and \n\\begin{itemize}\n\\item for any $\\phi \\in C^\\infty_c(\\mathbb{T}^n \\times [0, T); {\\mathbb R}^n)$ with ${\\rm div} \\, \\phi = 0$,\n\\begin{equation}\\label{weak incomp euler eqn}\n\\int^\\infty_0 \\int_{\\mathbb{T}^n} {\\bf u} \\cdot \\partial_t \\phi + {\\bf u} \\otimes {\\bf u} : \\nabla \\phi \\,dxdt = -\\int_{\\mathbb{T}^n} {\\bf u}^0 \\cdot \\phi(\\cdot, 0) \\,dx.\n\\end{equation}\n\\end{itemize}\n\\end{Definition}\n\n\\begin{Definition}\n\\label{definition of weak solutions compressible} \nFor $\\gamma>1$, we say $(\\rho_{\\delta}, \\rho_{\\delta} \\v_{\\delta})$ is a weak solution of \\eqref{scalling system 111} with initial value $(\\rho^0_{\\delta},\\m_{\\delta}^0) \\in L^\\gamma \\times L^2$ if the following holds true.\n\\begin{itemize}\n\\item $\\rho_{\\delta}\\in L^{\\infty}(0,T;L^{\\gamma}(\\mathbb{T}^n)), \\;\\;\\sqrt{\\rho_{\\delta}} \\v_{\\delta}\\in L^{\\infty}(0,T;L^2(\\mathbb{T}^n))$;\n\\item for any $\\varphi \\in C^\\infty_c(\\mathbb{T}^3 \\times [0, T))$,\n\\[\n\\int_0^\\infty \\int_{\\mathbb{T}^n} \\left( \\rho_{\\delta} \\, \\partial_t \\varphi + \\rho_{\\delta} \\v_{\\delta} \\cdot \\nabla \\varphi \\right) \\, dx \\, dt + \\int_{\\mathbb{T}^n} \\rho_{\\delta}^0(x) \\varphi(0,x) \\, dx = 0;\n\\]\n\n\\smallskip\n\nOur main result is stated as follows.\n\n\\begin{Definition}\n\\label{definition of weak solutions compressible} \nFor $\\gamma>1$, we say $(\\rho_{\\delta}, \\rho_{\\delta} \\v_{\\delta})$ is a weak solution of \\eqref{scalling system 111} with initial value $(\\rho^0_{\\delta},\\m_{\\delta}^0) \\in L^\\gamma \\times L^2$ if the following holds true.\n\\begin{itemize}\n\\item $\\rho_{\\delta}\\in L^{\\infty}(0,T;L^{\\gamma}(\\mathbb{T}^n)), \\;\\;\\sqrt{\\rho_{\\delta}} \\v_{\\delta}\\in L^{\\infty}(0,T;L^2(\\mathbb{T}^n))$;\n\\item for any $\\varphi \\in C^\\infty_c(\\mathbb{T}^3 \\times [0, T))$,\n\\[\n\\int_0^\\infty \\int_{\\mathbb{T}^n} \\left( \\rho_{\\delta} \\, \\partial_t \\varphi + \\rho_{\\delta} \\v_{\\delta} \\cdot \\nabla \\varphi \\right) \\, dx \\, dt + \\int_{\\mathbb{T}^n} \\rho_{\\delta}^0(x) \\varphi(0,x) \\, dx = 0;\n\\]\n\n\\smallskip\n\nThis theorem establishes a rigorous density result: the set of incompressible solutions attainable as low-Mach limits of convex integration solutions is the \\emph{entire} $L^2$-space. It affirms that any large-scale, divergence-free flow field can emerge as the effective, averaged description of a compressible fluid in the low Mach number regime, irrespective of the fine-scale, non-unique oscillatory dynamics modeled by convex integration. This provides a mathematical justification for the robustness of the incompressible approximation, confirming the physical intuition formalized in \\cite{KM2, Majda, MaS}.\n\nOnce this lemma is established, we are ready to construct a subsolution to the compressible Euler equations \\eqref{scalling system 111}. For $\\varepsilon, \\delta, K > 0$, define\n\\begin{subequations}\\label{eq varrho}\n\\begin{equation}\\label{eq def varrho}\n\\varrho_{\\delta,K}(t,x) := \\frac{(\\delta^2 \\pi^\\varepsilon + K)^{1/\\gamma} - 1}{\\delta^2},\n\\end{equation}\nwhere $\\pi^\\varepsilon$ is from Lemma \\ref{incom smooth}. Since $\\pi^\\varepsilon \\in C^\\infty([0, T-\\varepsilon] \\times \\T^n)$, we know that for any $\\delta > 0$ there exists a sufficiently large $K > 0$ such that $\\delta^2 \\pi^\\varepsilon + K > 0$, and hence $\\varrho_{\\delta, K} \\in C^\\infty([0, T-\\varepsilon] \\times \\T^n)$. Considering $\\delta$ sufficiently small such that $\\| \\delta^2 \\pi^\\eps \\|_{L^\\infty} \\ll1$, among those choices of $K$, a simple intermediate value theorem further implies that for any $t \\in [0, T-\\varepsilon]$ there exists some $K_\\ast = K_\\ast(\\delta, t) > 0$ such that\n\\begin{equation}\\label{eq ave varrho}\nnt_{\\T^n} \\varrho_{\\delta, K_\\ast} \\,dx = 0. \n\\end{equation}\n\\end{subequations}\n\\begin{Lemma}\\label{lem subsoln}\n\\label{subsolution}\nFor any $\\varepsilon>0$, consider $(\\u^\\varepsilon, \\pi^\\varepsilon, R^\\varepsilon)$ as in Lemma \\ref{incom smooth}. There exists a $\\delta_0 = \\delta_0(\\eps) >0$ such that for any $\\delta\\leq \\delta_0$, there exist $\\v_\\delta \\in C^{\\infty}([0,T-\\eps]\\times \\mathbb{T}^n)$ and $\\tilde{R}_{\\delta} \\in C^\\infty([0, T - \\varepsilon] \\times \\mathbb{T}^n; \\mathbb{S}^{n \\times n})$ with $\\tilde{R}_{\\delta} > 0$, solving \n\\begin{equation}\\label{eq subsoln}\n\\left\\{\n\\begin{split}\n&\\partial_t\\rho_{\\delta}+\\Dv(\\rho_{\\delta} \\v_{\\delta})=0,\n\\\\&\\partial_t(\\rho_{\\delta} \\v_{\\delta})+\\Dv(\\rho_{\\delta} \\v_{\\delta}\\otimes \\v_{\\delta})+\\nabla\\frac{\\rho_{\\delta}^\\gamma}{\\delta^2}+\\Dv(R^{\\varepsilon}+\\tilde{R}_{\\delta})=0,\n\\end{split}\\right.\n\\end{equation}\nwhere \n\\begin{equation}\\label{eq def rho}\n\\rho_\\delta(t,x) := 1 + \\delta^2 \\varrho_{\\delta, K_\\ast}(t,x) \n\\end{equation}\nwith $\\varrho_{\\delta, K_\\ast}$ being given in \\eqref{eq varrho}. Moreover we have\n\\begin{equation}\\label{eq subsoln est}\n\\begin{split}\n\\|\\rho_{\\delta}-1\\|_{C^0([0,T-\\varepsilon] \\times\\mathbb{T}^n)} + \\|\\rho_{\\delta}\\v_{\\delta}-\\u^{\\varepsilon}\\|_{L^2([0,T-\\varepsilon] \\times \\mathbb{T}^n)} + \\|\\tilde{R_{\\delta}}\\|_{L^\\infty([0,T-\\varepsilon] \\times \\mathbb{T}^n)}\\leq \\varepsilon.\n\\end{split}\n\\end{equation}\n\\end{Lemma}\n\\begin{proof}\nFrom \\eqref{eq varrho} and \\eqref{eq def rho} we have \n\\[\n\\rho_\\delta = 1 + \\delta^2 \\varrho_{\\delta, K_\\ast} = (\\delta^2 \\pi^\\varepsilon + K_\\ast)^{1/\\gamma}.\n\\]\nSo for $\\delta$ sufficiently small we know that $\\rho_\\delta > 0$, and \n\\begin{equation}\\label{eq prop rho}\n\\rho_\\delta \\in C^\\infty([0, T-\\varepsilon] \\times \\T^n), \\quad \\left\\langle \\rho_\\delta \\right\\rangle = 1, \\quad \\text{and} \\quad \\|\\rho_{\\delta} - K_\\ast^{1/\\gamma}\\|_{C^0([0,T-\\varepsilon] \\times\\mathbb{T}^n)} \\le C(\\varepsilon) \\delta^2.\n\\end{equation}\nThe time derivative of $ \\rho_{\\delta} $ is given by\n\\[\n\\partial_t \\rho_{\\delta} = \\frac{1}{\\gamma} \\left(\\delta^2 \\pi^{\\varepsilon} + K_\\ast \\right)^{\\frac{1}{\\gamma} - 1} (\\delta^2 \\partial_t \\pi^{\\varepsilon} + \\partial_t K_\\ast).\n\\]\n\nTo prove our main theorem, we will rely on the following lemma.\n\\begin{Lemma}\n\\label{global existence}\nFor any $\\eps > 0$, there exists a $\\delta_0 = \\delta_0(\\eps) > 0$ such that for any $0 < \\delta \\le \\delta_0$, there exist infinitely many weak solutions $(\\rho_{\\delta},\\rho_{\\delta}\\widehat{\\v}_{\\delta})$ to \\eqref{scalling system 111}on $[0, T- \\eps]$, satisfying \n$$\n\\|\\widehat{\\v}_{\\delta} - \\v_{\\delta}\\|_{L^2([0,T - \\eps] \\times \\mathbb{T}^n)}^2\\leq C \\varepsilon,\n$$\nwhere $(\\rho_\\delta, \\rho_\\delta \\v_\\delta, R = R^\\varepsilon + \\tilde{R}_\\delta)$ is the subsolution solving \\eqref{eq subsoln}. \n\\end{Lemma}\n\\begin{proof}\nRecalling Lemma \\ref{subsolution}, we have\n\\[\n\\rho_{\\delta} \\v_{\\delta} = \\m_{\\delta} + \\u^{\\varepsilon}, \\quad \\text{and} \\quad \\rho_\\delta(t,x) = 1 + \\delta^2 \\varrho_{\\delta, K_\\ast}(t,x), \n\\]\nwhere $\\varrho_{\\delta, K_\\ast}$ is given in \\eqref{eq varrho}. From Lemma \\ref{lem conv int} it follows that there exist infinitely many weak solutions to \\eqref{scalling system 111} of the form $({\\rho}_{\\delta}, {\\rho}_{\\delta} \\widehat{\\v}_{\\delta})$.\n\nEquipped with the preceding lemmas, we are now prepared to establish our main result. \n\\begin{proof}[Proof of Theorem \\ref{main result}]\nGiven any $L^2$-bounded weak solution $\\u$ to the incompressible Euler equations \\eqref{incom Euler} on the time interval $[0, T)$, for any $0 < T' < T$ there exists $\\eps > 0$ with $T' < T - \\eps$. We apply Lemma \\ref{smooth euler}, Lemma \\ref{subsolution}, and Lemma \\ref{global existence} to construct a family of weak solutions $(\\rho_{\\delta},\\rho_{\\delta}\\widehat{\\v}_{\\delta})$ to the scaled compressible Euler equations \\eqref{scalling system 111} on the time interval $[0, T-\\eps]$, and hence on $[0, T']$. Note that here $\\delta = \\delta(\\eps)$ with the property that $\\delta \\to 0 \\ \\Leftrightarrow \\ \\eps \\to 0$. \nWe can take the following initial data \n$$\n(\\rho_{\\delta}^0, \\m_\\delta^0)(x) := (\\rho_{\\delta}, \\rho_{\\delta}\\widehat{\\v}_{\\delta})(0,x).\n$$\n\nFor any $\\phi \\in C^\\infty_c([0, T) \\times \\mathbb{T}^n; \\R^n)$ with $\\Dv\\phi=0$, say the time support of $\\phi$ is in $[0, T']$ for some $T' < T$. Then\n\\begin{equation*}\n\\begin{split}\n\\int^{T'}_0 \\int_{\\mathbb{T}^3} \\rho_{\\delta}\\widehat{\\v}_{\\delta} \\cdot \\partial_t \\phi + \\sqrt{\\rho_{\\delta}}\\widehat{\\v}_{\\delta} \\otimes\\sqrt{\\rho_{\\delta}}\\widehat{\\v}_{\\delta} : \\nabla \\phi \\,dxdt +\\int_0^{T'} \\int_{\\mathbb{T}^3}\\frac{\\rho_{\\delta}^{\\gamma}}{\\delta^2}\\Dv \\phi\\,dx\\,dt\n= -\\int_{\\mathbb{T}^3} \\m_{\\delta}^0 \\cdot \\phi(\\cdot, 0) \\,dx,\n\\end{split}\n\\end{equation*}\nUsing the convergence results in the above, we find that the left-hand side of the above equality converges, as $\\delta \\to 0$, to\n$$\n\\int^{T'}_0 \\int_{\\mathbb{T}^3} \\u \\cdot \\partial_t \\phi + \\u \\otimes \\u : \\nabla \\phi \\,dxdt, \n$$\nwhich, recalling the fact that $\\u$ is a weak solution to the incompressible Euler equation, is equal to\n\\[\n-\\int_{\\mathbb{T}^3} \\u^0 \\cdot \\phi(\\cdot, 0) \\,dx.\n\\]\nTherefore, we must have \n$$\n\\int_{\\mathbb{T}^3} \\m_{\\delta}^0 \\cdot \\phi(\\cdot, 0) \\,dx\\to \\int_{\\mathbb{T}^3} \\u^0 \\cdot \\phi(\\cdot, 0) \\,dx \\qquad \\text{as }\\ \\delta \\to 0.\n$$\nPutting together all of the above, we have proved the main result, Theorem \\ref{main result}.\n\\end{proof}", "post_theorem_intro_text_len": 2006, "post_theorem_intro_text": "\\begin{Remark}[On the adiabatic exponent $\\gamma$]\nThe convex integration construction in Theorem \\ref{main result} is valid for all $\\gamma > 1$. This is in contrast to the result in \\cite[Theorem 1.1]{CVY}, where the constraint $1 < \\gamma \\le 1 + \\tfrac2n$ arises from the requirement that the constructed solutions satisfy the energy inequality. Since the present work does not impose the energy condition, our result holds in the full range $\\gamma > 1$. \n\\end{Remark}\n\nThis theorem establishes a rigorous density result: the set of incompressible solutions attainable as low-Mach limits of convex integration solutions is the \\emph{entire} $L^2$-space. It affirms that any large-scale, divergence-free flow field can emerge as the effective, averaged description of a compressible fluid in the low Mach number regime, irrespective of the fine-scale, non-unique oscillatory dynamics modeled by convex integration. This provides a mathematical justification for the robustness of the incompressible approximation, confirming the physical intuition formalized in \\cite{KM2, Majda, MaS}.\n\n\\subsection*{Outline of the Paper}\nThe rest of the paper is structured as follows. In Section \\ref{sec sub}, we detail the construction of subsolutions, from the regularization of the incompressible solution (Lemma \\ref{incom smooth}) to its lifting to a compressible subsolution for \\eqref{scalling system 111} (Lemma \\ref{lem subsoln}). Section \\ref{sec ci} is dedicated to the convex integration scheme. We present the core discretization and $L^1$-coercivity argument under the new constraint \\eqref{eq new constr} (Lemma \\ref{L1-coercivity}), prove the uniform estimate \\eqref{eq unif} (Proposition \\ref{proposition points}), and establish the existence of infinitely many solutions via a Bair\\'e category argument (Lemma \\ref{lem conv int}). Finally, in Section \\ref{sec pf}, we put these elements together to prove Theorem \\ref{main result}, confirming the convergence in the low Mach number limit.\n\n\\bigskip", "sketch": "To prove Theorem~\\ref{main result}, the paper proceeds as follows: in Section~\\ref{sec sub} it constructs subsolutions, starting from a \\emph{regularization of the incompressible solution} (Lemma~\\ref{incom smooth}) and then \\emph{lifting it to a compressible subsolution} for \\eqref{scalling system 111} (Lemma~\\ref{lem subsoln}). Section~\\ref{sec ci} carries out the \\emph{convex integration scheme}: it gives the \\emph{core discretization and $L^1$-coercivity argument} under the new constraint \\eqref{eq new constr} (Lemma~\\ref{L1-coercivity}), proves the \\emph{uniform estimate} \\eqref{eq unif} (Proposition~\\ref{proposition points}), and then obtains \\emph{infinitely many solutions} by a \\emph{Bair\\'e category argument} (Lemma~\\ref{lem conv int}). Finally, Section~\\ref{sec pf} \\emph{puts these elements together} to prove Theorem~\\ref{main result}, in particular \\emph{confirming the convergence in the low Mach number limit}.", "expanded_sketch": "To prove the main theorem, the paper proceeds as follows: next it constructs subsolutions, starting from a \\emph{regularization of the incompressible solution}. We first recall the following lemma.\n\n\\begin{Lemma}\n\\label{incom smooth} \nFor any weak solutions $\\u$ to the incompressible Euler equations \\eqref{incom Euler} and any $\\eps > 0$,\nthere exist $\\u^{\\varepsilon}, \\pi^\\varepsilon \\in C^{\\infty}([0,T-\\eps] \\times \\mathbb{T}^n)$ and $R^\\varepsilon \\in C^\\infty([0, T - \\varepsilon] \\times \\mathbb{T}^n; \\mathbb{S}^{n \\times n})$, where $\\mathbb{S}^{n \\times n}$ is the set of symmetric matrices in dimension $n$, such that\n\\begin{itemize}\n\\item\n$(\\u^{\\varepsilon}, \\pi^{\\varepsilon}, R^\\varepsilon)$ is a weak solution to \n\\begin{equation}\n\\label{smooth euler}\n\\left\\{\n\\begin{split}\n& \\partial_t \\u^{\\varepsilon} + \\u^{\\varepsilon}\\cdot\\nabla \\u^{\\varepsilon}+\\nabla \\pi^{\\varepsilon}+\\Dv R^{\\varepsilon}=0,\n\\\\& \\Dv {\\u^{\\varepsilon}}=0.\n\\end{split}\n\\right.\n\\end{equation}\n\\item $R^{\\varepsilon}>\\frac{\\varepsilon}{4} \\id_n$ is a symmetric matrix, where $\\id_n$ is the $n \\times n$ identity matrix.\n\\item There exists a constant $C>0$ independent of $\\eps$ such that\n\\begin{equation}\\label{eq reg est}\n\\|\\u^{\\varepsilon}-\\u\\|_{L^2([0,T - \\eps] \\times \\mathbb{T}^n)}+\\|R^{\\varepsilon}\\|_{L^1([0,T - \\eps] \\times \\mathbb{T}^n)}\\leq C \\varepsilon.\n\\end{equation}\n\\end{itemize}\n\\end{Lemma}\n\nIt then \\emph{lifts it to a compressible subsolution} for\n\\begin{equation}\n\\label{scalling system 111}\n\\left\\{\n\\begin{split}\n&\\partial_t\\rho_{\\delta}+\\Dv(\\rho_{\\delta} \\v_{\\delta})=0,\n\\\\&\\partial_t(\\rho_{\\delta} \\v_{\\delta})+\\Dv(\\rho_{\\delta} \\v_{\\delta}\\otimes \\v_{\\delta})+\\nabla\\frac{\\rho_{\\delta}^\\gamma}{\\delta^2}=0.\n\\end{split}\n\\right.\n\\end{equation}\nvia the following lemma.\n\n\\begin{Lemma}\\label{lem subsoln}\n\\label{subsolution}\nFor any $\\varepsilon>0$, consider $(\\u^\\varepsilon, \\pi^\\varepsilon, R^\\varepsilon)$ as in Lemma \\ref{incom smooth}. There exists a $\\delta_0 = \\delta_0(\\eps) >0$ such that for any $\\delta\\leq \\delta_0$, there exist $\\v_\\delta \\in C^{\\infty}([0,T-\\eps]\\times \\mathbb{T}^n)$ and $\\tilde{R}_{\\delta} \\in C^\\infty([0, T - \\varepsilon] \\times \\mathbb{T}^n; \\mathbb{S}^{n \\times n})$ with $\\tilde{R}_{\\delta} > 0$, solving \n\\begin{equation}\\label{eq subsoln}\n\\left\\{\n\\begin{split}\n&\\partial_t\\rho_{\\delta}+\\Dv(\\rho_{\\delta} \\v_{\\delta})=0,\n\\\\&\\partial_t(\\rho_{\\delta} \\v_{\\delta})+\\Dv(\\rho_{\\delta} \\v_{\\delta}\\otimes \\v_{\\delta})+\\nabla\\frac{\\rho_{\\delta}^\\gamma}{\\delta^2}+\\Dv(R^{\\varepsilon}+\\tilde{R}_{\\delta})=0,\n\\end{split}\\right.\n\\end{equation}\nwhere \n\\begin{equation}\\label{eq def rho}\n\\rho_\\delta(t,x) := 1 + \\delta^2 \\varrho_{\\delta, K_\\ast}(t,x) \n\\end{equation}\nwith $\\varrho_{\\delta, K_\\ast}$ being given in \\eqref{eq varrho}. Moreover we have\n\\begin{equation}\\label{eq subsoln est}\n\\begin{split}\n\\|\\rho_{\\delta}-1\\|_{C^0([0,T-\\varepsilon] \\times\\mathbb{T}^n)} + \\|\\rho_{\\delta}\\v_{\\delta}-\\u^{\\varepsilon}\\|_{L^2([0,T-\\varepsilon] \\times \\mathbb{T}^n)} + \\|\\tilde{R_{\\delta}}\\|_{L^\\infty([0,T-\\varepsilon] \\times \\mathbb{T}^n)}\\leq \\varepsilon.\n\\end{split}\n\\end{equation}\n\\end{Lemma}\n\nAfter that, it carries out the \\emph{convex integration scheme}: it gives the \\emph{core discretization and $L^1$-coercivity argument} under the new constraint\n\\begin{equation}\\label{eq new constr}\n\\left| \\int \\frac{V_\\delta \\cdot \\widetilde{V}}{\\rho_\\delta} \\,dxdt \\right| \\le \\frac18 \\int \\frac{|\\widetilde{V}|^2}{\\rho_\\delta}\\,dxdt,\n\\end{equation}\nusing the following lemma.\n\n\\begin{Lemma}[$L^1$-coercivity]\n\\label{L1-coercivity}\nFor any $(\\widetilde{V}, \\widetilde{U}) \\in X_0$, where $X_0$ is defined in \\eqref{defn X_0}, there exists a sequence $\\{(\\widetilde{V}_i, \\widetilde{U}_i)\\} \\subset X_0$ that converges weakly-$*$ to $(\\widetilde{V}, \\widetilde{U})$, such that\n$$\n\\|\\widetilde{V}_i - \\widetilde{V}\\|_{L^1(P)} \\geq \\frac{c_0}{\\Lambda} \\int_{P} \\trace \\, M \\, dx \\, dt,\n$$\nfor some positive geometric constant $c_0$, where $\\Lambda$ is given in \\eqref{bounds rho}.\n\\end{Lemma}\n\nIt then proves the \\emph{uniform estimate}\n\\begin{equation}\\label{eq unif}\n\\int \\frac{|\\widetilde{V}|^2}{\\rho_\\delta}\\,dxdt \\le C \\int \\trace R_\\delta \\,dxdt.\n\\end{equation}\nvia the following proposition.\n\n\\begin{Proposition}\n\\label{proposition points}\nLet $(\\widehat{V},\\widehat{U})\\in X$ be a point of continuity of the identity map $\\id$ from $(X,d)$ to $L^2(\\R^n\\times\\R)$. Then, $(\\widehat{V},\\widehat{U})$ satisfies \\eqref{restriction =}. In addition, there exists a positive number $C_0>0$, such that \n $$\n \\int_0^T\\int_{\\mathbb{T}^n}\\frac{|\\widehat{V}|^2}{\\rho_0}\\,dx\\,dt\\leq C_0\\int_0^T\\int_{\\mathbb{T}^n} \\trace R_0\\,dx\\,dt.\n $$\n\\end{Proposition}\n\nFinally, it obtains \\emph{infinitely many solutions} by a \\emph{Bair\\'e category argument} using the following lemma.\n\n\\begin{Lemma}\\label{lem conv int}\nLet $(\\rho_0, V_0, R_0) \\in C^0(\\R \\times \\R^n; \\R \\times \\R^n \\times \\mathbb{S}_0^{n\\times n})$\n be given with $\\rho_0$ satisfying \\eqref{bounds rho} and $R_0$ being positive definite in some open set $P \\subset \\R \\times \\R^n$. Let $U_0$ be given as in \\eqref{U0 condition}. There exist infinitely many $(\\widetilde{V},\\widetilde{U}) \\in L^\\infty(\\R \\times \\R^n; \\R^n \\times \\mathbb{S}_0^{n\\times n})$ which are compactly supported in $P$ and satisfy \\eqref{ci system} and \\eqref{restriction =}. In addition, there exists a positive number $C_0>0$, such that \n $$\\int_0^T\\int_{\\mathbb{T}^n}\\frac{|\\widetilde{V}|^2}{\\rho_0}\\,dx\\,dt\\leq C_0\\int_0^T\\int_{\\mathbb{T}^n} \\tr R_0\\,dx\\,dt.$$\n\\end{Lemma}\n\nIn establishing the main theorem, it finally \\emph{puts these elements together}, in particular \\emph{confirming the convergence in the low Mach number limit}.", "expanded_theorem": "\\label{main result}\nLet $n = 2$ or $3$ and $\\gamma > 1$. For any given $T > 0$, consider any weak solution ${\\bf u}\\in L^\\infty(0,T;L^2(\\mathbb{T}^n))$ to the incompressible Euler equation \n\\begin{equation}\\label{incom Euler} \n\\left\\{\n\\begin{split}&\n\\partial_t \\u + \\u\\cdot\\nabla\\u+\\nabla \\pi=0,\n\\\\& \\Dv \\u=0.\n\\end{split}\\right.\n\\end{equation}\nwith initial data ${\\bf u}^0\\in L^2(\\mathbb{T}^n)$, as defined in\n\\begin{Definition}\n\\label{definition of weak solution incom}\nWe say $\\u \\in L^{\\infty}(0,T;L^2(\\mathbb{T}^n)) $ is a weak solution of \\eqref{incom Euler} with intial data $\\u^0 \\in L^2$ if it is divergence-free in the sense of distribution and \n\\begin{itemize}\n\\item for any $\\phi \\in C^\\infty_c(\\mathbb{T}^n \\times [0, T); \\R^n)$ with $\\Dv \\, \\phi = 0$,\n\\begin{equation}\\label{weak incomp euler eqn}\n\\int^\\infty_0 \\int_{\\mathbb{T}^n} \\u \\cdot \\partial_t \\phi + \\u \\otimes \\u : \\nabla \\phi \\,dxdt = -\\int_{\\mathbb{T}^n} \\u^0 \\cdot \\phi(\\cdot, 0) \\,dx.\n\\end{equation}\n\\end{itemize}\n\\end{Definition}\nThen for any $T' < T$, \n there exists a small number $\\delta_0>0$ such that for any Mach number $\\delta<\\delta_0$, there exists a family of initial data \n$$\n(\\rho_{\\delta}^0,\\m_{\\delta}^0)\\in C^0(\\mathbb{T}^n)\\times L^2(\\mathbb{T}^n)\n$$\nwith the following properties.\n\\begin{itemize}\n\\item For any fixed $\\delta>0$, there exist infinitely many weak solutions $(\\rho_{\\delta},\\rho_{\\delta}\\widehat{{\\bf v}}_{\\delta})$ on $(0, T')$ to the rescaled compressible Euler equations\n\\begin{equation}\n\\label{scalling system 111}\n\\left\\{\n\\begin{split}\n&\\partial_t\\rho_{\\delta}+\\Dv(\\rho_{\\delta} \\v_{\\delta})=0,\n\\\\&\\partial_t(\\rho_{\\delta} \\v_{\\delta})+\\Dv(\\rho_{\\delta} \\v_{\\delta}\\otimes \\v_{\\delta})+\\nabla\\frac{\\rho_{\\delta}^\\gamma}{\\delta^2}=0.\n\\end{split}\n\\right.\n\\end{equation}\nemanating from the above data $(\\rho_{\\delta}^0,\\m_{\\delta}^0)$, in the sense of\n\\begin{Definition}\n\\label{definition of weak solutions compressible} \nFor $\\gamma>1$, we say $(\\rho_{\\delta}, \\rho_{\\delta} \\v_{\\delta})$ is a weak solution of \\eqref{scalling system 111} with initial value $(\\rho^0_{\\delta},\\m_{\\delta}^0) \\in L^\\gamma \\times L^2$ if the following holds true.\n\\begin{itemize}\n\\item $\\rho_{\\delta}\\in L^{\\infty}(0,T;L^{\\gamma}(\\mathbb{T}^n)), \\;\\;\\sqrt{\\rho_{\\delta}} \\v_{\\delta}\\in L^{\\infty}(0,T;L^2(\\mathbb{T}^n))$;\n\\item for any $\\varphi \\in C^\\infty_c(\\mathbb{T}^3 \\times [0, T))$,\n\\[\n\\int_0^\\infty \\int_{\\mathbb{T}^n} \\left( \\rho_{\\delta} \\, \\partial_t \\varphi + \\rho_{\\delta} \\v_{\\delta} \\cdot \\nabla \\varphi \\right) \\, dx \\, dt + \\int_{\\mathbb{T}^n} \\rho_{\\delta}^0(x) \\varphi(0,x) \\, dx = 0;\n\\]\n\n \\item for any $\\phi \\in C^\\infty_c(\\mathbb{T}^n \\times [0, T); \\R^n)$,\n\\begin{equation*}\n\\begin{split}\n\\int^\\infty_0 \\int_{\\mathbb{T}^n} \\rho_{\\delta} \\v_{\\delta} \\cdot \\partial_t \\phi &+ \\rho_{\\delta} \\v_{\\delta} \\otimes \\v_{\\delta} : \\nabla \\phi \\,dxdt +\\int_0^\\infty\\int_{\\mathbb{T}^n}\\frac{\\rho_{\\delta}^{\\gamma}}{\\delta^2}\\Dv \\phi\\,dx\\,dt\n\\\\&= -\\int_{\\mathbb{T}^n}\\m_{\\delta}^0 \\cdot \\phi(\\cdot, 0) \\,dx.\n\\end{split}\n\\end{equation*}\n\\end{itemize}\n\\end{Definition}\n\\item The following asymptotic limits hold:\n\\begin{equation*}\n\\rho_{\\delta}\\to 1\\;\\;\\;\\text{ in } C^0([0,T']\\times \\mathbb{T}^n),\\;\\;\\;\\;\\; \\rho_{\\delta}\\widehat{{\\bf v}}_{\\delta}\\to {\\bf u} \\;\\text{ in } L^{2}(0,T';L^2(\\mathbb{T}^n)),\n\\end{equation*}\nand the initial data \n$$\n\\m_{\\delta}^0\\to {\\bf u}^0\\;\\;\\;\\text{ weakly in }\\;\\; L^2(\\mathbb{T}^n),\n$$\nas $\\delta$ tends to zero.\n\\end{itemize}", "theorem_type": ["Existential–Universal", "Asymptotic or Limit"], "mcq": {"question": "Let $n\\in\\{2,3\\}$, let $\\gamma>1$, let $T>0$, and let ${\\bf u}\\in L^\\infty(0,T;L^2(\\mathbb T^n))$ be any weak solution of the incompressible Euler system on the torus $\\mathbb T^n$,\n\\[\n\\partial_t {\\bf u}+{\\bf u}\\cdot\\nabla {\\bf u}+\\nabla \\pi=0,\\qquad \\operatorname{div}{\\bf u}=0,\n\\]\nwith initial data ${\\bf u}^0\\in L^2(\\mathbb T^n)$. Consider also, for Mach number $\\delta>0$, the rescaled compressible Euler system\n\\[\n\\partial_t\\rho_\\delta+\\operatorname{div}(\\rho_\\delta {\\bf v}_\\delta)=0,\n\\qquad\n\\partial_t(\\rho_\\delta {\\bf v}_\\delta)+\\operatorname{div}(\\rho_\\delta {\\bf v}_\\delta\\otimes {\\bf v}_\\delta)+\\nabla\\!\\left(\\frac{\\rho_\\delta^\\gamma}{\\delta^2}\\right)=0.\n\\]\nWhich statement holds for every shorter time interval $[0,T']$ with $T'0$ such that for every $0<\\delta<\\delta_0$ there exist initial data $(\\rho_\\delta^0,\\mathbf m_\\delta^0)\\in C^0(\\mathbb T^n)\\times L^2(\\mathbb T^n)$ with the following properties: for that fixed $\\delta$, there exist infinitely many weak solutions $(\\rho_\\delta,\\rho_\\delta\\widehat{\\mathbf v}_\\delta)$ on $(0,T')$ of the rescaled compressible Euler system with initial data $(\\rho_\\delta^0,\\mathbf m_\\delta^0)$; moreover, as $\\delta\\to0$,\n\\[\n\\rho_\\delta\\to1\\quad\\text{in }C^0([0,T']\\times\\mathbb T^n),\n\\qquad\n\\rho_\\delta\\widehat{\\mathbf v}_\\delta\\to\\mathbf u\\quad\\text{in }L^2(0,T';L^2(\\mathbb T^n)),\n\\]\nand the initial momenta satisfy\n\\[\n\\mathbf m_\\delta^0\\rightharpoonup \\mathbf u^0\\quad\\text{weakly in }L^2(\\mathbb T^n).\n\\]"}, "choices": [{"label": "B", "text": "For every $T'0$ such that for every $0<\\delta<\\delta_0$ there exist initial data $(\\rho_\\delta^0,\\mathbf m_\\delta^0)\\in C^0(\\mathbb T^n)\\times L^2(\\mathbb T^n)$ with the following properties: for that fixed $\\delta$, there exist infinitely many weak solutions $(\\rho_\\delta,\\rho_\\delta\\widehat{\\mathbf v}_\\delta)$ on $(0,T')$ of the rescaled compressible Euler system with initial data $(\\rho_\\delta^0,\\mathbf m_\\delta^0)$; moreover, as $\\delta\\to0$,\n\\[\n\\rho_\\delta\\to1\\quad\\text{in }L^\\infty(0,T';L^\\gamma(\\mathbb T^n)),\n\\qquad\n\\rho_\\delta\\widehat{\\mathbf v}_\\delta\\to\\mathbf u\\quad\\text{weakly in }L^2(0,T';L^2(\\mathbb T^n)),\n\\]\nand the initial momenta satisfy\n\\[\n\\mathbf m_\\delta^0\\to \\mathbf u^0\\quad\\text{strongly in }L^2(\\mathbb T^n).\n\\]"}, {"label": "C", "text": "For every $T'0$ such that for every $0<\\delta<\\delta_0$ there exist initial data $(\\rho_\\delta^0,\\mathbf m_\\delta^0)\\in C^0(\\mathbb T^n)\\times L^2(\\mathbb T^n)$ such that, for that fixed $\\delta$, there exists at least one weak solution $(\\rho_\\delta,\\rho_\\delta\\widehat{\\mathbf v}_\\delta)$ on $(0,T')$ of the rescaled compressible Euler system with initial data $(\\rho_\\delta^0,\\mathbf m_\\delta^0)$, and as $\\delta\\to0$,\n\\[\n\\rho_\\delta\\to1\\quad\\text{in }C^0([0,T']\\times\\mathbb T^n),\n\\qquad\n\\rho_\\delta\\widehat{\\mathbf v}_\\delta\\to\\mathbf u\\quad\\text{in }L^2(0,T';L^2(\\mathbb T^n)),\n\\]\nwhile\n\\[\n\\mathbf m_\\delta^0\\rightharpoonup \\mathbf u^0\\quad\\text{weakly in }L^2(\\mathbb T^n).\n\\]"}, {"label": "D", "text": "For every $T'0$ every weak solution $(\\rho_\\delta,\\rho_\\delta\\widehat{\\mathbf v}_\\delta)$ on $(0,T')$ of the rescaled compressible Euler system with that initial data satisfies\n\\[\n\\rho_\\delta\\to1\\quad\\text{in }C^0([0,T']\\times\\mathbb T^n),\n\\qquad\n\\rho_\\delta\\widehat{\\mathbf v}_\\delta\\to\\mathbf u\\quad\\text{in }L^2(0,T';L^2(\\mathbb T^n)),\n\\]\nand\n\\[\n\\mathbf m_\\delta^0\\rightharpoonup \\mathbf u^0\\quad\\text{weakly in }L^2(\\mathbb T^n)\n\\]\nas $\\delta\\to0$."}, {"label": "E", "text": "For every $T'0$ such that for every $0<\\delta<\\delta_0$ there exist initial data $(\\rho_\\delta^0,\\mathbf m_\\delta^0)\\in C^0(\\mathbb T^n)\\times L^2(\\mathbb T^n)$ with the following properties: for that fixed $\\delta$, there exist infinitely many weak solutions $(\\rho_\\delta,\\rho_\\delta\\widehat{\\mathbf v}_\\delta)$ on $(0,T')$ of the rescaled compressible Euler system with initial data $(\\rho_\\delta^0,\\mathbf m_\\delta^0)$; moreover, as $\\delta\\to0$,\n\\[\n\\rho_\\delta\\to1\\quad\\text{in }C^0([0,T]\\times\\mathbb T^n),\n\\qquad\n\\rho_\\delta\\widehat{\\mathbf v}_\\delta\\to\\mathbf u\\quad\\text{in }L^2(0,T;L^2(\\mathbb T^n)),\n\\]\nand the initial momenta satisfy\n\\[\n\\mathbf m_\\delta^0\\rightharpoonup \\mathbf u^0\\quad\\text{weakly in }L^2(\\mathbb T^n).\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": "strong-uniform convergence replaced by weaker topology for solutions but stronger topology for initial momentum", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped infinitude of weak solutions, retaining existence and convergence", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "quantifier scope on initial data and universality over every weak solution", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "time interval restricted to every fixed $T'\\frac{n-2}{2}$ if $(M^n,g)$ is asymptotically flat of order $\\tau$ and $f=O_2(r^{-\\tau})$.\n\t\\end{defn}", "def-f-schwarzschild": "\\begin{defn}\\label{def-f-schwarzschild}\n\t\tFix $m>0$, and let $r_0=\\big(\\frac{m}{2}\\big)^{\\frac{1}{n-2}}$ and\n\t\t$M_m=\\R^n\\setminus B_{r_0}(0)$.\n\t\tGiven any $f\\in C^\\infty(M_m)$ with $f\\to 0$ as $r\\to\\infty$, define the metric\n\t\t\\[\n\t\tg_{m,f}\n\t\t=e^{\\frac{2}{n-1}f}\\Big(1+\\frac{m}{2\\,r^{\\,n-2}}\\Big)^{\\!\\frac{4}{n-2}}\\delta\n\t\t\\quad\\text{on }M_m,\n\t\t\\]\n\t\twhere $\\delta$ denotes the standard Euclidean metric.\n\n\t\tWe call $(M_m,g_{m,f},e^{-f}dV_g)$ an \\emph{$f$-Schwarzschild manifold of weighted mass $m$}.\n\t\\end{defn}", "eq-weightedPenrose": "\\begin{equation}\\label{eq-weightedPenrose}\n\t\t\tm_f(g)\\;\\ge\\;\\frac12\\left(\\frac{A_f(\\Sigma)}{\\omega_{n-1}}\\right)^{\\!\\frac{n-2}{n-1}}.\n\t\t\\end{equation}", "def-weighted-min": "\\begin{defn} \\label{def-weighted-min}\n\t\tWe say a hypersurface in a weighted manifold $(M^n,g,e^{-f}dV_g)$ is \\emph{$f$-minimal} if the weighted mean curvature $H_f=H-\\partial_\\nu f$ of the hypersurface vanishes. We further say that a closed hypersurface $\\Sigma$ in a weighted manifold is \\emph{$f$-outer-minimising} if it minimises the weighted area over all hypersurfaces enclosing $\\Sigma$.\n\t\\end{defn}"}, "pre_theorem_intro_text_len": 2630, "pre_theorem_intro_text": "The positive mass theorem is a celebrated cornerstone of mathematical general relativity, proven by Schoen and Yau \\cite{SchoenYau79} using minimal surface techniques, and by Witten \\cite{Witten81} using a spinor argument. Ideas from both proofs, as well as the positive mass theorem itself, have motivated a considerable amount of research in the decades that followed, including many generalisations and extensions. Some generalisations are driven by physical motivations, such as the Penrose inequality or the various inequalities between mass, charge and angular momentum, while other geometric notions of mass have also been studied in relation to scalar curvature problems. One such definition, which we consider here, is the mass of a weighted manifold (manifold with density) introduced by Baldauf and Ozuch \\cite{BaldaufOzuch22}.\n\n\tA weighted manifold $(M^n,g,e^{-f}dV_g)$ is a Riemannian manifold $(M^n,g)$ equipped with a smooth function $f$ defining the measure $e^{-f}dV_g$. The weighted scalar curvature is given by\n\t\\begin{equation*}\n\t\tR_f \\;=\\; R + 2\\Delta f - |\\nabla f|^2,\n\t\\end{equation*}\t\n\twhere we use the convention $\\Delta=\\nabla_i \\nabla^i$. Throughout, we assume all manifolds are smooth, connected, and oriented. Baldauf and Ozuch define the \\emph{weighted mass} of an asymptotically flat weighted manifold as\n\t\\begin{equation}\\label{eq-weightedmass}\n\t\t\\m_f(g) \\;=\\; \\m_{ADM}(g) \\;+\\; \\frac{1}{(n-1)\\,\\omega_{n-1}}\\lim_{\\rho\\to\\infty}\\int_{S_\\rho} \\nabla f_i \\,\\nu^i\\,e^{-f}\\,dS,\n\t\\end{equation}\n\twhere $S_\\rho$ are large coordinate spheres in an asymptotically flat end with outward unit normal $\\nu$, $\\omega_{n-1}$ is the volume of the unit $(n-1)$-sphere, and\n\t\\begin{equation}\\label{eq-ADMmass}\n\t\t\\m_{ADM}(g)=\\frac{1}{2(n-1)\\,\\omega_{n-1}}\\lim_{\\rho\\to\\infty}\\int_{S_\\rho} \\left( \\mathring \\nabla^jg_{ij}-\\onabla_i\\tr_{\\mathring g}(g)\\right)\\,\\nu^idS\n\t\\end{equation}\n\tis the ADM mass. Here and throughout we use $\\mathring g$ to denote the Euclidean metric pulled back to the asymptotic end, and $\\mathring\\nabla$ is the associated flat connection. Note that our definition of the weighted mass differs from that of \\cite{BaldaufOzuch22} by a multiplicative constant to align with the standard definition of the ADM mass, so that the resultant Penrose inequality takes the familiar form. Using a weighted Witten identity, they proved a weighted positive mass theorem for spin manifolds \\cite{BaldaufOzuch22}, and subsequently Chu and Zhu proved the weighted positive mass theorem in the non-spin case for $3\\le n\\le 7$ in the spirit of the Schoen--Yau minimal hypersurfaces argument \\cite{ChuZhu24}.", "context": "The positive mass theorem is a celebrated cornerstone of mathematical general relativity, proven by Schoen and Yau \\cite{SchoenYau79} using minimal surface techniques, and by Witten \\cite{Witten81} using a spinor argument. Ideas from both proofs, as well as the positive mass theorem itself, have motivated a considerable amount of research in the decades that followed, including many generalisations and extensions. Some generalisations are driven by physical motivations, such as the Penrose inequality or the various inequalities between mass, charge and angular momentum, while other geometric notions of mass have also been studied in relation to scalar curvature problems. One such definition, which we consider here, is the mass of a weighted manifold (manifold with density) introduced by Baldauf and Ozuch \\cite{BaldaufOzuch22}.\n\nA weighted manifold $(M^n,g,e^{-f}dV_g)$ is a Riemannian manifold $(M^n,g)$ equipped with a smooth function $f$ defining the measure $e^{-f}dV_g$. The weighted scalar curvature is given by\n \\begin{equation*}\n R_f \\;=\\; R + 2\\Delta f - |\\nabla f|^2,\n \\end{equation*} \n where we use the convention $\\Delta=\\nabla_i \\nabla^i$. Throughout, we assume all manifolds are smooth, connected, and oriented. Baldauf and Ozuch define the \\emph{weighted mass} of an asymptotically flat weighted manifold as\n \\begin{equation}\\label{eq-weightedmass}\n \\m_f(g) \\;=\\; \\m_{ADM}(g) \\;+\\; \\frac{1}{(n-1)\\,\\omega_{n-1}}\\lim_{\\rho\\to\\infty}\\int_{S_\\rho} \\nabla f_i \\,\\nu^i\\,e^{-f}\\,dS,\n \\end{equation}\n where $S_\\rho$ are large coordinate spheres in an asymptotically flat end with outward unit normal $\\nu$, $\\omega_{n-1}$ is the volume of the unit $(n-1)$-sphere, and\n \\begin{equation}\\label{eq-ADMmass}\n \\m_{ADM}(g)=\\frac{1}{2(n-1)\\,\\omega_{n-1}}\\lim_{\\rho\\to\\infty}\\int_{S_\\rho} \\left( \\mathring \\nabla^jg_{ij}-\\onabla_i\\tr_{\\mathring g}(g)\\right)\\,\\nu^idS\n \\end{equation}\n is the ADM mass. Here and throughout we use $\\mathring g$ to denote the Euclidean metric pulled back to the asymptotic end, and $\\mathring\\nabla$ is the associated flat connection. Note that our definition of the weighted mass differs from that of \\cite{BaldaufOzuch22} by a multiplicative constant to align with the standard definition of the ADM mass, so that the resultant Penrose inequality takes the familiar form. Using a weighted Witten identity, they proved a weighted positive mass theorem for spin manifolds \\cite{BaldaufOzuch22}, and subsequently Chu and Zhu proved the weighted positive mass theorem in the non-spin case for $3\\le n\\le 7$ in the spirit of the Schoen--Yau minimal hypersurfaces argument \\cite{ChuZhu24}.", "full_context": "The positive mass theorem is a celebrated cornerstone of mathematical general relativity, proven by Schoen and Yau \\cite{SchoenYau79} using minimal surface techniques, and by Witten \\cite{Witten81} using a spinor argument. Ideas from both proofs, as well as the positive mass theorem itself, have motivated a considerable amount of research in the decades that followed, including many generalisations and extensions. Some generalisations are driven by physical motivations, such as the Penrose inequality or the various inequalities between mass, charge and angular momentum, while other geometric notions of mass have also been studied in relation to scalar curvature problems. One such definition, which we consider here, is the mass of a weighted manifold (manifold with density) introduced by Baldauf and Ozuch \\cite{BaldaufOzuch22}.\n\nA weighted manifold $(M^n,g,e^{-f}dV_g)$ is a Riemannian manifold $(M^n,g)$ equipped with a smooth function $f$ defining the measure $e^{-f}dV_g$. The weighted scalar curvature is given by\n \\begin{equation*}\n R_f \\;=\\; R + 2\\Delta f - |\\nabla f|^2,\n \\end{equation*} \n where we use the convention $\\Delta=\\nabla_i \\nabla^i$. Throughout, we assume all manifolds are smooth, connected, and oriented. Baldauf and Ozuch define the \\emph{weighted mass} of an asymptotically flat weighted manifold as\n \\begin{equation}\\label{eq-weightedmass}\n \\m_f(g) \\;=\\; \\m_{ADM}(g) \\;+\\; \\frac{1}{(n-1)\\,\\omega_{n-1}}\\lim_{\\rho\\to\\infty}\\int_{S_\\rho} \\nabla f_i \\,\\nu^i\\,e^{-f}\\,dS,\n \\end{equation}\n where $S_\\rho$ are large coordinate spheres in an asymptotically flat end with outward unit normal $\\nu$, $\\omega_{n-1}$ is the volume of the unit $(n-1)$-sphere, and\n \\begin{equation}\\label{eq-ADMmass}\n \\m_{ADM}(g)=\\frac{1}{2(n-1)\\,\\omega_{n-1}}\\lim_{\\rho\\to\\infty}\\int_{S_\\rho} \\left( \\mathring \\nabla^jg_{ij}-\\onabla_i\\tr_{\\mathring g}(g)\\right)\\,\\nu^idS\n \\end{equation}\n is the ADM mass. Here and throughout we use $\\mathring g$ to denote the Euclidean metric pulled back to the asymptotic end, and $\\mathring\\nabla$ is the associated flat connection. Note that our definition of the weighted mass differs from that of \\cite{BaldaufOzuch22} by a multiplicative constant to align with the standard definition of the ADM mass, so that the resultant Penrose inequality takes the familiar form. Using a weighted Witten identity, they proved a weighted positive mass theorem for spin manifolds \\cite{BaldaufOzuch22}, and subsequently Chu and Zhu proved the weighted positive mass theorem in the non-spin case for $3\\le n\\le 7$ in the spirit of the Schoen--Yau minimal hypersurfaces argument \\cite{ChuZhu24}.\n\n\\noindent Then $(M^n,g,e^{-f}dV_g)$ is $f$-Schwarzschild of mass $m=\\m_f(g)$.\n \\end{thm}\n \\newpage\n Many other geometric quantities have natural weighted versions that fit together within a coherent weighted manifold framework (see Table 1 of \\cite{BaldaufOzuch22}, for example). This leads to natural definitions of weighted minimal surfaces, weighted outer-minimising surfaces (see Definition \\ref{def-weighted-min}), and to conjecture a weighted Riemannian Penrose inequality. However, we show this follows immediately from the conformal transformation discussed above.\n \\begin{thm}\\label{thm-weightedPenrose-intro}\n Let $(M^n,g,e^{-f}dV_g)$ be a weighted asymptotically flat manifold of dimension $3\\le n\\le 7$ containing a closed $f$-outer-minimising $f$-minimal hypersurface $\\Sigma$. Assume $S_f\\ge 0$, then\n \\begin{equation}\\label{eq-weightedPenrose}\n m_f(g)\\;\\ge\\;\\frac12\\left(\\frac{A_f(\\Sigma)}{\\omega_{n-1}}\\right)^{\\!\\frac{n-2}{n-1}}.\n \\end{equation}\n Furthermore, if we also assume that $M$ is spin\\footnote{Recall $n=3$ implies spin automatically.}, then equality holds in \\eqref{eq-weightedPenrose} if and only if $(M^n,g,e^{-f}dV_g)$ is an $f$-Schwarzschild manifold.\n \\end{thm}\n\nLet $(M^n,g,e^{-f}dV_g)$ be a weighted $n$-dimensional asymptotically flat manifold with decay $\\tau>\\frac{n-2}{2}$. We define a densitised curvature map\n \\[\n \\Phi(g,f)= S_f\\,e^{-f}dV_g,\n \\]\n where\n \\begin{equation*}\n S_f=R_f+\\frac{1}{n-1}|\\nabla f|^2=R+2\\Delta(f)- \\frac{n-2}{n-1}|\\nabla f|^2,\n \\end{equation*}\n motivated by Theorem \\ref{thm-LLS}. Following Michel's formalism on geometric mass invariants \\cite{Michel11}, there is $1$-form valued differential operator $\\mathbb U$ defined by\n \\begin{equation}\\label{eq-Michel-formula1}\n V\\,D\\Phi_{(\\mathring{g},\\mathring{f})}(h, \\varphi)-\\langle(h,\\varphi),D\\Phi_{(\\mathring{g},\\mathring{f})}^*(V)\\rangle=\\div_{\\mathring g}\\mathbb U(V,h,\\varphi)dV_{\\mathring g},\n \\end{equation}\n where $h=g-\\mathring g$, $\\varphi=f-\\mathring f$, $V$ is a scalar function, and $\\langle \\cdot,\\cdot,\\rangle$ is the natural pairing. We note here that instead of using the densitised map $\\Phi$, one could instead define the adjoint map with respect to a weighted $L^2$ inner product. Although we include the background weight function $\\mathring f$ in the expression above for clarity, we will be taking it to vanish identically in the definition of the weighted mass. Choosing $V$ in the kernel of $D\\Phi_{(\\mathring{g},\\mathring{f}\\equiv 0)}^*$ and integrating $\\div_{\\mathring g}\\mathbb U$ over $M$ gives a flux integral at infinity, which corresponds to a geometric invariant \\`a la Michel \\cite{Michel11}. Although for the mass definition we need only compute the variation around the base point $(\\mathring g,0)$, the linearisation and its adjoint at a general point will be useful in what follows. We compute\n \\begin{align}\\begin{split}\n D\\Phi_{(g,f)}(h,\\varphi) \\,=\\,& \\Big(\n \\big( \\nabla^i \\nabla^j h_{ij} - \\Delta(h^i{}_i) \\big) \n - 2\\big( \\nabla^j h_{ij} - \\frac{1}{2}\\nabla_i(h^k{}_k) \\big)\\nabla^i f \\\\\n &\\quad - h_{ij}\\Big( \\ric^{ij} + 2\\nabla^i \\nabla^j f - \\frac{n-2}{n-1}\\,\\nabla^i f \\nabla^j f \\Big) \\\\\n &\\qquad +\\left(2 \\Delta \\varphi - \\frac{2(n-2)}{n-1}\\,\\nabla^i f \\,\\nabla_i \\varphi\\right)\\\\\n &\\qquad+S_f(g)\\left( \\frac12 \\tr_g(h)-\\varphi \\right)\\Big)e^{-f}dV_g,\\end{split}\n \\end{align}\n with formal adjoint\n\nFrom this we see that linearising around $(\\mathring g, \\mathring f\\equiv0)$ gives\n \\begin{equation}\n D\\Phi_{(\\mathring g,0)}(h,\\varphi)=\\left( \\onabla^i \\onabla^j h_{ij} - \\oDelta(\\tr_{\\mathring g}(h)) +2\\oDelta(\\varphi)\\right)dV_{\\mathring g}\n \\end{equation}\n and\n \\begin{equation}\\begin{split}\n D_g\\Phi_{(\\mathring g,0)}^*(V)^{ij}=\\;& \\Big(\\onabla^i \\onabla^j V - (\\oDelta V)\\, \\mathring g^{ij}\\Big)dV_{\\mathring g},\\\\\n D_f\\Phi_{(\\mathring g,0)}^*(V)=\\;& 2 \\oDelta V dV_{\\mathring g}.\n \\end{split}\\end{equation}\n We see immediately that the kernel of $D\\Phi^*_{(\\mathring g,0)}$ consists of affine functions. Taking $V\\equiv 1$, and using $h=g-\\mathring g$ and $\\varphi=f$, we have\n \\begin{equation*}\n \\mathbb U(1,g-\\mathring g,f)=\\div_{\\mathring g}(g)-\\onabla\\tr_{\\mathring g}(g)+2\\onabla f.\n \\end{equation*}\n We then immediately see that integrating this over large spheres recovers the weighted mass\n \\begin{equation}\n \\lim_{\\rho\\to\\infty}\\int_{S_\\rho}\\mathbb U_i(1,g-\\mathring g,f)dS^i=2(n-1)\\omega_{n-1}\\m_f(g),\n \\end{equation}\n making use of the fact that $e^{-f}=1+O(r^{-\\tau})$. That is, the weighted mass is the geometric mass invariant associated with $S_fe^{-f}dV_g$ and we have established Theorem \\ref{thm-Michel}. An important aspect of this formalism for mass invariants is that it gives a somewhat universal framework to prove that they are indeed geometric quantities, independent of the choice of coordinates at infinity \\cite{Michel11}. However, for the weighted mass this readily follows from the coordinate invariance of the ADM mass \\cite{BaldaufOzuch22,LawLopezSantiago24}, so in the interest of brevity we do not explicitly present this here.\n\n\\begin{align}\\begin{split}\n \\mathfrak c^a_f(g)&=\\frac{1}{\\m_f(g)}\\lim_{\\rho\\to\\infty}\\int_{S_\\rho}\\mathbb U_i(x^a,g-\\mathring g,f)dS^i\\\\\n &=\\frac{1}{\\m_f(g)}\\lim_{\\rho\\to\\infty}\\int_{S_\\rho}x^a\\Big(\\onabla^{j}g_{ij}-\\onabla_i\\big(\\tr_{\\mathring g}g\\big)+2\\,\\onabla_if\\Big)\n \\\\ &\\qquad -(g-\\mathring g)_{ia}\n +\\Big(\\tr_{\\mathring g}(g-\\mathring g)-2f\\Big)\\delta_i^a\\, dS^i,\n \\end{split}\n \\end{align}\n where $x^a$ is a coordinate function. We next show that this is simply the same as the standard centre of mass for the conformal metric $\\widetilde g = e^{-\\frac{2}{n-1}f}\\,g$.\n \\begin{prop}\n Let $(M^n,g,e^{-f}dV_g)$ be a smooth asymptotically flat weighted manifold of order $\\tau>\\frac{n-2}{2}$, and let $\\widetilde g = e^{-\\frac{2}{n-1}f}\\,g$. Then\n \\[\n \\mathfrak c^a_f(g)=\\mathfrak c^a_{\\mathrm{ADM}}(\\widetilde g),\n \\]\n where $\\mathfrak c_{ADM}$ is the usual centre of mass for an asymptotically flat manifold given by \n \\begin{equation}\\begin{split}\\label{eq-CoM}\n \\mathfrak c_{ADM}^a(\\widetilde g)&=\\lim_{\\rho\\to\\infty}\\frac{1}{\\m_{ADM}(\\widetilde g)}\\int_{S_\\rho}\n x^a\\Big(\\mathring\\nabla^j(\\widetilde g)_{ij}-\\mathring\\nabla_i\\tr_{\\mathring g}(\\widetilde g)\\Big)\\\\ &\\qquad\n -(\\widetilde g -\\mathring g)_{ia}+\\tr_{\\mathring g}(\\widetilde g-\\mathring g)\\,\\delta_i^{\\,a}dS^i,\\end{split}\n \\end{equation}\n whenever $\\mathfrak c_{ADM}(\\widetilde g)$ is well-defined.\\footnote{ To ensure the centre of mass is well-defined requires an additional parity assumption \\cite{ReggeTeitelboim74}.}\n \\end{prop}\n \\begin{proof}\n The decay conditions allow us to write\n \\begin{equation}\\label{eq-conf-expansion-com}\n \\widetilde g\n = g - \\frac{2}{n-1}f\\,\\mathring g + O_2(r^{-2\\tau}).\n \\end{equation}\n Inserting this into \\eqref{eq-CoM} and recalling $\\m_{ADM}(\\widetilde g)=\\m_f(g)$ one quickly arrives at\n \\[\n \\mathfrak c^a_f(g)=\\mathfrak c^a_{\\mathrm{ADM}}(\\widetilde g).\n \\]\n \\end{proof}\n\n\\begin{thm}\\label{thm-weighted-hawking-bound}\n Let $(M^3,g,e^{-f}dV_g)$ be asymptotically flat and assume\n \\[\n S_f \\ge\\ 0.\n \\]\n Let $\\Sigma\\subset M$ be a connected closed surface that is $f$-outer-minimising in $(M^3,g,e^{-f}dV_g)$.\n Then\n \\begin{equation}\\label{eq-weighted-hawking-bound}\n \\m_f(g)\\ \\ge\\ \\m_{H,f}(\\Sigma;g).\n \\end{equation}\n \\end{thm}", "post_theorem_intro_text_len": 5695, "post_theorem_intro_text": "The reader is directed to definition \\ref{def-AF} for the assumed decay rates for asymptotically flat weighted manifolds.\n\n\tSomewhat surprisingly, Law, Lopez, and Santiago \\cite{LawLopezSantiago24} showed that after a particular conformal transformation, the weighted positive mass theorem reduces precisely to the usual (unweighted) positive mass theorem, which can then be applied directly. In fact, the result they obtain is actually stronger than the previously established more technical proofs.\n\t\\begin{thm}[Law--Lopez--Santiago \\cite{LawLopezSantiago24}]\\label{thm-LLS}\n\t\tLet $(M^n,g,e^{-f}dV_g)$ be an asymptotically flat weighted manifold with $f=o_2(|x|^{-(n-2)/2})$ and $R_f\\in L^1(M)$. Set\n\t\t\\[\n\t\t\\widetilde g \\;=\\; e^{-\\frac{2}{n-1}f}\\,g.\n\t\t\\]\n\t\tThen the scalar curvature of $\\widetilde g$, $\\widetilde R$ is given by\n\t\t\\[\n\t\t\\widetilde R \\;=\\; e^{\\frac{2}{n-1}f}\\Big(R_f + \\frac{1}{n-1}\\,|\\nabla f|^2\\Big),\n\t\t\\]\n\t\tand\n\t\t\\[\n\t\t\\m_f(g) \\;=\\; \\mathfrak{m}(\\widetilde g).\n\t\t\\]\n\t\tIn particular, if $R_f\\ge -\\frac{1}{n-1}|\\nabla f|^2$ then $R_{\\widetilde g}\\ge 0$ and hence $\\m_f(g)\\ge 0$.\n\t\\end{thm}\n\tWithout this insight, one may be tempted to pursue a proof of a weighted Riemannian Penrose inequality by adapting the proofs of Huisken and Ilmanen \\cite{HuiskenIlmanen01} or Bray \\cite{Bray01} to the weighted setting\\footnote{The present author at least was, prior to learning of the article \\cite{LawLopezSantiago24}.}. However, it turns out that the expected weighted Riemannian Penrose inequality follows from the same conformal transformation then applying the standard Riemannian Penrose inequality to the conformal metric. In this note, we show this weighted Riemannian Penrose inequality and some related results on weighted manifolds that follow from this same conformal picture. In order to better understand why this approach works so well, we investigate the quantity $$S_f=R_f+\\frac{1}{n-1}|\\nabla f|^2.$$ In particular, we study the geometric mass invariant associated with $S_f$ via Michel's mass invariant formalism \\cite{Michel11}, and establish that it is precisely the weighted mass. \n\t\\begin{thm}\\label{thm-Michel}\n\t\tLet $(M^n,g,e^{-f}dV_g)$ be an asymptotically flat weighted manifold and define the densitised curvature operator $\\Phi(g,f)=S_f(g)\\,e^{-f}dV_g$. The kernel of $D\\Phi^*_{(\\mathring g,0)}$ consists of affine functions $V$ defining geometric mass invariants in the sense of Michel \\cite{Michel11}\n\t\t\\begin{equation}\n\t\t\t\\mathfrak{m}(g,f,V)=\\lim_{\\rho\\to\\infty}\\int_{S_\\rho}\\mathbb U_i(g,f,V)\\,\\nu^idS,\n\t\t\\end{equation}\n\t\twhere $\\mathbb U(g,f,V)$ satisfies\n\t\t\\begin{equation*}\n\t\t\t\\div_{\\mathring g}\\mathbb U(g,f,V)=V\\, D\\Phi_{(\\mathring g,0)}.\n\t\t\\end{equation*}\n\t\tFurthermore, for $V=1$ and assuming $S_f\\in L^1$ then we have\n\t\t\\begin{equation*}\n\t\t\t\\mathfrak{m}(g,f,1)=\\m_f(g).\n\t\t\\end{equation*}\n\t\\end{thm}\n\n\tWith this in mind, it is not surprising that we can establish so many properties of the weighted mass via the conformal metric $\\widetilde g$, whose scalar curvature is $S_f$. This framework also naturally leads to a definition of centre of mass for weighted manifolds, which agrees with the standard (unweighted) centre of mass of $\\widetilde g$.\n\n\tExamining the linearisation of $S_f$ and its formal adjoint leads to a natural definition of weighted static metrics, which can be understood as candidates for minimisers of the weighted mass. We prove that this definition of weighted staticity also can be understood via the conformal metric $\\widetilde g$. From this we prove the following weighted static uniqueness theorem for what we call the $f$-Schwarzschild family of metrics, which differ from regular Schwarzschild metrics by the conformal factor $e^{-\\frac{2f}{n-1}}$ (see Definition \\ref{def-f-schwarzschild}).\n\t\\begin{thm}\\label{thm-fstatic-bh-rigidity-intro}\n\t\tLet $(M^n,g,e^{-f}dV_g)$ be an asymptotically flat weighted manifold with compact boundary\n\t\t$\\Sigma=\\partial M$, and assume $f\\to 0$ at infinity.\n\t\tAssume either $3\\le n\\le 7$ or $M$ is spin, and for $n>3$ further assume $\\Sigma$ is connected.\n\n\\noindent \t\tSuppose there exists an $f$-static potential $V\\in C^\\infty(M)$ satisfying\n\t\t\\[\n\t\tV>0\\ \\text{on }M\\setminus\\Sigma,\\qquad V=0\\ \\text{on }\\Sigma,\\qquad V\\to 1\\ \\text{at infinity}.\n\t\t\\]\n\n\t\\noindent \tThen $(M^n,g,e^{-f}dV_g)$ is $f$-Schwarzschild of mass $m=\\m_f(g)$.\n\t\\end{thm}\n\t\\newpage\n\tMany other geometric quantities have natural weighted versions that fit together within a coherent weighted manifold framework (see Table 1 of \\cite{BaldaufOzuch22}, for example). This leads to natural definitions of weighted minimal surfaces, weighted outer-minimising surfaces (see Definition \\ref{def-weighted-min}), and to conjecture a weighted Riemannian Penrose inequality. However, we show this follows immediately from the conformal transformation discussed above.\n\t\\begin{thm}\\label{thm-weightedPenrose-intro}\n\t\tLet $(M^n,g,e^{-f}dV_g)$ be a weighted asymptotically flat manifold of dimension $3\\le n\\le 7$ containing a closed $f$-outer-minimising $f$-minimal hypersurface $\\Sigma$. Assume $S_f\\ge 0$, then\n\t\t\\begin{equation}\\label{eq-weightedPenrose}\n\t\t\tm_f(g)\\;\\ge\\;\\frac12\\left(\\frac{A_f(\\Sigma)}{\\omega_{n-1}}\\right)^{\\!\\frac{n-2}{n-1}}.\n\t\t\\end{equation}\n\t\tFurthermore, if we also assume that $M$ is spin\\footnote{Recall $n=3$ implies spin automatically.}, then equality holds in \\eqref{eq-weightedPenrose} if and only if $(M^n,g,e^{-f}dV_g)$ is an $f$-Schwarzschild manifold.\n\t\\end{thm}\n\n\tWe remark that it appears as though much of what one might want to study in connection to this weighted mass ultimately can be reduced to the unweighted situation by means of this straightforward conformal transformation.", "sketch": "Law, Lopez, and Santiago \\cite{LawLopezSantiago24} show that “after a particular conformal transformation, the weighted positive mass theorem reduces precisely to the usual (unweighted) positive mass theorem, which can then be applied directly.” Concretely, they set \\(\\widetilde g=e^{-\\frac{2}{n-1}f}g\\), compute that the scalar curvature satisfies \\(\\widetilde R=e^{\\frac{2}{n-1}f}\\big(R_f+\\frac{1}{n-1}|\\nabla f|^2\\big)\\), and identify the masses by \\(\\m_f(g)=\\mathfrak{m}(\\widetilde g)\\). Thus, under the curvature condition (e.g. \\(R_f\\ge 0\\), or more generally \\(R_f\\ge-\\frac{1}{n-1}|\\nabla f|^2\\)), one has \\(\\widetilde R\\ge 0\\), and then the standard positive mass theorem applied to \\(\\widetilde g\\) yields \\(\\m_f(g)\\ge 0\\) (i.e. Theorem~\\ref{thm-BO-CZ}).", "expanded_sketch": "Law, Lopez, and Santiago \\cite{LawLopezSantiago24} show that “after a particular conformal transformation, the weighted positive mass theorem reduces precisely to the usual (unweighted) positive mass theorem, which can then be applied directly.” Concretely, they set \\(\\widetilde g=e^{-\\frac{2}{n-1}f}g\\), compute that the scalar curvature satisfies \\(\\widetilde R=e^{\\frac{2}{n-1}f}\\big(R_f+\\frac{1}{n-1}|\\nabla f|^2\\big)\\), and identify the masses by \\(\\m_f(g)=\\mathfrak{m}(\\widetilde g)\\). Thus, under the curvature condition (e.g. \\(R_f\\ge 0\\), or more generally \\(R_f\\ge-\\frac{1}{n-1}|\\nabla f|^2\\)), one has \\(\\widetilde R\\ge 0\\), and then the standard positive mass theorem applied to \\(\\widetilde g\\) yields \\(\\m_f(g)\\ge 0\\), thereby establishing the main theorem.", "expanded_theorem": "[Baldauf--Ozuch \\cite{BaldaufOzuch22}; Chu--Zhu \\cite{ChuZhu24}]\\label{thm-BO-CZ}\n\t\tLet $(M^n,g,e^{-f}dV_g)$ be an asymptotically flat weighted manifold with $R_f\\in L^1(M)$. Assume either $3\\le n\\le 7$ or $M$ is spin.\n\n\t\tThen if $R_f\\ge 0$, $\\m_f(g)\\ge 0$ with equality if and only if $(M^n,g)$ is isometric to Euclidean space and $f\\equiv0$.", "theorem_type": ["Implication", "Biconditional or Equivalence"], "mcq": {"question": "Let $(M^n,g,e^{-f}dV_g)$ be an asymptotically flat weighted manifold, meaning a Riemannian manifold $(M^n,g)$ equipped with a smooth function $f$ and measure $e^{-f}dV_g$. Its weighted scalar curvature is $R_f=R+2\\Delta f-|\\nabla f|^2$, and its weighted mass is\n\\[\n m_f(g)=m_{ADM}(g)+\\frac{1}{(n-1)\\omega_{n-1}}\\lim_{\\rho\\to\\infty}\\int_{S_\\rho} \\nabla f_i\\,\\nu^i\\,e^{-f}\\,dS,\n\\]\nwhere $m_{ADM}(g)$ is the ADM mass. Assume that $R_f\\in L^1(M)$ and that either $3\\le n\\le 7$ or $M$ is spin. If moreover $R_f\\ge 0$ on $M$, which statement about the weighted mass holds?", "correct_choice": {"label": "A", "text": "One has $m_f(g)\\ge 0$, with equality if and only if $(M^n,g)$ is isometric to Euclidean space and $f\\equiv 0$."}, "choices": [{"label": "B", "text": "One has $m_f(g)\\ge 0$, with equality if and only if $(M^n,g)$ is isometric to Euclidean space, with no further restriction on $f$."}, {"label": "C", "text": "One has $m_f(g)\\ge 0$."}, {"label": "D", "text": "One has $m_f(g)\\ge 0$, and equality holds if and only if $(M^n,g)$ is conformal to Euclidean space via $g=e^{\\frac{2}{n-1}f}\\delta$ on $M$."}, {"label": "E", "text": "One has $m_f(g)\\ge 0$ provided the stronger dimensional and topological hypotheses both hold, namely $3\\le n\\le 7$ and $M$ is spin; with equality if and only if $(M^n,g)$ is isometric to Euclidean space and $f\\equiv 0$."}], "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": "rigidity requires both flat metric and vanishing density", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped the rigidity/equality characterization", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "equality case for the conformal reduction", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "either/or hypothesis replaced by simultaneous requirement", "template_used": "quantifier_dependence"}]}} +{"id": "2601.20187v1", "paper_link": "http://arxiv.org/abs/2601.20187v1", "theorems_cnt": 4, "theorem": {"env_name": "theorem", "content": "[Deligne~\\cite{deligne_weil1}]\n\\label{theorem:rh}\nAssume \\(X_0\\) is smooth and proper over \\(\\mathbf{F}_q\\).\nSuppose \\(\\alpha\\) is an eigenvalue of the geometric Frobenius \\(F\\)\nacting on\n\\(\\mathrm{H}^i_{c}(X;\\overline{\\mathbf{Q}}_{\\ell})=\\mathrm{H}^i(X;\\overline{\\mathbf{Q}}_{\\ell})\\).\nThen for any isomorphism\n\\(\\iota\\colon \\overline{\\mathbf{Q}}_{\\ell} \\to \\mathbf{C}\\), we have\n\\(|\\iota(\\alpha)|=q^{\\frac{i}{2}}\\).", "start_pos": 4349, "end_pos": 4795, "label": "theorem:rh"}, "ref_dict": {"theorem:rh": "\\begin{theorem}[Deligne~\\cite{deligne_weil1}]\n\\label{theorem:rh}\nAssume \\(X_0\\) is smooth and proper over \\(\\mathbf{F}_q\\).\nSuppose \\(\\alpha\\) is an eigenvalue of the geometric Frobenius \\(F\\)\nacting on\n\\(\\mathrm{H}^i_{c}(X;\\overline{\\mathbf{Q}}_{\\ell})=\\mathrm{H}^i(X;\\overline{\\mathbf{Q}}_{\\ell})\\).\nThen for any isomorphism\n\\(\\iota\\colon \\overline{\\mathbf{Q}}_{\\ell} \\to \\mathbf{C}\\), we have\n\\(|\\iota(\\alpha)|=q^{\\frac{i}{2}}\\).\n\\end{theorem}", "theorem:katz": "\\begin{theorem}\n\\label{theorem:katz}\nLet \\(X_{0} \\subset \\mathbf{P}^{n}\\) be a smooth hypersurface. Then\nthe Riemann Hypothesis holds for \\(X_{0}\\).\n\\end{theorem}", "lemma:perverse-degeneration-lemma": "\\begin{lemma}[cf.~{\\cite[Proposition~9]{katz_perverse-origin}, \\cite[Lemma~3.1]{wan-zhang_betti-number-bounds-for-varieties-and-exponential-sums0}}]\n\\label{lemma:perverse-degeneration-lemma}\nIn Situation~\\ref{situation:curve-with-partition}, let\n\\(\\mathcal{F} \\in {}^{\\mathrm{p}}\\mathrm{D}^{\\geq0}_c(U,\\Sigma)\\) be perverse coconnective with\nrespect to \\((U, \\Sigma)\\). Then for any \\(s \\in \\Sigma\\), there is a\nnatural injective map\n\\[\n\\mathrm{H}^{-1}(\\mathcal{F}_s) \\hookrightarrow \\bigl(\\mathrm{R}^0 j_{\\ast} \\mathcal{H}^{-1}(j^{\\ast}\\mathcal{F})\\bigr)_s.\n\\]\nIn particular, the dimension of \\(\\mathrm{H}^{-1}(\\mathcal{F}_s)\\) does not exceed the rank of the local system \\(\\mathcal{H}^{-1}(j^{\\ast}\\mathcal{F})\\) on \\(U\\).\n\\end{lemma}", "theorem:wrh": "\\begin{theorem}\\label{theorem:wrh}\nLet \\(X_{0}\\) be a separated scheme of finite type over\n\\(\\mathbf{F}_{q}\\). Then for every integer \\(i\\), all eigenvalues of Frobenius acting on\n\\(\\mathrm{H}^{i}_{c}(X;\\overline{\\mathbf{Q}}_{\\ell})\\) have\n\\(q\\)-weight \\(\\leq i\\).\n\\end{theorem}"}, "pre_theorem_intro_text_len": 1779, "pre_theorem_intro_text": "Throughout this paper, we work over a finite base field\n\\(\\mathbf{F}_q\\). Our notation is as follows: capital Latin letters\nwith a subscript \\(0\\), such as \\(X_0, S_0\\), denote separated schemes\nof finite type defined over \\(\\mathbf{F}_q\\). Removing the subscript\n(e.g., \\(X, S\\)) indicates the base change of these schemes to a fixed\nalgebraic closure \\(\\overline{\\mathbf{F}}_q\\).\n\nWe fix a prime number \\(\\ell\\) which is invertible in\n\\(\\mathbf{F}_q\\), and throughout, we consider only\n\\(\\overline{\\mathbf{Q}}_{\\ell}\\)-sheaves. Script letters with\nsubscript \\(0\\), like \\(\\mathcal{F}_0, \\mathcal{G}_0\\), refer to\nsheaves on \\(X_0, S_0\\). Omitting the subscript denotes their pullback\nto the corresponding scheme over \\(\\overline{\\mathbf{F}}_q\\).\n\n\\medskip\nLet \\(X_0\\) be a separated scheme of finite type over \\(\\mathbf{F}_q\\).\nWe define the zeta function of \\(X_0\\) as\n\\[\nZ(X_0/\\mathbf{F}_q, t) = \\exp\\left \\{ \\sum_{e \\geqslant 1} \\# X_0(\\mathbf{F}_{q^e})\\frac{t^e}{e} \\right \\},\n\\]\nwhich is an element of \\(1 + t\\mathbf{Z}[\\![t]\\!]\\).\nGrothendieck's trace formula expresses the zeta function in terms of the action of the geometric Frobenius on the \\(\\ell\\)-adic cohomology of \\(X\\):\n\\[\nZ(X_0/\\mathbf{F}_q, t) = \\prod_{i=0}^{2\\dim X_0}\n\\det\\left(1 - tF, \\mathrm{H}^i_c(X; \\overline{\\mathbf{Q}}_\\ell)\\right)^{(-1)^{i+1}}.\n\\]\nRecall that \\(F\\), the \\emph{geometric} Frobenius, is the inverse of\nthe arithmetic Frobenius automorphism \\(a \\mapsto a^q\\) in\n\\(\\mathrm{Gal}(\\overline{\\mathbf{F}}_q/\\mathbf{F}_q)\\), acting on the\n\\(\\ell\\)-adic cohomology groups by transport of structure.\n\nOur purpose is to provide an alternative proof of the following\ncelebrated theorem of Deligne, known as the \\emph{Riemann Hypothesis}\nfor smooth proper varieties defined over a finite field:", "context": "Throughout this paper, we work over a finite base field\n\\(\\mathbf{F}_q\\). Our notation is as follows: capital Latin letters\nwith a subscript \\(0\\), such as \\(X_0, S_0\\), denote separated schemes\nof finite type defined over \\(\\mathbf{F}_q\\). Removing the subscript\n(e.g., \\(X, S\\)) indicates the base change of these schemes to a fixed\nalgebraic closure \\(\\overline{\\mathbf{F}}_q\\).\n\nWe fix a prime number \\(\\ell\\) which is invertible in\n\\(\\mathbf{F}_q\\), and throughout, we consider only\n\\(\\overline{\\mathbf{Q}}_{\\ell}\\)-sheaves. Script letters with\nsubscript \\(0\\), like \\(\\mathcal{F}_0, \\mathcal{G}_0\\), refer to\nsheaves on \\(X_0, S_0\\). Omitting the subscript denotes their pullback\nto the corresponding scheme over \\(\\overline{\\mathbf{F}}_q\\).\n\n\\medskip\nLet \\(X_0\\) be a separated scheme of finite type over \\(\\mathbf{F}_q\\).\nWe define the zeta function of \\(X_0\\) as\n\\[\nZ(X_0/\\mathbf{F}_q, t) = \\exp\\left \\{ \\sum_{e \\geqslant 1} \\# X_0(\\mathbf{F}_{q^e})\\frac{t^e}{e} \\right \\},\n\\]\nwhich is an element of \\(1 + t\\mathbf{Z}[\\![t]\\!]\\).\nGrothendieck's trace formula expresses the zeta function in terms of the action of the geometric Frobenius on the \\(\\ell\\)-adic cohomology of \\(X\\):\n\\[\nZ(X_0/\\mathbf{F}_q, t) = \\prod_{i=0}^{2\\dim X_0}\n\\det\\left(1 - tF, \\mathrm{H}^i_c(X; \\overline{\\mathbf{Q}}_\\ell)\\right)^{(-1)^{i+1}}.\n\\]\nRecall that \\(F\\), the \\emph{geometric} Frobenius, is the inverse of\nthe arithmetic Frobenius automorphism \\(a \\mapsto a^q\\) in\n\\(\\mathrm{Gal}(\\overline{\\mathbf{F}}_q/\\mathbf{F}_q)\\), acting on the\n\\(\\ell\\)-adic cohomology groups by transport of structure.\n\nOur purpose is to provide an alternative proof of the following\ncelebrated theorem of Deligne, known as the \\emph{Riemann Hypothesis}\nfor smooth proper varieties defined over a finite field:", "full_context": "Throughout this paper, we work over a finite base field\n\\(\\mathbf{F}_q\\). Our notation is as follows: capital Latin letters\nwith a subscript \\(0\\), such as \\(X_0, S_0\\), denote separated schemes\nof finite type defined over \\(\\mathbf{F}_q\\). Removing the subscript\n(e.g., \\(X, S\\)) indicates the base change of these schemes to a fixed\nalgebraic closure \\(\\overline{\\mathbf{F}}_q\\).\n\nWe fix a prime number \\(\\ell\\) which is invertible in\n\\(\\mathbf{F}_q\\), and throughout, we consider only\n\\(\\overline{\\mathbf{Q}}_{\\ell}\\)-sheaves. Script letters with\nsubscript \\(0\\), like \\(\\mathcal{F}_0, \\mathcal{G}_0\\), refer to\nsheaves on \\(X_0, S_0\\). Omitting the subscript denotes their pullback\nto the corresponding scheme over \\(\\overline{\\mathbf{F}}_q\\).\n\n\\medskip\nLet \\(X_0\\) be a separated scheme of finite type over \\(\\mathbf{F}_q\\).\nWe define the zeta function of \\(X_0\\) as\n\\[\nZ(X_0/\\mathbf{F}_q, t) = \\exp\\left \\{ \\sum_{e \\geqslant 1} \\# X_0(\\mathbf{F}_{q^e})\\frac{t^e}{e} \\right \\},\n\\]\nwhich is an element of \\(1 + t\\mathbf{Z}[\\![t]\\!]\\).\nGrothendieck's trace formula expresses the zeta function in terms of the action of the geometric Frobenius on the \\(\\ell\\)-adic cohomology of \\(X\\):\n\\[\nZ(X_0/\\mathbf{F}_q, t) = \\prod_{i=0}^{2\\dim X_0}\n\\det\\left(1 - tF, \\mathrm{H}^i_c(X; \\overline{\\mathbf{Q}}_\\ell)\\right)^{(-1)^{i+1}}.\n\\]\nRecall that \\(F\\), the \\emph{geometric} Frobenius, is the inverse of\nthe arithmetic Frobenius automorphism \\(a \\mapsto a^q\\) in\n\\(\\mathrm{Gal}(\\overline{\\mathbf{F}}_q/\\mathbf{F}_q)\\), acting on the\n\\(\\ell\\)-adic cohomology groups by transport of structure.\n\nOur purpose is to provide an alternative proof of the following\ncelebrated theorem of Deligne, known as the \\emph{Riemann Hypothesis}\nfor smooth proper varieties defined over a finite field:\n\nOur purpose is to provide an alternative proof of the following\ncelebrated theorem of Deligne, known as the \\emph{Riemann Hypothesis}\nfor smooth proper varieties defined over a finite field:\n\nIn terms of the definition below, for a smooth proper variety\n\\(X_0\\) over \\(\\mathbf{F}_q\\), the Riemann Hypothesis asserts that\nevery eigenvalue of the geometric Frobenius acting on\n\\(\\mathrm{H}^i(X;\\overline{\\mathbf{Q}}_{\\ell})\\) has \\(q\\)-weight\nexactly equal to \\(i\\).\n\n\\begin{theorem}\\label{theorem:wrh}\nLet \\(X_{0}\\) be a separated scheme of finite type over\n\\(\\mathbf{F}_{q}\\). Then for every integer \\(i\\), all eigenvalues of Frobenius acting on\n\\(\\mathrm{H}^{i}_{c}(X;\\overline{\\mathbf{Q}}_{\\ell})\\) have\n\\(q\\)-weight \\(\\leq i\\).\n\\end{theorem}\n\n\\begin{proof}[Proof that Theorem~\\ref{theorem:wrh} \\(\\Rightarrow\\) Theorem~\\ref{theorem:rh}]\nLet \\(0\\leq i\\leq 2d\\). Then by Theorem~\\ref{theorem:wrh}, any\nFrobenius eigenvalue \\(\\alpha\\) of\n\\(\\mathrm{H}^{i}(X;\\overline{\\mathbf{Q}}_{\\ell})\\) satisfies the\ninequality\n\\[\n|\\iota(\\alpha)| \\leq q^{\\frac{i}{2}}.\n\\]\nOn the other hand, Poincaré\nduality gives an isomorphism\n\\[\n\\mathrm{H}^i(X; \\overline{\\mathbf{Q}}_\\ell) \\simeq \\mathrm{H}^{2d-i}(X; \\overline{\\mathbf{Q}}_\\ell)^{\\vee} \\otimes \\overline{\\mathbf{Q}}_\\ell(-d).\n\\]\nTherefore, \\(q^{d}/\\alpha\\) is also a Frobenius eigenvalue of \\(\\mathrm{H}^{2d-i}(X;\\overline{\\mathbf{Q}}_{\\ell})\\).\nHence\n\\[\n\\left| \\iota\\left( \\frac{q^{d}}{\\alpha} \\right) \\right| \\leq q^{d-\\frac{i}{2}},\n\\]\nor equivalently \\(|\\iota(\\alpha)| \\geq q^{\\frac{i}{2}}\\).\nThis completes the proof.\n\\end{proof}\n\n\\begin{lemma}[Perverse weak Lefschetz theorem]\n\\label{lemma:gysin}\nLet \\(X\\) be an algebraic variety over an algebraically closed field\n\\(k\\), and let \\(\\iota \\colon X \\to \\mathbf{P}^{N}\\) be a quasi-finite\nmorphism. Suppose\n\\(\\mathcal{F} \\in {}^{\\mathrm{p}}\\mathrm{D}^{\\leq0}_{c}(X)\\) is\nperverse connective. Then, for a general hyperplane\n\\(B \\subset \\mathbf{P}^{N}\\), the Gysin map\n\\[\n\\mathrm{H}^{i-2}_{c}\\left(X \\times_{\\mathbf{P}^{N}} B; \\mathcal{F}(-1)\\right) \\longrightarrow \\mathrm{H}^{i}_{c}(X; \\mathcal{F})\n\\]\nis surjective for \\(i = 1\\), and is an isomorphism for all \\(i \\geq 2\\).\n\\end{lemma}\n\n\\begin{lemma}[Trivial bounds~{\\cite[Proposition~1.4.6]{deligne_weil2}}]\n\\label{lemma:simple}\nLet \\(X_0\\) be a separated scheme of finite type over\n\\(\\mathbf{F}_{q}\\) of dimension \\(\\leq d\\). Suppose\n\\(\\mathcal{F}_{0}\\) is a \\(\\overline{\\mathbf{Q}}_{\\ell}\\)-sheaf on\n\\(X_{0}\\) of pointwise weight \\(\\leq w\\). Then for any isomorphism\n\\(\\iota\\colon\\overline{\\mathbf{Q}}_{\\ell}\\xrightarrow{\\sim}\\mathbf{C}\\),\nthe L-function \\(\\iota L(\\mathcal{F}_0,t)\\) is convergent for all\n\\(|t| < q^{-w/2-d}\\), and has no zeros nor poles in this region.\n\\end{lemma}\n\n\\begin{proof}\nWithout loss of generality we can assume \\(X_0\\) is affine,\ngeometrically reduced, geometrically irreducible, and has dimension\nequal to \\(d\\). After passing to a finite extension, we can find a\nfinite morphism \\(X_0 \\to \\mathbf{A}^{d}_{\\mathbf{F}_{q}}\\). Then\nthere is a constant \\(C\\) independent of \\(e\\), such that\n\\[\n\\# X(\\mathbf{F}_{q^e}) \\leq C (q^{e})^{d}, \\quad e =1,2,\\ldots.\n\\]\nLet\n\\(r=\\max\\{\\dim_{\\overline{\\mathbf{Q}}_{\\ell}}\\mathcal{F}_{\\overline{x}}\\}\\)\nwhere \\(\\overline{x}\\) runs through all geometric points of \\(X_{0}\\).\nThen for any closed point \\(x\\) of \\(X_0\\), we have\n\\[\n|\\iota \\mathrm{Tr}(F_x)| \\leq r \\cdot (q^{\\deg x})^{\\frac{w}{2}}\n\\]\nTherefore, the series\n\\[\n\\iota \\left(\\frac{L^{\\prime}(\\mathcal{F}_0,t)}{L(\\mathcal{F}_0,t)}\\right)\n= \\sum_{e\\geq 1} \\sum_{x \\in X_0(\\mathbf{F}_{q^e})} \\iota \\mathrm{Tr}(F_{x}) \\cdot t^{e-1}\n\\]\nhas a majorant\n\\(rC \\cdot \\sum_{e\\geq 1} (q^{d+\\frac{w}{2}})^{e}\\cdot t^{e-1}\\). Hence,\n\\(\\iota \\left( L^{\\prime}(\\mathcal{F}_0,t) / L(\\mathcal{F}_0,t)\n\\right)\\) is convergent on the disk \\(|t| < q^{-w/2-d}\\), so is\n\\(\\log L(\\mathcal{F}_0,t)\\). The lemma then follows after taking\nexponentiation.\n\\end{proof}\n\nChoose an open embedding\n \\(j\\colon V_0 \\hookrightarrow \\mathbf{A}^1_{\\mathbf{F}_q}\\)\nof a dense Zariski open subset such that\n\\begin{itemize}\n\\item for all \\(m\\),\nthe cohomology sheaves \\(\\mathrm{R}^m \\pi_!(\\overline{\\mathbf{Q}}_{\\ell,\\mathscr{X}_0})\\) of\n\\(\\mathrm{R}\\pi_!(\\overline{\\mathbf{Q}}_{\\ell,\\mathscr{X}_0})\\) are local systems on\n\\(V_0\\), and\n\\item for every geometric point \\(t \\in V_0\\), the\nhypersurface given by \\(tg + (1-t)f = 0\\) has smooth projective\nclosure.\n\\end{itemize}\nBy Lemma~\\ref{lemma:hype} and the proper base change\ntheorem, the sheaf\n\\(\\mathrm{R}^n\\pi_!(\\overline{\\mathbf{Q}}_{\\ell,\\mathscr{X}_0})|_{V_0}\\) is of\npointwise weight \\(\\leq n\\), that is, for\nall closed points \\(x\\) of \\(V_0\\), the eigenvalues of Frobenius\n\\(F_x\\) acting on the stalks are of \\(\\#\\kappa(x)\\)-weight at most\n\\(n\\).\n\n\\begin{theorem}[Deligne~\\cite{deligne_weil1}]\n\\label{theorem:rh}\nAssume \\(X_0\\) is smooth and proper over \\(\\mathbf{F}_q\\).\nSuppose \\(\\alpha\\) is an eigenvalue of the geometric Frobenius \\(F\\)\nacting on\n\\(\\mathrm{H}^i_{c}(X;\\overline{\\mathbf{Q}}_{\\ell})=\\mathrm{H}^i(X;\\overline{\\mathbf{Q}}_{\\ell})\\).\nThen for any isomorphism\n\\(\\iota\\colon \\overline{\\mathbf{Q}}_{\\ell} \\to \\mathbf{C}\\), we have\n\\(|\\iota(\\alpha)|=q^{\\frac{i}{2}}\\).\n\\end{theorem}", "post_theorem_intro_text_len": 6477, "post_theorem_intro_text": "In terms of the definition below, for a smooth proper variety\n\\(X_0\\) over \\(\\mathbf{F}_q\\), the Riemann Hypothesis asserts that\nevery eigenvalue of the geometric Frobenius acting on\n\\(\\mathrm{H}^i(X;\\overline{\\mathbf{Q}}_{\\ell})\\) has \\(q\\)-weight\nexactly equal to \\(i\\).\n\n\\begin{definition}[Weight]\nLet \\(\\iota\\colon \\overline{\\mathbf{Q}}_{\\ell} \\to \\mathbf{C}\\) be an\nisomorphism of fields. Let \\(q > 0\\) be a real number, and let\n\\(\\alpha \\in \\overline{\\mathbf{Q}}_{\\ell}\\). We say that \\(\\alpha\\)\nhas \\(q\\)-\\emph{weight} (or simply \\emph{weight} when the number \\(q\\)\nis understood from the context) \\(\\leqslant w\\) (with respect to\nthe isomorphism \\(\\iota\\)), if\n\\[\n|\\iota(\\alpha)| \\leqslant q^{w/2}.\n\\]\nWe say \\(\\alpha\\) has \\(q\\)-weight equal to \\(w\\), if\n\\(|\\iota(\\alpha)| = q^{w/2}\\)\n\\end{definition}\n\nKatz~\\cite{katz_riemann-hypothesis-for-curves-and-hypersurfaces}\nprovides an (arguably) elementary and accessible proof of the Riemann\nHypothesis for smooth projective hypersurfaces in a projective space.\nWe shall reduce Theorem~\\ref{theorem:rh}\nto this case.\n\n\\begin{theorem}\n\\label{theorem:katz}\nLet \\(X_{0} \\subseteq \\mathbf{P}^{n}\\) be a smooth hypersurface. Then\nthe Riemann Hypothesis holds for \\(X_{0}\\).\n\\end{theorem}\n\nKatz's proof of Theorem~\\ref{theorem:katz} is in effect a\nspecialization argument. Building on Deligne's interpretation of\nRankin's method, Katz showed that to establish the Riemann Hypothesis\nfor \\emph{all} smooth hypersurfaces of degree \\(d\\) in\n\\(\\mathbf{P}^n\\), it is enough to find just \\emph{one} smooth\nhypersurface of degree \\(d\\) in \\(\\mathbf{P}^n\\) for which the Riemann\nHypothesis holds. Depending on whether the degree \\(d\\) is coprime to\nthe characteristic of the base field, we can construct the following\nexamples:\n\\begin{itemize}\n\\item If \\(d\\) is coprime to the characteristic of \\(\\mathbf{F}_q\\),\nWeil~\\cite{weil_equations-in-finite-fields} proved, using only basic\nproperties of Gauss sums, that the so-called \\emph{diagonal hypersurfaces}\n\\[\na_{0} T_{0}^{d} + \\cdots + a_{n} T_{n}^{d} = 0, \\qquad (a_{0}, \\ldots,\na_{n} \\in \\mathbf{F}_q^{\\ast})\n\\]\nsatisfy the Riemann Hypothesis.\n\\item If \\(d\\) is divisible by the characteristic, Katz then used\nproperties of Gauss sums to verify the Riemann Hypothesis for the\nso-called \\emph{Gabber hypersurface}:\n\\[\nT_{0}^{d} + T_{0} T_{1}^{d-1} + \\cdots + T_{n-1} T_{n}^{d-1} = 0.\n\\]\n\\end{itemize}\n\nIn this paper, we will use a degeneration argument to reduce the\ngeneral case of the Riemann Hypothesis for smooth proper varieties\nto the hypersurface case covered by Theorem~\\ref{theorem:katz}.\nEarlier, Scholl~\\cite{scholl_hypersurfaces-and-weil-conjectures}\ndeveloped a reduction method relying on alterations and the weight\nspectral sequence of Steenbrink--Rapoport--Zink. In contrast, the\nengine of our approach is Artin's vanishing\ntheorem~\\cite[Exposé~XIV,~3.1]{sga4} and an elementary ``perverse\ndegeneration lemma'' (\\cite[Proposition~9]{katz_perverse-origin},\n\\cite[Lemma~3.1]{wan-zhang_betti-number-bounds-for-varieties-and-exponential-sums0}).\nSince Artin's vanishing theorem is closely related to the perverse\nt-structure, we will also use some properties of perverse sheaves in a\nsuperficial way.\n\nIn fact, we will prove a slightly more general assertion, which is a\ndirect consequence of the main theorem of Weil\nII~\\cite{deligne_weil2}:\n\n\\begin{theorem}\\label{theorem:wrh}\nLet \\(X_{0}\\) be a separated scheme of finite type over\n\\(\\mathbf{F}_{q}\\). Then for every integer \\(i\\), all eigenvalues of Frobenius acting on\n\\(\\mathrm{H}^{i}_{c}(X;\\overline{\\mathbf{Q}}_{\\ell})\\) have\n\\(q\\)-weight \\(\\leqslant i\\).\n\\end{theorem}\n\n\\begin{proof}[Proof that Theorem~\\ref{theorem:wrh} \\(\\Rightarrow\\) Theorem~\\ref{theorem:rh}]\nLet \\(0\\leqslant i\\leqslant 2d\\). Then by Theorem~\\ref{theorem:wrh}, any\nFrobenius eigenvalue \\(\\alpha\\) of\n\\(\\mathrm{H}^{i}(X;\\overline{\\mathbf{Q}}_{\\ell})\\) satisfies the\ninequality\n\\[\n|\\iota(\\alpha)| \\leqslant q^{\\frac{i}{2}}.\n\\]\nOn the other hand, Poincaré\nduality gives an isomorphism\n\\[\n\\mathrm{H}^i(X; \\overline{\\mathbf{Q}}_\\ell) \\simeq \\mathrm{H}^{2d-i}(X; \\overline{\\mathbf{Q}}_\\ell)^{\\vee} \\otimes \\overline{\\mathbf{Q}}_\\ell(-d).\n\\]\nTherefore, \\(q^{d}/\\alpha\\) is also a Frobenius eigenvalue of \\(\\mathrm{H}^{2d-i}(X;\\overline{\\mathbf{Q}}_{\\ell})\\).\nHence\n\\[\n\\left| \\iota\\left( \\frac{q^{d}}{\\alpha} \\right) \\right| \\leqslant q^{d-\\frac{i}{2}},\n\\]\nor equivalently \\(|\\iota(\\alpha)| \\geqslant q^{\\frac{i}{2}}\\).\nThis completes the proof.\n\\end{proof}\n\nThe paper is organized as follows. In\nSection~\\ref{sec:perv-degeneration}, we give a detailed review of the\nperverse t-structure on a curve and present the ``perverse\ndegeneration lemma'', which is the main technical tool to our proof.\nSection~\\ref{sec:perv} summarizes key properties of perverse sheaves\non general varieties that will be used in our approach. In\nSection~\\ref{sec:trivial}, we discuss Deligne's sheaf-theoretic\nperspective on trivial bounds of character sums, and explain how these\nlead to the lower semicontinuity of weights. While the results of\nSection~\\ref{sec:trivial} are already clearly stated in Weil\nII~\\cite{deligne_weil2}, we include complete proofs here both for\nclarity and to emphasize their fundamental role in the proof.\nFinally, in Section~\\ref{sec:proof}, we assemble these ingredients to\nprove Theorem~\\ref{theorem:wrh}.\n\n\\subsection*{Acknowledgments}\nThis proof arose from teaching a course on the Weil conjectures in\nFall 2025. At first, I planned to cover Scholl's reduction, but I\nnoticed that the perverse degeneration\nlemma~\\ref{lemma:perverse-degeneration-lemma} could be used to give an\narguably less technical proof. The other key input, i.e., Artin's\nvanishing theorem, was needed in\n\\cite{deligne_weil1,scholl_hypersurfaces-and-weil-conjectures} anyway.\nI am grateful to the students in my class for their questions and\nfeedback.\n\nI worked out the details while at the Tianyuan Mathematical Research\nCenter, during the November event ``Exponential sums: Theory,\nComputation, and Applications'' organized by Daqing Wan and Ping Xi.\nI thank the organizers for inviting me, and the Center for a wonderful\nworking environment.\n\nI would like to thank Shizhang Li and Daqing Wan for carefully reading\nthe manuscript and for their suggestions that improved the exposition\nof this paper. I am also grateful to Tony Scholl for sharing the\nhistorical background and motivation behind his\npaper~\\cite{scholl_hypersurfaces-and-weil-conjectures}.", "sketch": "We shall reduce Theorem~\\ref{theorem:rh} to the hypersurface case covered by Theorem~\\ref{theorem:katz}. Theorem~\\ref{theorem:katz} (Riemann Hypothesis for smooth hypersurfaces) is proved by Katz via a specialization argument: “to establish the Riemann Hypothesis for \\emph{all} smooth hypersurfaces of degree \\(d\\) in \\(\\mathbf{P}^n\\), it is enough to find just \\emph{one} smooth hypersurface of degree \\(d\\) in \\(\\mathbf{P}^n\\) for which the Riemann Hypothesis holds,” using diagonal hypersurfaces when \\(d\\) is coprime to \\(\\mathrm{char}(\\mathbf{F}_q)\\) and the Gabber hypersurface when \\(d\\) is divisible by the characteristic.\n\nThe paper’s approach then uses “a degeneration argument to reduce the general case of the Riemann Hypothesis for smooth proper varieties to the hypersurface case,” with Artin’s vanishing theorem and a “perverse degeneration lemma” as the engine.\n\nIt proves a more general statement, Theorem~\\ref{theorem:wrh}, and shows Theorem~\\ref{theorem:wrh} \\(\\Rightarrow\\) Theorem~\\ref{theorem:rh}: from Theorem~\\ref{theorem:wrh}, any Frobenius eigenvalue \\(\\alpha\\) on \\(\\mathrm{H}^i(X;\\overline{\\mathbf{Q}}_\\ell)\\) satisfies \\(|\\iota(\\alpha)|\\le q^{i/2}\\); by Poincar\\'{e} duality \\(\\mathrm{H}^i\\simeq \\mathrm{H}^{2d-i\\,\\vee}\\otimes \\overline{\\mathbf{Q}}_\\ell(-d)\\), \\(q^d/\\alpha\\) is an eigenvalue on \\(\\mathrm{H}^{2d-i}\\), giving \\(\\big|\\iota(q^d/\\alpha)\\big|\\le q^{d-i/2}\\), i.e. \\(|\\iota(\\alpha)|\\ge q^{i/2}\\). Hence \\(|\\iota(\\alpha)|=q^{i/2}\\), proving Theorem~\\ref{theorem:rh}.", "expanded_sketch": "We shall reduce the main theorem to the hypersurface case covered by the following theorem.\n\n\\begin{theorem}\n\\label{theorem:katz}\nLet \\(X_{0} \\subset \\mathbf{P}^{n}\\) be a smooth hypersurface. Then\nthe Riemann Hypothesis holds for \\(X_{0}\\).\n\\end{theorem}\n\nThe preceding theorem (Riemann Hypothesis for smooth hypersurfaces) is proved by Katz via a specialization argument: “to establish the Riemann Hypothesis for \\emph{all} smooth hypersurfaces of degree \\(d\\) in \\(\\mathbf{P}^n\\), it is enough to find just \\emph{one} smooth hypersurface of degree \\(d\\) in \\(\\mathbf{P}^n\\) for which the Riemann Hypothesis holds,” using diagonal hypersurfaces when \\(d\\) is coprime to \\(\\mathrm{char}(\\mathbf{F}_q)\\) and the Gabber hypersurface when \\(d\\) is divisible by the characteristic.\n\nThe paper’s approach then uses “a degeneration argument to reduce the general case of the Riemann Hypothesis for smooth proper varieties to the hypersurface case,” with Artin’s vanishing theorem and a “perverse degeneration lemma” as the engine.\n\nIt proves a more general statement, namely:\n\n\\begin{theorem}\\label{theorem:wrh}\nLet \\(X_{0}\\) be a separated scheme of finite type over\n\\(\\mathbf{F}_{q}\\). Then for every integer \\(i\\), all eigenvalues of Frobenius acting on\n\\(\\mathrm{H}^{i}_{c}(X;\\overline{\\mathbf{Q}}_{\\ell})\\) have\n\\(q\\)-weight \\(\\leq i\\).\n\\end{theorem}\n\nIt then shows that the preceding theorem implies the main theorem: from the preceding theorem, any Frobenius eigenvalue \\(\\alpha\\) on \\(\\mathrm{H}^i(X;\\overline{\\mathbf{Q}}_\\ell)\\) satisfies \\(|\\iota(\\alpha)|\\le q^{i/2}\\); by Poincar\\'{e} duality \\(\\mathrm{H}^i\\simeq \\mathrm{H}^{2d-i\\,\\vee}\\otimes \\overline{\\mathbf{Q}}_\\ell(-d)\\), \\(q^d/\\alpha\\) is an eigenvalue on \\(\\mathrm{H}^{2d-i}\\), giving \\(\\big|\\iota(q^d/\\alpha)\\big|\\le q^{d-i/2}\\), i.e. \\(|\\iota(\\alpha)|\\ge q^{i/2}\\). Hence \\(|\\iota(\\alpha)|=q^{i/2}\\), completing the proof of the main theorem.", "expanded_theorem": "[Deligne~\\cite{deligne_weil1}]\n\\label{theorem:rh}\nAssume \\(X_0\\) is smooth and proper over \\(\\mathbf{F}_q\\).\nSuppose \\(\\alpha\\) is an eigenvalue of the geometric Frobenius \\(F\\)\nacting on\n\\(\\mathrm{H}^i_{c}(X;\\overline{\\mathbf{Q}}_{\\ell})=\\mathrm{H}^i(X;\\overline{\\mathbf{Q}}_{\\ell})\\).\nThen for any isomorphism\n\\(\\iota\\colon \\overline{\\mathbf{Q}}_{\\ell} \\to \\mathbf{C}\\), we have\n\\(|\\iota(\\alpha)|=q^{\\frac{i}{2}}\\).,", "theorem_type": ["Universal", "Equality"], "mcq": {"question": "Let \\(X_0\\) be a smooth proper scheme over the finite field \\(\\mathbf{F}_q\\), and let \\(X\\) denote its base change to a fixed algebraic closure \\(\\overline{\\mathbf{F}}_q\\). Let \\(F\\) be the geometric Frobenius acting on \\(\\ell\\)-adic cohomology, where \\(\\ell\\) is invertible in \\(\\mathbf{F}_q\\). For an integer \\(i\\), let \\(\\alpha\\) be an eigenvalue of \\(F\\) acting on \\(\\mathrm{H}^i_c(X;\\overline{\\mathbf{Q}}_\\ell)=\\mathrm{H}^i(X;\\overline{\\mathbf{Q}}_\\ell)\\). Which statement holds for every isomorphism \\(\\iota:\\overline{\\mathbf{Q}}_\\ell\\xrightarrow{\\sim}\\mathbf{C}\\)?", "correct_choice": {"label": "A", "text": "\\(|\\iota(\\alpha)|=q^{i/2}\\)."}, "choices": [{"label": "B", "text": "\\(|\\iota(\\alpha)|\\le q^{i/2}\\)."}, {"label": "C", "text": "\\(|\\iota(\\alpha)|\\le q^{i}\\)."}, {"label": "D", "text": "There exists an isomorphism \\(\\iota:\\overline{\\mathbf{Q}}_\\ell\\xrightarrow{\\sim}\\mathbf{C}\\) such that \\(|\\iota(\\alpha)|=q^{i/2}\\)."}, {"label": "E", "text": "\\(|\\iota(\\alpha)|=q^{(2\\dim X-i)/2}\\)."}], "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": "drop_duality_lower_bound", "template_used": "property_confusion"}, {"label": "C", "sketch_hook_type": "trace_identity", "tampered_component": "replace_exact_weight_i_over_2_by_coarser_bound", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "trace_identity", "tampered_component": "for_all_iota_to_exists_iota", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "trace_identity", "tampered_component": "misapply_poincare_duality_degree", "template_used": "wildcard"}]}} +{"id": "2601.20237v1", "paper_link": "http://arxiv.org/abs/2601.20237v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{thm:main}\nFix $\\varepsilon > 0$. Then for $n = \\Omega(1/\\varepsilon)$ there is a $Q = O(\\sqrt{n}/\\varepsilon)$ such that we get quantum state transfer between the chains ends with fidelity at least $1-\\varepsilon$ and in time $t_0 < O(n/\\varepsilon)$.", "start_pos": 7106, "end_pos": 7353, "label": "thm:main"}, "ref_dict": {"fig:us": "\\label{fig:us} Transfer fidelity as a function of time in our protocol for a chain of length $N=501$ and $Q = 80$ The second image is zoomed in to near the optimal transfer time. Note that our protoco", "fig:feder": "\\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{feder_zoom.png}\n\\end{minipage}\n \\caption{\\label{fig:feder} Transfer fidelity as a function of time in the Chen et al. protocol for a chain of length $N=501$. The second image is zoomed in to near the optimal transfer time.}\n\\end{center}\n\\end{figure}\n\nOur goal here is to present a simplification of their construction that addresses all the aforementioned issues: we are able to provide completely rigorous mathematical analysis, remove the sensitivity of the readout time (see Figure~\\ref{fig:us}), and even allow for calibration of the fidelity--time tradeoff.\n\n\\subsection{Results}\\label{sec:results}\n\nWe consider a quantum $XX$ spin chain on $n$ nodes with uniform couplings, and apply a magnetic field of strength $Q$ at the 2nd and 2nd-to-last nodes. Then the Hamiltonian restricted to the 1-excitation subspace becomes \n\\[\nH = A_{\\text{path}} +\nQ \\cdot D_{2,n-1}\n\\] where $D_{2,n-1}$ denotes the diagonal matrix whose entries in the 2nd and $n-1$st rows are 1, the rest are 0. \n\n\\begin{definition}\n The continuous time quantum walk of the system is given by $U(t) = e^{it H}$. If $|U(t_0)_{1,n}| > 1- \\ep$, we say that there is quantum state transfer between the ends of the chain at time $t_0$ with fidelity at least $1-\\ep$. \n\\end{definition}\n\n\\begin{theorem}\\label{thm:main}\nFix $\\ep > 0$. Then for $n = \\Omega(1/\\ep)$ there is a $Q = O(\\sqrt{n}/\\ep)$ such that we get quantum state transfer between the chains ends with fidelity at least $1-\\ep$ and in time $t_0 < O(n/\\ep)$.\n\\end{theorem}\n\nIn order to prove this result, in Section~\\ref{sec:vectors} we describe the exact form of the eigenvectors of $H$. Then, in Section~\\ref{sec:fidelity} we will show that for suitably chosen $Q$, exactly two of the eigenvectors will have most of their weight supported on nodes 1 and $n$. Thus by the symmetry of the system they will be approximately $(1/\\sqrt{2}, 0, 0, \\dots, 0, 1/\\sqrt{2})$ and $(1/\\sqrt{2}, 0, 0, \\dots, 0, -1/\\sqrt{2})$. If the corresponding eigenvalues are denoted by $\\lambda_1$ and $\\lambda_2$, then we get strong state transfer at time $t_0 = \\pi/(\\lambda_1-\\lambda_2)$. \n\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{us_large.png}\n\\end{minipage}"}, "pre_theorem_intro_text_len": 3579, "pre_theorem_intro_text": "Quantum state transfer is an important phenomenon that enables transmission of quantum information over physical distances via networks of spin particles. The simplest such network is the spin chain, represented by the path graph. The study of state transfer on it has been initiated by Bose~\\cite{bose2003} and has since received a lot of attention from both the physics and mathematics communities. We refer the reader to the surveys~\\cite{kay2010, QST} for further literature from both of these perspectives.\n\nThe most important features of the state transfer are its fidelity (how likely is the information transferred?) and the transfer time (how long do you have to wait for the transfer to happen?). It has been long known that without some inhomogeneity in the chain, the transfer strength cannot typically be arbitrarily close to 1. One such way of introducing inhomogeneity is to apply magnetic fields at certain positions of the chain. This has been explored extensively, for example in~\\cite{christandl2005,stolze2012,casaccino2009,path}, for the case of the magnetic field being applied to the first and last particles of the chain. It turns out that one can indeed achieve $1-\\varepsilon$ fidelity state transfer using magnetic fields of strength $O(1/\\varepsilon)$, but the transfer time will become exponentially large in the length of the chain - a side-effect that makes this approach impractical. \n\nHowever, Chen et al.~\\cite{chen2016} observed a rather unexpected phenomenon: high fidelity state transfer can be achieved in polynomial time by applying inhomogeneities near, but not exactly at, the two ends. \nYet, \\cite{chen2016} has a shortcoming: they place the inhomogeneity at the 3rd nodes of the chain. This results in the analysis becoming too complex to rigorously carry out, hence they fall back to numerical evidence coupled with a heuristic argument. More importantly, their protocol results in a highly sensitive readout time in the sense that the transfer fidelity fluctuates rapidly near the optimal time. See Figure~\\ref{fig:feder}.\n\n\\begin{figure}\n\\begin{center}\n\\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{feder_large.png}\n\\end{minipage}\n\\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{feder_zoom.png}\n\\end{minipage}\n \\caption{\\label{fig:feder} Transfer fidelity as a function of time in the Chen et al. protocol for a chain of length $N=501$. The second image is zoomed in to near the optimal transfer time.}\n\\end{center}\n\\end{figure}\n\nOur goal here is to present a simplification of their construction that addresses all the aforementioned issues: we are able to provide completely rigorous mathematical analysis, remove the sensitivity of the readout time (see Figure~\\ref{fig:us}), and even allow for calibration of the fidelity--time tradeoff.\n\n\\subsection{Results}\\label{sec:results}\n\nWe consider a quantum $XX$ spin chain on $n$ nodes with uniform couplings, and apply a magnetic field of strength $Q$ at the 2nd and 2nd-to-last nodes. Then the Hamiltonian restricted to the 1-excitation subspace becomes \n\\[\nH = A_{\\text{path}} +\nQ \\cdot D_{2,n-1}\n\\] where $D_{2,n-1}$ denotes the diagonal matrix whose entries in the 2nd and $n-1$st rows are 1, the rest are 0. \n\n\\begin{definition}\n The continuous time quantum walk of the system is given by $U(t) = e^{it H}$. If $|U(t_0)_{1,n}| > 1- \\varepsilon$, we say that there is quantum state transfer between the ends of the chain at time $t_0$ with fidelity at least $1-\\varepsilon$. \n\\end{definition}", "context": "Quantum state transfer is an important phenomenon that enables transmission of quantum information over physical distances via networks of spin particles. The simplest such network is the spin chain, represented by the path graph. The study of state transfer on it has been initiated by Bose~\\cite{bose2003} and has since received a lot of attention from both the physics and mathematics communities. We refer the reader to the surveys~\\cite{kay2010, QST} for further literature from both of these perspectives.\n\nThe most important features of the state transfer are its fidelity (how likely is the information transferred?) and the transfer time (how long do you have to wait for the transfer to happen?). It has been long known that without some inhomogeneity in the chain, the transfer strength cannot typically be arbitrarily close to 1. One such way of introducing inhomogeneity is to apply magnetic fields at certain positions of the chain. This has been explored extensively, for example in~\\cite{christandl2005,stolze2012,casaccino2009,path}, for the case of the magnetic field being applied to the first and last particles of the chain. It turns out that one can indeed achieve $1-\\varepsilon$ fidelity state transfer using magnetic fields of strength $O(1/\\varepsilon)$, but the transfer time will become exponentially large in the length of the chain - a side-effect that makes this approach impractical.\n\nHowever, Chen et al.~\\cite{chen2016} observed a rather unexpected phenomenon: high fidelity state transfer can be achieved in polynomial time by applying inhomogeneities near, but not exactly at, the two ends. \nYet, \\cite{chen2016} has a shortcoming: they place the inhomogeneity at the 3rd nodes of the chain. This results in the analysis becoming too complex to rigorously carry out, hence they fall back to numerical evidence coupled with a heuristic argument. More importantly, their protocol results in a highly sensitive readout time in the sense that the transfer fidelity fluctuates rapidly near the optimal time. See Figure~\\ref{fig:feder}.\n\n\\begin{figure}\n\\begin{center}\n\\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{feder_large.png}\n\\end{minipage}\n\\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{feder_zoom.png}\n\\end{minipage}\n \\caption{\\label{fig:feder} Transfer fidelity as a function of time in the Chen et al. protocol for a chain of length $N=501$. The second image is zoomed in to near the optimal transfer time.}\n\\end{center}\n\\end{figure}\n\nWe consider a quantum $XX$ spin chain on $n$ nodes with uniform couplings, and apply a magnetic field of strength $Q$ at the 2nd and 2nd-to-last nodes. Then the Hamiltonian restricted to the 1-excitation subspace becomes \n\\[\nH = A_{\\text{path}} +\nQ \\cdot D_{2,n-1}\n\\] where $D_{2,n-1}$ denotes the diagonal matrix whose entries in the 2nd and $n-1$st rows are 1, the rest are 0.\n\n\\begin{definition}\n The continuous time quantum walk of the system is given by $U(t) = e^{it H}$. If $|U(t_0)_{1,n}| > 1- \\varepsilon$, we say that there is quantum state transfer between the ends of the chain at time $t_0$ with fidelity at least $1-\\varepsilon$. \n\\end{definition}\n\n\\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{feder_zoom.png}\n\\end{minipage}\n \\caption{\\label{fig:feder} Transfer fidelity as a function of time in the Chen et al. protocol for a chain of length $N=501$. The second image is zoomed in to near the optimal transfer time.}\n\\end{center}\n\\end{figure}\n\nOur goal here is to present a simplification of their construction that addresses all the aforementioned issues: we are able to provide completely rigorous mathematical analysis, remove the sensitivity of the readout time (see Figure~\\ref{fig:us}), and even allow for calibration of the fidelity--time tradeoff.\n\n\\subsection{Results}\\label{sec:results}\n\nWe consider a quantum $XX$ spin chain on $n$ nodes with uniform couplings, and apply a magnetic field of strength $Q$ at the 2nd and 2nd-to-last nodes. Then the Hamiltonian restricted to the 1-excitation subspace becomes \n\\[\nH = A_{\\text{path}} +\nQ \\cdot D_{2,n-1}\n\\] where $D_{2,n-1}$ denotes the diagonal matrix whose entries in the 2nd and $n-1$st rows are 1, the rest are 0. \n\n\\begin{definition}\n The continuous time quantum walk of the system is given by $U(t) = e^{it H}$. If $|U(t_0)_{1,n}| > 1- \\ep$, we say that there is quantum state transfer between the ends of the chain at time $t_0$ with fidelity at least $1-\\ep$. \n\\end{definition}\n\n\\begin{theorem}\\label{thm:main}\nFix $\\ep > 0$. Then for $n = \\Omega(1/\\ep)$ there is a $Q = O(\\sqrt{n}/\\ep)$ such that we get quantum state transfer between the chains ends with fidelity at least $1-\\ep$ and in time $t_0 < O(n/\\ep)$.\n\\end{theorem}\n\nIn order to prove this result, in Section~\\ref{sec:vectors} we describe the exact form of the eigenvectors of $H$. Then, in Section~\\ref{sec:fidelity} we will show that for suitably chosen $Q$, exactly two of the eigenvectors will have most of their weight supported on nodes 1 and $n$. Thus by the symmetry of the system they will be approximately $(1/\\sqrt{2}, 0, 0, \\dots, 0, 1/\\sqrt{2})$ and $(1/\\sqrt{2}, 0, 0, \\dots, 0, -1/\\sqrt{2})$. If the corresponding eigenvalues are denoted by $\\lambda_1$ and $\\lambda_2$, then we get strong state transfer at time $t_0 = \\pi/(\\lambda_1-\\lambda_2)$. \n\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{us_large.png}\n\\end{minipage}", "full_context": "Quantum state transfer is an important phenomenon that enables transmission of quantum information over physical distances via networks of spin particles. The simplest such network is the spin chain, represented by the path graph. The study of state transfer on it has been initiated by Bose~\\cite{bose2003} and has since received a lot of attention from both the physics and mathematics communities. We refer the reader to the surveys~\\cite{kay2010, QST} for further literature from both of these perspectives.\n\nThe most important features of the state transfer are its fidelity (how likely is the information transferred?) and the transfer time (how long do you have to wait for the transfer to happen?). It has been long known that without some inhomogeneity in the chain, the transfer strength cannot typically be arbitrarily close to 1. One such way of introducing inhomogeneity is to apply magnetic fields at certain positions of the chain. This has been explored extensively, for example in~\\cite{christandl2005,stolze2012,casaccino2009,path}, for the case of the magnetic field being applied to the first and last particles of the chain. It turns out that one can indeed achieve $1-\\varepsilon$ fidelity state transfer using magnetic fields of strength $O(1/\\varepsilon)$, but the transfer time will become exponentially large in the length of the chain - a side-effect that makes this approach impractical.\n\nHowever, Chen et al.~\\cite{chen2016} observed a rather unexpected phenomenon: high fidelity state transfer can be achieved in polynomial time by applying inhomogeneities near, but not exactly at, the two ends. \nYet, \\cite{chen2016} has a shortcoming: they place the inhomogeneity at the 3rd nodes of the chain. This results in the analysis becoming too complex to rigorously carry out, hence they fall back to numerical evidence coupled with a heuristic argument. More importantly, their protocol results in a highly sensitive readout time in the sense that the transfer fidelity fluctuates rapidly near the optimal time. See Figure~\\ref{fig:feder}.\n\n\\begin{figure}\n\\begin{center}\n\\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{feder_large.png}\n\\end{minipage}\n\\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{feder_zoom.png}\n\\end{minipage}\n \\caption{\\label{fig:feder} Transfer fidelity as a function of time in the Chen et al. protocol for a chain of length $N=501$. The second image is zoomed in to near the optimal transfer time.}\n\\end{center}\n\\end{figure}\n\nWe consider a quantum $XX$ spin chain on $n$ nodes with uniform couplings, and apply a magnetic field of strength $Q$ at the 2nd and 2nd-to-last nodes. Then the Hamiltonian restricted to the 1-excitation subspace becomes \n\\[\nH = A_{\\text{path}} +\nQ \\cdot D_{2,n-1}\n\\] where $D_{2,n-1}$ denotes the diagonal matrix whose entries in the 2nd and $n-1$st rows are 1, the rest are 0.\n\n\\begin{definition}\n The continuous time quantum walk of the system is given by $U(t) = e^{it H}$. If $|U(t_0)_{1,n}| > 1- \\varepsilon$, we say that there is quantum state transfer between the ends of the chain at time $t_0$ with fidelity at least $1-\\varepsilon$. \n\\end{definition}\n\n\\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{feder_zoom.png}\n\\end{minipage}\n \\caption{\\label{fig:feder} Transfer fidelity as a function of time in the Chen et al. protocol for a chain of length $N=501$. The second image is zoomed in to near the optimal transfer time.}\n\\end{center}\n\\end{figure}\n\nOur goal here is to present a simplification of their construction that addresses all the aforementioned issues: we are able to provide completely rigorous mathematical analysis, remove the sensitivity of the readout time (see Figure~\\ref{fig:us}), and even allow for calibration of the fidelity--time tradeoff.\n\n\\subsection{Results}\\label{sec:results}\n\nWe consider a quantum $XX$ spin chain on $n$ nodes with uniform couplings, and apply a magnetic field of strength $Q$ at the 2nd and 2nd-to-last nodes. Then the Hamiltonian restricted to the 1-excitation subspace becomes \n\\[\nH = A_{\\text{path}} +\nQ \\cdot D_{2,n-1}\n\\] where $D_{2,n-1}$ denotes the diagonal matrix whose entries in the 2nd and $n-1$st rows are 1, the rest are 0. \n\n\\begin{definition}\n The continuous time quantum walk of the system is given by $U(t) = e^{it H}$. If $|U(t_0)_{1,n}| > 1- \\ep$, we say that there is quantum state transfer between the ends of the chain at time $t_0$ with fidelity at least $1-\\ep$. \n\\end{definition}\n\n\\begin{theorem}\\label{thm:main}\nFix $\\ep > 0$. Then for $n = \\Omega(1/\\ep)$ there is a $Q = O(\\sqrt{n}/\\ep)$ such that we get quantum state transfer between the chains ends with fidelity at least $1-\\ep$ and in time $t_0 < O(n/\\ep)$.\n\\end{theorem}\n\nIn order to prove this result, in Section~\\ref{sec:vectors} we describe the exact form of the eigenvectors of $H$. Then, in Section~\\ref{sec:fidelity} we will show that for suitably chosen $Q$, exactly two of the eigenvectors will have most of their weight supported on nodes 1 and $n$. Thus by the symmetry of the system they will be approximately $(1/\\sqrt{2}, 0, 0, \\dots, 0, 1/\\sqrt{2})$ and $(1/\\sqrt{2}, 0, 0, \\dots, 0, -1/\\sqrt{2})$. If the corresponding eigenvalues are denoted by $\\lambda_1$ and $\\lambda_2$, then we get strong state transfer at time $t_0 = \\pi/(\\lambda_1-\\lambda_2)$. \n\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{us_large.png}\n\\end{minipage}\n\nHowever, Chen et al.~\\cite{chen2016} observed a rather unexpected phenomenon: high fidelity state transfer can be achieved in polynomial time by applying inhomogeneities near, but not exactly at, the two ends. \nYet, \\cite{chen2016} has a shortcoming: they place the inhomogeneity at the 3rd nodes of the chain. This results in the analysis becoming too complex to rigorously carry out, hence they fall back to numerical evidence coupled with a heuristic argument. More importantly, their protocol results in a highly sensitive readout time in the sense that the transfer fidelity fluctuates rapidly near the optimal time. See Figure~\\ref{fig:feder}.\n\n\\begin{definition}\n The continuous time quantum walk of the system is given by $U(t) = e^{it H}$. If $|U(t_0)_{1,n}| > 1- \\ep$, we say that there is quantum state transfer between the ends of the chain at time $t_0$ with fidelity at least $1-\\ep$. \n\\end{definition}\n\nIn order to prove this result, in Section~\\ref{sec:vectors} we describe the exact form of the eigenvectors of $H$. Then, in Section~\\ref{sec:fidelity} we will show that for suitably chosen $Q$, exactly two of the eigenvectors will have most of their weight supported on nodes 1 and $n$. Thus by the symmetry of the system they will be approximately $(1/\\sqrt{2}, 0, 0, \\dots, 0, 1/\\sqrt{2})$ and $(1/\\sqrt{2}, 0, 0, \\dots, 0, -1/\\sqrt{2})$. If the corresponding eigenvalues are denoted by $\\lambda_1$ and $\\lambda_2$, then we get strong state transfer at time $t_0 = \\pi/(\\lambda_1-\\lambda_2)$.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{us_large.png}\n\\end{minipage}\n\\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{us_zoom.png}\n\\end{minipage}\n \\caption{\\label{fig:us} Transfer fidelity as a function of time in our protocol for a chain of length $N=501$ and $Q = 80$ The second image is zoomed in to near the optimal transfer time. Note that our protocol has transfer fidelity consistently above 0.95 in a large time window.}\n\\end{center}\n\\end{figure}\n\n\\begin{rmk}\n Our goal is to arrange $Q$ and $k$ such that one of these eigenvectors has almost all of its weight supported on the first and last nodes. This requires $\\lambda$ (and hence $\\cos \\theta)$ to be very small in absolute value. At the same time $\\cos(k\\theta/2 )$ should not be too small. Specifically, as we will later establish, we need \n \\[|\\lambda| \\leq \\left(\\tfrac{\\sqrt{\\ep}}{\\sqrt{k+1}} |\\cos(k\\theta/2)|\\right)\\] \n in order for this eigenvector to help with $1-\\ep$ fidelity state transfer.\n\\end{rmk}\n\n\\begin{lemma}\\label{lem:transfer_fidelity}\n Let $\\psi_1, \\dots, \\psi_n$ be an orthonormal eigenbasis for $H$ with corresponding eigenvalues $\\lambda_1, \\dots, \\lambda_n$. Suppose $\\psi_1$ is symmetric and $\\psi_2$ is alternating, and that both $\\psi_1(1)^2, \\psi_2(1)^2 \\geq \\tfrac{1}{2}-\\tfrac{\\ep}{4}$. Then the transfer fidelity between the endpoints at time $t_0 = \\tfrac{\\pi}{|\\lambda_1-\\lambda_2|}$ is at least $1-\\ep$.\n\\end{lemma}\n\nCombining Lemmas~\\ref{lem:sym_case},~\\ref{lem:alt_case}, and~\\ref{lem:transfer_fidelity} with Claim~\\ref{cl:psi_bound} leads to the following. \n\\begin{corollary}\\label{cor:transfer}\n If $\\theta_1, \\theta_2$ are such that \n \\begin{enumerate}\n \\item $S(\\theta_1) = Q = A(\\theta_2)$\n \\item $2|\\cos\\theta_1| \\leq \\tfrac{\\sqrt{\\ep}}{\\sqrt{k+1}} |\\cos(k\\theta_1/2)|$\n \\item $2|\\cos \\theta_2| \\leq \\tfrac{\\sqrt{\\ep}}{\\sqrt{k+1}} |\\sin(k\\theta_2/2)|$\n \\end{enumerate} then $H$ admits quantum state transfer between the endpoints at time $\\tfrac{\\pi}{2|\\cos \\theta_1 - \\cos \\theta_2|}$ with fidelity at least $1-\\ep$.\n\\end{corollary}\n\nWe are ready to prove Theorem~\\ref{thm:main}. Let $\\ep > 0$ be given. Since $k = n-3 > C/\\ep$, we can choose an $m$ such that \n\\[ k + \\frac{\\sqrt{k \\ep}}{10}+\\frac{1}{2} \\leq 8m+1 \\leq k + \\frac{\\sqrt{k \\ep}}{9}- \\frac{1}{2},\\] hence \\eqref{eq:m_cond} is satisfied with $c= \\sqrt{\\ep}\\cdot \\pi/20$. Then, by Theorem~\\ref{thm:picktheta} there exists $\\theta_1, \\theta_2$ such that \n\\[ S(\\theta_1) = A\n(\\theta_2) = Q = F\\left(\\tfrac{8m+1}{2k}\\pi\\right),\\]\n\\[ \\frac{\\pi}{2} < \\frac{16m+1}{4k}\\pi < \\theta_1 < \\theta_2 < \\frac{16m+3}{4k}\\pi < \\frac{\\pi}{2}+\\frac{\\sqrt{\\ep}}{18 \\sqrt{k}}\\]\n \\[ 2m\\pi + \\frac{\\pi}{8} < \\frac{k}{2}\\theta_1 < \\frac{k}{2}\\theta_2 < 2m\\pi + \\frac{3\\pi}{8},\\] and \n \\[ \\theta_2 - \\theta_1 \\geq \\frac{\\sqrt{2}-1}{3k} \\frac{\\pi^2\\ep}{400}\\]\nHence both $\\cos(\\tfrac{k}{2}\\theta_1)$ and $\\sin(\\tfrac{k}{2}\\theta_2)$ are at least $\\sin(\\pi/8) = \\tfrac{\\sqrt{2-\\sqrt{2}}}{2}$.\nOn the other hand, $\\cos(\\theta_i): (i=1,2)$ is at most $\\tfrac{\\pi \\sqrt{\\ep}}{18\\sqrt{k}}$. Hence\n\\[ 2|\\cos(\\theta_1)| \\leq \\frac{\\pi\\sqrt{\\ep}}{9 \\sqrt{k}} < \\frac{\\sqrt{\\ep}}{\\sqrt{k+1}}\\frac{\\sqrt{2-\\sqrt{2}}}{2} \\leq \\frac{\\sqrt{\\ep}}{\\sqrt{k+1}}\\cos\\left(\\tfrac{k}{2}\\theta_1\\right),\\] and similarly \n\\[ 2|\\cos(\\theta_2)| \\leq \\frac{\\pi\\sqrt{\\ep}}{9 \\sqrt{k}} < \\frac{\\sqrt{\\ep}}{\\sqrt{k+1}}\\frac{\\sqrt{2-\\sqrt{2}}}{2} \\leq \\frac{\\sqrt{\\ep}}{\\sqrt{k+1}}\\sin\\left(\\tfrac{k}{2}\\theta_2\\right),\\] so all the conditions of Corollary~\\ref{cor:transfer} are fulfilled. Thus there is transfer between the endpoints of the chain with fidelity at least $1-\\ep$ and in time \n\\[ \\frac{\\pi}{2|\\cos(\\theta_1)-\\cos(\\theta_2)|} \\approx \\frac{\\pi}{2|\\theta_1 - \\theta_2|} \\geq \\Omega(k/\\ep).\\]\n\n\\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{feder_zoom.png}\n\\end{minipage}\n \\caption{\\label{fig:feder} Transfer fidelity as a function of time in the Chen et al. protocol for a chain of length $N=501$. The second image is zoomed in to near the optimal transfer time.}\n\\end{center}\n\\end{figure}\n\nOur goal here is to present a simplification of their construction that addresses all the aforementioned issues: we are able to provide completely rigorous mathematical analysis, remove the sensitivity of the readout time (see Figure~\\ref{fig:us}), and even allow for calibration of the fidelity--time tradeoff.\n\n\\subsection{Results}\\label{sec:results}\n\nWe consider a quantum $XX$ spin chain on $n$ nodes with uniform couplings, and apply a magnetic field of strength $Q$ at the 2nd and 2nd-to-last nodes. Then the Hamiltonian restricted to the 1-excitation subspace becomes \n\\[\nH = A_{\\text{path}} +\nQ \\cdot D_{2,n-1}\n\\] where $D_{2,n-1}$ denotes the diagonal matrix whose entries in the 2nd and $n-1$st rows are 1, the rest are 0. \n\n\\begin{definition}\n The continuous time quantum walk of the system is given by $U(t) = e^{it H}$. If $|U(t_0)_{1,n}| > 1- \\ep$, we say that there is quantum state transfer between the ends of the chain at time $t_0$ with fidelity at least $1-\\ep$. \n\\end{definition}\n\n\\begin{theorem}\\label{thm:main}\nFix $\\ep > 0$. Then for $n = \\Omega(1/\\ep)$ there is a $Q = O(\\sqrt{n}/\\ep)$ such that we get quantum state transfer between the chains ends with fidelity at least $1-\\ep$ and in time $t_0 < O(n/\\ep)$.\n\\end{theorem}\n\nIn order to prove this result, in Section~\\ref{sec:vectors} we describe the exact form of the eigenvectors of $H$. Then, in Section~\\ref{sec:fidelity} we will show that for suitably chosen $Q$, exactly two of the eigenvectors will have most of their weight supported on nodes 1 and $n$. Thus by the symmetry of the system they will be approximately $(1/\\sqrt{2}, 0, 0, \\dots, 0, 1/\\sqrt{2})$ and $(1/\\sqrt{2}, 0, 0, \\dots, 0, -1/\\sqrt{2})$. If the corresponding eigenvalues are denoted by $\\lambda_1$ and $\\lambda_2$, then we get strong state transfer at time $t_0 = \\pi/(\\lambda_1-\\lambda_2)$. \n\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{us_large.png}\n\\end{minipage}", "post_theorem_intro_text_len": 1173, "post_theorem_intro_text": "In order to prove this result, in Section~\\ref{sec:vectors} we describe the exact form of the eigenvectors of $H$. Then, in Section~\\ref{sec:fidelity} we will show that for suitably chosen $Q$, exactly two of the eigenvectors will have most of their weight supported on nodes 1 and $n$. Thus by the symmetry of the system they will be approximately $(1/\\sqrt{2}, 0, 0, \\dots, 0, 1/\\sqrt{2})$ and $(1/\\sqrt{2}, 0, 0, \\dots, 0, -1/\\sqrt{2})$. If the corresponding eigenvalues are denoted by $\\lambda_1$ and $\\lambda_2$, then we get strong state transfer at time $t_0 = \\pi/(\\lambda_1-\\lambda_2)$. \n\n\\begin{figure}[h!]\n\\begin{center}\n\\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{us_large.png}\n\\end{minipage}\n\\begin{minipage}{0.35\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{us_zoom.png}\n\\end{minipage}\n \\caption{\\label{fig:us} Transfer fidelity as a function of time in our protocol for a chain of length $N=501$ and $Q = 80$ The second image is zoomed in to near the optimal transfer time. Note that our protocol has transfer fidelity consistently above 0.95 in a large time window.}\n\\end{center}\n\\end{figure}", "sketch": "To prove Theorem~\\ref{thm:main}, the introduction outlines the following steps. First, in Section~\\ref{sec:vectors} the authors “describe the exact form of the eigenvectors of $H$.” Next, in Section~\\ref{sec:fidelity} they “show that for suitably chosen $Q$, exactly two of the eigenvectors will have most of their weight supported on nodes 1 and $n$.” By symmetry, these two eigenvectors are approximately\n\\[\n(1/\\sqrt{2}, 0, 0, \\dots, 0, 1/\\sqrt{2})\\quad\\text{and}\\quad (1/\\sqrt{2}, 0, 0, \\dots, 0, -1/\\sqrt{2}).\n\\]\nIf their eigenvalues are $\\lambda_1$ and $\\lambda_2$, then “we get strong state transfer at time $t_0 = \\pi/(\\lambda_1-\\lambda_2)$.”", "expanded_sketch": "To prove the main theorem, the introduction outlines the following steps. First, the authors “describe the exact form of the eigenvectors of $H$.” Next, they “show that for suitably chosen $Q$, exactly two of the eigenvectors will have most of their weight supported on nodes 1 and $n$.” By symmetry, these two eigenvectors are approximately\n\\[\n(1/\\sqrt{2}, 0, 0, \\dots, 0, 1/\\sqrt{2})\\quad\\text{and}\\quad (1/\\sqrt{2}, 0, 0, \\dots, 0, -1/\\sqrt{2}).\n\\]\nIf their eigenvalues are $\\lambda_1$ and $\\lambda_2$, then “we get strong state transfer at time $t_0 = \\pi/(\\lambda_1-\\lambda_2)$.”", "expanded_theorem": "\\label{thm:main}\nFix $\\varepsilon > 0$. Then for $n = \\Omega(1/\\varepsilon)$ there is a $Q = O(\\sqrt{n}/\\varepsilon)$ such that we get quantum state transfer between the chains ends with fidelity at least $1-\\varepsilon$ and in time $t_0 < O(n/\\varepsilon)$.,", "theorem_type": ["Existential–Universal", "Asymptotic or Limit"], "mcq": {"question": "Consider a quantum XX spin chain on n nodes with uniform couplings, with Hamiltonian on the 1-excitation subspace given by H = A_path + Q D_{2,n-1}, where A_path is the adjacency matrix of the path graph on n vertices and D_{2,n-1} is the diagonal matrix with 1 in the 2nd and (n-1)st diagonal entries and 0 elsewhere. Let U(t) = e^{itH}. Say that there is quantum state transfer between the ends of the chain at time t0 with fidelity at least 1-ε if |U(t0)_{1,n}| > 1-ε. Fix ε > 0. Which statement holds for chain lengths n satisfying n = Ω(1/ε)?", "correct_choice": {"label": "A", "text": "There exists a magnetic field strength Q = O(√n/ε) such that the resulting evolution has quantum state transfer between vertices 1 and n with fidelity at least 1-ε at some time t0 satisfying t0 < O(n/ε)."}, "choices": [{"label": "B", "text": "There exists a magnetic field strength $Q = O(\\sqrt{n}/\\varepsilon)$ such that the resulting evolution has quantum state transfer between vertices $1$ and $n$ with fidelity at least $1-\\varepsilon$ at some time $t_0$ satisfying $t_0 < O(\\sqrt{n}/\\varepsilon)$."}, {"label": "C", "text": "There exists a magnetic field strength $Q$ such that the resulting evolution has quantum state transfer between vertices $1$ and $n$ with fidelity at least $1-\\varepsilon$ at some time $t_0$ satisfying $t_0 < O(n/\\varepsilon)$."}, {"label": "D", "text": "There exists an absolute constant $Q = O(1/\\varepsilon)$ such that for every $n = \\Omega(1/\\varepsilon)$ the resulting evolution has quantum state transfer between vertices $1$ and $n$ with fidelity at least $1-\\varepsilon$ at some time $t_0$ satisfying $t_0 < O(n/\\varepsilon)$."}, {"label": "E", "text": "There exists a magnetic field strength $Q = O(\\sqrt{n}/\\varepsilon)$ such that the resulting evolution has perfect quantum state transfer between vertices $1$ and $n$, i.e. $|U(t_0)_{1,n}| = 1$, at some time $t_0$ satisfying $t_0 < O(n/\\varepsilon)$."}], "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": "eigenvalue-gap/time-scale from $t_0=\\pi/(\\lambda_1-\\lambda_2)$", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "explicit asymptotic bound on $Q$", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "dependence of $Q$ on $n$", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "approximate two-eigenvector support yields high fidelity, not exact perfect transfer", "template_used": "wildcard"}]}} +{"id": "2601.20684v2", "paper_link": "http://arxiv.org/abs/2601.20684v2", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{Main}\nLet $(\\bar{v},\\bar{u},\\bar{\\phi})$ be the shock profile described in Lemma~\\ref{SP}, and let $L$ denote the linear operator defined in \\eqref{Ldef}--\\eqref{calAB}. There exists a constant $\\delta_1>0$ such that if the shock strength $\\delta_S$ is less than $\\delta_1$, then the shock profile is strongly spectrally stable in $L^2$. That is,\n\\begin{equation} \\label{sstability}\n\\sigma(L) \\cap \\{ \\lambda \\in \\mathbb{C}: \\operatorname{Re}{\\lambda} \\geq 0 \\} = \\{0\\},\n\\end{equation}\nwhere $\\sigma(L)$ denotes the spectrum of $L$ on $L^2(\\mathbb{R}) \\times L^2(\\mathbb{R})$. Moreover, $\\lambda =0$ is an eigenvalue of $L$ with algebraic multiplicity one.", "start_pos": 13851, "end_pos": 14544, "label": "Main"}, "ref_dict": {"waveeq": "\\begin{subequations} \\label{waveeq}\n\\begin{align}\n&\\label{waveeq_1}- s \\bar{v}_x - \\bar{u}_x =0, \\\\\n&\\label{waveeq2}- s \\bar{u}_x + \\left(\\frac{T}{\\bar{v}} \\right)_x = \\left(\\frac{\\nu \\bar{u}_x}{\\bar{v}} \\right)_x - \\frac{\\bar{\\phi}_x}{\\bar{v}}, \\\\\n&\\label{waveeq3}- \\varepsilon^2 \\left( \\frac{\\bar{\\phi}_x}{\\bar{v}}\\right)_x = 1 - \\bar{v} e^{\\bar{\\phi}},\n\\end{align}\n\\end{subequations}", "Rem_E-cond": "\\begin{remark} \\label{Rem_E-cond}\nThe statement in Theorem~\\ref{Main} that $\\lambda=0$ has algebraic multiplicity one is understood in the sense that the (extended) Evans function associated with the eigenvalue problem for $L$ has a simple zero at $\\lambda=0$; see Remark~\\ref{agree}. With this interpretation, the result in Theorem~\\ref{Main} may be rewritten as the following conditions for the Evans function $\\mathcal D(\\lambda)$:\n\\smallskip\n\\begin{enumerate}\n\\item[$(\\textbf{D1})$] $\\mathcal{D}(\\cdot)$ has no zeros in $\\{ \\operatorname{Re} \\lambda \\geq 0 \\} \\setminus \\{ 0 \\}$.\n\\item[$(\\textbf{D2})$] $(d/d\\lambda) \\mathcal{D}(0) \\neq 0$.\n\\end{enumerate}\n\\smallskip\nHere $(\\textbf{D1})$ is implied by \\eqref{sstability}, and $(\\textbf{D2})$ encodes the above simplicity statement at $\\lambda=0$. We note that the Evans-function conditions $(\\textbf{D1})$--$(\\textbf{D2})$ are necessary and sufficient conditions for linearized and nonlinear stability of shock profiles for a general system of viscous conservation laws in one space dimension \\cite[Section~1.4.2]{ZJL}; see also \\cite{MZ3, MZ4, MZ5}. As shown in Proposition~\\ref{D'rel}, the condition $(\\textbf{D2})$ is equivalent to\n\\begin{equation}\n\\Gamma \\Delta \\neq 0,\n\\end{equation}\nwhere $\\Gamma$ measures transversality of the connection between $(v_-,u_-,\\phi_-)$ and $(v_+,u_+,\\phi_+)$ given by the profile $(\\bar{v},\\bar{u},\\bar{\\phi})$, and $\\Delta$ is the Liu--Majda determinant for the corresponding shock of the associated hyperbolic system, namely the quasi-neutral Euler system:\n\\begin{equation*}\n\\begin{split}\n& v_t - u_y =0, \\\\\n& u_t + (T+1) \\left( \\frac{1}{v} \\right)_y =0,\\\\\n& \\phi = - \\ln v.\n\\end{split}\n\\end{equation*}\nThis system can be obtained by taking the formal limits $\\varepsilon \\to 0$ and $\\nu \\to 0$ in \\eqref{NSP}. For further discussion of the relation \\eqref{gDel} and the associated transversality coefficient in the context of viscous shock profiles, we refer the reader to \\cite[Section~1.4]{MZ2}.\n\\end{remark}", "sstability": "\\begin{equation} \\label{sstability}\n\\sigma(L) \\cap \\{ \\lambda \\in \\mathbb{C}: \\operatorname{Re}{\\lambda} \\geq 0 \\} = \\{0\\},\n\\end{equation}", "5": "\\begin{equation}\\label{5}\nW' = \\mathbb{A}(x,\\lambda) W, \\qquad x\\in\\mathbb{R},\\ \\lambda\\in\\mathbb{C},\n\\end{equation}", "ffcond": "\\begin{equation} \\label{ffcond}\n\\lim_{x \\rightarrow \\pm \\infty} (\\bar{v},\\bar{u},\\bar{\\phi})(x) = (v_\\pm,u_\\pm,\\phi_\\pm), \\quad \\phi_\\pm=-\\ln v_\\pm,\n\\end{equation}", "waveeq2": "\\begin{align}\n&\\label{waveeq_1}- s \\bar{v}_x - \\bar{u}_x =0, \\\\\n&\\label{waveeq2}- s \\bar{u}_x + \\left(\\frac{T}{\\bar{v}} \\right)_x = \\left(\\frac{\\nu \\bar{u}_x}{\\bar{v}} \\right)_x - \\frac{\\bar{\\phi}_x}{\\bar{v}}, \\\\\n&\\label{waveeq3}- \\varepsilon^2 \\left( \\frac{\\bar{\\phi}_x}{\\bar{v}}\\right)_x = 1 - \\bar{v} e^{\\bar{\\phi}},\n\\end{align}", "SP": "\\begin{lemma}[\\cite{DLZ}, Proposition~1.9] \\label{SP}\nFor given $(v_-,u_-)$ with $v_->0$, there exists a positive constant $\\delta_0>0$ such that for any $(v_+,u_+,s)$ satisfying \\eqref{RH}, \\eqref{Lax}, and\n\\begin{equation*}\n|v_+-v_-| =: \\delta_S < \\delta_0,\n\\end{equation*}\nthe ODE system \\eqref{waveeq} admits a unique (up to a shift) shock profile $(\\bar{v},\\bar{u},\\bar{\\phi})(x)$ satisfying \\eqref{ffcond} and\n\\begin{equation} \\label{sprel}\ns\\bar{v}_x = -\\bar{u}_x >0, \\quad \\underline{C} \\bar{u}_x \\leq \\bar{\\phi}_x \\leq \\overline{C} \\bar{u}_x\n\\end{equation}\nfor some positive constants $\\underline{C}$, $\\overline{C}$. Moreover, the unique solution satisfying $\\textstyle \\bar{v}(0)=\\frac{v_-+v_+}{2}$ verifies the derivative bounds\n\\begin{equation} \\label{spbound}\n\\begin{cases}\n\\left| \\frac{d^k}{dx^k} (\\bar{v}-v_+, \\bar{u}-u_+,\\bar{\\phi}-\\phi_+) \\right| \\leq C_k \\delta_S^{k+1} e^{-\\theta \\delta_S |x|}, & x>0 \\\\\n\\left| \\frac{d^k}{dx^k} (\\bar{v}-v_-, \\bar{u}-u_-,\\bar{\\phi}-\\phi_-) \\right| \\leq C_k \\delta_S^{k+1} e^{-\\theta \\delta_S |x|}, & x<0\n\\end{cases}\n\\end{equation}\nfor $k \\in \\mathbb{N} \\cup \\{0 \\}$, where $C_k>0$ and $\\theta>0$ are generic constants.\n\\end{lemma}", "App_B": "\\label{App_B}\n\nIn this appendix, we prove solvability of the linearized Poisson equation. This allows to define the solution operator $\\mathcal{A}^{-1}\\mathcal{B}$ in \\eqref{phisol}.\n\n\\begin{lemma} \\l", "Lax": "\\begin{equation} \\label{Lax}\nv_+ > v_-.\n\\end{equation}", "RH": "\\begin{equation} \\label{RH}\n\\begin{split}\n& -s (v_+-v_-)-(u_+-u_-)=0, \\\\\n& -s (u_+-u_-)+(T+1)\\left(\\frac{1}{v_+} - \\frac{1}{v_-} \\right) = 0,\n\\end{split}\n\\end{equation}", "Main": "\\begin{theorem} \\label{Main}\nLet $(\\bar{v},\\bar{u},\\bar{\\phi})$ be the shock profile described in Lemma~\\ref{SP}, and let $L$ denote the linear operator defined in \\eqref{Ldef}--\\eqref{calAB}. There exists a constant $\\delta_1>0$ such that if the shock strength $\\delta_S$ is less than $\\delta_1$, then the shock profile is strongly spectrally stable in $L^2$. That is,\n\\begin{equation} \\label{sstability}\n\\sigma(L) \\cap \\{ \\lambda \\in \\mathbb{C}: \\operatorname{Re}{\\lambda} \\geq 0 \\} = \\{0\\},\n\\end{equation}\nwhere $\\sigma(L)$ denotes the spectrum of $L$ on $L^2(\\mathbb{R}) \\times L^2(\\mathbb{R})$. Moreover, $\\lambda =0$ is an eigenvalue of $L$ with algebraic multiplicity one.\n\\end{theorem}", "Ldef": "\\begin{equation} \\label{Ldef}\nLU := \\left(A U \\right)_x + \\left( B U_x \\right)_x + \\left(D ( \\mathcal{A}^{-1}\\mathcal{B} U )_x \\right)_x + \\left(E ( \\mathcal{A}^{-1}\\mathcal{B} U )_{xx} \\right)_x,\n\\end{equation}", "NSP": "\\begin{subequations} \\label{NSP}\n\\begin{align}\n& \\label{NSP11} v_t - u_y = 0,\\\\\n& \\label{NSP22} u_t + \\left( \\frac{T}{v} \\right)_y = \\left( \\frac{\\nu u_y}{v} \\right)_y - \\frac{\\phi_y}{v}, \\\\\n& \\label{NSP33} - \\varepsilon^2 \\bigg( \\frac{\\phi_y}{v} \\bigg)_y = 1 - ve^{\\phi}\n\\end{align}\n\\end{subequations}", "ABDE": "\\begin{equation} \\label{ABDE}\n\\begin{split}\nA(x) &= \\begin{pmatrix}\ns & 1 \\\\\n\\frac{T+1}{\\bar{v}^2} - \\frac{\\nu\\bar{u}_x}{\\bar{v}^2} - \\frac{\\varepsilon^2 \\bar{\\phi}_x^2}{\\bar{v}^3} + \\frac{\\varepsilon^2}{\\bar{v}} \\left( \\frac{\\bar{\\phi}_x}{\\bar{v}^2} \\right)_x + \\frac{\\varepsilon^2}{\\bar{v}^2} \\left( \\frac{\\bar{\\phi}_x}{\\bar{v}} \\right)_x & s\n\\end{pmatrix} \\\\\nB(x) &= \\begin{pmatrix}\n0 & 0 \\\\\n\\frac{\\varepsilon^2 \\bar{\\phi}_x}{\\bar{v}^3} & \\frac{\\nu}{\\bar{v}}\n\\end{pmatrix}, \\quad D(x) = \\begin{pmatrix}\n0 & 0 \\\\\n\\frac{\\varepsilon^2 \\bar{\\phi}_x}{\\bar{v}^2} + \\frac{\\varepsilon^2\\bar{v}_x}{\\bar{v}^3} & 0\n\\end{pmatrix}, \\quad E(x) = \\begin{pmatrix}\n0 & 0 \\\\\n-\\frac{\\varepsilon^2}{\\bar{v}^2} & 0\n\\end{pmatrix}.\n\\end{split}\n\\end{equation}", "App_trans": "\\begin{split}\n|\\RNum{1}| & \\leq \\eta \\left( \\operatorname{Re} \\lambda \\int_\\mathbb{R} |v_x|^2 \\, dx + \\int_\\mathbb{R} |v_x|^2 \\, dx \\right) + \\frac{C}{\\eta} \\left( \\operatorname{Re} \\lambda \\int_\\mathbb{R} |u|^2 \\, dx + \\int_\\mathbb{R} |u_x|^2 \\, dx \\right) \\\\\n& \\quad + C \\delta_S \\int_\\mathbb{R} \\left( |v_x|^2 + |u|^2 + |u_x|^2 \\right) \\, dx.\n\\end{split}\n\\end{equation*}\nWe apply Young's inequality to estimate $\\RNum{2}$:\n\\begin{equation*}\n|\\RNum{2}| \\leq \\eta \\int_\\mathbb{R} |v_x|^2 \\, dx + \\frac{C}{\\eta} \\int_\\mathbb{R} \\left( |u|^2 + |u_x|^2 + |\\phi_x|^2 \\right) \\, dx.\n\\end{equation*}\nLastly, using the bound \\eqref{spbound}, we have\n\\begin{equation*}\n|\\RNum{3}| \\leq C \\delta_S \\int_\\mathbb{R} \\left( |v|^2 + |v_x|^2 + |u_x|^2 \\right) \\, dx.\n\\end{equation*}\nCollecting the above estimates and choosing $\\eta$ small enough, we get\n\\begin{equation*}\n\\operatorname{Re}{\\lambda} \\lVert v_x \\rVert_{L^2}^2 + \\lVert v_x \\rVert_{L^2}^2 \\leq C \\delta_S \\lVert v \\rVert_{H^1}^2 + C \\left( \\operatorname{Re}{\\lambda} \\lVert u \\rVert_{L^2}^2 + \\lVert u \\rVert_{H^1}^2 + \\lVert \\phi_x \\rVert_{L^2}^2 \\right).\n\\end{equation*}\nHere, to control the terms $\\operatorname{Re}{\\lambda} \\lVert u \\rVert_{L^2}^2 + \\lvert u_x \\rVert_{L^2}^2$, we make use of \\eqref{uL2}. Then, we have\n\\begin{equation*}\n\\operatorname{Re}{\\lambda} \\lVert v_x \\rVert_{L^2}^2 + \\lVert v_x \\rVert_{L^2}^2 \\leq C \\delta_S \\lVert v \\rVert_{H^1}^2 + C \\left( \\lVert v \\rVert_{L^2}^2 + \\lVert u \\rVert_{L^2}^2 + \\lVert \\phi_x \\rVert_{L^2}^2 \\right).\n\\end{equation*}\nFinally, using the smallness of $\\delta_S$, the desired inequality follows.\n\\end{proof}\n\n\\section{Transversality of the shock profiles} \\label{App_trans}\n\nIn this appendix, we provide the proof of Lemma~\\ref{trans}, establishing the transversality of the small-amplitude shock profiles described in Lemma~\\ref{SP}.\n\n\\begin{proof}[Proof of Lemma~\\ref{trans}]\n\nWhen $\\lambda=0$, integrating the first and third relations in \\eqref{coordinate} from $-\\infty$ to $x$ yields\n\\begin{equation} \\label{Aconstraints}\nv = - \\frac{1}{s}u, \\quad \\tilde{u}' = (b_1' - A_{21})v + (b_2'-s)u - D_{21}\\phi' - E_{21}\\phi''.\n\\end{equation}\nThese constraints reduce \\eqref{5} to a three-dimensional ODE system, namely,\n\\begin{equation} \\label{v_eq}\nV'=\\widetilde{A}(x;s)V,\n\\end{equation}\nwhere $V = (u,\\phi,\\phi')^{\\mathrm{tr}}$. Note that $\\bar{V}' := (\\bar{u}',\\bar{\\phi}',\\bar{\\phi}'')^{\\mathrm{tr}}$ solves \\eqref{v_eq} and is uniformly bounded on $\\mathbb{R}$ by the bound \\eqref{spbound}. A standard fact (see, e.g., \\cite[Section~10.4]{ZH} and \\cite[Section~1.4]{MZ2}) is that the intersection of the manifolds of solutions decaying as $x \\to -\\infty$ and as $x \\to +\\infty$ is transverse if and only if the (variational) equation \\eqref{v_eq} admits a one-dimensional space of solutions bounded on $\\mathbb{R}$ necessarily spanned by $\\bar{V}'$. Thus, it suffices to show that \\eqref{v_eq} has no other globally bounded solution.\n\nLet\n\\begin{equation*}\nA_0 := \\lim_{x \\to -\\infty} \\widetilde{A}(x;v_-^{-1}\\sqrt{T+1}) = \\begin{pmatrix}\n- \\frac{1}{\\nu\\sqrt{T+1}} & \\frac{1}{\\nu} & 0 \\\\\n0 & 0 & 1 \\\\\n-\\frac{v_-}{\\varepsilon^2 \\sqrt{T+1}} & \\frac{v_-}{\\varepsilon^2} & 0\n\\end{pmatrix}\n\\end{equation*}\nThe eigenvalues of $A_0$ are given by\n\\begin{equation*}\n\\begin{split}\n\\sigma_- & = -\\frac{1}{2\\nu\\sqrt{T+1}} - \\frac{1}{2} \\sqrt{\\frac{1}{\\nu^2(T+1)} + \\frac{4v_-}{\\varepsilon^2}} <0, \\\\\n\\sigma_0 & = 0, \\\\\n\\sigma_+ & = -\\frac{1}{2\\nu\\sqrt{T+1}} + \\frac{1}{2} \\sqrt{\\frac{1}{\\nu^2(T+1)} + \\frac{4v_-}{\\varepsilon^2}} >0.\n\\end{split}", "NSP22": "\\begin{align}\n& \\label{NSP11} v_t - u_y = 0,\\\\\n& \\label{NSP22} u_t + \\left( \\frac{T}{v} \\right)_y = \\left( \\frac{\\nu u_y}{v} \\right)_y - \\frac{\\phi_y}{v}, \\\\\n& \\label{NSP33} - \\varepsilon^2 \\bigg( \\frac{\\phi_y}{v} \\bigg)_y = 1 - ve^{\\phi}\n\\end{align}", "calAB": "\\begin{equation} \\label{calAB}\n\\begin{split}\n\\mathcal{A} & =\\bar{v} e^{\\bar{\\phi}}+ \\frac{\\varepsilon^2 \\bar{v}_x}{\\bar{v}^2} \\partial_x-\\frac{\\varepsilon^2}{\\bar{v}} \\partial_{xx}, \\quad \\mathcal{B} = -e^{\\bar{\\phi}} -\\frac{\\varepsilon^2 \\bar{\\phi}_{xx}}{\\bar{v}^2} + \\frac{2\\varepsilon^2 \\bar{v}_x \\bar{\\phi}_x}{\\bar{v}^3}-\\frac{\\varepsilon^2 \\bar{\\phi}_x}{\\bar{v}^2} \\partial_x.\n\\end{split}\n\\end{equation}", "divfor2": "\\begin{align}\n&\\label{divfor1} v_t - sv_x - u_x =0, \\\\\n&\\label{divfor2} u_t - su_x + (T+1) \\bigg( \\frac{1}{v} \\bigg)_x = \\nu \\left( \\frac{u_x}{v} \\right)_x - \\varepsilon^2 \\bigg[ \\frac{1}{v} \\bigg( \\frac{\\phi_x}{v} \\bigg)_x -\\frac{1}{2} \\bigg( \\frac{\\phi_x}{v} \\bigg)^2 \\bigg]_x, \\\\\n&\\label{divfor3} -\\varepsilon^2 \\left( \\frac{\\phi_x}{v} \\right)_x = 1- ve^{\\phi}.\n\\end{align}", "NSP33": "\\begin{align}\n& \\label{NSP11} v_t - u_y = 0,\\\\\n& \\label{NSP22} u_t + \\left( \\frac{T}{v} \\right)_y = \\left( \\frac{\\nu u_y}{v} \\right)_y - \\frac{\\phi_y}{v}, \\\\\n& \\label{NSP33} - \\varepsilon^2 \\bigg( \\frac{\\phi_y}{v} \\bigg)_y = 1 - ve^{\\phi}\n\\end{align}", "waveeq3": "\\begin{align}\n&\\label{waveeq_1}- s \\bar{v}_x - \\bar{u}_x =0, \\\\\n&\\label{waveeq2}- s \\bar{u}_x + \\left(\\frac{T}{\\bar{v}} \\right)_x = \\left(\\frac{\\nu \\bar{u}_x}{\\bar{v}} \\right)_x - \\frac{\\bar{\\phi}_x}{\\bar{v}}, \\\\\n&\\label{waveeq3}- \\varepsilon^2 \\left( \\frac{\\bar{\\phi}_x}{\\bar{v}}\\right)_x = 1 - \\bar{v} e^{\\bar{\\phi}},\n\\end{align}", "agree": "\\begin{remark} \\label{agree}\nOn any simply connected subset of the domain of consistent splitting, zeros of the Evans function correspond to eigenvalues of $L$, counted with algebraic multiplicity; see \\cite{GJ1,GJ2}. The analytic extension constructed above (cf. Lemma~\\ref{lemma0.2}) preserves this property. Indeed, it can be interpreted within the \\emph{extended spectral theory}, in which the order of vanishing of $\\mathcal D(\\lambda)$ agrees with the algebraic multiplicity of $\\lambda$ as an \\emph{effective} eigenvalue; see \\cite[Section~6]{ZH}.\n\\end{remark}", "App_closed": "\\begin{equation}\\label{Biform}\nB[\\phi,\\psi]=\\int_{\\mathbb R} f\\psi\\,dx \\quad \\text{for all} \\ \\psi\\in H^1(\\mathbb R).\n\\end{equation}\nThis implies that the unique weak solution $\\phi \\in H^1$ of \\eqref{Biform} solves \\eqref{5.1} in the distributional sense. Moreover, $\\phi \\in C^2 \\cap H^2$ follows from \\eqref{5.1}, together with the regularity of $v,\\bar{v},\\bar{\\phi}$ and the bound \\eqref{vphibd}.\n\\end{proof}\n\n\\section{Domain and closedness of the linearized operator} \\label{App_closed}\n\nWe show that the linearized operator $L$ defined by \\eqref{Ldef}--\\eqref{ABDE} is closed and densely defined on $L^2(\\mathbb{R}) \\times L^2(\\mathbb{R})$. We define the domain $\\mathcal{H}$ of the operator $L$ as $\\mathcal{H} :=\\{ U =(v,u)^{\\mathrm{tr}}: v \\in H^1(\\mathbb{R}), u \\in H^2(\\mathbb{R}) \\}$.\n\n\\begin{lemma}\nThere exists a constant $C>0$ such that\n\\begin{equation} \\label{2.1}\n\\lVert v \\rVert_{H^1(\\mathbb{R})} + \\lVert u \\rVert_{H^2(\\mathbb{R})} \\leq C \\left( \\lVert LU \\rVert_{L^2(\\mathbb{R})} +\\lVert U \\rVert_{L^2(\\mathbb{R})} \\right)\n\\end{equation}", "D'rel": "\\begin{proposition} \\label{D'rel}\nAssume that the hypotheses of Theorem~\\ref{Main} hold. Let $\\mathcal D(\\lambda)$ be the Evans function for \\eqref{5} defined by \\eqref{Evans}--\\eqref{relabel}. Then, we have\n\\begin{equation}\\label{gDel}\n\\mathcal D'(0) = \\Gamma \\Delta,\n\\end{equation}\nwhere $\\Gamma$ is a constant measuring transversality of the intersection of the stable and unstable manifolds spanned by the bases $\\{\\varphi_1^+, \\varphi_2^+\\}$ and $\\{\\varphi_3^-,\\varphi_4^-,\\varphi_5^- \\}$, respectively, and $\\Delta$ is defined by\n\\begin{equation} \\label{LMdet}\n\\Delta := \\det \\left( U_+ - U_-, r_2^- \\right).\n\\end{equation}\nIn particular, $\\Gamma\\neq 0$ if and only if this intersection is transverse.\n\\end{proposition}", "Outline": "\\begin{split}\n& v_t - u_y =0, \\\\\n& u_t + (T+1) \\left( \\frac{1}{v} \\right)_y =0,\\\\\n& \\phi = - \\ln v.\n\\end{split}\n\\end{equation*}\nThis system can be obtained by taking the formal limits $\\varepsilon \\to 0$ and $\\nu \\to 0$ in \\eqref{NSP}. For further discussion of the relation \\eqref{gDel} and the associated transversality coefficient in the context of viscous shock profiles, we refer the reader to \\cite[Section~1.4]{MZ2}.\n\\end{remark}\n\n\\subsection{Main ideas of the proof} \\label{Outline}\nWe adopt the definitions of the essential spectrum $\\sigma_{\\mathrm{ess}}(L)$ and the point spectrum $\\sigma_{\\mathrm{pt}}(L)$ from \\cite[Chapter~2]{KP} for the closed, densely defined operator $L$. Then the spectrum of the linearized operator consists of the essential and point spectra, i.e., $\\sigma(L) = \\sigma_{\\mathrm{ess}}(L) \\cup \\sigma_{\\mathrm{pt}}(L)$, where $\\sigma_{\\mathrm{pt}}(L)$ coincides with the set of discrete isolated eigenvalues with finite multiplicity. Theorem~\\ref{Main} is proved by combining bounds on $\\sigma_{\\mathrm{ess}}(L)$ and $\\sigma_{\\mathrm{pt}}(L)$ with a verification that the eigenvalue at the origin is simple.\n\nWe first prove the stability condition \\eqref{sstability} for the essential spectrum. By standard arguments of \\cite{Henry, KP}, the bounds on $\\sigma_{\\mathrm{ess}}(L)$ are determined by the essential spectra of the limiting constant-coefficient operators $L_\\pm$, characterized by the associated dispersion relations.\n\nBuilding on the above result, we next show $\\sigma_{\\mathrm{pt}}(L) \\subset \\mathbb{C} \\setminus \\Omega$, where $\\Omega:=\\{\\lambda\\in\\mathbb{C}: \\operatorname{Re}\\lambda\\ge 0\\}\\setminus\\{0\\}$. We consider an integrated linear operator $\\mathcal{L}$, obtained by introducing anti-derivative variables, whose point spectrum coincides with that of $L$ on $\\Omega$. We then establish the desired bounds on the point spectrum via spectral energy estimates for the eigenvalue equation associated with $\\mathcal{L}$, combined with a contradiction argument.\n\nFinally, we prove that the eigenvalue $\\lambda=0$ is simple. Note that the essential spectrum bounds obtained above guarantee that a neighborhood $B_+(0,r)=B(0,r)\\cap\\{\\operatorname{Re}\\lambda>0\\}$ lies in the domain of consistent splitting for the associated first-order system \\eqref{5}. Thus we may define the Evans function on $B_+(0,r)$. For this, we apply the conjugation lemma to construct analytic bases of solutions decaying as $x\\to+\\infty$ and as $x\\to-\\infty$ on $B(0,r)$. We first define the Evans function $\\mathcal{D}(\\lambda)$ on $B_+(0,r)$ using these bases, and then extend it analytically to $B(0,r)$. A direct computation yields\n\\begin{equation*}\n\\mathcal{D}'(0)=\\Gamma\\Delta,\n\\end{equation*}\nwhere $\\Gamma$ and $\\Delta$ are as described in Remark~\\ref{Rem_E-cond}. The Rankine--Hugoniot relation \\eqref{RH} implies $\\Delta\\neq0$, and thus it remains to prove $\\Gamma\\neq0$. Here, we note that the transversality of the small-amplitude shock profiles in Lemma~\\ref{SP} follows essentially from their construction in \\cite[Section~2]{DLZ}. There the profiles are constructed via a center manifold reduction. In particular, for an augmented profile ODE system, the dynamics near a reference equilibrium reduces to a one-dimensional equation on the center manifold, with a heteroclinic orbit that is unique up to translation. Moreover, the linearized flow in the complementary subspace admits an exponential dichotomy. These two observations imply the transverse connection between corresponding stable and unstable manifolds at infinity, and hence $\\Gamma \\neq 0$ for sufficiently small amplitudes. The detailed proof is given in Appendix~\\ref{App_trans}. Since $\\Delta \\neq 0$ and $\\Gamma \\neq 0$, we conclude $\\mathcal{D}'(0)\\neq0$, which implies that $\\lambda=0$ has algebraic multiplicity one.\n\n\\medskip\n\\emph{Plan of the paper.} In Section~\\ref{sec:lin}, we derive the linearized eigenvalue problem about the shock profiles. In Sections~\\ref{sec:Ess} and~\\ref{sec:Point}, we obtain the bounds on the essential spectrum and point spectrum, respectively. These bounds establish the strong spectral stability of the shock profiles stated in Theorem~\\ref{Main}. Section~\\ref{sec:Simplicity} proves the simplicity of the zero eigenvalue by showing that the Evans function has a simple zero at $\\lambda=0$. The appendices collect background material and proofs: we recall the conjugation lemma of \\cite{MeZ}, and we prove solvability of the linearized Poisson equation, the domain and closedness of the linearized operator, higher-order energy estimates deferred from Section~\\ref{sec:Point}, and transversality of the shock profiles.\n\n\\section{Linearization about the profile} \\label{sec:lin}\nWe rewrite the NSP system \\eqref{NSP} in the moving frame $(t,x)$, where $x=y-st$, so that the shock profile defined in Lemma~\\ref{SP} becomes a standing wave within the resulting system. The NSP system in the coordinates $(t,x)$ reads\n\\begin{subequations} \\label{movingNSP}\n\\begin{align}\n& \\label{movingNSP1} v_t - sv_x - u_x =0, \\\\\n& \\label{movingNSP2} u_t - su_x + T \\left( \\frac{1}{v} \\right)_x = \\nu \\left( \\frac{u_x}{v} \\right)_x - \\frac{\\phi_x}{v}, \\\\\n& \\label{movingNSP3} -\\varepsilon^2 \\left( \\frac{\\phi_x}{v} \\right)_x = 1- ve^{\\phi}.\n\\end{align}\n\\end{subequations}\nFrom the Poisson equation \\eqref{movingNSP3}, we derive the following identity:\n\\begin{equation*}\n\\frac{\\phi_x}{v} = \\left[ \\frac{1}{v} + \\frac{\\varepsilon^2}{v} \\left( \\frac{\\phi_x}{v} \\right)_x - \\frac{\\varepsilon^2}{2} \\left( \\frac{\\phi_x}{v} \\right)^2 \\right]_x.\n\\end{equation*}\nSubstituting this into \\eqref{movingNSP2}, the system \\eqref{movingNSP} is rewritten in divergence form as\n\\begin{subequations} \\label{divfor}\n\\begin{align}\n&\\label{divfor1} v_t - sv_x - u_x =0, \\\\\n&\\label{divfor2} u_t - su_x + (T+1) \\bigg( \\frac{1}{v} \\bigg)_x = \\nu \\left( \\frac{u_x}{v} \\right)_x - \\varepsilon^2 \\bigg[ \\frac{1}{v} \\bigg( \\frac{\\phi_x}{v} \\bigg)_x -\\frac{1}{2} \\bigg( \\frac{\\phi_x}{v} \\bigg)^2 \\bigg]_x, \\\\\n&\\label{divfor3} -\\varepsilon^2 \\left( \\frac{\\phi_x}{v} \\right)_x = 1- ve^{\\phi}.\n\\end{align}\n\\end{subequations}\nWe note that this reformulation highlights the structure of the NSP system. In the momentum equation \\eqref{divfor2}, we find that $\\textstyle(\\frac{1}{v})_x$ on the left-hand side contributes to the hyperbolic part of the system, coming from the electric force in \\eqref{movingNSP2}, while the forcing term due to the non-neutral plasma density, depending on $\\varepsilon$, remains separated on the right-hand side.\n\nDefine perturbations around the shock profile $(\\bar{v},\\bar{u},\\bar{\\phi})(x)$ as\n\\begin{equation} \\label{defpert}\n(\\tilde{v},\\tilde{u},\\tilde{\\phi})(t,x) := (v,u,\\phi)(t,x) - (\\bar{v},\\bar{u},\\bar{\\phi})(x).\n\\end{equation}\nThen, the perturbation equations are given by\n\\begin{equation*}\n\\begin{split}\n& \\tilde{v}_t - s\\tilde{v}_x - \\tilde{u}_x = 0, \\\\\n& \\tilde{u}_t - s\\tilde{u}_x + (T+1) \\left( \\frac{1}{v} - \\frac{1}{\\bar{v}} \\right)_x - \\nu \\left( \\frac{u_x}{v} - \\frac{\\bar{u}_x}{\\bar{v}} \\right)_x \\\\\n& \\qquad \\qquad \\qquad = - \\varepsilon^2 \\bigg[ \\frac{1}{v} \\left( \\frac{\\phi_x}{v} \\right)_x - \\frac{1}{\\bar{v}} \\left( \\frac{\\bar{\\phi}_x}{\\bar{v}} \\right)_x \\bigg]_x + \\frac{\\varepsilon^2}{2} \\bigg[ \\left( \\frac{\\phi_x}{v} \\right)^2 - \\left(\\frac{\\bar{\\phi}_x}{\\bar{v}}\\right)^2 \\bigg]_x, \\\\\n& -\\varepsilon^2 \\left( \\frac{\\phi_x}{v}- \\frac{\\bar{\\phi}_x}{\\bar{v}} \\right)_x = \\bar{v}e^{\\bar{\\phi}} - v e^\\phi,\n\\end{split}", "gDel": "\\begin{equation}\\label{gDel}\n\\mathcal D'(0) = \\Gamma \\Delta,\n\\end{equation}"}, "pre_theorem_intro_text_len": 10793, "pre_theorem_intro_text": "We consider the one-dimensional compressible Navier--Stokes--Poisson (NSP) system, which serves as a model for the dynamics of ions in an isothermal plasma in the collision-dominated regime \\cite{GGKS}. In Lagrangian mass coordinates, the NSP system is written as\n\\begin{subequations} \\label{NSP}\n\\begin{align}\n& \\label{NSP11} v_t - u_y = 0,\\\\\n& \\label{NSP22} u_t + \\left( \\frac{T}{v} \\right)_y = \\left( \\frac{\\nu u_y}{v} \\right)_y - \\frac{\\phi_y}{v}, \\\\\n& \\label{NSP33} - \\varepsilon^2 \\bigg( \\frac{\\phi_y}{v} \\bigg)_y = 1 - ve^{\\phi}\n\\end{align}\n\\end{subequations}\nfor $t>0$ and $y \\in \\mathbb{R}$. Here $v=\\tfrac{1}{n}$ is the specific volume for $n>0$, the density of ions, and $\\phi$ is the electric potential. The constants $T>0$, $\\nu>0$ and $\\varepsilon>0$ represent the absolute temperature, viscosity coefficient and Debye length, respectively. In the Poisson equation \\eqref{NSP33}, we have assumed that the electron density $n_e$ is determined by the Boltzmann relation, $n_e = e^\\phi$, which is justified by the physical observation that electrons reach the equilibrium state much faster than ions for varying potential in a plasma \\cite{Ch}.\n\nThe NSP system \\eqref{NSP} admits a smooth traveling-wave solution, called a shock profile, connecting two distinct constant far-field states \\cite{DLZ}. In this paper, we consider small-amplitude shock profiles of \\eqref{NSP} and prove that the linearized operator about the profile has no spectrum in the closed right half-plane except at the origin. Moreover, the zero eigenvalue arising from translation invariance is simple. These two spectral stability conditions are recognized as necessary and sufficient conditions for linear and nonlinear orbital stability within the general stability theory for Lax-type viscous and relaxation shocks \\cite{MZ1, MZ2, MZ3, MZ4, MZ5}.\n\nAs in the general case of viscous and relaxation shocks, the zero eigenvalue and the essential spectrum of the linearized operator are not separated, i.e., there is no spectral gap between them. This motivates us to employ the Evans-function framework of Zumbrun--Howard \\cite{ZH} and Mascia--Zumbrun \\cite{MZ1, MZ2}, based on the gap/conjugation lemma \\cite{GZ, KS, MeZ}. This enables us to characterize the zero eigenvalue by analyzing an \\emph{extended} Evans function at the origin. As a result, the first derivative of the Evans function at the origin is factored into a transversality coefficient and the Liu--Majda determinant for the corresponding quasi-neutral Euler shock; see Remark~\\ref{Rem_E-cond}. This factorization is consistent with corresponding results for viscous conservation laws and relaxation systems \\cite{Z1}. In the low-frequency regime, compared with the Navier--Stokes equations, the (nonlocal) electric force in the momentum equation \\eqref{NSP22} introduces two additional fast modes in the associated eigenvalue problem. These modes consequently contribute to the transversality coefficient at the origin. We then show transversality via an analysis of the variational equation along the profile under the small-amplitude assumption. Further details of the overall argument are provided in Section~\\ref{Outline}.\n\nTo put the present work in perspective, we briefly recall earlier stability results for one-dimensional shock profiles of the NSP system. Under a zero-mass type assumption on the initial perturbation, Duan, Liu, and Zhang \\cite{DLZ} proved asymptotic stability using the classical energy method; see also \\cite{LMY, Zh} for stability results within this zero-mass framework in different perturbation settings. The zero-mass assumption is closely tied to the use of anti-derivative variables and restricts the class of admissible perturbations, in particular excluding perturbations with nonzero total mass that lead to nontrivial modulation of the shock location.\n\nMore recently, Kang, Kwon, and the present author \\cite{KKSh} applied a relative entropy method to establish asymptotic stability up to a time-dependent shift, without imposing the zero-mass restriction; see also \\cite{Sh} for an extension to composite waves. In that approach, the shift is shown to be asymptotically negligible relative to the shock speed, thereby yielding asymptotic orbital stability. However, the estimates obtained in \\cite{KKSh} do not suffice to show that the shift converges to a constant, so a limiting shock location is not identified.\n\nThe nonlinear stability results reviewed above are largely formulated at the level of energy estimates and therefore yield limited quantitative information on the long-time dynamics. In particular, they neither quantify the influence of the Poisson coupling nor provide sharp decay rates for the perturbations under consideration. The spectral stability conditions established in this paper complement these works by characterizing the spectral structure of the associated linearized operator, including the additional modes in the low-frequency regime induced by the nonlocal Poisson coupling. Moreover, these spectral conditions place the stability problem for NSP shocks within the scope of the pointwise semigroup theory of Zumbrun--Howard \\cite{ZH} and Mascia--Zumbrun \\cite{MZ1, MZ2}, in which such conditions yield linear and nonlinear orbital stability with sharp decay rates and identification of the limiting shock location. The corresponding analyses of linear and nonlinear stability for NSP shocks are left to future work.\n\n\\subsection{Shock profiles for the Navier--Stokes--Poisson system}\nWe introduce shock profiles for the NSP system \\eqref{NSP} and their basic properties. A shock profile is a smooth traveling-wave solution of the form $(v,u,\\phi)(t,y)= (\\bar{v},\\bar{u},\\bar{\\phi})(x)$, where $x=y-st$ with shock speed $s$, connecting two distinct constant far-field states $(v_-,u_-,\\phi_-)$ and $(v_+,u_+,\\phi_+)$. Substituting this ansatz into \\eqref{NSP}, we obtain the governing ODEs for the shock profiles:\n\\begin{subequations} \\label{waveeq}\n\\begin{align}\n&\\label{waveeq_1}- s \\bar{v}_x - \\bar{u}_x =0, \\\\\n&\\label{waveeq2}- s \\bar{u}_x + \\left(\\frac{T}{\\bar{v}} \\right)_x = \\left(\\frac{\\nu \\bar{u}_x}{\\bar{v}} \\right)_x - \\frac{\\bar{\\phi}_x}{\\bar{v}}, \\\\\n&\\label{waveeq3}- \\varepsilon^2 \\left( \\frac{\\bar{\\phi}_x}{\\bar{v}}\\right)_x = 1 - \\bar{v} e^{\\bar{\\phi}},\n\\end{align}\n\\end{subequations}\nwith the far-field condition\n\\begin{equation} \\label{ffcond}\n\\lim_{x \\rightarrow \\pm \\infty} (\\bar{v},\\bar{u},\\bar{\\phi})(x) = (v_\\pm,u_\\pm,\\phi_\\pm), \\quad \\phi_\\pm=-\\ln v_\\pm,\n\\end{equation}\nwhere the quasi-neutral relation $\\phi_\\pm=-\\ln v_\\pm$ follows by taking the formal limit $x \\to \\pm \\infty$ in \\eqref{waveeq3}. The shock speed $s$ is determined by the Rankine--Hugoniot condition:\n\\begin{equation} \\label{RH}\n\\begin{split}\n& -s (v_+-v_-)-(u_+-u_-)=0, \\\\\n& -s (u_+-u_-)+(T+1)\\left(\\frac{1}{v_+} - \\frac{1}{v_-} \\right) = 0,\n\\end{split}\n\\end{equation}\nwhich yields\n\\begin{equation} \\label{RH1}\ns = s_\\pm = \\pm \\sqrt{\\frac{T+1}{v_+v_-}}.\n\\end{equation}\nThe second relation in \\eqref{RH} is obtained by rewriting \\eqref{waveeq2} in divergence form and integrating the resulting equation; see \\eqref{divfor2}. In this paper, without loss of generality, we restrict ourselves to the \\emph{2-shock profile} with $s=s_+$, which satisfies the Lax entropy condition\n\\begin{equation} \\label{Lax}\nv_+ > v_-.\n\\end{equation}\nThe existence and uniqueness of the small-amplitude 2-shock profile have been treated in \\cite{DLZ}:\n\n\\begin{lemma}[\\cite{DLZ}, Proposition~1.9] \\label{SP}\nFor given $(v_-,u_-)$ with $v_->0$, there exists a positive constant $\\delta_0>0$ such that for any $(v_+,u_+,s)$ satisfying \\eqref{RH}, \\eqref{Lax}, and\n\\begin{equation*}\n|v_+-v_-| =: \\delta_S < \\delta_0,\n\\end{equation*}\nthe ODE system \\eqref{waveeq} admits a unique (up to a shift) shock profile $(\\bar{v},\\bar{u},\\bar{\\phi})(x)$ satisfying \\eqref{ffcond} and\n\\begin{equation} \\label{sprel}\ns\\bar{v}_x = -\\bar{u}_x >0, \\quad \\underline{C} \\bar{u}_x \\leq \\bar{\\phi}_x \\leq \\overline{C} \\bar{u}_x\n\\end{equation}\nfor some positive constants $\\underline{C}$, $\\overline{C}$. Moreover, the unique solution satisfying $\\textstyle \\bar{v}(0)=\\frac{v_-+v_+}{2}$ verifies the derivative bounds\n\\begin{equation} \\label{spbound}\n\\begin{cases}\n\\left| \\frac{d^k}{dx^k} (\\bar{v}-v_+, \\bar{u}-u_+,\\bar{\\phi}-\\phi_+) \\right| \\leq C_k \\delta_S^{k+1} e^{-\\theta \\delta_S |x|}, & x>0 \\\\\n\\left| \\frac{d^k}{dx^k} (\\bar{v}-v_-, \\bar{u}-u_-,\\bar{\\phi}-\\phi_-) \\right| \\leq C_k \\delta_S^{k+1} e^{-\\theta \\delta_S |x|}, & x<0\n\\end{cases}\n\\end{equation}\nfor $k \\in \\mathbb{N} \\cup \\{0 \\}$, where $C_k>0$ and $\\theta>0$ are generic constants.\n\\end{lemma}\n\n\\subsection{Main result}\nTo state our result, we define the linearized operator $L$ about the profile $(\\bar{v},\\bar{u},\\bar{\\phi})$ on $L^2(\\mathbb{R})\\times L^2(\\mathbb{R})$:\n\\begin{equation} \\label{Ldef}\nLU := \\left(A U \\right)_x + \\left( B U_x \\right)_x + \\left(D ( \\mathcal{A}^{-1}\\mathcal{B} U )_x \\right)_x + \\left(E ( \\mathcal{A}^{-1}\\mathcal{B} U )_{xx} \\right)_x,\n\\end{equation}\nwhere\n\\begin{equation} \\label{ABDE}\n\\begin{split}\nA(x) &= \\begin{pmatrix}\ns & 1 \\\\\n\\frac{T+1}{\\bar{v}^2} - \\frac{\\nu\\bar{u}_x}{\\bar{v}^2} - \\frac{\\varepsilon^2 \\bar{\\phi}_x^2}{\\bar{v}^3} + \\frac{\\varepsilon^2}{\\bar{v}} \\left( \\frac{\\bar{\\phi}_x}{\\bar{v}^2} \\right)_x + \\frac{\\varepsilon^2}{\\bar{v}^2} \\left( \\frac{\\bar{\\phi}_x}{\\bar{v}} \\right)_x & s\n\\end{pmatrix} \\\\\nB(x) &= \\begin{pmatrix}\n0 & 0 \\\\\n\\frac{\\varepsilon^2 \\bar{\\phi}_x}{\\bar{v}^3} & \\frac{\\nu}{\\bar{v}}\n\\end{pmatrix}, \\quad D(x) = \\begin{pmatrix}\n0 & 0 \\\\\n\\frac{\\varepsilon^2 \\bar{\\phi}_x}{\\bar{v}^2} + \\frac{\\varepsilon^2\\bar{v}_x}{\\bar{v}^3} & 0\n\\end{pmatrix}, \\quad E(x) = \\begin{pmatrix}\n0 & 0 \\\\\n-\\frac{\\varepsilon^2}{\\bar{v}^2} & 0\n\\end{pmatrix}.\n\\end{split}\n\\end{equation}\nThe nonlocal operator $\\mathcal{A}^{-1}\\mathcal{B}$ denotes the solution operator of the linearized Poisson equation (cf. Appendix~\\ref{App_B}), where\n\\begin{equation} \\label{calAB}\n\\begin{split}\n\\mathcal{A} & =\\bar{v} e^{\\bar{\\phi}}+ \\frac{\\varepsilon^2 \\bar{v}_x}{\\bar{v}^2} \\partial_x-\\frac{\\varepsilon^2}{\\bar{v}} \\partial_{xx}, \\quad \\mathcal{B} = -e^{\\bar{\\phi}} -\\frac{\\varepsilon^2 \\bar{\\phi}_{xx}}{\\bar{v}^2} + \\frac{2\\varepsilon^2 \\bar{v}_x \\bar{\\phi}_x}{\\bar{v}^3}-\\frac{\\varepsilon^2 \\bar{\\phi}_x}{\\bar{v}^2} \\partial_x.\n\\end{split}\n\\end{equation}\nFor $U=(v,u)^{\\mathrm{tr}}$ we use the notational extension $\\mathcal{A}^{-1}\\mathcal{B}U := (\\mathcal{A}^{-1}\\mathcal{B}v,0)^{\\mathrm{tr}}$, since $D$ and $E$ in \\eqref{ABDE} have vanishing second column, and hence only the first component is used in \\eqref{Ldef}. We note that, as shown in Appendix~\\ref{App_closed}, the operator $L$ is closed and densely defined on $L^2(\\mathbb{R})\\times L^2(\\mathbb{R})$.\n\nThe main result of this paper is as follows.", "context": "\\subsection{Shock profiles for the Navier--Stokes--Poisson system}\nWe introduce shock profiles for the NSP system \\eqref{NSP} and their basic properties. A shock profile is a smooth traveling-wave solution of the form $(v,u,\\phi)(t,y)= (\\bar{v},\\bar{u},\\bar{\\phi})(x)$, where $x=y-st$ with shock speed $s$, connecting two distinct constant far-field states $(v_-,u_-,\\phi_-)$ and $(v_+,u_+,\\phi_+)$. Substituting this ansatz into \\eqref{NSP}, we obtain the governing ODEs for the shock profiles:\n\\begin{subequations} \\label{waveeq}\n\\begin{align}\n&\\label{waveeq_1}- s \\bar{v}_x - \\bar{u}_x =0, \\\\\n&\\label{waveeq2}- s \\bar{u}_x + \\left(\\frac{T}{\\bar{v}} \\right)_x = \\left(\\frac{\\nu \\bar{u}_x}{\\bar{v}} \\right)_x - \\frac{\\bar{\\phi}_x}{\\bar{v}}, \\\\\n&\\label{waveeq3}- \\varepsilon^2 \\left( \\frac{\\bar{\\phi}_x}{\\bar{v}}\\right)_x = 1 - \\bar{v} e^{\\bar{\\phi}},\n\\end{align}\n\\end{subequations}\nwith the far-field condition\n\\begin{equation} \\label{ffcond}\n\\lim_{x \\rightarrow \\pm \\infty} (\\bar{v},\\bar{u},\\bar{\\phi})(x) = (v_\\pm,u_\\pm,\\phi_\\pm), \\quad \\phi_\\pm=-\\ln v_\\pm,\n\\end{equation}\nwhere the quasi-neutral relation $\\phi_\\pm=-\\ln v_\\pm$ follows by taking the formal limit $x \\to \\pm \\infty$ in \\eqref{waveeq3}. The shock speed $s$ is determined by the Rankine--Hugoniot condition:\n\\begin{equation} \\label{RH}\n\\begin{split}\n& -s (v_+-v_-)-(u_+-u_-)=0, \\\\\n& -s (u_+-u_-)+(T+1)\\left(\\frac{1}{v_+} - \\frac{1}{v_-} \\right) = 0,\n\\end{split}\n\\end{equation}\nwhich yields\n\\begin{equation} \\label{RH1}\ns = s_\\pm = \\pm \\sqrt{\\frac{T+1}{v_+v_-}}.\n\\end{equation}\nThe second relation in \\eqref{RH} is obtained by rewriting \\eqref{waveeq2} in divergence form and integrating the resulting equation; see \\eqref{divfor2}. In this paper, without loss of generality, we restrict ourselves to the \\emph{2-shock profile} with $s=s_+$, which satisfies the Lax entropy condition\n\\begin{equation} \\label{Lax}\nv_+ > v_-.\n\\end{equation}\nThe existence and uniqueness of the small-amplitude 2-shock profile have been treated in \\cite{DLZ}:\n\n\\begin{lemma}[\\cite{DLZ}, Proposition~1.9] \\label{SP}\nFor given $(v_-,u_-)$ with $v_->0$, there exists a positive constant $\\delta_0>0$ such that for any $(v_+,u_+,s)$ satisfying \\eqref{RH}, \\eqref{Lax}, and\n\\begin{equation*}\n|v_+-v_-| =: \\delta_S < \\delta_0,\n\\end{equation*}\nthe ODE system \\eqref{waveeq} admits a unique (up to a shift) shock profile $(\\bar{v},\\bar{u},\\bar{\\phi})(x)$ satisfying \\eqref{ffcond} and\n\\begin{equation} \\label{sprel}\ns\\bar{v}_x = -\\bar{u}_x >0, \\quad \\underline{C} \\bar{u}_x \\leq \\bar{\\phi}_x \\leq \\overline{C} \\bar{u}_x\n\\end{equation}\nfor some positive constants $\\underline{C}$, $\\overline{C}$. Moreover, the unique solution satisfying $\\textstyle \\bar{v}(0)=\\frac{v_-+v_+}{2}$ verifies the derivative bounds\n\\begin{equation} \\label{spbound}\n\\begin{cases}\n\\left| \\frac{d^k}{dx^k} (\\bar{v}-v_+, \\bar{u}-u_+,\\bar{\\phi}-\\phi_+) \\right| \\leq C_k \\delta_S^{k+1} e^{-\\theta \\delta_S |x|}, & x>0 \\\\\n\\left| \\frac{d^k}{dx^k} (\\bar{v}-v_-, \\bar{u}-u_-,\\bar{\\phi}-\\phi_-) \\right| \\leq C_k \\delta_S^{k+1} e^{-\\theta \\delta_S |x|}, & x<0\n\\end{cases}\n\\end{equation}\nfor $k \\in \\mathbb{N} \\cup \\{0 \\}$, where $C_k>0$ and $\\theta>0$ are generic constants.\n\\end{lemma}\n\n\\subsection{Main result}\nTo state our result, we define the linearized operator $L$ about the profile $(\\bar{v},\\bar{u},\\bar{\\phi})$ on $L^2(\\mathbb{R})\\times L^2(\\mathbb{R})$:\n\\begin{equation} \\label{Ldef}\nLU := \\left(A U \\right)_x + \\left( B U_x \\right)_x + \\left(D ( \\mathcal{A}^{-1}\\mathcal{B} U )_x \\right)_x + \\left(E ( \\mathcal{A}^{-1}\\mathcal{B} U )_{xx} \\right)_x,\n\\end{equation}\nwhere\n\\begin{equation} \\label{ABDE}\n\\begin{split}\nA(x) &= \\begin{pmatrix}\ns & 1 \\\\\n\\frac{T+1}{\\bar{v}^2} - \\frac{\\nu\\bar{u}_x}{\\bar{v}^2} - \\frac{\\varepsilon^2 \\bar{\\phi}_x^2}{\\bar{v}^3} + \\frac{\\varepsilon^2}{\\bar{v}} \\left( \\frac{\\bar{\\phi}_x}{\\bar{v}^2} \\right)_x + \\frac{\\varepsilon^2}{\\bar{v}^2} \\left( \\frac{\\bar{\\phi}_x}{\\bar{v}} \\right)_x & s\n\\end{pmatrix} \\\\\nB(x) &= \\begin{pmatrix}\n0 & 0 \\\\\n\\frac{\\varepsilon^2 \\bar{\\phi}_x}{\\bar{v}^3} & \\frac{\\nu}{\\bar{v}}\n\\end{pmatrix}, \\quad D(x) = \\begin{pmatrix}\n0 & 0 \\\\\n\\frac{\\varepsilon^2 \\bar{\\phi}_x}{\\bar{v}^2} + \\frac{\\varepsilon^2\\bar{v}_x}{\\bar{v}^3} & 0\n\\end{pmatrix}, \\quad E(x) = \\begin{pmatrix}\n0 & 0 \\\\\n-\\frac{\\varepsilon^2}{\\bar{v}^2} & 0\n\\end{pmatrix}.\n\\end{split}\n\\end{equation}\nThe nonlocal operator $\\mathcal{A}^{-1}\\mathcal{B}$ denotes the solution operator of the linearized Poisson equation (cf. Appendix~\\ref{App_B}), where\n\\begin{equation} \\label{calAB}\n\\begin{split}\n\\mathcal{A} & =\\bar{v} e^{\\bar{\\phi}}+ \\frac{\\varepsilon^2 \\bar{v}_x}{\\bar{v}^2} \\partial_x-\\frac{\\varepsilon^2}{\\bar{v}} \\partial_{xx}, \\quad \\mathcal{B} = -e^{\\bar{\\phi}} -\\frac{\\varepsilon^2 \\bar{\\phi}_{xx}}{\\bar{v}^2} + \\frac{2\\varepsilon^2 \\bar{v}_x \\bar{\\phi}_x}{\\bar{v}^3}-\\frac{\\varepsilon^2 \\bar{\\phi}_x}{\\bar{v}^2} \\partial_x.\n\\end{split}\n\\end{equation}\nFor $U=(v,u)^{\\mathrm{tr}}$ we use the notational extension $\\mathcal{A}^{-1}\\mathcal{B}U := (\\mathcal{A}^{-1}\\mathcal{B}v,0)^{\\mathrm{tr}}$, since $D$ and $E$ in \\eqref{ABDE} have vanishing second column, and hence only the first component is used in \\eqref{Ldef}. We note that, as shown in Appendix~\\ref{App_closed}, the operator $L$ is closed and densely defined on $L^2(\\mathbb{R})\\times L^2(\\mathbb{R})$.\n\nThe main result of this paper is as follows.", "full_context": "\\subsection{Shock profiles for the Navier--Stokes--Poisson system}\nWe introduce shock profiles for the NSP system \\eqref{NSP} and their basic properties. A shock profile is a smooth traveling-wave solution of the form $(v,u,\\phi)(t,y)= (\\bar{v},\\bar{u},\\bar{\\phi})(x)$, where $x=y-st$ with shock speed $s$, connecting two distinct constant far-field states $(v_-,u_-,\\phi_-)$ and $(v_+,u_+,\\phi_+)$. Substituting this ansatz into \\eqref{NSP}, we obtain the governing ODEs for the shock profiles:\n\\begin{subequations} \\label{waveeq}\n\\begin{align}\n&\\label{waveeq_1}- s \\bar{v}_x - \\bar{u}_x =0, \\\\\n&\\label{waveeq2}- s \\bar{u}_x + \\left(\\frac{T}{\\bar{v}} \\right)_x = \\left(\\frac{\\nu \\bar{u}_x}{\\bar{v}} \\right)_x - \\frac{\\bar{\\phi}_x}{\\bar{v}}, \\\\\n&\\label{waveeq3}- \\varepsilon^2 \\left( \\frac{\\bar{\\phi}_x}{\\bar{v}}\\right)_x = 1 - \\bar{v} e^{\\bar{\\phi}},\n\\end{align}\n\\end{subequations}\nwith the far-field condition\n\\begin{equation} \\label{ffcond}\n\\lim_{x \\rightarrow \\pm \\infty} (\\bar{v},\\bar{u},\\bar{\\phi})(x) = (v_\\pm,u_\\pm,\\phi_\\pm), \\quad \\phi_\\pm=-\\ln v_\\pm,\n\\end{equation}\nwhere the quasi-neutral relation $\\phi_\\pm=-\\ln v_\\pm$ follows by taking the formal limit $x \\to \\pm \\infty$ in \\eqref{waveeq3}. The shock speed $s$ is determined by the Rankine--Hugoniot condition:\n\\begin{equation} \\label{RH}\n\\begin{split}\n& -s (v_+-v_-)-(u_+-u_-)=0, \\\\\n& -s (u_+-u_-)+(T+1)\\left(\\frac{1}{v_+} - \\frac{1}{v_-} \\right) = 0,\n\\end{split}\n\\end{equation}\nwhich yields\n\\begin{equation} \\label{RH1}\ns = s_\\pm = \\pm \\sqrt{\\frac{T+1}{v_+v_-}}.\n\\end{equation}\nThe second relation in \\eqref{RH} is obtained by rewriting \\eqref{waveeq2} in divergence form and integrating the resulting equation; see \\eqref{divfor2}. In this paper, without loss of generality, we restrict ourselves to the \\emph{2-shock profile} with $s=s_+$, which satisfies the Lax entropy condition\n\\begin{equation} \\label{Lax}\nv_+ > v_-.\n\\end{equation}\nThe existence and uniqueness of the small-amplitude 2-shock profile have been treated in \\cite{DLZ}:\n\n\\begin{lemma}[\\cite{DLZ}, Proposition~1.9] \\label{SP}\nFor given $(v_-,u_-)$ with $v_->0$, there exists a positive constant $\\delta_0>0$ such that for any $(v_+,u_+,s)$ satisfying \\eqref{RH}, \\eqref{Lax}, and\n\\begin{equation*}\n|v_+-v_-| =: \\delta_S < \\delta_0,\n\\end{equation*}\nthe ODE system \\eqref{waveeq} admits a unique (up to a shift) shock profile $(\\bar{v},\\bar{u},\\bar{\\phi})(x)$ satisfying \\eqref{ffcond} and\n\\begin{equation} \\label{sprel}\ns\\bar{v}_x = -\\bar{u}_x >0, \\quad \\underline{C} \\bar{u}_x \\leq \\bar{\\phi}_x \\leq \\overline{C} \\bar{u}_x\n\\end{equation}\nfor some positive constants $\\underline{C}$, $\\overline{C}$. Moreover, the unique solution satisfying $\\textstyle \\bar{v}(0)=\\frac{v_-+v_+}{2}$ verifies the derivative bounds\n\\begin{equation} \\label{spbound}\n\\begin{cases}\n\\left| \\frac{d^k}{dx^k} (\\bar{v}-v_+, \\bar{u}-u_+,\\bar{\\phi}-\\phi_+) \\right| \\leq C_k \\delta_S^{k+1} e^{-\\theta \\delta_S |x|}, & x>0 \\\\\n\\left| \\frac{d^k}{dx^k} (\\bar{v}-v_-, \\bar{u}-u_-,\\bar{\\phi}-\\phi_-) \\right| \\leq C_k \\delta_S^{k+1} e^{-\\theta \\delta_S |x|}, & x<0\n\\end{cases}\n\\end{equation}\nfor $k \\in \\mathbb{N} \\cup \\{0 \\}$, where $C_k>0$ and $\\theta>0$ are generic constants.\n\\end{lemma}\n\n\\subsection{Main result}\nTo state our result, we define the linearized operator $L$ about the profile $(\\bar{v},\\bar{u},\\bar{\\phi})$ on $L^2(\\mathbb{R})\\times L^2(\\mathbb{R})$:\n\\begin{equation} \\label{Ldef}\nLU := \\left(A U \\right)_x + \\left( B U_x \\right)_x + \\left(D ( \\mathcal{A}^{-1}\\mathcal{B} U )_x \\right)_x + \\left(E ( \\mathcal{A}^{-1}\\mathcal{B} U )_{xx} \\right)_x,\n\\end{equation}\nwhere\n\\begin{equation} \\label{ABDE}\n\\begin{split}\nA(x) &= \\begin{pmatrix}\ns & 1 \\\\\n\\frac{T+1}{\\bar{v}^2} - \\frac{\\nu\\bar{u}_x}{\\bar{v}^2} - \\frac{\\varepsilon^2 \\bar{\\phi}_x^2}{\\bar{v}^3} + \\frac{\\varepsilon^2}{\\bar{v}} \\left( \\frac{\\bar{\\phi}_x}{\\bar{v}^2} \\right)_x + \\frac{\\varepsilon^2}{\\bar{v}^2} \\left( \\frac{\\bar{\\phi}_x}{\\bar{v}} \\right)_x & s\n\\end{pmatrix} \\\\\nB(x) &= \\begin{pmatrix}\n0 & 0 \\\\\n\\frac{\\varepsilon^2 \\bar{\\phi}_x}{\\bar{v}^3} & \\frac{\\nu}{\\bar{v}}\n\\end{pmatrix}, \\quad D(x) = \\begin{pmatrix}\n0 & 0 \\\\\n\\frac{\\varepsilon^2 \\bar{\\phi}_x}{\\bar{v}^2} + \\frac{\\varepsilon^2\\bar{v}_x}{\\bar{v}^3} & 0\n\\end{pmatrix}, \\quad E(x) = \\begin{pmatrix}\n0 & 0 \\\\\n-\\frac{\\varepsilon^2}{\\bar{v}^2} & 0\n\\end{pmatrix}.\n\\end{split}\n\\end{equation}\nThe nonlocal operator $\\mathcal{A}^{-1}\\mathcal{B}$ denotes the solution operator of the linearized Poisson equation (cf. Appendix~\\ref{App_B}), where\n\\begin{equation} \\label{calAB}\n\\begin{split}\n\\mathcal{A} & =\\bar{v} e^{\\bar{\\phi}}+ \\frac{\\varepsilon^2 \\bar{v}_x}{\\bar{v}^2} \\partial_x-\\frac{\\varepsilon^2}{\\bar{v}} \\partial_{xx}, \\quad \\mathcal{B} = -e^{\\bar{\\phi}} -\\frac{\\varepsilon^2 \\bar{\\phi}_{xx}}{\\bar{v}^2} + \\frac{2\\varepsilon^2 \\bar{v}_x \\bar{\\phi}_x}{\\bar{v}^3}-\\frac{\\varepsilon^2 \\bar{\\phi}_x}{\\bar{v}^2} \\partial_x.\n\\end{split}\n\\end{equation}\nFor $U=(v,u)^{\\mathrm{tr}}$ we use the notational extension $\\mathcal{A}^{-1}\\mathcal{B}U := (\\mathcal{A}^{-1}\\mathcal{B}v,0)^{\\mathrm{tr}}$, since $D$ and $E$ in \\eqref{ABDE} have vanishing second column, and hence only the first component is used in \\eqref{Ldef}. We note that, as shown in Appendix~\\ref{App_closed}, the operator $L$ is closed and densely defined on $L^2(\\mathbb{R})\\times L^2(\\mathbb{R})$.\n\nThe main result of this paper is as follows.\n\nThe main result of this paper is as follows.\n\nFinally, we prove that the eigenvalue $\\lambda=0$ is simple. Note that the essential spectrum bounds obtained above guarantee that a neighborhood $B_+(0,r)=B(0,r)\\cap\\{\\operatorname{Re}\\lambda>0\\}$ lies in the domain of consistent splitting for the associated first-order system \\eqref{5}. Thus we may define the Evans function on $B_+(0,r)$. For this, we apply the conjugation lemma to construct analytic bases of solutions decaying as $x\\to+\\infty$ and as $x\\to-\\infty$ on $B(0,r)$. We first define the Evans function $\\mathcal{D}(\\lambda)$ on $B_+(0,r)$ using these bases, and then extend it analytically to $B(0,r)$. A direct computation yields\n\\begin{equation*}\n\\mathcal{D}'(0)=\\Gamma\\Delta,\n\\end{equation*}\nwhere $\\Gamma$ and $\\Delta$ are as described in Remark~\\ref{Rem_E-cond}. The Rankine--Hugoniot relation \\eqref{RH} implies $\\Delta\\neq0$, and thus it remains to prove $\\Gamma\\neq0$. Here, we note that the transversality of the small-amplitude shock profiles in Lemma~\\ref{SP} follows essentially from their construction in \\cite[Section~2]{DLZ}. There the profiles are constructed via a center manifold reduction. In particular, for an augmented profile ODE system, the dynamics near a reference equilibrium reduces to a one-dimensional equation on the center manifold, with a heteroclinic orbit that is unique up to translation. Moreover, the linearized flow in the complementary subspace admits an exponential dichotomy. These two observations imply the transverse connection between corresponding stable and unstable manifolds at infinity, and hence $\\Gamma \\neq 0$ for sufficiently small amplitudes. The detailed proof is given in Appendix~\\ref{App_trans}. Since $\\Delta \\neq 0$ and $\\Gamma \\neq 0$, we conclude $\\mathcal{D}'(0)\\neq0$, which implies that $\\lambda=0$ has algebraic multiplicity one.\n\n\\begin{lemma} \\label{Lemma:3.2}\nAssume that the hypotheses of Theorem~\\ref{Main} hold, and denote the shock profile $(\\bar{v},\\bar{u},\\bar{\\phi})$ by $(\\hat{v},\\hat{u},\\hat{\\phi})$. Suppose that $\\lambda \\in \\Omega $ is an eigenvalue of the integrated operator $\\mathcal{L}$ with the corresponding eigenfunction $(\\Phi,\\Psi) \\in H^2(\\mathbb{R}) \\times H^3(\\mathbb{R})$. Then it holds that\n\\begin{equation} \\label{3.4}\n\\begin{split}\n& \\operatorname{Re}{\\lambda} \\lVert (\\Phi,\\Psi) \\rVert_{H^2(\\mathbb{R})}^2 + c \\left( \\lVert \\lvert \\hat{v}_x \\rvert^{1/2} \\Psi \\rVert_{L^2(\\mathbb{R})}^2 + \\lVert \\Phi_x \\rVert_{H^1(\\mathbb{R})}^2 + \\lVert \\Psi_x \\rVert_{H^2(\\mathbb{R})}^2 \\right) \\leq 0\n\\end{split}\n\\end{equation}\nfor some constant $c>0$.\n\\end{lemma}\n\nIn this section, we prove simplicity of the zero eigenvalue of the linearized operator $L$ by computing the first-derivative of the associated Evans function at $\\lambda=0$. Following \\cite{GZ, MZ1, ZH}, we begin by rewriting the eigenvalue equation $(L-\\lambda I)U=0$ as the first-order ODE system\n\\begin{equation}\\label{5}\nW' = \\mathbb{A}(x,\\lambda) W, \\qquad x\\in\\mathbb{R},\\ \\lambda\\in\\mathbb{C},\n\\end{equation}\nin the coordinates:\n\\begin{equation} \\label{coord}\nW = (v, u, \\tilde{u}', \\phi, \\phi')^{\\mathrm{tr}},\n\\end{equation}\nwhere $\\tilde{u} = b_1 v + b_2 u$, $b_j$ denotes the $(2,j)$-component of the matrix $B$ defined in \\eqref{ABDE}, and $\\phi = \\mathcal{A}^{-1}\\mathcal{B}v$ with $\\mathcal{A}, \\mathcal{B}$ in \\eqref{calAB}. The coefficient matrix $\\mathbb{A}$ is defined by the relations\n\\begin{subequations} \\label{coordinate}\n\\begin{align}\n\\label{co1} v' & = \\frac{\\lambda}{s} v - \\frac{1}{s}u', \\\\\n\\label{co2} u' & = \\frac{1}{b_2} \\left( \\tilde{u}' -b_1' v - b_1v' - b_2'u \\right), \\\\\n\\label{co3} \\tilde{u}'' &= \\Big( (b_1'-A_{21})v + (b_2'-s)u - D_{21}\\phi' - E_{21}\\phi'' \\Big)' + \\lambda u, \\\\\n\\label{co4} \\phi'' & = \\bigg( \\frac{\\bar{\\phi}''}{\\bar{v}} - \\frac{2\\bar{v}'\\bar{\\phi}'}{\\bar{v}^2} + \\frac{\\bar{v}e^{\\bar{\\phi}}}{\\varepsilon^2} \\bigg) v + \\frac{\\bar{\\phi}'}{\\bar{v}} v' + \\frac{\\bar{v}^2e^{\\bar{\\phi}}}{\\varepsilon^2} \\phi + \\frac{\\bar{v}'}{\\bar{v}}\\phi',\n\\end{align}\n\\end{subequations}\nand the limits $ \\mathbb{A}_\\pm (\\lambda) := \\lim_{x \\rightarrow \\pm \\infty} \\mathbb{A}(x,\\lambda)$ are given by\n\\begin{equation} \\label{A_pm}\n\\mathbb{A}_\\pm (\\lambda) = \\begin{pmatrix}\n\\frac{\\lambda}{s} & 0 & - \\frac{v_\\pm}{s\\nu} & 0 & 0 \\\\\n0 & 0 & \\frac{v_\\pm}{\\nu} & 0 & 0 \\\\\n- \\frac{T \\lambda}{sv_\\pm^2} & \\lambda & \\frac{T}{s\\nu v_\\pm} - \\frac{sv_\\pm}{\\nu} & 0 & \\frac{1}{v_\\pm} \\\\\n0 & 0 & 0 & 0 & 1 \\\\\n\\frac{1}{\\varepsilon^2} & 0 & 0 & \\frac{v_\\pm}{\\varepsilon^2} & 0 \n\\end{pmatrix}.\n\\end{equation}\nBy Lemma~\\ref{SP}, the matrix $\\mathbb{A}(x,\\lambda)$ satisfies the assumption \\textup{(H0)} in Appendix~\\ref{Appendix_A}. We briefly indicate how the relations in \\eqref{coordinate} define the coefficient matrix $\\mathbb{A}(x,\\lambda)$. First, since $sb_2-b_1\\neq 0$, the first two relations \\eqref{co1}--\\eqref{co2} determine $(v',u')$ uniquely in terms of $W$. Next, differentiating \\eqref{co4} and using the resulting expression for $v'$ allows us to write $\\phi'''$ in terms of $W$ and $\\tilde{u}''$. Substituting these expressions for $v',u',\\phi'',$ and $\\phi'''$ into \\eqref{co3}, and using $\\textstyle \\frac{sb_2}{sb_2-b_1} \\neq 0$, we obtain that $\\tilde{u}''$ is uniquely determined by $W$. Consequently, \\eqref{coordinate} closes to a first-order system of the form \\eqref{5}.\n\nWe complete the construction of the Evans function by fixing a normalization at $\\lambda=0$. By the translation invariance of the shock, the linearized operator $L$ has a zero eigenvalue with the corresponding eigenfunction $\\bar{U}'=(\\bar{v}',\\bar{u}')^{\\mathrm{tr}}$. Equivalently, a direct computation shows that\n\\begin{equation*}\nW_0:= \\big(\\bar{v}',\\bar{u}',(b_1\\bar{v}' + b_2\\bar{u}')',\\bar{\\phi}',\\bar{\\phi}'' \\big)^{\\mathrm{tr}}\n\\end{equation*}\nsolves the first-order ODE system \\eqref{5} at $\\lambda=0$, i.e.,\n\\begin{equation*}\nW_0' = \\mathbb{A}(x,0) W_0.\n\\end{equation*}\nMoreover, $W_0(x)$ decays exponentially as $x\\to\\pm\\infty$ by Lemma~\\ref{SP}. At $\\lambda=0$, the modes $\\varphi_1^+$, $\\varphi_2^+$, $\\varphi_3^-$, and $\\varphi_4^-$ are fast,\nwhereas $\\varphi_5^-$ is a slow (center) mode with $\\mu_5^-(0)=0$. Therefore, any solution that decays exponentially as $x\\to-\\infty$ lies in the span of the fast modes $\\varphi_3^-$ and $\\varphi_4^-$, and similarly any solution that decays exponentially as $x\\to+\\infty$ lies in the span of $\\varphi_1^+$ and $\\varphi_2^+$. Since $W_0$ decays at both ends, we may perform an analytic change of basis within the fast modes and relabel the resulting basis so that\n\\begin{equation}\\label{relabel}\n\\varphi_1^+(x;0)\\equiv \\varphi_3^-(x;0)\\equiv W_0(x), \\quad x\\in\\mathbb{R}.\n\\end{equation}\nThe remaining fast modes $\\varphi_2^+$ and $\\varphi_4^-$ are modified accordingly, but still span the corresponding fast decaying manifolds, while $\\varphi_5^-$ is unchanged.", "post_theorem_intro_text_len": 6236, "post_theorem_intro_text": "\\begin{remark} \\label{Rem_E-cond}\nThe statement in Theorem~\\ref{Main} that $\\lambda=0$ has algebraic multiplicity one is understood in the sense that the (extended) Evans function associated with the eigenvalue problem for $L$ has a simple zero at $\\lambda=0$; see Remark~\\ref{agree}. With this interpretation, the result in Theorem~\\ref{Main} may be rewritten as the following conditions for the Evans function $\\mathcal D(\\lambda)$:\n\\smallskip\n\\begin{enumerate}\n\\item[$(\\textbf{D1})$] $\\mathcal{D}(\\cdot)$ has no zeros in $\\{ \\operatorname{Re} \\lambda \\geq 0 \\} \\setminus \\{ 0 \\}$.\n\\item[$(\\textbf{D2})$] $(d/d\\lambda) \\mathcal{D}(0) \\neq 0$.\n\\end{enumerate}\n\\smallskip\nHere $(\\textbf{D1})$ is implied by \\eqref{sstability}, and $(\\textbf{D2})$ encodes the above simplicity statement at $\\lambda=0$. We note that the Evans-function conditions $(\\textbf{D1})$--$(\\textbf{D2})$ are necessary and sufficient conditions for linearized and nonlinear stability of shock profiles for a general system of viscous conservation laws in one space dimension \\cite[Section~1.4.2]{ZJL}; see also \\cite{MZ3, MZ4, MZ5}. As shown in Proposition~\\ref{D'rel}, the condition $(\\textbf{D2})$ is equivalent to\n\\begin{equation}\n\\Gamma \\Delta \\neq 0,\n\\end{equation}\nwhere $\\Gamma$ measures transversality of the connection between $(v_-,u_-,\\phi_-)$ and $(v_+,u_+,\\phi_+)$ given by the profile $(\\bar{v},\\bar{u},\\bar{\\phi})$, and $\\Delta$ is the Liu--Majda determinant for the corresponding shock of the associated hyperbolic system, namely the quasi-neutral Euler system:\n\\begin{equation*}\n\\begin{split}\n& v_t - u_y =0, \\\\\n& u_t + (T+1) \\left( \\frac{1}{v} \\right)_y =0,\\\\\n& \\phi = - \\ln v.\n\\end{split}\n\\end{equation*}\nThis system can be obtained by taking the formal limits $\\varepsilon \\to 0$ and $\\nu \\to 0$ in \\eqref{NSP}. For further discussion of the relation \\eqref{gDel} and the associated transversality coefficient in the context of viscous shock profiles, we refer the reader to \\cite[Section~1.4]{MZ2}.\n\\end{remark}\n\n\\subsection{Main ideas of the proof} \\label{Outline}\nWe adopt the definitions of the essential spectrum $\\sigma_{\\mathrm{ess}}(L)$ and the point spectrum $\\sigma_{\\mathrm{pt}}(L)$ from \\cite[Chapter~2]{KP} for the closed, densely defined operator $L$. Then the spectrum of the linearized operator consists of the essential and point spectra, i.e., $\\sigma(L) = \\sigma_{\\mathrm{ess}}(L) \\cup \\sigma_{\\mathrm{pt}}(L)$, where $\\sigma_{\\mathrm{pt}}(L)$ coincides with the set of discrete isolated eigenvalues with finite multiplicity. Theorem~\\ref{Main} is proved by combining bounds on $\\sigma_{\\mathrm{ess}}(L)$ and $\\sigma_{\\mathrm{pt}}(L)$ with a verification that the eigenvalue at the origin is simple.\n\nWe first prove the stability condition \\eqref{sstability} for the essential spectrum. By standard arguments of \\cite{Henry, KP}, the bounds on $\\sigma_{\\mathrm{ess}}(L)$ are determined by the essential spectra of the limiting constant-coefficient operators $L_\\pm$, characterized by the associated dispersion relations.\n\nBuilding on the above result, we next show $\\sigma_{\\mathrm{pt}}(L) \\subset \\mathbb{C} \\setminus \\Omega$, where $\\Omega:=\\{\\lambda\\in\\mathbb{C}: \\operatorname{Re}\\lambda\\ge 0\\}\\setminus\\{0\\}$. We consider an integrated linear operator $\\mathcal{L}$, obtained by introducing anti-derivative variables, whose point spectrum coincides with that of $L$ on $\\Omega$. We then establish the desired bounds on the point spectrum via spectral energy estimates for the eigenvalue equation associated with $\\mathcal{L}$, combined with a contradiction argument.\n\nFinally, we prove that the eigenvalue $\\lambda=0$ is simple. Note that the essential spectrum bounds obtained above guarantee that a neighborhood $B_+(0,r)=B(0,r)\\cap\\{\\operatorname{Re}\\lambda>0\\}$ lies in the domain of consistent splitting for the associated first-order system \\eqref{5}. Thus we may define the Evans function on $B_+(0,r)$. For this, we apply the conjugation lemma to construct analytic bases of solutions decaying as $x\\to+\\infty$ and as $x\\to-\\infty$ on $B(0,r)$. We first define the Evans function $\\mathcal{D}(\\lambda)$ on $B_+(0,r)$ using these bases, and then extend it analytically to $B(0,r)$. A direct computation yields\n\\begin{equation*}\n\\mathcal{D}'(0)=\\Gamma\\Delta,\n\\end{equation*}\nwhere $\\Gamma$ and $\\Delta$ are as described in Remark~\\ref{Rem_E-cond}. The Rankine--Hugoniot relation \\eqref{RH} implies $\\Delta\\neq0$, and thus it remains to prove $\\Gamma\\neq0$. Here, we note that the transversality of the small-amplitude shock profiles in Lemma~\\ref{SP} follows essentially from their construction in \\cite[Section~2]{DLZ}. There the profiles are constructed via a center manifold reduction. In particular, for an augmented profile ODE system, the dynamics near a reference equilibrium reduces to a one-dimensional equation on the center manifold, with a heteroclinic orbit that is unique up to translation. Moreover, the linearized flow in the complementary subspace admits an exponential dichotomy. These two observations imply the transverse connection between corresponding stable and unstable manifolds at infinity, and hence $\\Gamma \\neq 0$ for sufficiently small amplitudes. The detailed proof is given in Appendix~\\ref{App_trans}. Since $\\Delta \\neq 0$ and $\\Gamma \\neq 0$, we conclude $\\mathcal{D}'(0)\\neq0$, which implies that $\\lambda=0$ has algebraic multiplicity one.\n\n\\medskip\n\\emph{Plan of the paper.} In Section~\\ref{sec:lin}, we derive the linearized eigenvalue problem about the shock profiles. In Sections~\\ref{sec:Ess} and~\\ref{sec:Point}, we obtain the bounds on the essential spectrum and point spectrum, respectively. These bounds establish the strong spectral stability of the shock profiles stated in Theorem~\\ref{Main}. Section~\\ref{sec:Simplicity} proves the simplicity of the zero eigenvalue by showing that the Evans function has a simple zero at $\\lambda=0$. The appendices collect background material and proofs: we recall the conjugation lemma of \\cite{MeZ}, and we prove solvability of the linearized Poisson equation, the domain and closedness of the linearized operator, higher-order energy estimates deferred from Section~\\ref{sec:Point}, and transversality of the shock profiles.", "sketch": "Theorem~\\ref{Main} is proved by combining bounds on $\\sigma_{\\mathrm{ess}}(L)$ and $\\sigma_{\\mathrm{pt}}(L)$ with a verification that the eigenvalue at the origin is simple.\n\n(1) **Essential spectrum.** “We first prove the stability condition \\eqref{sstability} for the essential spectrum.” By “standard arguments of \\cite{Henry, KP},” the bounds on $\\sigma_{\\mathrm{ess}}(L)$ are “determined by the essential spectra of the limiting constant-coefficient operators $L_\\pm$, characterized by the associated dispersion relations.”\n\n(2) **Point spectrum.** “Building on the above result, we next show $\\sigma_{\\mathrm{pt}}(L) \\subset \\mathbb{C} \\setminus \\Omega$, where $\\Omega:=\\{\\lambda\\in\\mathbb{C}: \\operatorname{Re}\\lambda\\ge 0\\}\\setminus\\{0\\}$.” The argument introduces “an integrated linear operator $\\mathcal{L}$, obtained by introducing anti-derivative variables, whose point spectrum coincides with that of $L$ on $\\Omega$,” and then derives point spectrum bounds “via spectral energy estimates for the eigenvalue equation associated with $\\mathcal{L}$, combined with a contradiction argument.”\n\n(3) **Simplicity of $\\lambda=0$ (Evans function).** The essential spectrum bounds ensure a neighborhood $B_+(0,r)$ lies in the “domain of consistent splitting,” so one can “define the Evans function on $B_+(0,r)$.” Using the conjugation lemma, the proof “construct[s] analytic bases of solutions decaying as $x\\to+\\infty$ and as $x\\to-\\infty$,” defines $\\mathcal D(\\lambda)$ on $B_+(0,r)$ and “extend[s] it analytically to $B(0,r)$.” Then “a direct computation yields $\\mathcal{D}'(0)=\\Gamma\\Delta$,” where “The Rankine--Hugoniot relation \\eqref{RH} implies $\\Delta\\neq0$,” so it remains to show “$\\Gamma\\neq0$.” This transversality is tied to the construction in \\cite[Section~2]{DLZ}: profiles arise via “a center manifold reduction,” giving a “one-dimensional equation on the center manifold, with a heteroclinic orbit that is unique up to translation,” and the complementary subspace has “an exponential dichotomy,” which together imply “the transverse connection between corresponding stable and unstable manifolds at infinity,” hence “$\\Gamma \\neq 0$ for sufficiently small amplitudes.” Therefore $\\mathcal{D}'(0)\\neq0$, so “$\\lambda=0$ has algebraic multiplicity one.”", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{Main}\nLet $(\\bar{v},\\bar{u},\\bar{\\phi})$ be the shock profile described in the following lemma.\n\n\\begin{lemma}[\\cite{DLZ}, Proposition~1.9] \\label{SP}\nFor given $(v_-,u_-)$ with $v_->0$, there exists a positive constant $\\delta_0>0$ such that for any $(v_+,u_+,s)$ satisfying \\eqref{RH}, \\eqref{Lax}, and\n\\begin{equation*}\n|v_+-v_-| =: \\delta_S < \\delta_0,\n\\end{equation*}\nthe ODE system \\eqref{waveeq} admits a unique (up to a shift) shock profile $(\\bar{v},\\bar{u},\\bar{\\phi})(x)$ satisfying \\eqref{ffcond} and\n\\begin{equation} \\label{sprel}\ns\\bar{v}_x = -\\bar{u}_x >0, \\quad \\underline{C} \\bar{u}_x \\leq \\bar{\\phi}_x \\leq \\overline{C} \\bar{u}_x\n\\end{equation}\nfor some positive constants $\\underline{C}$, $\\overline{C}$. Moreover, the unique solution satisfying $\\textstyle \\bar{v}(0)=\\frac{v_-+v_+}{2}$ verifies the derivative bounds\n\\begin{equation} \\label{spbound}\n\\begin{cases}\n\\left| \\frac{d^k}{dx^k} (\\bar{v}-v_+, \\bar{u}-u_+,\\bar{\\phi}-\\phi_+) \\right| \\leq C_k \\delta_S^{k+1} e^{-\\theta \\delta_S |x|}, & x>0 \\\\\n\\left| \\frac{d^k}{dx^k} (\\bar{v}-v_-, \\bar{u}-u_-,\\bar{\\phi}-\\phi_-) \\right| \\leq C_k \\delta_S^{k+1} e^{-\\theta \\delta_S |x|}, & x<0\n\\end{cases}\n\\end{equation}\nfor $k \\in \\mathbb{N} \\cup \\{0 \\}$, where $C_k>0$ and $\\theta>0$ are generic constants.\n\\end{lemma}\n\nLet $L$ denote the linear operator defined by\n\\begin{equation} \\label{Ldef}\nLU := \\left(A U \\right)_x + \\left( B U_x \\right)_x + \\left(D ( \\mathcal{A}^{-1}\\mathcal{B} U )_x \\right)_x + \\left(E ( \\mathcal{A}^{-1}\\mathcal{B} U )_{xx} \\right)_x,\n\\end{equation}\nwhere\n\\begin{equation} \\label{calAB}\n\\begin{split}\n\\mathcal{A} & =\\bar{v} e^{\\bar{\\phi}}+ \\frac{\\varepsilon^2 \\bar{v}_x}{\\bar{v}^2} \\partial_x-\\frac{\\varepsilon^2}{\\bar{v}} \\partial_{xx}, \\quad \\mathcal{B} = -e^{\\bar{\\phi}} -\\frac{\\varepsilon^2 \\bar{\\phi}_{xx}}{\\bar{v}^2} + \\frac{2\\varepsilon^2 \\bar{v}_x \\bar{\\phi}_x}{\\bar{v}^3}-\\frac{\\varepsilon^2 \\bar{\\phi}_x}{\\bar{v}^2} \\partial_x.\n\\end{split}\n\\end{equation}\nThere exists a constant $\\delta_1>0$ such that if the shock strength $\\delta_S$ is less than $\\delta_1$, then the shock profile is strongly spectrally stable in $L^2$. That is,\n\\begin{equation} \\label{sstability}\n\\sigma(L) \\cap \\{ \\lambda \\in \\mathbb{C}: \\operatorname{Re}{\\lambda} \\geq 0 \\} = \\{0\\},\n\\end{equation}\nwhere $\\sigma(L)$ denotes the spectrum of $L$ on $L^2(\\mathbb{R}) \\times L^2(\\mathbb{R})$. Moreover, $\\lambda =0$ is an eigenvalue of $L$ with algebraic multiplicity one.", "theorem_type": ["Existential–Universal", "Implication"], "mcq": {"question": "Let $(\\bar v,\\bar u,\\bar\\phi)(x)$ be a small-amplitude 2-shock profile connecting $(v_-,u_-,\\phi_-)$ and $(v_+,u_+,\\phi_+)$, where $v_->0$, $v_+>v_-$, $\\phi_\\pm=-\\ln v_\\pm$, and $\\delta_S:=|v_+-v_-|<\\delta_0$. Assume $(v_+,u_+,s)$ satisfy the Rankine--Hugoniot relations\n$$-s(v_+-v_-)-(u_+-u_-)=0,\\qquad -s(u_+-u_-)+(T+1)\\left(\\frac1{v_+}-\\frac1{v_-}\\right)=0,$$\nand that $(\\bar v,\\bar u,\\bar\\phi)$ solves\n$$-s\\bar v_x-\\bar u_x=0,$$\n$$-s\\bar u_x+\\left(\\frac{T}{\\bar v}\\right)_x=\\left(\\frac{\\nu\\bar u_x}{\\bar v}\\right)_x-\\frac{\\bar\\phi_x}{\\bar v},$$\n$$-\\varepsilon^2\\left(\\frac{\\bar\\phi_x}{\\bar v}\\right)_x=1-\\bar v e^{\\bar\\phi},$$\nwith far-field condition\n$$\\lim_{x\\to\\pm\\infty}(\\bar v,\\bar u,\\bar\\phi)(x)=(v_\\pm,u_\\pm,\\phi_\\pm).$$\nLet $L$ be the linearized operator on $L^2(\\mathbb R)\\times L^2(\\mathbb R)$ defined by\n$$LU=(AU)_x+(BU_x)_x+\\big(D(\\mathcal A^{-1}\\mathcal B U)_x\\big)_x+\\big(E(\\mathcal A^{-1}\\mathcal B U)_{xx}\\big)_x,$$\nwhere\n$$A(x)=\\begin{pmatrix}\n s & 1 \\\\\n \\dfrac{T+1}{\\bar v^2}-\\dfrac{\\nu\\bar u_x}{\\bar v^2}-\\dfrac{\\varepsilon^2\\bar\\phi_x^2}{\\bar v^3}+\\dfrac{\\varepsilon^2}{\\bar v}\\left(\\dfrac{\\bar\\phi_x}{\\bar v^2}\\right)_x+\\dfrac{\\varepsilon^2}{\\bar v^2}\\left(\\dfrac{\\bar\\phi_x}{\\bar v}\\right)_x & s\n\\end{pmatrix},$$\n$$B(x)=\\begin{pmatrix}0&0\\\\ \\dfrac{\\varepsilon^2\\bar\\phi_x}{\\bar v^3}&\\dfrac{\\nu}{\\bar v}\\end{pmatrix},\\quad D(x)=\\begin{pmatrix}0&0\\\\ \\dfrac{\\varepsilon^2\\bar\\phi_x}{\\bar v^2}+\\dfrac{\\varepsilon^2\\bar v_x}{\\bar v^3}&0\\end{pmatrix},\\quad E(x)=\\begin{pmatrix}0&0\\\\ -\\dfrac{\\varepsilon^2}{\\bar v^2}&0\\end{pmatrix},$$\nand\n$$\\mathcal A=\\bar v e^{\\bar\\phi}+\\frac{\\varepsilon^2\\bar v_x}{\\bar v^2}\\partial_x-\\frac{\\varepsilon^2}{\\bar v}\\partial_{xx},\\qquad \\mathcal B=-e^{\\bar\\phi}-\\frac{\\varepsilon^2\\bar\\phi_{xx}}{\\bar v^2}+\\frac{2\\varepsilon^2\\bar v_x\\bar\\phi_x}{\\bar v^3}-\\frac{\\varepsilon^2\\bar\\phi_x}{\\bar v^2}\\partial_x.$$ Under these assumptions, which statement about the spectrum of $L$ is valid when the shock strength is sufficiently small?", "correct_choice": {"label": "A", "text": "There exists a constant $\\delta_1>0$ such that, if $\\delta_S<\\delta_1$, then the shock profile is strongly spectrally stable in $L^2(\\mathbb R)\\times L^2(\\mathbb R)$, namely\n$$\\sigma(L)\\cap\\{\\lambda\\in\\mathbb C:\\operatorname{Re}\\lambda\\ge 0\\}=\\{0\\}.$$ Moreover, $\\lambda=0$ is an eigenvalue of $L$ with algebraic multiplicity one."}, "choices": [{"label": "B", "text": "There exists a constant $\\delta_1>0$ such that, if $\\delta_S<\\delta_1$, then the shock profile is strongly spectrally stable in $L^2(\\mathbb R)\\times L^2(\\mathbb R)$, namely\n$$\\sigma(L)\\cap\\{\\lambda\\in\\mathbb C:\\operatorname{Re}\\lambda\\ge 0\\}=\\varnothing.$$"}, {"label": "C", "text": "There exists a constant $\\delta_1>0$ such that, if $\\delta_S<\\delta_1$, then\n$$\\sigma(L)\\cap\\{\\lambda\\in\\mathbb C:\\operatorname{Re}\\lambda>0\\}=\\varnothing.$$"}, {"label": "D", "text": "There exists a constant $\\delta_1>0$ such that, if $\\delta_S<\\delta_1$, then the shock profile is strongly spectrally stable in $L^2(\\mathbb R)\\times L^2(\\mathbb R)$, namely\n$$\\sigma(L)\\cap\\{\\lambda\\in\\mathbb C:\\operatorname{Re}\\lambda\\ge 0\\}=\\{0\\}.$$ Moreover, $\\lambda=0$ is an eigenvalue of $L$ with geometric multiplicity one."}, {"label": "E", "text": "For every shock strength $\\delta_S<\\delta_0$, the shock profile is strongly spectrally stable in $L^2(\\mathbb R)\\times L^2(\\mathbb R)$, namely\n$$\\sigma(L)\\cap\\{\\lambda\\in\\mathbb C:\\operatorname{Re}\\lambda\\ge 0\\}=\\{0\\}.$$ Moreover, $\\lambda=0$ is an eigenvalue of $L$ with algebraic multiplicity one."}], "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": "translation_zero_mode_at_lambda_0", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "drops_the_boundary_point_lambda_0_and_multiplicity_claim", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "algebraic_vs_geometric_multiplicity", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "smallness_threshold_delta_1_replaced_by_full_delta_0_range", "template_used": "boundary_range"}]}} +{"id": "2601.20855v1", "paper_link": "http://arxiv.org/abs/2601.20855v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{the theorem in introduction}\nFor all $0 \\leq j \\leq \\ell \\leq k,\\; k\\geq 1$, there exists a strictly ergodic system which admits the following:\n\\begin{enumerate}\n \\item $Z_k$ is a nontrivial extension of $Z_{k-1}$\n \\item $Z_i$ is topologically isomorphic to $X_i$ for all $0 \\leq i \\leq j$\n \\item $Z_i$ is not measurably isomorphic to $X_i$ for all $j+1 \\leq i \\leq k$\n \\item $X_i$ is topologically isomorphic to $X_{i+1}$ for all $i \\geq \\ell$.\n\n\\end{enumerate}\nFurthermore, for every minimal, ergodic, and non-weak mixing $(X,\\mu,T)$ there exist $j,\\ell$, and $k$, such that $X$ satisfies properties (1)-(4).", "start_pos": 8837, "end_pos": 9496, "label": "the theorem in introduction"}, "ref_dict": {"proposition with one coboundary on k": "\\begin{proposition} \\label{proposition with one coboundary on k}\n Let $01$. Concrete examples of such systems can be found in, for instance, \\cite{{nontrivial Kronecker trivial MEF}, {Gla}, {Lehrer}, {Parry}}. We can illustrate these phenomena in the diagram below\n \\begin{center}\n\\begin{tikzpicture}\n \\node (X) {X};\n \\node[above right=0.5cm and 1.25cm of X] (Y) {$Z_k$};\n \\node[below right=0.5cm and 1.25cm of X] (Z) {$X_k$};\n \\node[right=1.25cm of Y] (A) {$Z_{k-1}$}; \n \\node[right=1.25cm of Z] (B) {$X_{k-1}$};\n \\node[right=1.25cm of A] (C) {$Z_1$};\n \\node[right=1.25cm of B] (D) {$X_1$};\n\n \\draw[->] (X) -- (Y);\n \\draw[->] (X) -- (Z);\n \\draw[->, dashed] (Y) -- (Z);\n \\draw[->] (Y) -- (A);\n \\draw[->] (Z) -- (B);\n \\draw[->, dashed] (A) -- (B);\n \\draw[->, dashed] (A) -- (C);\n \\draw[->, dashed] (B) -- (D);\n \\draw[->, dashed] (C) -- (D);\n\\end{tikzpicture}\n\\end{center}\n where \\( Z_i \\dasharrow X_i \\) indicates either a measurable isomorphism for all $i$ or a proper factor map for each $i$. We construct examples realizing all possible behaviors in between these two phenomena. Understanding when the measurable and topological nilfactors are different constitutes partly to the goal of this paper. We note in a weak mixing system all nilfactors are trivial (see, for example, Proposition 20 of Chapter 8 in \\cite{HK2}). We also note that nilsystems are classic examples of systems with $Z_k$ and $X_k$ isomorphic.", "context": "Nilsystems and nilfactors of systems arise in convergence and recurrence problems in ergodic theory and topological dynamics. Prime examples of these can be found in \\cite{topological characteristic factors} and \\cite{HK paper}. Numerous measure-theoretic and topological properties, such as transitivity, minimality, ergodicity, and unique ergodicity, are equivalent in nilsystems, combining results of \\cite{Auslander, Leibman, Lesigne, Parry2}. These equivalences are key in many recent developments in dynamics.\n\nTo describe the behaviors that may arise, we assume throughout that our dynamical systems are endowed with both measurable and topological structure. Every topological factor naturally has the structure of a measurable factor, but the converse does not hold in general. \nThe measurable $k$-step nilfactor of a system has been shown in works of Host and Kra in \\cite{HK paper} to control the dynamical behavior of certain multiple ergodic averages. The topological analog of these factors was first introduced by Host, Kra and Maass in \\cite{Host Kra and Maass} (see Section \\ref{sec: notations and preliminaries} for definitions). A more recent interplay of the measurable and topological nilfactors has appeared in \\cite{a model with topological pronilfactors} where Kra, Moreira, Richter and Robertson, solved a conjecture of Erd\\H{o}s on sumsets. There, it was shown how these nilfactors control certain infinite patterns in sets with positive density. \nThe well-known Jewett-Krieger theorem says that every ergodic system has a strictly ergodic (minimal and uniquely ergodic) topological model \\cite{{Jewett}, {Krieger}}. By a topological model, or just model, we mean a topological dynamical system equipped with an invariant probability measure that is measurably isomorphic to the original system.\nIn \\cite{maximal pronilfactors}, it is shown that every ergodic system has a strictly ergodic model where the measurable and topological $k$-step nilfactors, $Z_k$ and $X_k$, are isomorphic. An older result of Lehrer \\cite{Lehrer} obtaining topologically mixing models of ergodic systems is most relevant for this work. An important consequence of his theorem is the existence of systems where $Z_1$ and $X_1$ are not isomorphic; it then follows $Z_i$ and $X_i$ are not isomorphic as well for all $i>1$. Concrete examples of such systems can be found in, for instance, \\cite{{nontrivial Kronecker trivial MEF}, {Gla}, {Lehrer}, {Parry}}. We can illustrate these phenomena in the diagram below\n \\begin{center}\n\\begin{tikzpicture}\n \\node (X) {X};\n \\node[above right=0.5cm and 1.25cm of X] (Y) {$Z_k$};\n \\node[below right=0.5cm and 1.25cm of X] (Z) {$X_k$};\n \\node[right=1.25cm of Y] (A) {$Z_{k-1}$}; \n \\node[right=1.25cm of Z] (B) {$X_{k-1}$};\n \\node[right=1.25cm of A] (C) {$Z_1$};\n \\node[right=1.25cm of B] (D) {$X_1$};\n\n\\draw[->] (X) -- (Y);\n \\draw[->] (X) -- (Z);\n \\draw[->, dashed] (Y) -- (Z);\n \\draw[->] (Y) -- (A);\n \\draw[->] (Z) -- (B);\n \\draw[->, dashed] (A) -- (B);\n \\draw[->, dashed] (A) -- (C);\n \\draw[->, dashed] (B) -- (D);\n \\draw[->, dashed] (C) -- (D);\n\\end{tikzpicture}\n\\end{center}\n where \\( Z_i \\dasharrow X_i \\) indicates either a measurable isomorphism for all $i$ or a proper factor map for each $i$. We construct examples realizing all possible behaviors in between these two phenomena. Understanding when the measurable and topological nilfactors are different constitutes partly to the goal of this paper. We note in a weak mixing system all nilfactors are trivial (see, for example, Proposition 20 of Chapter 8 in \\cite{HK2}). We also note that nilsystems are classic examples of systems with $Z_k$ and $X_k$ isomorphic.", "full_context": "Nilsystems and nilfactors of systems arise in convergence and recurrence problems in ergodic theory and topological dynamics. Prime examples of these can be found in \\cite{topological characteristic factors} and \\cite{HK paper}. Numerous measure-theoretic and topological properties, such as transitivity, minimality, ergodicity, and unique ergodicity, are equivalent in nilsystems, combining results of \\cite{Auslander, Leibman, Lesigne, Parry2}. These equivalences are key in many recent developments in dynamics.\n\nTo describe the behaviors that may arise, we assume throughout that our dynamical systems are endowed with both measurable and topological structure. Every topological factor naturally has the structure of a measurable factor, but the converse does not hold in general. \nThe measurable $k$-step nilfactor of a system has been shown in works of Host and Kra in \\cite{HK paper} to control the dynamical behavior of certain multiple ergodic averages. The topological analog of these factors was first introduced by Host, Kra and Maass in \\cite{Host Kra and Maass} (see Section \\ref{sec: notations and preliminaries} for definitions). A more recent interplay of the measurable and topological nilfactors has appeared in \\cite{a model with topological pronilfactors} where Kra, Moreira, Richter and Robertson, solved a conjecture of Erd\\H{o}s on sumsets. There, it was shown how these nilfactors control certain infinite patterns in sets with positive density. \nThe well-known Jewett-Krieger theorem says that every ergodic system has a strictly ergodic (minimal and uniquely ergodic) topological model \\cite{{Jewett}, {Krieger}}. By a topological model, or just model, we mean a topological dynamical system equipped with an invariant probability measure that is measurably isomorphic to the original system.\nIn \\cite{maximal pronilfactors}, it is shown that every ergodic system has a strictly ergodic model where the measurable and topological $k$-step nilfactors, $Z_k$ and $X_k$, are isomorphic. An older result of Lehrer \\cite{Lehrer} obtaining topologically mixing models of ergodic systems is most relevant for this work. An important consequence of his theorem is the existence of systems where $Z_1$ and $X_1$ are not isomorphic; it then follows $Z_i$ and $X_i$ are not isomorphic as well for all $i>1$. Concrete examples of such systems can be found in, for instance, \\cite{{nontrivial Kronecker trivial MEF}, {Gla}, {Lehrer}, {Parry}}. We can illustrate these phenomena in the diagram below\n \\begin{center}\n\\begin{tikzpicture}\n \\node (X) {X};\n \\node[above right=0.5cm and 1.25cm of X] (Y) {$Z_k$};\n \\node[below right=0.5cm and 1.25cm of X] (Z) {$X_k$};\n \\node[right=1.25cm of Y] (A) {$Z_{k-1}$}; \n \\node[right=1.25cm of Z] (B) {$X_{k-1}$};\n \\node[right=1.25cm of A] (C) {$Z_1$};\n \\node[right=1.25cm of B] (D) {$X_1$};\n\n\\draw[->] (X) -- (Y);\n \\draw[->] (X) -- (Z);\n \\draw[->, dashed] (Y) -- (Z);\n \\draw[->] (Y) -- (A);\n \\draw[->] (Z) -- (B);\n \\draw[->, dashed] (A) -- (B);\n \\draw[->, dashed] (A) -- (C);\n \\draw[->, dashed] (B) -- (D);\n \\draw[->, dashed] (C) -- (D);\n\\end{tikzpicture}\n\\end{center}\n where \\( Z_i \\dasharrow X_i \\) indicates either a measurable isomorphism for all $i$ or a proper factor map for each $i$. We construct examples realizing all possible behaviors in between these two phenomena. Understanding when the measurable and topological nilfactors are different constitutes partly to the goal of this paper. We note in a weak mixing system all nilfactors are trivial (see, for example, Proposition 20 of Chapter 8 in \\cite{HK2}). We also note that nilsystems are classic examples of systems with $Z_k$ and $X_k$ isomorphic.\n\nTo describe the behaviors that may arise, we assume throughout that our dynamical systems are endowed with both measurable and topological structure. Every topological factor naturally has the structure of a measurable factor, but the converse does not hold in general. \nThe measurable $k$-step nilfactor of a system has been shown in works of Host and Kra in \\cite{HK paper} to control the dynamical behavior of certain multiple ergodic averages. The topological analog of these factors was first introduced by Host, Kra and Maass in \\cite{Host Kra and Maass} (see Section \\ref{sec: notations and preliminaries} for definitions). A more recent interplay of the measurable and topological nilfactors has appeared in \\cite{a model with topological pronilfactors} where Kra, Moreira, Richter and Robertson, solved a conjecture of Erd\\H{o}s on sumsets. There, it was shown how these nilfactors control certain infinite patterns in sets with positive density. \nThe well-known Jewett-Krieger theorem says that every ergodic system has a strictly ergodic (minimal and uniquely ergodic) topological model \\cite{{Jewett}, {Krieger}}. By a topological model, or just model, we mean a topological dynamical system equipped with an invariant probability measure that is measurably isomorphic to the original system.\nIn \\cite{maximal pronilfactors}, it is shown that every ergodic system has a strictly ergodic model where the measurable and topological $k$-step nilfactors, $Z_k$ and $X_k$, are isomorphic. An older result of Lehrer \\cite{Lehrer} obtaining topologically mixing models of ergodic systems is most relevant for this work. An important consequence of his theorem is the existence of systems where $Z_1$ and $X_1$ are not isomorphic; it then follows $Z_i$ and $X_i$ are not isomorphic as well for all $i>1$. Concrete examples of such systems can be found in, for instance, \\cite{{nontrivial Kronecker trivial MEF}, {Gla}, {Lehrer}, {Parry}}. We can illustrate these phenomena in the diagram below\n \\begin{center}\n\\begin{tikzpicture}\n \\node (X) {X};\n \\node[above right=0.5cm and 1.25cm of X] (Y) {$Z_k$};\n \\node[below right=0.5cm and 1.25cm of X] (Z) {$X_k$};\n \\node[right=1.25cm of Y] (A) {$Z_{k-1}$}; \n \\node[right=1.25cm of Z] (B) {$X_{k-1}$};\n \\node[right=1.25cm of A] (C) {$Z_1$};\n \\node[right=1.25cm of B] (D) {$X_1$};\n\n\\draw[->] (X) -- (Y);\n \\draw[->] (X) -- (Z);\n \\draw[->, dashed] (Y) -- (Z);\n \\draw[->] (Y) -- (A);\n \\draw[->] (Z) -- (B);\n \\draw[->, dashed] (A) -- (B);\n \\draw[->, dashed] (A) -- (C);\n \\draw[->, dashed] (B) -- (D);\n \\draw[->, dashed] (C) -- (D);\n\\end{tikzpicture}\n\\end{center}\n where \\( Z_i \\dasharrow X_i \\) indicates either a measurable isomorphism for all $i$ or a proper factor map for each $i$. We construct examples realizing all possible behaviors in between these two phenomena. Understanding when the measurable and topological nilfactors are different constitutes partly to the goal of this paper. We note in a weak mixing system all nilfactors are trivial (see, for example, Proposition 20 of Chapter 8 in \\cite{HK2}). We also note that nilsystems are classic examples of systems with $Z_k$ and $X_k$ isomorphic.\n\nTo prove Theorem \\ref{the theorem in introduction}, we draw inspiration from an example of a minimal but not uniquely ergodic system constructed by Furstenberg in \\cite{Fur61}. There, it is shown that for $\\alpha \\notin \\mathbb{Q}$, the system $$T(x,y)=(x+\\alpha,y+f(x)), \\;\\;T\\colon\\mathbb{T}^{2} \\rightarrow \\mathbb{T}^{2}$$ is minimal and not uniquely ergodic if and only if \n\\begin{equation}\\label{eq: coboundary furstenberg}\nnf \\equiv F(x+\\alpha)-F(x) \\mod 1\n\\end{equation}\nhas a measurable solution for some $n \\in \\mathbb{Z} \\setminus \\{0\\}$ but no continuous solution unless $n=0$. In \\cite{Fur61}, an $f$ which has a measurable solution for $n=1$ but no continuous solution except when $n=0$ is constructed. We refer to $f$ as a measurable but not continuous coboundary over the system $(\\mathbb{T},m_{\\mathbb{T}},T_{\\alpha})$. In Section \\ref{sec: flexibility}, we use these coboundaries to construct systems where the measurable and topological $k$-step nilfactors are not measurably isomorphic after some point.\n\nTo prove Theorem \\ref{the theorem in introduction}, we begin with a lemma which characterizes when the factors $Z_k$ and $X_k$ are isomorphic.\n \\begin{lemma} \\label{Zk and Xk measurably iso iff cont factor map}\n Let $(X,\\mu,T)$ be a minimal and ergodic system. Then the following are equivalent: \n \\begin{enumerate}\n \\item $Z_k$ is measurably isomorphic to $X_k$\n \\item $Z_k$ is topologically isomorphic to $X_k$\n \\item There exists a topological factor map from $\\pi\\colon X \\rightarrow Z_k$\n\\end{enumerate}\n\n\\begin{enumerate}\n \\item $Z_k$ is a nontrivial extension of $Z_{k-1}$\n \\item $Z_{i}$ and $X_{i}$ are topologically isomorphic, for all $0 \\leq i \\leq j$\n \\item $Z_i$ is not measurably isomorphic to $X_i$ for all $j+1 \\leq i \\leq k$\n \\item $X_i$ is topologically isomorphic to $X_{i+1}$ for all $i \\geq j$\n \\end{enumerate}\nwhere $Z_k$ is the maximal measurable nilfactor of $(\\mathbb{T}^{k},m_{\\mathbb{T}^{k}},T)$ and $X_j$ is the maximal topological nilfactor of $(\\mathbb{T}^{k},m_{\\mathbb{T}^{k}},T)$. Furthermore, for $j=0$, there exists a measurable but not continuous coboundary $g\\colon \\mathbb{T} \\rightarrow \\mathbb{T}$ over $(\\mathbb{T},m_{\\mathbb{T}},T_{\\alpha})$ such that the system $(\\mathbb{T}^{k+1},m_{\\mathbb{T}^{k+1}},R)$ where $$R(x_1,x_2,\\dots,x_{k+1})=(x_1+\\alpha,x_2+g(x_1)+\\beta,x_3+x_2,\\dots,x_{k+1}+x_k)$$ is strictly ergodic and has properties (1)-(4) as well.\n\\end{proposition}\n\n\\begin{proposition} \\label{proposition with two coboundaries on k}\n Let $1\\leq \\ell \\leq k$ and $\\alpha,\\beta \\in \\mathbb{T} \\setminus \\mathbb{Q}$ be rationally independent. Then there exists a measurable but not continuous coboundary $f\\colon\\mathbb{T} \\rightarrow \\mathbb{T}$ over $(\\mathbb{T},m_{\\mathbb{T}},T_{\\alpha})$ such that the system $(\\mathbb{T}^{k+1},m_{\\mathbb{T}^{k+1}},T)$ given by \n \\begin{align*}\n T(x_1,x_2,\\dots,x_{k+1})=\\Big(&x_1+\\alpha,x_2+x_1,\\dots,x_{\\ell}+x_{\\ell-1},x_{\\ell+1}+f(x_1)+x_{\\ell},\\\\ &x_{\\ell+2}+x_{\\ell+1},\\dots,x_{k}+x_{k-1},x_{k+1}+f(x_1)+\\beta\\Big)\n \\end{align*}\n is strictly ergodic and satisfies the following properties:\n\n\\begin{enumerate}\n \\item $Z_k$ is a nontrivial extension of $Z_{k-1}$\n \\item $Z_i$ is not measurably isomorphic to $X_i$ for all $i\\geq 1$\n \\item $X_i$ is topologically isomorphic to $X_{i+1}$ for all $i \\geq \\ell$\n \\end{enumerate}\nwhere $Z_k$ is the maximal measurable nilfactor of $(\\mathbb{T}^{k+1},m_{\\mathbb{T}^{k+1}},T)$ and $X_\\ell$ is the maximal topological nilfactor of $(\\mathbb{T}^{k+1},m_{\\mathbb{T}^{k+1}},T)$.\n\\end{proposition}\n\n\\begin{enumerate}\n \\item $Z_k$ is a nontrivial extension of $Z_{k-1}$\n \\item $Z_{i}$ and $X_{i}$ are topologically isomorphic, for all $0 \\leq i \\leq j$\n \\item $Z_i$ is not measurably isomorphic to $X_i$ for all $j+1 \\leq i \\leq k$\n \\item $X_i$ is topologically isomorphic to $X_{i+1}$ for all $i \\geq \\ell$\n \\end{enumerate}\nwhere $Z_k$ is the maximal measurable nilfactor and $X_\\ell$ is the maximal topological nilfactor of $(\\mathbb{T}^{2k},m_{\\mathbb{T}^{2k}},T)$ and $X_\\ell$ is the maximal topological nilfactor of $(\\mathbb{T}^{2k},m_{\\mathbb{T}^{2k}},T)$.\n\\end{proposition}\n\n\\begin{theorem}\\label{the theorem in introduction}\nFor all $0 \\leq j \\leq \\ell \\leq k,\\; k\\geq 1$, there exists a strictly ergodic system which admits the following:\n\\begin{enumerate}\n \\item $Z_k$ is a nontrivial extension of $Z_{k-1}$\n \\item $Z_i$ is topologically isomorphic to $X_i$ for all $0 \\leq i \\leq j$\n \\item $Z_i$ is not measurably isomorphic to $X_i$ for all $j+1 \\leq i \\leq k$\n \\item $X_i$ is topologically isomorphic to $X_{i+1}$ for all $i \\geq \\ell$.\n\n\\end{enumerate}\nFurthermore, for every minimal, ergodic, and non-weak mixing $(X,\\mu,T)$ there exist $j,\\ell$, and $k$, such that $X$ satisfies properties (1)-(4).\n \\end{theorem}", "post_theorem_intro_text_len": 1656, "post_theorem_intro_text": "To prove Theorem \\ref{the theorem in introduction}, we draw inspiration from an example of a minimal but not uniquely ergodic system constructed by Furstenberg in \\cite{Fur61}. There, it is shown that for $\\alpha \\notin \\mathbb{Q}$, the system $$T(x,y)=(x+\\alpha,y+f(x)), \\;\\;T\\colon\\mathbb{T}^{2} \\rightarrow \\mathbb{T}^{2}$$ is minimal and not uniquely ergodic if and only if \n\\begin{equation}\\label{eq: coboundary furstenberg}\nnf \\equiv F(x+\\alpha)-F(x) \\mod 1\n\\end{equation}\nhas a measurable solution for some $n \\in \\mathbb{Z} \\setminus \\{0\\}$ but no continuous solution unless $n=0$. In \\cite{Fur61}, an $f$ which has a measurable solution for $n=1$ but no continuous solution except when $n=0$ is constructed. We refer to $f$ as a measurable but not continuous coboundary over the system $(\\mathbb{T},m_{\\mathbb{T}},T_{\\alpha})$. In Section \\ref{sec: flexibility}, we use these coboundaries to construct systems where the measurable and topological $k$-step nilfactors are not measurably isomorphic after some point.\n\nIn Proposition \\ref{nontrivial measurable nils but trivial topological nils}, we exhibit another flexibility of the nilfactors where for each $k\\in \\mathbb{N}$, we build a strictly ergodic system which has $Z_k$ as a nontrivial extension of $Z_{k-1}$ yet has trivial topological nilfactors. Such examples are known to exist by Lehrer \\cite{Lehrer}, however, we provide a construction that is more explicit using a certain minimal Cantor system built by Durand, Frank, and Maass, in \\cite{nontrivial Kronecker trivial MEF} along with techniques we develop in this article from Proposition \\ref{proposition with one coboundary on k}.", "sketch": "To prove Theorem \\ref{the theorem in introduction}, the introduction says the proof is inspired by Furstenberg’s example \\cite{Fur61} of a minimal but not uniquely ergodic skew product $T(x,y)=(x+\\alpha,y+f(x))$ on $\\mathbb{T}^2$, where (for $\\alpha\\notin\\mathbb{Q}$) minimality/non-unique ergodicity is characterized by the coboundary equation\n\\[\n nf \\equiv F(x+\\alpha)-F(x) \\mod 1\n\\]\nhaving a measurable solution for some $n\\neq 0$ but no continuous solution unless $n=0$. An $f$ with a measurable solution for $n=1$ but no continuous solution (except $n=0$) is constructed in \\cite{Fur61}; the paper refers to such an $f$ as a “measurable but not continuous coboundary” over $(\\mathbb{T},m_{\\mathbb{T}},T_\\alpha)$. In Section \\ref{sec: flexibility}, these coboundaries are then used to construct systems “where the measurable and topological $k$-step nilfactors are not measurably isomorphic after some point.”\n\nThe introduction also points to a second ingredient: Proposition \\ref{nontrivial measurable nils but trivial topological nils}, which “exhibit[s] another flexibility of the nilfactors” by building, for each $k\\in\\mathbb{N}$, a strictly ergodic system with $Z_k$ a nontrivial extension of $Z_{k-1}$ but with “trivial topological nilfactors.” This construction is described as “more explicit,” using “a certain minimal Cantor system built by Durand, Frank, and Maass” \\cite{nontrivial Kronecker trivial MEF} together with techniques developed from Proposition \\ref{proposition with one coboundary on k}.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{the theorem in introduction}\nFor all $0 \\leq j \\leq \\ell \\leq k,\\; k\\geq 1$, there exists a strictly ergodic system which admits the following:\n\\begin{enumerate}\n \\item $Z_k$ is a nontrivial extension of $Z_{k-1}$\n \\item $Z_i$ is topologically isomorphic to $X_i$ for all $0 \\leq i \\leq j$\n \\item $Z_i$ is not measurably isomorphic to $X_i$ for all $j+1 \\leq i \\leq k$\n \\item $X_i$ is topologically isomorphic to $X_{i+1}$ for all $i \\geq \\ell$.\n\n\\end{enumerate}\nFurthermore, for every minimal, ergodic, and non-weak mixing $(X,\\mu,T)$ there exist $j,\\ell$, and $k$, such that $X$ satisfies properties (1)-(4).,", "theorem_type": ["Universal–Existential", "Existence"], "mcq": {"question": "Let Z_i denote the maximal measurable i-step nilfactor and X_i the maximal topological i-step nilfactor of a dynamical system carrying both measurable and topological structure, and recall that a system is strictly ergodic if it is minimal and uniquely ergodic. For integers satisfying \\(0 \\le j \\le \\ell \\le k\\) with \\(k\\ge 1\\), which statement is valid?", "correct_choice": {"label": "A", "text": "For every choice of integers \\(0 \\le j \\le \\ell \\le k\\) with \\(k\\ge 1\\), there exists a strictly ergodic system such that: (1) \\(Z_k\\) is a nontrivial extension of \\(Z_{k-1}\\); (2) \\(Z_i\\) is topologically isomorphic to \\(X_i\\) for all \\(0 \\le i \\le j\\); (3) \\(Z_i\\) is not measurably isomorphic to \\(X_i\\) for all \\(j+1 \\le i \\le k\\); and (4) \\(X_i\\) is topologically isomorphic to \\(X_{i+1}\\) for all \\(i \\ge \\ell\\). Furthermore, for every minimal, ergodic, non-weak-mixing system \\((X,\\mu,T)\\), there exist integers \\(j,\\ell,k\\) such that \\((X,\\mu,T)\\) satisfies properties (1)\\u2013(4)."}, "choices": [{"label": "B", "text": "For every choice of integers \\(0 \\le j \\le \\ell \\le k\\) with \\(k\\ge 1\\), there exists a strictly ergodic system such that: (1) \\(Z_k\\) is a nontrivial extension of \\(Z_{k-1}\\); (2) \\(Z_i\\) is topologically isomorphic to \\(X_i\\) for all \\(0 \\le i < j\\); (3) \\(Z_i\\) is not measurably isomorphic to \\(X_i\\) for all \\(j \\le i \\le k\\); and (4) \\(X_i\\) is topologically isomorphic to \\(X_{i+1}\\) for all \\(i > \\ell\\). Furthermore, for every minimal, ergodic, non-weak-mixing system \\((X,\\mu,T)\\), there exist integers \\(j,\\ell,k\\) such that \\((X,\\mu,T)\\) satisfies properties (1)–(4)."}, {"label": "C", "text": "For every choice of integers \\(0 \\le j \\le \\ell \\le k\\) with \\(k\\ge 1\\), there exists a strictly ergodic system such that: (1) \\(Z_k\\) is a nontrivial extension of \\(Z_{k-1}\\); (2) \\(Z_i\\) is topologically isomorphic to \\(X_i\\) for all \\(0 \\le i \\le j\\); (3) \\(Z_i\\) is not measurably isomorphic to \\(X_i\\) for all \\(j+1 \\le i \\le k\\)."}, {"label": "D", "text": "For every choice of integers \\(0 \\le j \\le \\ell \\le k\\) with \\(k\\ge 1\\), there exists a strictly ergodic system such that: (1) \\(Z_k\\) is a nontrivial extension of \\(Z_{k-1}\\); (2) \\(Z_i\\) is measurably isomorphic to \\(X_i\\) for all \\(0 \\le i \\le j\\); (3) \\(Z_i\\) is not topologically isomorphic to \\(X_i\\) for all \\(j+1 \\le i \\le k\\); and (4) \\(X_i\\) is topologically isomorphic to \\(X_i\\) for all \\(i \\ge \\ell\\). Furthermore, for every minimal, ergodic, non-weak-mixing system \\((X,\\mu,T)\\), there exist integers \\(j,\\ell,k\\) such that \\((X,\\mu,T)\\) satisfies properties (1)–(4)."}, {"label": "E", "text": "There exist integers \\(0 \\le j \\le \\ell \\le k\\) with \\(k\\ge 1\\) such that every strictly ergodic system satisfies: (1) \\(Z_k\\) is a nontrivial extension of \\(Z_{k-1}\\); (2) \\(Z_i\\) is topologically isomorphic to \\(X_i\\) for all \\(0 \\le i \\le j\\); (3) \\(Z_i\\) is not measurably isomorphic to \\(X_i\\) for all \\(j+1 \\le i \\le k\\); and (4) \\(X_i\\) is topologically isomorphic to \\(X_{i+1}\\) for all \\(i \\ge \\ell\\). Furthermore, for every minimal, ergodic, non-weak-mixing system \\((X,\\mu,T)\\), the same integers \\(j,\\ell,k\\) can be chosen so that \\((X,\\mu,T)\\) satisfies properties (1)–(4)."}], "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": "endpoint ranges for j and ell", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "dropped property (4)", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "type of isomorphism and factor identity in clauses (2)-(4)", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "quantifier order over j, ell, k and systems", "template_used": "quantifier_dependence"}]}} +{"id": "2601.21326v1", "paper_link": "http://arxiv.org/abs/2601.21326v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{t-su} (Sullivan \\cite{Su1}, \\cite{MS})\nGiven $N$ there exists $m=m(N)>0$ as follows. Let $f$ be an infinitely renormalizable real quadratic polynomial with combinatorics\nbounded by $N$.\nThen, for every $n\\ge n(f)$,\nthe corresponding renormalization $R^{q_n}(f):[-1,1]\\to[-1,1]$ admits a polynomial-like extension $R^{q_n}(f): V_n'\\to V_n$ of degree $2$ such that\n$\\mod(V_n\\setminus\\overline{V_n'})\\ge m$, for the modulus of the annulus $V_n\\setminus\\overline{V_n'}$.", "start_pos": 12653, "end_pos": 13155, "label": "t-su"}, "ref_dict": {"d-eps": "\\begin{definition}\\label{d-eps} (Epstein classes $E_\\beta(J)$, cf. \\cite{E}, \\cite{Su1}, \\cite{MS})\nLet $g: [-1,1]\\to [-1,1]$ be a real analytic map as follows: (1) $g$ is unimodal: $g'(x)=0$ iff $x=0$, (2) $g(-1)=g(1)=1$\nand $0\\in g([-1,1])$, i.e., $g(0)\\le 0$. Given a symmetric w.r.t. $0$ open interval $J$, we say that $g\\in E_1(J)$ if\nthere exists another open symmetric interval $J'$ such that $[-1,1]\\subset J'\\subset J$, $g$ extends to a real-analytic unimodal map $g:J'\\to J$ and there exists a representation\n$g=F\\circ Q$ where $Q(z)=z^2$ and $F: [0,1]\\to [g(0),1]$ is a diffeomorphism such that $F^{-1}$ extends to a univalent (i.e. holomorphic injective)\nmap defined on $\\C_J$.\nMore generally, given $\\beta\\in\\R\\setminus \\{0\\}$,\n$g\\in E_{\\beta}(J)$ (or simply $E_\\beta$) if $g=\\beta g_1 \\beta^{-1}$ for some $g_1\\in E_1(\\beta^{-1}J)$.\n\\end{definition}", "t-su": "\\begin{theorem}\\label{t-su} (Sullivan \\cite{Su1}, \\cite{MS})\nGiven $N$ there exists $m=m(N)>0$ as follows. Let $f$ be an infinitely renormalizable real quadratic polynomial with combinatorics\nbounded by $N$.\nThen, for every $n\\ge n(f)$,\nthe corresponding renormalization $R^{q_n}(f):[-1,1]\\to[-1,1]$ admits a polynomial-like extension $R^{q_n}(f): V_n'\\to V_n$ of degree $2$ such that\n$\\mod(V_n\\setminus\\overline{V_n'})\\ge m$, for the modulus of the annulus $V_n\\setminus\\overline{V_n'}$.\n\\end{theorem}", "p-infren-tower": "\\begin{prop}\\label{p-infren-tower}\nTo every sequence $\\mathcal{R}\\subset\\N$ one can associate its subsequence $S$ and a sequence of maps\n$g_m\\in E_{b_m}(J_m)$, where $J_m=(-L|b_m|, L|b_m|)$, $m=0,1,2,...$, as follows:\n\\begin{enumerate}\\label{e-t-subs}\n\\item $b_0=1$, i.e., $g_0\\in E_1(J_0)$, and the sequence of renormalizations $\\{R^{q_j}f\\}_{j\\in S}\\to g_0$ uniformly on compacts in $\\Omega_{g_0}$,\n\\item if $I_m=<-b_m,b_m>$, then $\\mu\\le \\frac{|I_{m+1}|}{|I_m|}\\le \\lambda$ and $J_m\\subset I_{m+1}$, $m=0,1,2,...$,\n\\item for every $m>1$ there exists an integer $p_m\\ge 2$, such that\n$$g_0(x)=g_m^{p_m}(x), \\mbox{ for all } x\\in [-1,1].$$\n\\end{enumerate}\n\\end{prop}", "s-m": "\\label{s-m}\nDefine inductively: $T_0=[-1,1]$ and $T_{n+1}=g_0^{-1}(T_n)$, $n=0,1,2,...$. As $g_0([-1,1])\\subset [-1,1]$, then\n$T_n=\\cup_{i=0}^n g_0^{-i}([-1,1])$ so that $\\{T_n\\}$ is an increasing seq", "t-m": "\\begin{theorem}\\label{t-m}\nLet $\\mathcal{G}=\\{g_n\\}_{n=0}^\\infty$ where $g_n\\in E_{\\beta_n}(J_n)$, $n=0,1,2,...$, and $\\beta_n$, $J_n$ are as follows:\n\\begin{enumerate}\\label{e-t}\n\\item $\\beta_0=1$, $|\\beta_n|<|\\beta_{n+1}|$ for $n=0,1,2,,,$, and $\\beta_n\\to\\infty$,\n\\item if $I_n=<-\\beta_n,\\beta_n>$, then $I_n\\subset J_n\\subset I_{n+1}$, $n=0,1,2,...$,\n\\item for every $n>0$ there exists an integer $p_n\\ge 2$, such that\n$$g_0(x)=g_n^{p_n}(x), \\mbox{ for all } x\\in [-1,1].$$\n\\end{enumerate}\nThen there exists a simply connected neighborhood $V\\subset C_{J_0}$ of $[-1,1]$, such that $\\overline{V'}\\subset V$ where $V'=g_0^{-1}(V)$.\nIn other words, $g_0: V'\\to V$ is a PL restriction of the map $g_0:\\Omega_{g_0}\\to \\C_{J_0}$.\n\nMoreover, assuming\n\\begin{equation}\\label{label-m}\nK_1\\le \\frac{|J_n|}{|I_n|}\\le K_2, \\\n|I_n|\\le\\Lambda^n, \\ p_n\\le N^n,\n\\end{equation}\nfor some $11$, $N\\ge 2$ and all $n$, the following uniform bounds holds:\nthere exists $m=m(K_1,K_2,\\Lambda,N)>0$\nsuch that $\\mod(V\\setminus \\overline{V'})\\ge m$, for any $\\mathcal{G}$ as above and for a choice of the domain $V$.\n\\end{theorem}", "r-tower": "\\begin{remark}\\label{r-tower}\nIn our application of Theorem \\ref{t-m}, $p_{n+1}/p_n\\ge 2$ and are integers (where $p_0=1)$, moreover, $g_n=g_{n+1}^{p_{n+1}/p_n}$ on $I_n$, $n=0,1,...$.\n\\footnote{It seems, the converse is also true: under the conditions of Theorem \\ref{t-m}, $p_{n+1}/p_n\\ge 2$ and are integers and $g_n=g_{n+1}^{p_{n+1}/p_n}$ on $I_n$, but we don't need this for the proof.}\nIn other words, $\\mathcal{G}$ is a one-sided infinite tower of maps from the Epstein class. Cf. \\cite{mcm1}\nwhere the notion of tower in the context of PL maps was introduced and greatly developed. Theorem \\ref{t-m} claims that\na tower of Epstein's maps is a tower of PL maps.\n\\end{remark}", "t-bi": "\\begin{theorem}\\label{t-bi}\nLet $g:U_{-1}\\to U_0$ be a proper holomorphic map of degree $d\\ge 2$ where $U_{-1}, U_0$ are open simply connected subsets of $\\C$ such that $U_{-1}\\subset U_0$.\nSuppose that $X\\subset U_0$ is a compact and full (i.e. $\\C\\setminus X$ is connected) set which is completely invariant, i.e.,\n$g^{-1}(X)=X$, and contains all critical points of $g$. Then there exists a polynomial-like restriction $g:V'\\to V$ where $V', V$ are neighborhoods of $X$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 1321, "pre_theorem_intro_text": "A real quadratic map $f(x)=x^2+c$ of the real line $\\mathbb{R}$ into itself is renormalizable of period $q\\ge 2$ if there exists a\nsymmetric closed $q$-periodic interval\n$I=I(q)$, that is, $f^q(I)\\subset I$, $f^q: I\\to I$ is unimodal, i.e., a folding map of an interval with a single turning point (which will usually assumed to be at $0$), and one of the end points $\\beta$ of $I$ is a fixed point of $f^q$.\nTo every such $f^q: I\\to I$ one associates a {\\it renormalization} (rescaling)\n$$R^q(f)=\\beta^{-1}\\circ f^q\\circ \\beta:[-1,1]\\to [-1,1].$$\n\nThe map $f:\\mathbb{R}\\to\\mathbb{R}$ is called infinitely renormalizable with combinatorics bounded by $N$,\nif $f$ is renormalizable of periods $2\\le q_1<...q_n1$, $N\\ge 2$ and all $n$, the following uniform bounds holds:\nthere exists $m=m(K_1,K_2,\\Lambda,N)>0$\nsuch that $\\mod(V\\setminus \\overline{V'})\\ge m$, for any $\\mathcal{G}$ as above and for a choice of the domain $V$.\n\\end{theorem}\nThe proof will be given in Sect. \\ref{s-m}.\n\\begin{remark}\\label{r-tower}\nIn our application of Theorem \\ref{t-m}, $p_{n+1}/p_n\\ge 2$ and are integers (where $p_0=1)$, moreover, $g_n=g_{n+1}^{p_{n+1}/p_n}$ on $I_n$, $n=0,1,...$.\n\\footnote{It seems, the converse is also true: under the conditions of Theorem \\ref{t-m}, $p_{n+1}/p_n\\ge 2$ and are integers and $g_n=g_{n+1}^{p_{n+1}/p_n}$ on $I_n$, but we don't need this for the proof.}\nIn other words, $\\mathcal{G}$ is a one-sided infinite tower of maps from the Epstein class. Cf. \\cite{mcm1}\nwhere the notion of tower in the context of PL maps was introduced and greatly developed. Theorem \\ref{t-m} claims that\na tower of Epstein's maps is a tower of PL maps.\n\\end{remark}\nNow we associate to every infinitely renormalizable real quadratic polynomial with bounded combinatorics a tower as above, see Proposition \\ref{p-infren-tower}. The main ingredient is the following real bounds for such maps.\n\nGiven a real interval $A\\ni 0$ and a number $t>1$, let $t.A=\\{tx: x\\in A\\}$, the $(t-1)$-neighborhood, or $t$-enlargement of $A$.\n\\begin{prop}\\label{p-real} (Real bounds \\cite{Su1}, \\cite{MS})\nThere exist an absolute constant $L>1$ and some $1<\\mu<\\lambda<\\infty$ which depend only on $N$, such that for every $n\\ge n(f)$:\n(a) if $J(q_n)=L.I(q_n)$, then $J(q_n)\\subset W_n$,\n(b) $\\mu\\le I(q_{n})/I(q_{n+1})\\le \\lambda$.\n\\end{prop}\nNote that these real bounds are known to hold in a much more generality, see \\cite{Su1}, \\cite{MS}, in particular, for infinitely renormalizable maps of bounded combinatorics from the Epstein class.\n\nBy Proposition \\ref{p-real} and the preceding discussion,\n$$\\tilde f_n:=f^{q_n}|_{I(q_n)}\\in E_{\\beta_n}(J(q_n)).$$\n\\begin{prop}\\label{p-infren-tower}\nTo every sequence $\\mathcal{R}\\subset\\N$ one can associate its subsequence $S$ and a sequence of maps\n$g_m\\in E_{b_m}(J_m)$, where $J_m=(-L|b_m|, L|b_m|)$, $m=0,1,2,...$, as follows:\n\\begin{enumerate}\\label{e-t-subs}\n\\item $b_0=1$, i.e., $g_0\\in E_1(J_0)$, and the sequence of renormalizations $\\{R^{q_j}f\\}_{j\\in S}\\to g_0$ uniformly on compacts in $\\Omega_{g_0}$,\n\\item if $I_m=<-b_m,b_m>$, then $\\mu\\le \\frac{|I_{m+1}|}{|I_m|}\\le \\lambda$ and $J_m\\subset I_{m+1}$, $m=0,1,2,...$,\n\\item for every $m>1$ there exists an integer $p_m\\ge 2$, such that\n$$g_0(x)=g_m^{p_m}(x), \\mbox{ for all } x\\in [-1,1].$$\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nFor a fixed $m>n(f)$, define a finite string of rescalings\n$$\\mathcal{G}_m=\\{g_{m,n}=\\beta_m^{-1}\\circ \\tilde f_n\\circ \\beta_m: n=m,m-1,...,n(f)\\}.$$\nIn other words, we rescale all $\\tilde f_n$, $n=m,m-1,...,n(f)$, by a single rescaling $z\\mapsto \\beta_m z$, which turns $I(q_m)$ into $[-1,1]$\nso that $g_{m,m}=R^{q_m}f$.\nThus, if $J_{m,n}=\\beta_m^{-1} J(q_n)$, $\\beta_{m,n}=\\beta_m^{-1}\\beta_n$, then\n$$g_{m,m}\\in E_1(J_{m,m}), \\ \\ g_{m,m-k}\\in E_{\\beta_{m,m-k}}(J_{m,m-k}), \\ k=1,...,m-n(f).$$\nDenote $I_{m,n}=<-\\beta_{m,n},\\beta_{m,n}>$ and $J_{m,n}= L.I_{m,n}$. Then\n$$I_{m,m}=[-1,1], \\ I_{m,n}\\subset J_{m,n}\\subset I_{m,n-1}, n=m,m-1,...,m-n(f).$$\nRecall that $a_n=q_n/q_{n-1}$. Hence,\n$$g_{m,n}(x)=g_{m,n-1}^{a_n}(x), x\\in I_{m,n},\\mbox{ and }\ng_{m,m}(x)=g_{m,n}^{p_{m,n}}(x), x\\in I_{m,m}=[-1,1],\n$$\nwhere $p_{m,n}=\\frac{q_m}{q_n}=a_m a_{m-1}...a_{n+1}, \\ n=m-1,...,n(f)$. Also:\n\\begin{equation}\\label{e-p}\n2^{m-n}\\le p_{m,n}\\le N^{m-n},\n\\end{equation}\n\\begin{equation}\\label{e-I}\n\\mu^{m-n}<|\\beta_{m,n}|<\\lambda^{m-n}, \\mbox{ i.e., } \\mu^{m-n}<|I_{m,n}|<\\lambda^{m-n}.\n\\end{equation}\nFix any infinite sequence of indices $\\mathcal{R}\\subset\\N$. Apply Lemma \\ref{l-eps}(2) to a sequence $\\{g_{m,m}\\in E_1((-L,L)): m\\in\\mathcal{R}\\}$ and find a converging subsequence $g_{m,m}\\to g_0$ along some $\\mathcal{R}_1\\subset\\mathcal{R}$,\nwhere $g_0\\in E_1((-L,L))$.\nThen apply Lemma \\ref{l-eps}(3) to $\\{g_{m,m-1}\\in E_{\\beta_{m,m-1}}(J_{m,m-1}): m\\in\\mathcal{R}_1\\}$ and find $\\mathcal{R}_2\\subset\\mathcal{R}_1$ such that $\\beta_{m,m-1}\\to b_1$ and $p_{m,m-1}=p_1$\n(which is possible in view of (\\ref{e-p})-(\\ref{e-I})), and\n$g_{m,m-1}\\to g_1$. Observe that $I_1:=<-b_1,b_1>\\supset L.[-1,1]=[-L,L]$, $p_1\\ge 2$, $g_1\\in E_{b_1}((-L|b_1|, L|b_1|))$ and\n$g_0=g_1^{p_1}$ on $[-1,1]$. Then consider $\\{g_{m,m-1}\\}_{m\\in\\mathcal{R}_2}$,\npass to a subsequence $\\mathcal{R}_3\\subset\\mathcal{R}_2$, and so on. Cantor's diagonal procedure finishes the proof.\n\\end{proof}\n\\section{Preliminaries: Poincare's neighborhoods}\\label{s-po}\nAs in \\cite{Su1}, we use Schwartz's lemma for maps of the slit complex plane and the corresponding Poincare's neighborhoods\n$D(A, \\theta)$.\nWe state here some known related to this results which are used in the proof of Theorem \\ref{t-m}.", "post_theorem_intro_text_len": 4689, "post_theorem_intro_text": "This theorem is proved in \\cite{Su1} in a more generality, for maps from Epstein's class, see Definition \\ref{d-eps}. Our proof also works in\nthis class.\n\nTheorem \\ref{t-su} was a key step in Sullivan's proof \\cite{Su1} that bounded type renormalizations converge to the fixed renormalization orbit,\nas well as in later\nproofs \\cite{mcm1}, \\cite{ly1} of other Feigenbaum's conjectures \\cite{feig}.\nSince \\cite{Su1}, complex bounds\nhave been obtained, essentially, for all real analytic\ninfinitely renormalizable maps\n\\cite{LS}, \\cite{grsw},\n\\cite{lyym},\n\\cite{shen}, \\cite{css}, and\ncrucially used e.g. in \\cite{LS}, \\cite{grsw}, \\cite{ly}, \\cite{kss1}, \\cite{kss2}.\n\nIt is usually a big deal to obtain complex bounds, and proofs\nin the works cited above are very technical.\nWe propose a new approach which\nis based on the main result of \\cite{L}, more precisely, on its very particular case as follows:\n\\begin{theorem}\\label{t-bi}\nLet $g:U_{-1}\\to U_0$ be a proper holomorphic map of degree $d\\ge 2$ where $U_{-1}, U_0$ are open simply connected subsets of $\\mathbb{C}$ such that $U_{-1}\\subset U_0$.\nSuppose that $X\\subset U_0$ is a compact and full (i.e. $\\mathbb{C}\\setminus X$ is connected) set which is completely invariant, i.e.,\n$g^{-1}(X)=X$, and contains all critical points of $g$. Then there exists a polynomial-like restriction $g:V'\\to V$ where $V', V$ are neighborhoods of $X$.\n\\end{theorem}\n\nThe idea of using Theorem \\ref{t-bi} to prove\ncomplex bounds is roughly as follows. It is well-known \\cite{Su1} that any limit of renormalizations $R^{q_n}f$ is a unimodal map\n$g:[-1,1]\\to [-1,1]$ which extends to a map $g: U_{-1}\\to U_0$ from the Epstein class, and the convergence is uniform on compacts in $U_{-1}$ assuming we start with $f$ from the Epstein class.\nHere $g: U_{-1}\\to U_0$ is a proper analytic map of degree $2$, where $U_{-1}, U_0$ are as in Theorem \\ref{t-bi}\n(in fact, $U_0$ is a slit complex plane, see below). Let $T$ be a minimal $g$-completely invariant set which contains the interval $[-1,1]$.\nThe main goal is to prove that the closure $\\mathcal{J}$ of $T$ is a proper subset of $U_0$. Then Theorem \\ref{t-bi} applies to the topological hull $\\widehat{\\mathcal{J}}$ of $\\mathcal{J}$, the union of $\\mathcal{J}$ with all connected components of its complement, and we end up with\na PL restriction $g:V'\\to V$ in a neighborhood of $\\widehat{\\mathcal{J}}$.\nSince PL maps are stable under small perturbations, a nearby PL map persists\nfor any nearby (to $g$) renormalization $R^{q_n}f$, and the modulus of this PL map is close to the limit modulus\n$\\mod(V\\setminus \\overline{V'})$.\nThat would mean complex bounds.\n\\begin{remark}\\label{r-bi}\nWe don't assume a priori that the compact $X$ in Theorem \\ref{t-bi} is connected although this holds a posteriori as $X$ turns out to be the non-escaping set\nof a PL maps with non-escaping critical points.\nHowever, it can be seen beforehand, from the construction, see Sect. \\ref{s-m}, that\nthe set $X=\\widehat{\\mathcal{J}}$ as above is connected.\nIn this case (of connected $X$) the proof of Theorem \\ref{t-bi}\nbecomes significantly simpler, see \\cite{L}.\n\\end{remark}\nWe realize the idea described above by giving a detailed proof\nfor infinitely renormalizable maps with bounded combinatorics.\nTo be more precise, we prove that $\\mathcal{J}$ is proper inside of $U_0$ by a contradiction. Assuming the contrary, first we show that $\\mathcal{J}$ must meet slits of the slit complex plane $U_0$, i.e., it is impossible for $\\mathcal{J}$ to be unbounded without touching the slits. This part of the proof holds for any combinatorics.\nOn the other hand, the bounded combinatorics along with real bounds imply that the function $g$ lies at the bottom of an infinite tower of maps from the Epstein class\n(cf. \\cite{mcm1}, see Theorem \\ref{t-m} and Remark \\ref{r-tower}). This allows us to extend $g:U_{-1}\\to U_0$ to a holomorphic function in bigger simply connected domains, somewhat similar to \\cite{EL}. That leads to a contradiction, see the proof of Theorem \\ref{t-m}. Then Theorem \\ref{t-su} is reduced to Theorem \\ref{t-m} with help of (basically, known) Proposition \\ref{p-infren-tower}.\n\n{\\it Notations}.\n\nFor $a,b\\in\\mathbb{R}$, $$ is the closed interval with end points $a,b$ (where $a>b$ is possible),\n\n$\\R_+=\\{x>0\\}$, $\\R_-=\\{x<0\\}$,\n\n$\\mathbb{H}^+=\\{\\Im(z)>0\\}$, $\\mathbb{H}^-=-\\mathbb{H}^+$, the upper and lower half-planes,\n\n$\\Pi=\\{\\Im(z)>0, \\Re(z)>0\\}$, $\\overline\\Pi=\\{\\Im(z)\\ge 0, \\Re(z)\\ge 0\\}$,\n\nfor an open interval $A\\subset\\mathbb{R}$,\n$\\C_{A}=(\\mathbb{C}\\setminus \\mathbb{R})\\cup A$, a slit complex plane.\n\n{\\bf Acknowledgment.} The author thanks Sebastian van Strien for helpful comments.", "sketch": "The proposed proof strategy for Theorem~\\ref{t-su} is to use Theorem~\\ref{t-bi} and show that complex bounds follow from a suitable polynomial-like (PL) restriction for a limit map of renormalizations.\n\n- One starts from the fact (\\cite{Su1}) that “any limit of renormalizations $R^{q_n}f$ is a unimodal map $g:[-1,1]\\to [-1,1]$ which extends to a map $g: U_{-1}\\to U_0$ from the Epstein class,” with “$g: U_{-1}\\to U_0$ … a proper analytic map of degree $2$,” where $U_0$ is “a slit complex plane.”\n\n- Let $T$ be “a minimal $g$-completely invariant set which contains the interval $[-1,1]$,” and let $\\mathcal{J}$ be “the closure $\\mathcal{J}$ of $T$.” The “main goal is to prove that the closure $\\mathcal{J}$ of $T$ is a proper subset of $U_0$.”\n\n- If $\\mathcal{J}\\subsetneq U_0$, then “Theorem~\\ref{t-bi} applies to the topological hull $\\widehat{\\mathcal{J}}$ of $\\mathcal{J}$,” yielding “a PL restriction $g:V'\\to V$ in a neighborhood of $\\widehat{\\mathcal{J}}$.” Using that “PL maps are stable under small perturbations,” the PL restriction “persists for any nearby (to $g$) renormalization $R^{q_n}f$,” and “the modulus of this PL map is close to the limit modulus $\\mod(V\\setminus \\overline{V'})$. That would mean complex bounds.”\n\n- To show $\\mathcal{J}$ is proper in $U_0$, they proceed “by a contradiction.” Assuming instead that $\\mathcal{J}$ is not proper, “first we show that $\\mathcal{J}$ must meet slits of the slit complex plane $U_0$, i.e., it is impossible for $\\mathcal{J}$ to be unbounded without touching the slits” (this part “holds for any combinatorics”).\n\n- For bounded combinatorics, “bounded combinatorics along with real bounds imply that the function $g$ lies at the bottom of an infinite tower of maps from the Epstein class,” which “allows us to extend $g:U_{-1}\\to U_0$ to a holomorphic function in bigger simply connected domains.” This “leads to a contradiction, see the proof of Theorem~\\ref{t-m}$.”\n\n- Finally, “Theorem~\\ref{t-su} is reduced to Theorem~\\ref{t-m} with help of (basically, known) Proposition~\\ref{p-infren-tower}.”", "expanded_sketch": "The proposed proof strategy, in establishing the main theorem, is to use the following theorem and show that complex bounds follow from a suitable polynomial-like (PL) restriction for a limit map of renormalizations.\n\nWe first use the following theorem. \\begin{theorem}\\label{t-bi}\nLet $g:U_{-1}\\to U_0$ be a proper holomorphic map of degree $d\\ge 2$ where $U_{-1}, U_0$ are open simply connected subsets of $\\C$ such that $U_{-1}\\subset U_0$.\nSuppose that $X\\subset U_0$ is a compact and full (i.e. $\\C\\setminus X$ is connected) set which is completely invariant, i.e.,\n$g^{-1}(X)=X$, and contains all critical points of $g$. Then there exists a polynomial-like restriction $g:V'\\to V$ where $V', V$ are neighborhoods of $X$.\n\\end{theorem}\n\n- One starts from the fact (Sullivan \\cite{Su1}) that “any limit of renormalizations $R^{q_n}f$ is a unimodal map $g:[-1,1]\\to [-1,1]$ which extends to a map $g: U_{-1}\\to U_0$ from the Epstein class,” with “$g: U_{-1}\\to U_0$ … a proper analytic map of degree $2$,” where $U_0$ is “a slit complex plane.”\n\n- Let $T$ be “a minimal $g$-completely invariant set which contains the interval $[-1,1]$,” and let $\\mathcal{J}$ be “the closure $\\mathcal{J}$ of $T$.” The “main goal is to prove that the closure $\\mathcal{J}$ of $T$ is a proper subset of $U_0$.”\n\n- If $\\mathcal{J}\\subsetneq U_0$, then the theorem above applies to the topological hull $\\widehat{\\mathcal{J}}$ of $\\mathcal{J}$, yielding “a PL restriction $g:V'\\to V$ in a neighborhood of $\\widehat{\\mathcal{J}}$.” Using that “PL maps are stable under small perturbations,” the PL restriction “persists for any nearby (to $g$) renormalization $R^{q_n}f$,” and “the modulus of this PL map is close to the limit modulus $\\mod(V\\setminus \\overline{V'})$. That would mean complex bounds.”\n\n- To show $\\mathcal{J}$ is proper in $U_0$, they proceed “by a contradiction.” Assuming instead that $\\mathcal{J}$ is not proper, “first we show that $\\mathcal{J}$ must meet slits of the slit complex plane $U_0$, i.e., it is impossible for $\\mathcal{J}$ to be unbounded without touching the slits” (this part “holds for any combinatorics”).\n\n- For bounded combinatorics, “bounded combinatorics along with real bounds imply that the function $g$ lies at the bottom of an infinite tower of maps from the Epstein class,” which “allows us to extend $g:U_{-1}\\to U_0$ to a holomorphic function in bigger simply connected domains.” This “leads to a contradiction, see the proof of the following theorem.”\n\nWe use the following theorem. \\begin{theorem}\\label{t-m}\nLet $\\mathcal{G}=\\{g_n\\}_{n=0}^\\infty$ where $g_n\\in E_{\\beta_n}(J_n)$, $n=0,1,2,...$, and $\\beta_n$, $J_n$ are as follows:\n\\begin{enumerate}\\label{e-t}\n\\item $\\beta_0=1$, $|\\beta_n|<|\\beta_{n+1}|$ for $n=0,1,2,,,$, and $\\beta_n\\to\\infty$,\n\\item if $I_n=<-\\beta_n,\\beta_n>$, then $I_n\\subset J_n\\subset I_{n+1}$, $n=0,1,2,...$,\n\\item for every $n>0$ there exists an integer $p_n\\ge 2$, such that\n$$g_0(x)=g_n^{p_n}(x), \\mbox{ for all } x\\in [-1,1].$$\n\\end{enumerate}\nThen there exists a simply connected neighborhood $V\\subset C_{J_0}$ of $[-1,1]$, such that $\\overline{V'}\\subset V$ where $V'=g_0^{-1}(V)$.\nIn other words, $g_0: V'\\to V$ is a PL restriction of the map $g_0:\\Omega_{g_0}\\to \\C_{J_0}$.\n\nMoreover, assuming\n\\begin{equation}\\label{label-m}\nK_1\\le \\frac{|J_n|}{|I_n|}\\le K_2, \\\n|I_n|\\le\\Lambda^n, \\ p_n\\le N^n,\n\\end{equation}\nfor some $11$, $N\\ge 2$ and all $n$, the following uniform bounds holds:\nthere exists $m=m(K_1,K_2,\\Lambda,N)>0$\nsuch that $\\mod(V\\setminus \\overline{V'})\\ge m$, for any $\\mathcal{G}$ as above and for a choice of the domain $V$.\n\\end{theorem}\n\n- Finally, the main theorem is reduced to the theorem above with help of the following proposition.\n\n\\begin{prop}\\label{p-infren-tower}\nTo every sequence $\\mathcal{R}\\subset\\N$ one can associate its subsequence $S$ and a sequence of maps\n$g_m\\in E_{b_m}(J_m)$, where $J_m=(-L|b_m|, L|b_m|)$, $m=0,1,2,...$, as follows:\n\\begin{enumerate}\\label{e-t-subs}\n\\item $b_0=1$, i.e., $g_0\\in E_1(J_0)$, and the sequence of renormalizations $\\{R^{q_j}f\\}_{j\\in S}\\to g_0$ uniformly on compacts in $\\Omega_{g_0}$,\n\\item if $I_m=<-b_m,b_m>$, then $\\mu\\le \\frac{|I_{m+1}|}{|I_m|}\\le \\lambda$ and $J_m\\subset I_{m+1}$, $m=0,1,2,...$,\n\\item for every $m>1$ there exists an integer $p_m\\ge 2$, such that\n$$g_0(x)=g_m^{p_m}(x), \\mbox{ for all } x\\in [-1,1].$$\n\\end{enumerate}\n\\end{prop}", "expanded_theorem": "\\label{t-su} (Sullivan \\cite{Su1}, \\cite{MS})\nGiven $N$ there exists $m=m(N)>0$ as follows. Let $f$ be an infinitely renormalizable real quadratic polynomial with combinatorics\nbounded by $N$.\nThen, for every $n\\ge n(f)$,\nthe corresponding renormalization $R^{q_n}(f):[-1,1]\\to[-1,1]$ admits a polynomial-like extension $R^{q_n}(f): V_n'\\to V_n$ of degree $2$ such that\n$\\mod(V_n\\setminus\\overline{V_n'})\\ge m$, for the modulus of the annulus $V_n\\setminus\\overline{V_n'}$.,\n", "theorem_type": ["Existential–Universal", "Universal"], "mcq": {"question": "Fix a number $N$. Let $f(x)=x^2+c$ be a real quadratic polynomial. Say that $f$ is infinitely renormalizable with combinatorics bounded by $N$ if it is renormalizable for periods $2\\le q_10$, depending only on $N$, such that for every infinitely renormalizable real quadratic polynomial $f$ with combinatorics bounded by $N$ and for every renormalization index $n\\ge n(f)$, the renormalization $R^{q_n}(f):[-1,1]\\to[-1,1]$ admits a polynomial-like extension of degree $2$, say $R^{q_n}(f):V_n'\\to V_n$, with\n$$\\operatorname{mod}(V_n\\setminus\\overline{V_n'})\\ge m.$$"}, "choices": [{"label": "B", "text": "There exists a constant $m=m(N,f)>0$ such that for every infinitely renormalizable real quadratic polynomial $f$ with combinatorics bounded by $N$ there is an index $n_0=n_0(f)$ for which, for every $n\\ge n_0$, the renormalization $R^{q_n}(f):[-1,1]\\to[-1,1]$ admits a polynomial-like extension of degree $2$, say $R^{q_n}(f):V_n'\\to V_n$, with\n$$\\operatorname{mod}(V_n\\setminus\\overline{V_n'})\\ge m.$$"}, {"label": "C", "text": "For every infinitely renormalizable real quadratic polynomial $f$ with combinatorics bounded by $N$ and every renormalization index $n\\ge n(f)$, the renormalization $R^{q_n}(f):[-1,1]\\to[-1,1]$ admits a polynomial-like extension of degree $2$, say $R^{q_n}(f):V_n'\\to V_n$."}, {"label": "D", "text": "There exists a constant $m=m(N)>0$, depending only on $N$, such that for every infinitely renormalizable real quadratic polynomial $f$ with combinatorics bounded by $N$ there exists an index $n(f)$ for which the single renormalization $R^{q_{n(f)}}(f):[-1,1]\\to[-1,1]$ admits a polynomial-like extension of degree $2$, say $R^{q_{n(f)}}(f):V'\\to V$, with\n$$\\operatorname{mod}(V\\setminus\\overline{V'})\\ge m.$$"}, {"label": "E", "text": "There exists a constant $m=m(N)>0$, depending only on $N$, such that for every infinitely renormalizable real quadratic polynomial $f$ with combinatorics bounded by $N$ and for every renormalization index $n\\ge n(f)$, the renormalization $R^{q_n}(f):[-1,1]\\to[-1,1]$ admits a proper holomorphic extension of degree $2$, say $R^{q_n}(f):V_n'\\to V_n$, between simply connected domains with\n$$\\operatorname{mod}(V_n\\setminus\\overline{V_n'})\\ge m,$$\nbut one need not require $V_n$ and $V_n'$ to be topological disks or $\\overline{V_n'}\\subset V_n$."}], "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": "uniform dependence of the modulus constant", "template_used": "quantifier_dependence"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "uniform lower modulus bound depending only on N", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "for every sufficiently deep renormalization, not merely one level", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "polynomial-like geometry requiring topological disks and nesting $\\overline{V_n'}\\subset V_n$", "template_used": "wildcard"}]}} +{"id": "2601.21326v1", "paper_link": "http://arxiv.org/abs/2601.21326v1", "theorems_cnt": 2, "theorem": {"env_name": "theorem", "content": "\\label{t-su} (Sullivan \\cite{Su1}, \\cite{MS})\nGiven $N$ there exists $m=m(N)>0$ as follows. Let $f$ be an infinitely renormalizable real quadratic polynomial with combinatorics\nbounded by $N$.\nThen, for every $n\\ge n(f)$,\nthe corresponding renormalization $R^{q_n}(f):[-1,1]\\to[-1,1]$ admits a polynomial-like extension $R^{q_n}(f): V_n'\\to V_n$ of degree $2$ such that\n$\\mod(V_n\\setminus\\overline{V_n'})\\ge m$, for the modulus of the annulus $V_n\\setminus\\overline{V_n'}$.", "start_pos": 12653, "end_pos": 13155, "label": "t-su"}, "ref_dict": {"d-eps": "\\begin{definition}\\label{d-eps} (Epstein classes $E_\\beta(J)$, cf. \\cite{E}, \\cite{Su1}, \\cite{MS})\nLet $g: [-1,1]\\to [-1,1]$ be a real analytic map as follows: (1) $g$ is unimodal: $g'(x)=0$ iff $x=0$, (2) $g(-1)=g(1)=1$\nand $0\\in g([-1,1])$, i.e., $g(0)\\le 0$. Given a symmetric w.r.t. $0$ open interval $J$, we say that $g\\in E_1(J)$ if\nthere exists another open symmetric interval $J'$ such that $[-1,1]\\subset J'\\subset J$, $g$ extends to a real-analytic unimodal map $g:J'\\to J$ and there exists a representation\n$g=F\\circ Q$ where $Q(z)=z^2$ and $F: [0,1]\\to [g(0),1]$ is a diffeomorphism such that $F^{-1}$ extends to a univalent (i.e. holomorphic injective)\nmap defined on $\\C_J$.\nMore generally, given $\\beta\\in\\R\\setminus \\{0\\}$,\n$g\\in E_{\\beta}(J)$ (or simply $E_\\beta$) if $g=\\beta g_1 \\beta^{-1}$ for some $g_1\\in E_1(\\beta^{-1}J)$.\n\\end{definition}", "t-su": "\\begin{theorem}\\label{t-su} (Sullivan \\cite{Su1}, \\cite{MS})\nGiven $N$ there exists $m=m(N)>0$ as follows. Let $f$ be an infinitely renormalizable real quadratic polynomial with combinatorics\nbounded by $N$.\nThen, for every $n\\ge n(f)$,\nthe corresponding renormalization $R^{q_n}(f):[-1,1]\\to[-1,1]$ admits a polynomial-like extension $R^{q_n}(f): V_n'\\to V_n$ of degree $2$ such that\n$\\mod(V_n\\setminus\\overline{V_n'})\\ge m$, for the modulus of the annulus $V_n\\setminus\\overline{V_n'}$.\n\\end{theorem}", "p-infren-tower": "\\begin{prop}\\label{p-infren-tower}\nTo every sequence $\\mathcal{R}\\subset\\N$ one can associate its subsequence $S$ and a sequence of maps\n$g_m\\in E_{b_m}(J_m)$, where $J_m=(-L|b_m|, L|b_m|)$, $m=0,1,2,...$, as follows:\n\\begin{enumerate}\\label{e-t-subs}\n\\item $b_0=1$, i.e., $g_0\\in E_1(J_0)$, and the sequence of renormalizations $\\{R^{q_j}f\\}_{j\\in S}\\to g_0$ uniformly on compacts in $\\Omega_{g_0}$,\n\\item if $I_m=<-b_m,b_m>$, then $\\mu\\le \\frac{|I_{m+1}|}{|I_m|}\\le \\lambda$ and $J_m\\subset I_{m+1}$, $m=0,1,2,...$,\n\\item for every $m>1$ there exists an integer $p_m\\ge 2$, such that\n$$g_0(x)=g_m^{p_m}(x), \\mbox{ for all } x\\in [-1,1].$$\n\\end{enumerate}\n\\end{prop}", "s-m": "\\label{s-m}\nDefine inductively: $T_0=[-1,1]$ and $T_{n+1}=g_0^{-1}(T_n)$, $n=0,1,2,...$. As $g_0([-1,1])\\subset [-1,1]$, then\n$T_n=\\cup_{i=0}^n g_0^{-i}([-1,1])$ so that $\\{T_n\\}$ is an increasing seq", "t-m": "\\begin{theorem}\\label{t-m}\nLet $\\mathcal{G}=\\{g_n\\}_{n=0}^\\infty$ where $g_n\\in E_{\\beta_n}(J_n)$, $n=0,1,2,...$, and $\\beta_n$, $J_n$ are as follows:\n\\begin{enumerate}\\label{e-t}\n\\item $\\beta_0=1$, $|\\beta_n|<|\\beta_{n+1}|$ for $n=0,1,2,,,$, and $\\beta_n\\to\\infty$,\n\\item if $I_n=<-\\beta_n,\\beta_n>$, then $I_n\\subset J_n\\subset I_{n+1}$, $n=0,1,2,...$,\n\\item for every $n>0$ there exists an integer $p_n\\ge 2$, such that\n$$g_0(x)=g_n^{p_n}(x), \\mbox{ for all } x\\in [-1,1].$$\n\\end{enumerate}\nThen there exists a simply connected neighborhood $V\\subset C_{J_0}$ of $[-1,1]$, such that $\\overline{V'}\\subset V$ where $V'=g_0^{-1}(V)$.\nIn other words, $g_0: V'\\to V$ is a PL restriction of the map $g_0:\\Omega_{g_0}\\to \\C_{J_0}$.\n\nMoreover, assuming\n\\begin{equation}\\label{label-m}\nK_1\\le \\frac{|J_n|}{|I_n|}\\le K_2, \\\n|I_n|\\le\\Lambda^n, \\ p_n\\le N^n,\n\\end{equation}\nfor some $11$, $N\\ge 2$ and all $n$, the following uniform bounds holds:\nthere exists $m=m(K_1,K_2,\\Lambda,N)>0$\nsuch that $\\mod(V\\setminus \\overline{V'})\\ge m$, for any $\\mathcal{G}$ as above and for a choice of the domain $V$.\n\\end{theorem}", "r-tower": "\\begin{remark}\\label{r-tower}\nIn our application of Theorem \\ref{t-m}, $p_{n+1}/p_n\\ge 2$ and are integers (where $p_0=1)$, moreover, $g_n=g_{n+1}^{p_{n+1}/p_n}$ on $I_n$, $n=0,1,...$.\n\\footnote{It seems, the converse is also true: under the conditions of Theorem \\ref{t-m}, $p_{n+1}/p_n\\ge 2$ and are integers and $g_n=g_{n+1}^{p_{n+1}/p_n}$ on $I_n$, but we don't need this for the proof.}\nIn other words, $\\mathcal{G}$ is a one-sided infinite tower of maps from the Epstein class. Cf. \\cite{mcm1}\nwhere the notion of tower in the context of PL maps was introduced and greatly developed. Theorem \\ref{t-m} claims that\na tower of Epstein's maps is a tower of PL maps.\n\\end{remark}", "t-bi": "\\begin{theorem}\\label{t-bi}\nLet $g:U_{-1}\\to U_0$ be a proper holomorphic map of degree $d\\ge 2$ where $U_{-1}, U_0$ are open simply connected subsets of $\\C$ such that $U_{-1}\\subset U_0$.\nSuppose that $X\\subset U_0$ is a compact and full (i.e. $\\C\\setminus X$ is connected) set which is completely invariant, i.e.,\n$g^{-1}(X)=X$, and contains all critical points of $g$. Then there exists a polynomial-like restriction $g:V'\\to V$ where $V', V$ are neighborhoods of $X$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 1321, "pre_theorem_intro_text": "A real quadratic map $f(x)=x^2+c$ of the real line $\\mathbb{R}$ into itself is renormalizable of period $q\\ge 2$ if there exists a\nsymmetric closed $q$-periodic interval\n$I=I(q)$, that is, $f^q(I)\\subset I$, $f^q: I\\to I$ is unimodal, i.e., a folding map of an interval with a single turning point (which will usually assumed to be at $0$), and one of the end points $\\beta$ of $I$ is a fixed point of $f^q$.\nTo every such $f^q: I\\to I$ one associates a {\\it renormalization} (rescaling)\n$$R^q(f)=\\beta^{-1}\\circ f^q\\circ \\beta:[-1,1]\\to [-1,1].$$\n\nThe map $f:\\mathbb{R}\\to\\mathbb{R}$ is called infinitely renormalizable with combinatorics bounded by $N$,\nif $f$ is renormalizable of periods $2\\le q_1<...q_n1$, $N\\ge 2$ and all $n$, the following uniform bounds holds:\nthere exists $m=m(K_1,K_2,\\Lambda,N)>0$\nsuch that $\\mod(V\\setminus \\overline{V'})\\ge m$, for any $\\mathcal{G}$ as above and for a choice of the domain $V$.\n\\end{theorem}\nThe proof will be given in Sect. \\ref{s-m}.\n\\begin{remark}\\label{r-tower}\nIn our application of Theorem \\ref{t-m}, $p_{n+1}/p_n\\ge 2$ and are integers (where $p_0=1)$, moreover, $g_n=g_{n+1}^{p_{n+1}/p_n}$ on $I_n$, $n=0,1,...$.\n\\footnote{It seems, the converse is also true: under the conditions of Theorem \\ref{t-m}, $p_{n+1}/p_n\\ge 2$ and are integers and $g_n=g_{n+1}^{p_{n+1}/p_n}$ on $I_n$, but we don't need this for the proof.}\nIn other words, $\\mathcal{G}$ is a one-sided infinite tower of maps from the Epstein class. Cf. \\cite{mcm1}\nwhere the notion of tower in the context of PL maps was introduced and greatly developed. Theorem \\ref{t-m} claims that\na tower of Epstein's maps is a tower of PL maps.\n\\end{remark}\nNow we associate to every infinitely renormalizable real quadratic polynomial with bounded combinatorics a tower as above, see Proposition \\ref{p-infren-tower}. The main ingredient is the following real bounds for such maps.\n\nGiven a real interval $A\\ni 0$ and a number $t>1$, let $t.A=\\{tx: x\\in A\\}$, the $(t-1)$-neighborhood, or $t$-enlargement of $A$.\n\\begin{prop}\\label{p-real} (Real bounds \\cite{Su1}, \\cite{MS})\nThere exist an absolute constant $L>1$ and some $1<\\mu<\\lambda<\\infty$ which depend only on $N$, such that for every $n\\ge n(f)$:\n(a) if $J(q_n)=L.I(q_n)$, then $J(q_n)\\subset W_n$,\n(b) $\\mu\\le I(q_{n})/I(q_{n+1})\\le \\lambda$.\n\\end{prop}\nNote that these real bounds are known to hold in a much more generality, see \\cite{Su1}, \\cite{MS}, in particular, for infinitely renormalizable maps of bounded combinatorics from the Epstein class.\n\nBy Proposition \\ref{p-real} and the preceding discussion,\n$$\\tilde f_n:=f^{q_n}|_{I(q_n)}\\in E_{\\beta_n}(J(q_n)).$$\n\\begin{prop}\\label{p-infren-tower}\nTo every sequence $\\mathcal{R}\\subset\\N$ one can associate its subsequence $S$ and a sequence of maps\n$g_m\\in E_{b_m}(J_m)$, where $J_m=(-L|b_m|, L|b_m|)$, $m=0,1,2,...$, as follows:\n\\begin{enumerate}\\label{e-t-subs}\n\\item $b_0=1$, i.e., $g_0\\in E_1(J_0)$, and the sequence of renormalizations $\\{R^{q_j}f\\}_{j\\in S}\\to g_0$ uniformly on compacts in $\\Omega_{g_0}$,\n\\item if $I_m=<-b_m,b_m>$, then $\\mu\\le \\frac{|I_{m+1}|}{|I_m|}\\le \\lambda$ and $J_m\\subset I_{m+1}$, $m=0,1,2,...$,\n\\item for every $m>1$ there exists an integer $p_m\\ge 2$, such that\n$$g_0(x)=g_m^{p_m}(x), \\mbox{ for all } x\\in [-1,1].$$\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nFor a fixed $m>n(f)$, define a finite string of rescalings\n$$\\mathcal{G}_m=\\{g_{m,n}=\\beta_m^{-1}\\circ \\tilde f_n\\circ \\beta_m: n=m,m-1,...,n(f)\\}.$$\nIn other words, we rescale all $\\tilde f_n$, $n=m,m-1,...,n(f)$, by a single rescaling $z\\mapsto \\beta_m z$, which turns $I(q_m)$ into $[-1,1]$\nso that $g_{m,m}=R^{q_m}f$.\nThus, if $J_{m,n}=\\beta_m^{-1} J(q_n)$, $\\beta_{m,n}=\\beta_m^{-1}\\beta_n$, then\n$$g_{m,m}\\in E_1(J_{m,m}), \\ \\ g_{m,m-k}\\in E_{\\beta_{m,m-k}}(J_{m,m-k}), \\ k=1,...,m-n(f).$$\nDenote $I_{m,n}=<-\\beta_{m,n},\\beta_{m,n}>$ and $J_{m,n}= L.I_{m,n}$. Then\n$$I_{m,m}=[-1,1], \\ I_{m,n}\\subset J_{m,n}\\subset I_{m,n-1}, n=m,m-1,...,m-n(f).$$\nRecall that $a_n=q_n/q_{n-1}$. Hence,\n$$g_{m,n}(x)=g_{m,n-1}^{a_n}(x), x\\in I_{m,n},\\mbox{ and }\ng_{m,m}(x)=g_{m,n}^{p_{m,n}}(x), x\\in I_{m,m}=[-1,1],\n$$\nwhere $p_{m,n}=\\frac{q_m}{q_n}=a_m a_{m-1}...a_{n+1}, \\ n=m-1,...,n(f)$. Also:\n\\begin{equation}\\label{e-p}\n2^{m-n}\\le p_{m,n}\\le N^{m-n},\n\\end{equation}\n\\begin{equation}\\label{e-I}\n\\mu^{m-n}<|\\beta_{m,n}|<\\lambda^{m-n}, \\mbox{ i.e., } \\mu^{m-n}<|I_{m,n}|<\\lambda^{m-n}.\n\\end{equation}\nFix any infinite sequence of indices $\\mathcal{R}\\subset\\N$. Apply Lemma \\ref{l-eps}(2) to a sequence $\\{g_{m,m}\\in E_1((-L,L)): m\\in\\mathcal{R}\\}$ and find a converging subsequence $g_{m,m}\\to g_0$ along some $\\mathcal{R}_1\\subset\\mathcal{R}$,\nwhere $g_0\\in E_1((-L,L))$.\nThen apply Lemma \\ref{l-eps}(3) to $\\{g_{m,m-1}\\in E_{\\beta_{m,m-1}}(J_{m,m-1}): m\\in\\mathcal{R}_1\\}$ and find $\\mathcal{R}_2\\subset\\mathcal{R}_1$ such that $\\beta_{m,m-1}\\to b_1$ and $p_{m,m-1}=p_1$\n(which is possible in view of (\\ref{e-p})-(\\ref{e-I})), and\n$g_{m,m-1}\\to g_1$. Observe that $I_1:=<-b_1,b_1>\\supset L.[-1,1]=[-L,L]$, $p_1\\ge 2$, $g_1\\in E_{b_1}((-L|b_1|, L|b_1|))$ and\n$g_0=g_1^{p_1}$ on $[-1,1]$. Then consider $\\{g_{m,m-1}\\}_{m\\in\\mathcal{R}_2}$,\npass to a subsequence $\\mathcal{R}_3\\subset\\mathcal{R}_2$, and so on. Cantor's diagonal procedure finishes the proof.\n\\end{proof}\n\\section{Preliminaries: Poincare's neighborhoods}\\label{s-po}\nAs in \\cite{Su1}, we use Schwartz's lemma for maps of the slit complex plane and the corresponding Poincare's neighborhoods\n$D(A, \\theta)$.\nWe state here some known related to this results which are used in the proof of Theorem \\ref{t-m}.", "post_theorem_intro_text_len": 4689, "post_theorem_intro_text": "This theorem is proved in \\cite{Su1} in a more generality, for maps from Epstein's class, see Definition \\ref{d-eps}. Our proof also works in\nthis class.\n\nTheorem \\ref{t-su} was a key step in Sullivan's proof \\cite{Su1} that bounded type renormalizations converge to the fixed renormalization orbit,\nas well as in later\nproofs \\cite{mcm1}, \\cite{ly1} of other Feigenbaum's conjectures \\cite{feig}.\nSince \\cite{Su1}, complex bounds\nhave been obtained, essentially, for all real analytic\ninfinitely renormalizable maps\n\\cite{LS}, \\cite{grsw},\n\\cite{lyym},\n\\cite{shen}, \\cite{css}, and\ncrucially used e.g. in \\cite{LS}, \\cite{grsw}, \\cite{ly}, \\cite{kss1}, \\cite{kss2}.\n\nIt is usually a big deal to obtain complex bounds, and proofs\nin the works cited above are very technical.\nWe propose a new approach which\nis based on the main result of \\cite{L}, more precisely, on its very particular case as follows:\n\\begin{theorem}\\label{t-bi}\nLet $g:U_{-1}\\to U_0$ be a proper holomorphic map of degree $d\\ge 2$ where $U_{-1}, U_0$ are open simply connected subsets of $\\mathbb{C}$ such that $U_{-1}\\subset U_0$.\nSuppose that $X\\subset U_0$ is a compact and full (i.e. $\\mathbb{C}\\setminus X$ is connected) set which is completely invariant, i.e.,\n$g^{-1}(X)=X$, and contains all critical points of $g$. Then there exists a polynomial-like restriction $g:V'\\to V$ where $V', V$ are neighborhoods of $X$.\n\\end{theorem}\n\nThe idea of using Theorem \\ref{t-bi} to prove\ncomplex bounds is roughly as follows. It is well-known \\cite{Su1} that any limit of renormalizations $R^{q_n}f$ is a unimodal map\n$g:[-1,1]\\to [-1,1]$ which extends to a map $g: U_{-1}\\to U_0$ from the Epstein class, and the convergence is uniform on compacts in $U_{-1}$ assuming we start with $f$ from the Epstein class.\nHere $g: U_{-1}\\to U_0$ is a proper analytic map of degree $2$, where $U_{-1}, U_0$ are as in Theorem \\ref{t-bi}\n(in fact, $U_0$ is a slit complex plane, see below). Let $T$ be a minimal $g$-completely invariant set which contains the interval $[-1,1]$.\nThe main goal is to prove that the closure $\\mathcal{J}$ of $T$ is a proper subset of $U_0$. Then Theorem \\ref{t-bi} applies to the topological hull $\\widehat{\\mathcal{J}}$ of $\\mathcal{J}$, the union of $\\mathcal{J}$ with all connected components of its complement, and we end up with\na PL restriction $g:V'\\to V$ in a neighborhood of $\\widehat{\\mathcal{J}}$.\nSince PL maps are stable under small perturbations, a nearby PL map persists\nfor any nearby (to $g$) renormalization $R^{q_n}f$, and the modulus of this PL map is close to the limit modulus\n$\\mod(V\\setminus \\overline{V'})$.\nThat would mean complex bounds.\n\\begin{remark}\\label{r-bi}\nWe don't assume a priori that the compact $X$ in Theorem \\ref{t-bi} is connected although this holds a posteriori as $X$ turns out to be the non-escaping set\nof a PL maps with non-escaping critical points.\nHowever, it can be seen beforehand, from the construction, see Sect. \\ref{s-m}, that\nthe set $X=\\widehat{\\mathcal{J}}$ as above is connected.\nIn this case (of connected $X$) the proof of Theorem \\ref{t-bi}\nbecomes significantly simpler, see \\cite{L}.\n\\end{remark}\nWe realize the idea described above by giving a detailed proof\nfor infinitely renormalizable maps with bounded combinatorics.\nTo be more precise, we prove that $\\mathcal{J}$ is proper inside of $U_0$ by a contradiction. Assuming the contrary, first we show that $\\mathcal{J}$ must meet slits of the slit complex plane $U_0$, i.e., it is impossible for $\\mathcal{J}$ to be unbounded without touching the slits. This part of the proof holds for any combinatorics.\nOn the other hand, the bounded combinatorics along with real bounds imply that the function $g$ lies at the bottom of an infinite tower of maps from the Epstein class\n(cf. \\cite{mcm1}, see Theorem \\ref{t-m} and Remark \\ref{r-tower}). This allows us to extend $g:U_{-1}\\to U_0$ to a holomorphic function in bigger simply connected domains, somewhat similar to \\cite{EL}. That leads to a contradiction, see the proof of Theorem \\ref{t-m}. Then Theorem \\ref{t-su} is reduced to Theorem \\ref{t-m} with help of (basically, known) Proposition \\ref{p-infren-tower}.\n\n{\\it Notations}.\n\nFor $a,b\\in\\mathbb{R}$, $$ is the closed interval with end points $a,b$ (where $a>b$ is possible),\n\n$\\R_+=\\{x>0\\}$, $\\R_-=\\{x<0\\}$,\n\n$\\mathbb{H}^+=\\{\\Im(z)>0\\}$, $\\mathbb{H}^-=-\\mathbb{H}^+$, the upper and lower half-planes,\n\n$\\Pi=\\{\\Im(z)>0, \\Re(z)>0\\}$, $\\overline\\Pi=\\{\\Im(z)\\ge 0, \\Re(z)\\ge 0\\}$,\n\nfor an open interval $A\\subset\\mathbb{R}$,\n$\\C_{A}=(\\mathbb{C}\\setminus \\mathbb{R})\\cup A$, a slit complex plane.\n\n{\\bf Acknowledgment.} The author thanks Sebastian van Strien for helpful comments.", "sketch": "The proposed proof strategy for Theorem~\\ref{t-su} is to use Theorem~\\ref{t-bi} and show that complex bounds follow from a suitable polynomial-like (PL) restriction for a limit map of renormalizations.\n\n- One starts from the fact (\\cite{Su1}) that “any limit of renormalizations $R^{q_n}f$ is a unimodal map $g:[-1,1]\\to [-1,1]$ which extends to a map $g: U_{-1}\\to U_0$ from the Epstein class,” with “$g: U_{-1}\\to U_0$ … a proper analytic map of degree $2$,” where $U_0$ is “a slit complex plane.”\n\n- Let $T$ be “a minimal $g$-completely invariant set which contains the interval $[-1,1]$,” and let $\\mathcal{J}$ be “the closure $\\mathcal{J}$ of $T$.” The “main goal is to prove that the closure $\\mathcal{J}$ of $T$ is a proper subset of $U_0$.”\n\n- If $\\mathcal{J}\\subsetneq U_0$, then “Theorem~\\ref{t-bi} applies to the topological hull $\\widehat{\\mathcal{J}}$ of $\\mathcal{J}$,” yielding “a PL restriction $g:V'\\to V$ in a neighborhood of $\\widehat{\\mathcal{J}}$.” Using that “PL maps are stable under small perturbations,” the PL restriction “persists for any nearby (to $g$) renormalization $R^{q_n}f$,” and “the modulus of this PL map is close to the limit modulus $\\mod(V\\setminus \\overline{V'})$. That would mean complex bounds.”\n\n- To show $\\mathcal{J}$ is proper in $U_0$, they proceed “by a contradiction.” Assuming instead that $\\mathcal{J}$ is not proper, “first we show that $\\mathcal{J}$ must meet slits of the slit complex plane $U_0$, i.e., it is impossible for $\\mathcal{J}$ to be unbounded without touching the slits” (this part “holds for any combinatorics”).\n\n- For bounded combinatorics, “bounded combinatorics along with real bounds imply that the function $g$ lies at the bottom of an infinite tower of maps from the Epstein class,” which “allows us to extend $g:U_{-1}\\to U_0$ to a holomorphic function in bigger simply connected domains.” This “leads to a contradiction, see the proof of Theorem~\\ref{t-m}$.”\n\n- Finally, “Theorem~\\ref{t-su} is reduced to Theorem~\\ref{t-m} with help of (basically, known) Proposition~\\ref{p-infren-tower}.”", "expanded_sketch": "The proposed proof strategy, in establishing the main theorem, is to use the following theorem and show that complex bounds follow from a suitable polynomial-like (PL) restriction for a limit map of renormalizations.\n\nWe first use the following theorem. \\begin{theorem}\\label{t-bi}\nLet $g:U_{-1}\\to U_0$ be a proper holomorphic map of degree $d\\ge 2$ where $U_{-1}, U_0$ are open simply connected subsets of $\\C$ such that $U_{-1}\\subset U_0$.\nSuppose that $X\\subset U_0$ is a compact and full (i.e. $\\C\\setminus X$ is connected) set which is completely invariant, i.e.,\n$g^{-1}(X)=X$, and contains all critical points of $g$. Then there exists a polynomial-like restriction $g:V'\\to V$ where $V', V$ are neighborhoods of $X$.\n\\end{theorem}\n\n- One starts from the fact (Sullivan \\cite{Su1}) that “any limit of renormalizations $R^{q_n}f$ is a unimodal map $g:[-1,1]\\to [-1,1]$ which extends to a map $g: U_{-1}\\to U_0$ from the Epstein class,” with “$g: U_{-1}\\to U_0$ … a proper analytic map of degree $2$,” where $U_0$ is “a slit complex plane.”\n\n- Let $T$ be “a minimal $g$-completely invariant set which contains the interval $[-1,1]$,” and let $\\mathcal{J}$ be “the closure $\\mathcal{J}$ of $T$.” The “main goal is to prove that the closure $\\mathcal{J}$ of $T$ is a proper subset of $U_0$.”\n\n- If $\\mathcal{J}\\subsetneq U_0$, then the theorem above applies to the topological hull $\\widehat{\\mathcal{J}}$ of $\\mathcal{J}$, yielding “a PL restriction $g:V'\\to V$ in a neighborhood of $\\widehat{\\mathcal{J}}$.” Using that “PL maps are stable under small perturbations,” the PL restriction “persists for any nearby (to $g$) renormalization $R^{q_n}f$,” and “the modulus of this PL map is close to the limit modulus $\\mod(V\\setminus \\overline{V'})$. That would mean complex bounds.”\n\n- To show $\\mathcal{J}$ is proper in $U_0$, they proceed “by a contradiction.” Assuming instead that $\\mathcal{J}$ is not proper, “first we show that $\\mathcal{J}$ must meet slits of the slit complex plane $U_0$, i.e., it is impossible for $\\mathcal{J}$ to be unbounded without touching the slits” (this part “holds for any combinatorics”).\n\n- For bounded combinatorics, “bounded combinatorics along with real bounds imply that the function $g$ lies at the bottom of an infinite tower of maps from the Epstein class,” which “allows us to extend $g:U_{-1}\\to U_0$ to a holomorphic function in bigger simply connected domains.” This “leads to a contradiction, see the proof of the following theorem.”\n\nWe use the following theorem. \\begin{theorem}\\label{t-m}\nLet $\\mathcal{G}=\\{g_n\\}_{n=0}^\\infty$ where $g_n\\in E_{\\beta_n}(J_n)$, $n=0,1,2,...$, and $\\beta_n$, $J_n$ are as follows:\n\\begin{enumerate}\\label{e-t}\n\\item $\\beta_0=1$, $|\\beta_n|<|\\beta_{n+1}|$ for $n=0,1,2,,,$, and $\\beta_n\\to\\infty$,\n\\item if $I_n=<-\\beta_n,\\beta_n>$, then $I_n\\subset J_n\\subset I_{n+1}$, $n=0,1,2,...$,\n\\item for every $n>0$ there exists an integer $p_n\\ge 2$, such that\n$$g_0(x)=g_n^{p_n}(x), \\mbox{ for all } x\\in [-1,1].$$\n\\end{enumerate}\nThen there exists a simply connected neighborhood $V\\subset C_{J_0}$ of $[-1,1]$, such that $\\overline{V'}\\subset V$ where $V'=g_0^{-1}(V)$.\nIn other words, $g_0: V'\\to V$ is a PL restriction of the map $g_0:\\Omega_{g_0}\\to \\C_{J_0}$.\n\nMoreover, assuming\n\\begin{equation}\\label{label-m}\nK_1\\le \\frac{|J_n|}{|I_n|}\\le K_2, \\\n|I_n|\\le\\Lambda^n, \\ p_n\\le N^n,\n\\end{equation}\nfor some $11$, $N\\ge 2$ and all $n$, the following uniform bounds holds:\nthere exists $m=m(K_1,K_2,\\Lambda,N)>0$\nsuch that $\\mod(V\\setminus \\overline{V'})\\ge m$, for any $\\mathcal{G}$ as above and for a choice of the domain $V$.\n\\end{theorem}\n\n- Finally, the main theorem is reduced to the theorem above with help of the following proposition.\n\n\\begin{prop}\\label{p-infren-tower}\nTo every sequence $\\mathcal{R}\\subset\\N$ one can associate its subsequence $S$ and a sequence of maps\n$g_m\\in E_{b_m}(J_m)$, where $J_m=(-L|b_m|, L|b_m|)$, $m=0,1,2,...$, as follows:\n\\begin{enumerate}\\label{e-t-subs}\n\\item $b_0=1$, i.e., $g_0\\in E_1(J_0)$, and the sequence of renormalizations $\\{R^{q_j}f\\}_{j\\in S}\\to g_0$ uniformly on compacts in $\\Omega_{g_0}$,\n\\item if $I_m=<-b_m,b_m>$, then $\\mu\\le \\frac{|I_{m+1}|}{|I_m|}\\le \\lambda$ and $J_m\\subset I_{m+1}$, $m=0,1,2,...$,\n\\item for every $m>1$ there exists an integer $p_m\\ge 2$, such that\n$$g_0(x)=g_m^{p_m}(x), \\mbox{ for all } x\\in [-1,1].$$\n\\end{enumerate}\n\\end{prop}", "expanded_theorem": "\\label{t-su} (Sullivan \\cite{Su1}, \\cite{MS})\nGiven $N$ there exists $m=m(N)>0$ as follows. Let $f$ be an infinitely renormalizable real quadratic polynomial with combinatorics\nbounded by $N$.\nThen, for every $n\\ge n(f)$,\nthe corresponding renormalization $R^{q_n}(f):[-1,1]\\to[-1,1]$ admits a polynomial-like extension $R^{q_n}(f): V_n'\\to V_n$ of degree $2$ such that\n$\\mod(V_n\\setminus\\overline{V_n'})\\ge m$, for the modulus of the annulus $V_n\\setminus\\overline{V_n'}$.,\n", "theorem_type": ["Existential–Universal", "Universal"], "mcq": {"question": "Let \\(N\\ge 2\\). For a real quadratic polynomial \\(f(x)=x^2+c\\), say that \\(f\\) is infinitely renormalizable with combinatorics bounded by \\(N\\) if there are renormalization periods \\(2\\le q_10\\) such that for every infinitely renormalizable real quadratic polynomial \\(f(x)=x^2+c\\) with combinatorics bounded by \\(N\\), and for every renormalization level \\(n\\ge n(f)\\), the renormalization \\(R^{q_n}(f):[-1,1]\\to[-1,1]\\) admits a polynomial-like extension \\(R^{q_n}(f):V_n'\\to V_n\\) of degree \\(2\\) with\n\\[\n\\operatorname{mod}\\bigl(V_n\\setminus \\overline{V_n'}\\bigr)\\ge m.\n\\]"}, "choices": [{"label": "B", "text": "There exists a constant \\(m=m(N)>0\\) such that for every infinitely renormalizable real quadratic polynomial \\(f(x)=x^2+c\\) with combinatorics bounded by \\(N\\), there is a renormalization level \\(n_0=n_0(f)\\) for which, for every \\(n\\ge n_0\\), the renormalization \\(R^{q_n}(f):[-1,1]\\to[-1,1]\\) admits a polynomial-like extension \\(R^{q_n}(f):V_n'\\to V_n\\) of degree \\(2\\) with\n\\[\n\\operatorname{mod}\\bigl(V_n\\setminus \\overline{V_n'}\\bigr)= m.\n\\]"}, {"label": "C", "text": "For every infinitely renormalizable real quadratic polynomial \\(f(x)=x^2+c\\) with combinatorics bounded by \\(N\\), and for every renormalization level \\(n\\ge n(f)\\), the renormalization \\(R^{q_n}(f):[-1,1]\\to[-1,1]\\) admits a polynomial-like extension \\(R^{q_n}(f):V_n'\\to V_n\\) of degree \\(2\\)."}, {"label": "D", "text": "For every infinitely renormalizable real quadratic polynomial \\(f(x)=x^2+c\\) with combinatorics bounded by \\(N\\), and for every renormalization level \\(n\\ge n(f)\\), there exists a constant \\(m_n=m_n(f)>0\\) such that the renormalization \\(R^{q_n}(f):[-1,1]\\to[-1,1]\\) admits a polynomial-like extension \\(R^{q_n}(f):V_n'\\to V_n\\) of degree \\(2\\) with\n\\[\n\\operatorname{mod}\\bigl(V_n\\setminus \\overline{V_n'}\\bigr)\\ge m_n.\n\\]"}, {"label": "E", "text": "There exists a constant \\(m=m(N)>0\\) such that for every infinitely renormalizable real quadratic polynomial \\(f(x)=x^2+c\\) with combinatorics bounded by \\(N\\), and for every renormalization level \\(n\\ge n(f)\\), the renormalization \\(R^{q_n}(f):[-1,1]\\to[-1,1]\\) admits a polynomial-like extension \\(R^{q_n}(f):V_n'\\to V_n\\) with\n\\[\n\\operatorname{mod}\\bigl(V_n\\setminus \\overline{V_n'}\\bigr)\\ge m,\n\\]\nand the extension can be chosen so that \\(V_n'\\) and \\(V_n\\) are simply connected neighborhoods of \\([-1,1]\\) on which \\(R^{q_n}(f)\\) is univalent."}], "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": "uniform lower bound replaced by exact fixed modulus", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "finiteness", "tampered_component": "dropped the uniform modulus lower bound", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "finiteness", "tampered_component": "uniformity of the modulus constant in f and n", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "geometric_construction", "tampered_component": "proper degree-2 polynomial-like restriction confused with univalent extension", "template_used": "property_confusion"}]}} +{"id": "2601.21865v1", "paper_link": "http://arxiv.org/abs/2601.21865v1", "theorems_cnt": 2, "theorem": {"env_name": "mtheorem", "content": "\\label{mainthmA}\n$\\CH_c(n)$ grows at least as fast as $n^{2}/4$. More precisely,\n\\begin{equation}\\label{eq:liminf}\n\\liminf_{n\\to\\infty} \\frac{\\CH_c(n)}{n^{2}} \\ge \\frac{1}{4}.\n\\end{equation}\nFurthermore, if $\\CH_c(n) < \\infty$ for some $n \\in \\mathbb{N}$, then $\\CH_c(n+1) \\ge 1+\\CH_c(n).$", "start_pos": 11162, "end_pos": 11482, "label": "mainthmA"}, "ref_dict": {"mainthmA": "\\begin{mtheorem}\\label{mainthmA}\n$\\CH_c(n)$ grows at least as fast as $n^{2}/4$. More precisely,\n\\begin{equation}\\label{eq:liminf}\n\\liminf_{n\\to\\infty} \\frac{\\CH_c(n)}{n^{2}} \\ge \\frac{1}{4}.\n\\end{equation}\nFurthermore, if $\\CH_c(n) < \\infty$ for some $n \\in \\mathbb{N}$, then $\\CH_c(n+1) \\ge 1+\\CH_c(n).$\n\\end{mtheorem}", "mainthmB": "\\begin{mtheorem}\\label{mainthmB}\n\t$\\widehat{\\CH}_c(n)$ grows at least as fast as $(n\\log n)/(2\\log 2)$. More precisely,\n\t\\begin{equation}\\label{liminfHhat}\n\t\\liminf_{n\\to \\infty} \\frac{\\widehat{\\CH}_c(n)}{n\\, \\log \\, n} \\ge \\frac{1}{2\\log 2}.\n\t\\end{equation}\n\\end{mtheorem}"}, "pre_theorem_intro_text_len": 6776, "pre_theorem_intro_text": "Since its formulation in 1900, Hilbert’s Sixteenth Problem has played a key role in driving advances in the theory of planar vector fields. It concerns bounding the amount of limit cycles that may arise in planar polynomial vector fields of a given degree. When such a bound exists for a degree $n$, it is referred to as the $n$th Hilbert number $\\CH(n)$. More specifically, the $n$th Hilbert number is defined by\n\\[\n\\CH(n):=\\sup\\{\\pi(P,Q): \\deg(P)\\le n,\\ \\deg(Q)\\le n\\},\n\\]\nwith $\\pi(P,Q)$ being the number of limit cycles of the polynomial vector field $X=(P,Q)$. Accordingly, the problem reduces to showing that $\\CH(n)$ is finite for each $n\\in\\mathbb{N}$.\n\nExcept for the linear case ($n=1$), where limit cycles cannot exist, determining the finiteness of $\\CH(n)$ has proven to be a highly challenging problem. Indeed, the question whether $\\CH(n)$ is finite remains open already for $n = 2$. The main existing results concern lower bounds for the Hilbert number. For instance, $\\CH(2)\\geq 4$ \\cite{CW79,1980A}, $\\CH(3)\\geq 13$ \\cite{LLY09}, and $\\CH(4)\\geq 28$ \\cite{PT19}. Lower bounds for several other small values of $n$ are reported in \\cite{PT19}. The standard approach to improve these bounds, for a fixed degree $n$, is to construct explicit examples exhibiting more limit cycles than previously known. For arbitrary values of $n$, the challenge is to generate limit cycles within families of planar polynomial vector fields in which the degree is treated as a parameter. In the early stages of research, a quadratic lower bound for the growth of $\\CH(n)$ was established (see Otrokov \\cite{O54}, Il’yashenko \\cite{I91}, and Basarab-Horwath and Lloyd \\cite{BL88}). Later, Christopher and Lloyd \\cite{CL95} proved a lower bound of order $n^{2}\\log n $ for the growth of $\\CH(n)$, a result that has since been revisited and refined by several authors (see, for instance, \\cite{HL12,IS00,LCC02,ACMP20}).\n\nThe key idea of the methodology developed by Christopher and Lloyd~\\cite{CL95} is a recursive construction based on singular transformations of kind $(x,y)\\mapsto(x^2-A,y^2-A)$. Starting from a planar polynomial vector field $X$ possessing a given number of limit cycles, this transformation produces a new vector field whose degree is doubled and which contains four diffeomorphic copies of $X$, one in each quadrant, and consequently four copies of all its limit cycles. Moreover, new centers created along the coordinate axes after the transformation give rise to additional limit cycles, whose number is proportional to the square of the degree of $X$. By iterating this process, they construct a sequence $(X_k)_{k\\in\\mathbb{N}}$ of polynomial vector fields of degree $2^k - 1$ and admitting at least $S_{k+1} = 4S_k + (2^k - 1)^2$ limit cycles, which yields a lower bound for the growth of $\\CH(n)$ of order $n^2\\log n$.\n\nThe most recent advance in understanding the Hilbert number was achieved by Gasull and Santana~\\cite{GS25}, who established the strict monotonicity of the function $\\CH(n)$, provided it is finite. More precisely, they first showed that $\\CH(n)$, if finite, can be realized by hyperbolic limit cycles, and subsequently that $\\CH(n+1) \\geq 1+\\CH(n)$. Their approach begins with a polynomial vector field $X$ of degree $n$ having exactly $\\CH(n)$ hyperbolic limit cycles. They then perform a singular reparametrization of time that preserves these limit cycles, increases the degree of $X$ by one, and creates a line of singularities. Next, a first linear perturbation is applied to destroy this line of singularities and generate a Hopf point. Finally, a second linear perturbation is introduced so that the system undergoes a Hopf bifurcation, giving rise to one additional hyperbolic limit cycle beyond the existing $\\CH(n)$.\n\nOver the past few decades, discontinuous differential systems have attracted increasing attention, driven both by their broad range of applications and by the mathematical challenges they pose. The second part of Hilbert's Sixteenth Problem has also been investigated within this framework (see, for instance, \\cite{BCT24,CarFerNov2022b,CNT19,GouTor20,LL23,NOVAES2022133523}). In this paper, our main objective is to extend the aforementioned results to the case of crossing limit cycles in piecewise polynomial vector fields, that is, limit cycles that cross transversely the set of discontinuity. As expected, the strategies employed in the smooth case do not apply directly to this discontinuous context. Nevertheless, we are able to develop suitable adaptations of these methods to address the presence of discontinuities along the switching line, thereby establishing analogous results for crossing limit cycles of piecewise polynomial vector fields.\n\nLet $ Z= (P, Q) $ be a piecewise polynomial planar vector field defined by \n\\[\nP(x,y) =\n\\begin{cases}\nP^+(x,y), & x > 0,\\\\\nP^-(x,y), & x < 0,\n\\end{cases}\n\\qquad\nQ(x,y) =\n\\begin{cases}\nQ^+(x,y), & x > 0,\\\\\nQ^-(x,y), & x < 0,\n\\end{cases}\n\\]\nwhere $ P^{\\pm} $ and $ Q^{\\pm} $ are polynomials. The trajectories of $Z$ is given by the Filippov convention \\cite{Filippov88}. We define $\\deg P:=\\max\\{\\deg P^+,\\deg P^-\\}$ (analogously for $\\deg Q$) and denote by $ \\pi_c(P,Q) $ the number of crossing limit cycles of $ Z $.\nThe analogue of the Hilbert number in this discontinuous setting is defined as\n\\begin{equation*}\\label{hilbert}\n\\CH_c(n) := \\sup \\bigl\\{\\, \\pi_c(P,Q) : \\deg P,\\deg Q \\le n \\,\\bigr\\}.\n\\end{equation*}\n\nAs in the polynomial case, determining upper bounds for $\\CH_c(n)$ remains a challenging problem, even for $n=1$, that is, for piecewise linear vector fields. The finiteness of $\\CH_c(1)$ was established only in recent years by Carmona et al.\\cite{CarFerNov2022b}, using an integral characterization of Poincaré half-maps \\cite{CarmonaEtAl19} and fewnomial theory~\\cite{kho}. We refer the interested reader to~\\cite{CarmonaEtAl19b,CFSN-center,Carmona2022-mc}, where this technique has been further employed in other contexts. Most known results in this piecewise setting also provide lower bounds, such as $\\CH_c(1)\\ge 3$~\\cite{HuanYang12,LNT15,LP12}, $\\CH_c(2)\\ge 12$~\\cite{BCT24}, and $\\CH_c(3)\\ge 24$~\\cite{GouTor20}, while the best general lower bound currently available is $\\CH_c(n)\\ge 2n-1$, established by Buzzi et al.~\\cite{BFJ18}, yielding a linear lower estimate for the growth of $\\CH_c(n)$. Although this lower bound is strictly increasing, this does not by itself imply that $\\CH_c(n)$ is strictly increasing, a fact which, to date, has not been established.\n\nOur first main result sharpens the known linear estimate by showing that the growth rate of \n$ \\CH_c(n) $ is at least of order $ n^{2} $. In addition, we establish that $ \\CH_c(n) $ is indeed strictly increasing, provided it is finite, as stated below:", "context": "Since its formulation in 1900, Hilbert’s Sixteenth Problem has played a key role in driving advances in the theory of planar vector fields. It concerns bounding the amount of limit cycles that may arise in planar polynomial vector fields of a given degree. When such a bound exists for a degree $n$, it is referred to as the $n$th Hilbert number $\\CH(n)$. More specifically, the $n$th Hilbert number is defined by\n\\[\n\\CH(n):=\\sup\\{\\pi(P,Q): \\deg(P)\\le n,\\ \\deg(Q)\\le n\\},\n\\]\nwith $\\pi(P,Q)$ being the number of limit cycles of the polynomial vector field $X=(P,Q)$. Accordingly, the problem reduces to showing that $\\CH(n)$ is finite for each $n\\in\\mathbb{N}$.\n\nThe key idea of the methodology developed by Christopher and Lloyd~\\cite{CL95} is a recursive construction based on singular transformations of kind $(x,y)\\mapsto(x^2-A,y^2-A)$. Starting from a planar polynomial vector field $X$ possessing a given number of limit cycles, this transformation produces a new vector field whose degree is doubled and which contains four diffeomorphic copies of $X$, one in each quadrant, and consequently four copies of all its limit cycles. Moreover, new centers created along the coordinate axes after the transformation give rise to additional limit cycles, whose number is proportional to the square of the degree of $X$. By iterating this process, they construct a sequence $(X_k)_{k\\in\\mathbb{N}}$ of polynomial vector fields of degree $2^k - 1$ and admitting at least $S_{k+1} = 4S_k + (2^k - 1)^2$ limit cycles, which yields a lower bound for the growth of $\\CH(n)$ of order $n^2\\log n$.\n\nThe most recent advance in understanding the Hilbert number was achieved by Gasull and Santana~\\cite{GS25}, who established the strict monotonicity of the function $\\CH(n)$, provided it is finite. More precisely, they first showed that $\\CH(n)$, if finite, can be realized by hyperbolic limit cycles, and subsequently that $\\CH(n+1) \\geq 1+\\CH(n)$. Their approach begins with a polynomial vector field $X$ of degree $n$ having exactly $\\CH(n)$ hyperbolic limit cycles. They then perform a singular reparametrization of time that preserves these limit cycles, increases the degree of $X$ by one, and creates a line of singularities. Next, a first linear perturbation is applied to destroy this line of singularities and generate a Hopf point. Finally, a second linear perturbation is introduced so that the system undergoes a Hopf bifurcation, giving rise to one additional hyperbolic limit cycle beyond the existing $\\CH(n)$.\n\nLet $ Z= (P, Q) $ be a piecewise polynomial planar vector field defined by \n\\[\nP(x,y) =\n\\begin{cases}\nP^+(x,y), & x > 0,\\\\\nP^-(x,y), & x < 0,\n\\end{cases}\n\\qquad\nQ(x,y) =\n\\begin{cases}\nQ^+(x,y), & x > 0,\\\\\nQ^-(x,y), & x < 0,\n\\end{cases}\n\\]\nwhere $ P^{\\pm} $ and $ Q^{\\pm} $ are polynomials. The trajectories of $Z$ is given by the Filippov convention \\cite{Filippov88}. We define $\\deg P:=\\max\\{\\deg P^+,\\deg P^-\\}$ (analogously for $\\deg Q$) and denote by $ \\pi_c(P,Q) $ the number of crossing limit cycles of $ Z $.\nThe analogue of the Hilbert number in this discontinuous setting is defined as\n\\begin{equation*}\\label{hilbert}\n\\CH_c(n) := \\sup \\bigl\\{\\, \\pi_c(P,Q) : \\deg P,\\deg Q \\le n \\,\\bigr\\}.\n\\end{equation*}\n\nAs in the polynomial case, determining upper bounds for $\\CH_c(n)$ remains a challenging problem, even for $n=1$, that is, for piecewise linear vector fields. The finiteness of $\\CH_c(1)$ was established only in recent years by Carmona et al.\\cite{CarFerNov2022b}, using an integral characterization of Poincaré half-maps \\cite{CarmonaEtAl19} and fewnomial theory~\\cite{kho}. We refer the interested reader to~\\cite{CarmonaEtAl19b,CFSN-center,Carmona2022-mc}, where this technique has been further employed in other contexts. Most known results in this piecewise setting also provide lower bounds, such as $\\CH_c(1)\\ge 3$~\\cite{HuanYang12,LNT15,LP12}, $\\CH_c(2)\\ge 12$~\\cite{BCT24}, and $\\CH_c(3)\\ge 24$~\\cite{GouTor20}, while the best general lower bound currently available is $\\CH_c(n)\\ge 2n-1$, established by Buzzi et al.~\\cite{BFJ18}, yielding a linear lower estimate for the growth of $\\CH_c(n)$. Although this lower bound is strictly increasing, this does not by itself imply that $\\CH_c(n)$ is strictly increasing, a fact which, to date, has not been established.\n\nOur first main result sharpens the known linear estimate by showing that the growth rate of \n$ \\CH_c(n) $ is at least of order $ n^{2} $. In addition, we establish that $ \\CH_c(n) $ is indeed strictly increasing, provided it is finite, as stated below:", "full_context": "Since its formulation in 1900, Hilbert’s Sixteenth Problem has played a key role in driving advances in the theory of planar vector fields. It concerns bounding the amount of limit cycles that may arise in planar polynomial vector fields of a given degree. When such a bound exists for a degree $n$, it is referred to as the $n$th Hilbert number $\\CH(n)$. More specifically, the $n$th Hilbert number is defined by\n\\[\n\\CH(n):=\\sup\\{\\pi(P,Q): \\deg(P)\\le n,\\ \\deg(Q)\\le n\\},\n\\]\nwith $\\pi(P,Q)$ being the number of limit cycles of the polynomial vector field $X=(P,Q)$. Accordingly, the problem reduces to showing that $\\CH(n)$ is finite for each $n\\in\\mathbb{N}$.\n\nThe key idea of the methodology developed by Christopher and Lloyd~\\cite{CL95} is a recursive construction based on singular transformations of kind $(x,y)\\mapsto(x^2-A,y^2-A)$. Starting from a planar polynomial vector field $X$ possessing a given number of limit cycles, this transformation produces a new vector field whose degree is doubled and which contains four diffeomorphic copies of $X$, one in each quadrant, and consequently four copies of all its limit cycles. Moreover, new centers created along the coordinate axes after the transformation give rise to additional limit cycles, whose number is proportional to the square of the degree of $X$. By iterating this process, they construct a sequence $(X_k)_{k\\in\\mathbb{N}}$ of polynomial vector fields of degree $2^k - 1$ and admitting at least $S_{k+1} = 4S_k + (2^k - 1)^2$ limit cycles, which yields a lower bound for the growth of $\\CH(n)$ of order $n^2\\log n$.\n\nThe most recent advance in understanding the Hilbert number was achieved by Gasull and Santana~\\cite{GS25}, who established the strict monotonicity of the function $\\CH(n)$, provided it is finite. More precisely, they first showed that $\\CH(n)$, if finite, can be realized by hyperbolic limit cycles, and subsequently that $\\CH(n+1) \\geq 1+\\CH(n)$. Their approach begins with a polynomial vector field $X$ of degree $n$ having exactly $\\CH(n)$ hyperbolic limit cycles. They then perform a singular reparametrization of time that preserves these limit cycles, increases the degree of $X$ by one, and creates a line of singularities. Next, a first linear perturbation is applied to destroy this line of singularities and generate a Hopf point. Finally, a second linear perturbation is introduced so that the system undergoes a Hopf bifurcation, giving rise to one additional hyperbolic limit cycle beyond the existing $\\CH(n)$.\n\nLet $ Z= (P, Q) $ be a piecewise polynomial planar vector field defined by \n\\[\nP(x,y) =\n\\begin{cases}\nP^+(x,y), & x > 0,\\\\\nP^-(x,y), & x < 0,\n\\end{cases}\n\\qquad\nQ(x,y) =\n\\begin{cases}\nQ^+(x,y), & x > 0,\\\\\nQ^-(x,y), & x < 0,\n\\end{cases}\n\\]\nwhere $ P^{\\pm} $ and $ Q^{\\pm} $ are polynomials. The trajectories of $Z$ is given by the Filippov convention \\cite{Filippov88}. We define $\\deg P:=\\max\\{\\deg P^+,\\deg P^-\\}$ (analogously for $\\deg Q$) and denote by $ \\pi_c(P,Q) $ the number of crossing limit cycles of $ Z $.\nThe analogue of the Hilbert number in this discontinuous setting is defined as\n\\begin{equation*}\\label{hilbert}\n\\CH_c(n) := \\sup \\bigl\\{\\, \\pi_c(P,Q) : \\deg P,\\deg Q \\le n \\,\\bigr\\}.\n\\end{equation*}\n\nAs in the polynomial case, determining upper bounds for $\\CH_c(n)$ remains a challenging problem, even for $n=1$, that is, for piecewise linear vector fields. The finiteness of $\\CH_c(1)$ was established only in recent years by Carmona et al.\\cite{CarFerNov2022b}, using an integral characterization of Poincaré half-maps \\cite{CarmonaEtAl19} and fewnomial theory~\\cite{kho}. We refer the interested reader to~\\cite{CarmonaEtAl19b,CFSN-center,Carmona2022-mc}, where this technique has been further employed in other contexts. Most known results in this piecewise setting also provide lower bounds, such as $\\CH_c(1)\\ge 3$~\\cite{HuanYang12,LNT15,LP12}, $\\CH_c(2)\\ge 12$~\\cite{BCT24}, and $\\CH_c(3)\\ge 24$~\\cite{GouTor20}, while the best general lower bound currently available is $\\CH_c(n)\\ge 2n-1$, established by Buzzi et al.~\\cite{BFJ18}, yielding a linear lower estimate for the growth of $\\CH_c(n)$. Although this lower bound is strictly increasing, this does not by itself imply that $\\CH_c(n)$ is strictly increasing, a fact which, to date, has not been established.\n\nOur first main result sharpens the known linear estimate by showing that the growth rate of \n$ \\CH_c(n) $ is at least of order $ n^{2} $. In addition, we establish that $ \\CH_c(n) $ is indeed strictly increasing, provided it is finite, as stated below:\n\nTheorem \\ref{mainthmA} extend some of the main developments previously obtained for Hilbert's number in the polynomial setting to the piecewise polynomial context.\n\n\\begin{mtheorem}\\label{mainthmB}\n $\\widehat{\\CH}_c(n)$ grows at least as fast as $(n\\log n)/(2\\log 2)$. More precisely,\n \\begin{equation}\\label{liminfHhat}\n \\liminf_{n\\to \\infty} \\frac{\\widehat{\\CH}_c(n)}{n\\, \\log \\, n} \\ge \\frac{1}{2\\log 2}.\n \\end{equation}\n\\end{mtheorem}\n\nNow, define the \\emph{displacement function}\n\\[\n\\de_{0} : (0,1)\\times(-\\tilde{\\e},\\tilde{\\e}) \\rightarrow \\mathbb{R}, \\quad\n\\de_0(y, \\e) := y_1^{+}(y,\\e) - y_1^{-}(y,\\e),\n\\]\nwhose zeros correspond to periodic orbits of $Z_{0,\\e}$. Furthermore, isolated (resp. simple) zeros of $\\de_0$ correspond to (resp. hyperbolic) crossing limit cycles of $Z_{0,\\e}$. \n Notice that\n\\begin{equation}\\label{eq:disp0}\n\\de_0(y, \\e)= \\e\\,\\mathcal{M}_0(y) + \\mathcal{O}(\\e^2),\\quad\\text{where}\\quad \\mathcal{M}_0(y) =2(A_{0,1} + A_{0,3}y^2),\n\\end{equation}\nand $A_{0,i} = a^{+}_{0,i} - a^{-}_{0,i}$ for $i = 1,3$. Thus, consider $\\Delta_0 : [0, 1\n ]\\times(-\\tilde{\\e},\\tilde{\\e}) \\rightarrow \\mathbb{R}$ defined by:\n\\[\n\\Delta_0(y, \\e) := \\frac{\\de(y,\\e)}{\\e} = \\mathcal{M}_0(y)+ \\mathcal{O}(\\e).\n\\]\nThere are coefficients $a_{0,i}^{\\pm}$, $i=0,\\ldots,3$, for which $\\mathcal{M}_0$ has a unique simple zero $y_0\\in (0,1)$. For these coefficients,\n \\begin{equation}\\label{eq: A0}\n \\Delta_0(y_0, 0) = 0\\ \\ \\mbox{and}\\ \\ \\frac{\\partial\\Delta_0\n }{\\partial y} (y_0,0)\\neq 0,\n \\end{equation}\nand, therefor, the Implicit Function Theorem provides the existence of $\\bar{\\e}\\in(0,\\tilde{\\e})$ and a unique function $y^*:(-\\bar{\\e},\\bar{\\e})\\to (0,1)$ such that \n$y^*(0)=y_0$ and $\\Delta_0(y^*(\\e), \\e) = 0$ for all $\\e\\in(-\\bar{\\e},\\bar{\\e}).$\n\nAssume that, for a fixed $E^*_k = (\\e_1^*, \\dots, \\e_k^*) \\in \\mathbb{R}^k$ and every $\\e \\in (0, \\hat{\\e}]$, the vector field $Z_{k, \\e, E^*_k}$ possesses $c_k$ hyperbolic crossing limit cycles contained in $\\mathbb{R} \\times (-2, \\infty)$. For each $\\e \\in (0, \\hat{\\e}]$, denote by $(0, y_i(\\e))$ and $(0, z_i(\\e))$ the intersection points of the $i$-th limit cycle with the $y$-axis, where $y_i(\\e) > z_i(\\e)$ and $y_i(\\e), z_i(\\e) \\neq 0$ for $i\\in\\{1, \\dots, c_k\\}$. Define $ y_i(0) := \\check{y}_i$ and assume that $\\check{y}_i \\neq 0$ for all $ i\\in\\{1, \\dots, c_k\\}$. This assumption ensures that each of these limit cycles converges, as $\\e \\to 0$, to a periodic orbit of $ Z_{k,0,E^*_k}$. Such a condition will be explicitly incorporated into our inductive hypothesis below. Accordingly, for each $ i\\in\\{1, \\dots, c_k\\}$, consider the displacement function defined in a neighborhood $ V_i$ of $\\check{y}_i$, $\\delta^i: V_i \\rightarrow \\mathbb{R}$, by\n\\begin{equation}\\label{deltai}\n \\delta^i(y,\\e)= R^+_i(y,\\e)-R^-_i(y,\\e)\n\\end{equation} \nwhere $R_i^{\\pm}(y, \\e)$ denotes the half-return point to the $y$-axis, in forward or backward time respectively, of the orbit of $Z_{k,\\e,E^*_k}$ passing through $(0, y)$. Hence, for all $\\e \\in [0, \\hat{\\e}]$, $\\delta^i(y_i(\\e), \\e) = 0.$\n\nLet us analyze the point $(0, \\tilde{y}_i)$. The analysis for $(0, -\\tilde{y}_i)$ is completely analogous. \nNotice that there exists a small neighborhood $\\widetilde{V}_i \\subset \\mathbb{R}_+$ of $\\tilde{y}_i$ such that, for $\\varepsilon_{k+1}$ sufficiently small, the vector field $Z_{k+1,\\varepsilon,(E_k^*,\\varepsilon_{k+1})}$ admits a well-defined displacement function on $\\widetilde{V}_i$. Moreover, taking \\eqref{eq:Hk} into account, this displacement function can be written as\n\\begin{equation*}\\label{tildedisp}\n \\tilde{\\delta}^i(y, \\varepsilon, \\varepsilon_{k+1}) \n = \\widetilde{R}^{+}_i(y, \\varepsilon) - \\widetilde{R}^{-}_i(y, \\varepsilon) \n + \\mathcal{O}(\\varepsilon \\, \\varepsilon_{k+1}),\n\\end{equation*}\nwhere $\\widetilde{R}^{\\pm}_i(y, \\varepsilon)$ denote the return points on the $y$-axis of the orbits of \n$Z^{\\pm}_{k+1, \\varepsilon, (E^*_k, 0)}$ passing through $(0, y)$, for $y \\in \\widetilde{V}_i$. \nMoreover, $$\\widetilde{R}^{\\pm}_i(y, \\varepsilon)=\\sqrt{R^{\\pm}_{i}(y^2 - 2, \\varepsilon) + 2},$$ where $R_i^{\\pm}(y, \\varepsilon)$ are the half-return maps defined in a neighborhood of $\\check{y}_i$ by the orbits of $Z_{k, \\varepsilon, E^*_k}$ passing through $(0, y)$, as established in \\eqref{deltai}. Hence,\n\\begin{equation*}\\label{deltatil}\n \\tilde{\\delta}^i(y, \\varepsilon, \\varepsilon_{k+1}) \n = \\sqrt{R^{+}_{i}(y^2 - 2, \\varepsilon) + 2} \n - \\sqrt{R^{-}_{i}(y^2 - 2, \\varepsilon) + 2} \n + \\mathcal{O}(\\varepsilon \\, \\varepsilon_{k+1}).\n\\end{equation*} \nThus, expanding $\\tilde{\\delta}^i$ at $\\e=0$ and taking into account that $R_i^+(y,0)=R_i^-(y,0)$ for $y\\in V_i$, we obtain\n \\begin{equation*}\n \\begin{aligned}\n \\tilde{\\delta}^i(y,\\e, \\e_{k+1})& =\\ \\tilde{\\delta}^i(y, 0, \\e_{k+1}) + \\e \\frac{\\partial }{\\partial \\e}\\tilde{\\delta}^i(y, \\e, \\e_{k+1})\\big|_{\\e=0} +\\mathcal{O}(\\e^2)\\\\\n &=\\e \\left( \\frac{\\ \\displaystyle{\\frac{\\partial\\delta^{i}}{\\partial\\e}}(y^2-2,0)}{2\\sqrt{R^{+}_{i}(y^2-2,0)+2}}\\right)+ \\mathcal{O}(\\e\\e_{k+1})+\\mathcal{O}(\\e^2).\n \\end{aligned}\n \\end{equation*}\n Consider\n \\begin{equation*}\n \\tilde{\\Delta}^{i}(y,\\e, \\e_{k+1}) := \\frac{\\tilde{\\delta}^{i}(y, \\e, \\e_{k+1})}{\\e} = \\frac{\\ \\displaystyle{\\frac{\\partial\\delta^{i}}{\\partial\\e}}(y^2-2,0)}{2\\sqrt{R^{+}_{i}(y^2-2,0)+2}} + \\mathcal{O}(\\e_{k+1})+\\mathcal{O}(\\e).\n \\end{equation*}\nNote that, for each $i\\in\\{1,\\ldots,c_k\\}$,\n \\begin{equation*}\n \\vspace{-0.3cm}\\tilde{\\Delta}^{i}(\\tilde{y}_i, 0, 0)= \\frac{ \\displaystyle{\\frac{\\partial\\delta^{i}}{\\partial\\e}}(\\tilde{y}_i^2-2,0)}{2\\sqrt{R^{+}_{i}(\\tilde{y}_i^2-2,0)+2}}\\stackrel{\\tilde{y}_i= \\sqrt{\\check y_i+2}}{=}\\frac{ \\overbrace{\\frac{\\partial^2\\delta^{i}}{\\partial y\\partial\\e}(y_i,0)}^{\\quad\\quad\\ = 0\\ (A_k)}}{2\\sqrt{R^{+}_{i}(y_i,0)+2}} = 0,\n \\end{equation*}\n and\n \\begin{equation*}\n \\begin{aligned}\n \\frac{\\partial\\tilde{\\Delta}^i}{\\partial y}(\\tilde{y}_i,0,0)&= \\frac{\\partial}{\\partial y}\\left(\\frac{\\ \\displaystyle{\\frac{\\partial\\delta^i}{\\partial \\e}}(\\tilde{y}_i^2-2,0)}{2\\sqrt{R^+_i(\\tilde{y}_i^2-2,0)+2}}\\right) \\stackrel{\\tilde{y}_i= \\sqrt{\\check y_i+2}}{=} \\frac{\\overbrace{\\frac{\\partial^2 \\delta^i}{\\partial y\\partial\\e} (y_i,0)}^{\\quad \\quad \\neq 0\\ (A_k)}\\cdot\\ 2\\sqrt{R^{+}_i(y_i,0)+2}}{4(R^+_i(y_i,0)+2)} \\neq 0. \\\\\n \\end{aligned}\n \\end{equation*}\n\nWe now define the displacement function\n\\[\n\\delta_{k+1}: I_{k+1}\\times[-\\tilde{\\e},\\tilde{\\e}]\\times [0,\\e_{k+1}^*]\\rightarrow \\mathbb{R}\n\\]\nby\n\\begin{equation*}\\label{eq:dk1}\n \\delta_{k+1}(y,\\e,\\e_{k+1})\n := y^+_{1,k+1}(y,\\e,\\e_{k+1})-y^-_{1,k+1}(y,\\e,\\e_{k+1}).\n\\end{equation*}\nThen, from \\eqref{ep:yqk}, it follows that\n\\[\n\\delta_{k+1}(y,\\e,\\e_{k+1})\n= \\e\\,\\mathcal{M}_{k+1}(y)+\\mathcal{O}(\\e^2),\n\\]\nwhere\n\\[\n\\mathcal{M}_{k+1}(y)\n= \\e_{k+1}\\frac{\\sum_{j=1}^{d_k} A_j\\,y^{2j-2}}{2^k \\prod_{i=1}^{k+1}\\phi^{i}(y)},\n\\qquad\nA_j = a^+_{k+1,\\,2j-1}-a^-_{k+1,\\,2j-1}.\n\\]\nNotice that we may choose the parameters $A_j$ so that $\\mathcal{M}_{k+1}$ has $d_k-1$ simple zeros in $\\operatorname{Int}(I_{k+1})$, denoted by $\\{y_i \\mid i=1,\\ldots,d_{k-1}\\}.$ Accordingly, we define\n\\[\n\\Delta_{k+1}: I_{k+1}\\times[-\\tilde{\\e},\\tilde{\\e}]\\times [0,\\e_{k+1}^*]\\rightarrow \\mathbb{R}\n\\]\nby\n\\[\n\\Delta_{k+1}(y,\\e,\\e_{k+1})\n:= \\frac{\\delta_{k+1}(y,\\e,\\e_{k+1})}{\\e}\n= \\mathcal{M}_{k+1}(y)+\\mathcal{O}(\\e).\n\\]\nSince\n\\[\n\\Delta_{k+1}(y_i,0,\\e_{k+1})=0\n\\quad \\text{and} \\quad\n\\frac{\\partial}{\\partial \\e}\\Delta_{k+1}(y_i,0,\\e_{k+1})\\neq 0,\n\\]\nand since $\\{1,\\ldots, d_{k-1}\\}$ is finite and $[0,\\e_{k+1}^*]$ is compact, the Implicit Function Theorem implies the existence of $\\bar{\\e}\\in(0,\\tilde{\\e})$ and, for each $i\\in\\{1,\\ldots,d_{k-1}\\}$, a unique function $y_i^*(\\e,\\e_{k+1})$ such that\n\\[\n\\Delta_{k+1}\\bigl(y_i^*(\\e,\\e_{k+1}), \\e, \\e_{k+1}\\bigr)=0,\n\\qquad\ny_i^*(0,\\e_{k+1})=y_i,\n\\]\nfor all $\\e_{k+1}\\in[0,\\e_{k+1}^*]$ and $\\e\\in(0,\\bar{\\e})$.", "post_theorem_intro_text_len": 2341, "post_theorem_intro_text": "Theorem \\ref{mainthmA} extend some of the main developments previously obtained for Hilbert's number in the polynomial setting to the piecewise polynomial context.\n\nThe number of crossing limit cycles has also been investigated for the more restrictive class of piecewise polynomial Hamiltonian systems, that is, under the assumption that the polynomial vector fields $(P^+,Q^+)$ and $(P^-,Q^-)$ are both Hamiltonian. In this context, we denote\n\\begin{equation*}\\label{hilbert}\n\\widehat{\\CH}_c(n) := \\sup \\bigl\\{\\pi_c(P,Q) : (P^{\\pm},Q^{\\pm}) \\text{ are Hamiltonian v.f. and } \\deg P,\\deg Q \\le n \\bigr\\}.\n\\end{equation*}\n\nAlthough no general uniform upper bound has been established for the number of crossing limit cycles in this setting, the finiteness problem can be approached by means of algebraic equations provided by the Hamiltonians. For instance, in \\cite{LL23} it is shown that there exist at most $[n^2/2]$ crossing limit cycles with only two pieces in piecewise polynomial Hamiltonian vector fields of degree $n$. For a general uniform upper bound, one must take into account the possible configurations of crossing limit cycles. Lower bounds have also been addressed in several works, see, for example, \\cite{LL23,YANG20111026}. The best general lower bound depending on the degree of the piecewise polynomial Hamiltonian system known so far is $\\widehat{\\CH}_c(n) \\ge n-1$. Our second main result strengthens this estimate by showing that the growth rate of $\\widehat{\\CH}_c(n)$ is at least of order $n \\log n$, as stated below:\n\n\\begin{mtheorem}\\label{mainthmB}\n\t$\\widehat{\\CH}_c(n)$ grows at least as fast as $(n\\log n)/(2\\log 2)$. More precisely,\n\t\\begin{equation}\\label{liminfHhat}\n\t\\liminf_{n\\to \\infty} \\frac{\\widehat{\\CH}_c(n)}{n\\, \\log \\, n} \\ge \\frac{1}{2\\log 2}.\n\t\\end{equation}\n\\end{mtheorem}\n\nThe paper is structured as follows. In Section~\\ref{sec:proofB}, we present the proof of Theorem~\\ref{mainthmA}. The overall strategy follows the approach used in \\cite{GS25} for the smooth case; however, several technical aspects must be adapted to accommodate the piecewise setting, for instance by incorporating the notion of pseudo-Hopf bifurcation. In Section~\\ref{sec:proofA}, we provide the proof of Theorem~\\ref{mainthmB}, where the Christopher–Lloyd methodology is suitably modified for the piecewise framework.", "sketch": "The proof of Theorem~\\ref{mainthmA} is given in Section~\\ref{sec:proofB}. The overall strategy follows the approach used in \\cite{GS25} for the smooth case; however, several technical aspects must be adapted to accommodate the piecewise setting, for instance by incorporating the notion of pseudo-Hopf bifurcation.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{mainthmA}\n$\\CH_c(n)$ grows at least as fast as $n^{2}/4$. More precisely,\n\\begin{equation}\\label{eq:liminf}\n\\liminf_{n\\to\\infty} \\frac{\\CH_c(n)}{n^{2}} \\ge \\frac{1}{4}.\n\\end{equation}\nFurthermore, if $\\CH_c(n) < \\infty$ for some $n \\in \\mathbb{N}$, then $\\CH_c(n+1) \\ge 1+\\CH_c(n).$", "theorem_type": ["Asymptotic or Limit", "Implication"], "mcq": {"question": "Let\n\\[\nZ=(P,Q),\\qquad P(x,y)=\\begin{cases}P^+(x,y),&x>0,\\\\ P^-(x,y),&x<0,\\end{cases}\n\\qquad\nQ(x,y)=\\begin{cases}Q^+(x,y),&x>0,\\\\ Q^-(x,y),&x<0,\\end{cases}\n\\]\nbe a piecewise polynomial planar vector field, where $P^{\\pm}$ and $Q^{\\pm}$ are polynomials and trajectories are understood in the Filippov sense. Define\n\\[\n\\deg P:=\\max\\{\\deg P^+,\\deg P^-\\},\\qquad \\deg Q:=\\max\\{\\deg Q^+,\\deg Q^-\\},\n\\]\nand let $\\pi_c(P,Q)$ denote the number of crossing limit cycles of $Z$. The discontinuous Hilbert number is\n\\[\n\\CH_c(n):=\\sup\\{\\pi_c(P,Q):\\deg P\\le n,\\ \\deg Q\\le n\\}.\n\\]\nWhich of the following conclusions about $\\CH_c(n)$ holds?", "correct_choice": {"label": "A", "text": "$\\CH_c(n)$ has at least quadratic growth in the sense that\n\\[\n\\liminf_{n\\to\\infty}\\frac{\\CH_c(n)}{n^2}\\ge \\frac14.\n\\]\nMoreover, whenever $\\CH_c(n)<\\infty$ for some $n\\in\\mathbb N$, one has\n\\[\n\\CH_c(n+1)\\ge 1+\\CH_c(n).\n\\]"}, "choices": [{"label": "B", "text": "$\\CH_c(n)$ has at least quadratic growth in the sense that\n\\[\n\\liminf_{n\\to\\infty}\\frac{\\CH_c(n)}{n^2}> \\frac14.\n\\]\nMoreover, whenever $\\CH_c(n)<\\infty$ for some $n\\in\\mathbb N$, one has\n\\[\n\\CH_c(n+1)\\ge 2+\\CH_c(n).\n\\]"}, {"label": "C", "text": "$\\CH_c(n)$ has at least quadratic growth in the sense that\n\\[\n\\liminf_{n\\to\\infty}\\frac{\\CH_c(n)}{n^2}\\ge \\frac14.\n\\]"}, {"label": "D", "text": "$\\CH_c(n)$ has at least quadratic growth in the sense that\n\\[\n\\limsup_{n\\to\\infty}\\frac{\\CH_c(n)}{n^2}\\ge \\frac14.\n\\]\nMoreover, whenever $\\CH_c(n)<\\infty$ for some $n\\in\\mathbb N$, one has\n\\[\n\\CH_c(n+1)>\\CH_c(n).\n\\]"}, {"label": "E", "text": "$\\CH_c(n)$ has at least quadratic growth in the sense that\n\\[\n\\liminf_{n\\to\\infty}\\frac{\\CH_c(n)}{n^2}\\ge \\frac14.\n\\]\nMoreover, if $\\CH_c(n)<\\infty$ for every $n\\in\\mathbb N$, then there exists a constant $C>0$, independent of $n$, such that\n\\[\n\\CH_c(n+1)\\ge \\CH_c(n)+C\n\\]\nfor all $n\\in\\mathbb 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": "sharp increment and boundary constant", "template_used": "stronger_trap"}, {"label": "C", "sketch_hook_type": "other", "tampered_component": "strict monotonicity clause dropped", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "liminf replaced by weaker limsup; quantified conclusion weakened", "template_used": "property_confusion"}, {"label": "E", "sketch_hook_type": "other", "tampered_component": "pointwise finiteness hypothesis replaced by global uniform additive bound", "template_used": "wildcard"}]}} +{"id": "2601.21958v1", "paper_link": "http://arxiv.org/abs/2601.21958v1", "theorems_cnt": 5, "theorem": {"env_name": "theorem", "content": "[Theorem \\ref{e+4}]\n Let $e\\geqslant 0$ and let $r\\leqslant e+4$. Conjecture \\ref{conj3} holds for $\\mb F_{e,r}$.", "start_pos": 8350, "end_pos": 8496, "label": null}, "ref_dict": {"conj3": "\\begin{conjecture}\\label{conj3}\n\t\tThe following holds on $\\Fer$:\n\t\t\\begin{enumerate}[label=(\\alph*)]\n\t\t\t\\item\\label{conj3a} Effective nef divisors on $\\Fer$ are non-special.\n\t\t\t\\item\\label{conj3b} Every integral curve $C$ on $\\Fer$ with $C^2<0$ is either a $(-1)$-curve or the strict transform of $C_e$ on $\\Fer$.\n\t\t\\end{enumerate}\n\t\\end{conjecture}", "conj1": "\\begin{conjecture}\\cite[Conjecture 2.6]{Laf2002}\\label{conj1}\n\t\tLet $D$ be a divisor as in \\eqref{cond} and assume it is effective. \n\t\tThen $D$ is special if and only if it is $(-1)$-special.\n\t\\end{conjecture}", "e+4": "\\begin{theorem}\\label{e+4}\n Let $e\\geqslant 0$ and let $r\\leqslant e+4$. Then \n Conjecture \\ref{conj3} holds on $\\Fer$. \n \\end{theorem}", "conj3a": "\\begin{enumerate}[label=(\\alph*)]\n\t\t\t\\item\\label{conj3a} Effective nef divisors on $\\Fer$ are non-special.\n\t\t\t\\item\\label{conj3b} Every integral curve $C$ on $\\Fer$ with $C^2<0$ is either a $(-1)$-curve or the strict transform of $C_e$ on $\\Fer$.\n\t\t\\end{enumerate}", "algo": "\\begin{algorithm}\\label{algo}\n\t\tLet $D$ be as in \\eqref{cond} and effective. \n\t\t\\begin{enumerate}\n\t\t\t\\item \\label{algo1} If $E$ is a $(-1)$-curve with $-t =D\\cdot E<0$, replace $D$ by $D - tE$. \n\t\t\t\\item \\label{algo2} If $D \\cdot \\widetilde{C}_e < 0$, replace $D$ by $D -\\widetilde{C}_e$, where $\\widetilde{C}_e$ is the strict transform of $C_e$.\n\t\t\t\\item \\label{algo3} If $ D \\cdot \\widetilde{C}_e \\geqslant 0$ and $D\\cdot E\\geqslant 0$ for every $(-1)$-curve $E$, then stop. Else, go to Step \\ref{algo1}.\n\t\t\\end{enumerate}\n\t\\end{algorithm}", "conj3b": "\\begin{enumerate}[label=(\\alph*)]\n\t\t\t\\item\\label{conj3a} Effective nef divisors on $\\Fer$ are non-special.\n\t\t\t\\item\\label{conj3b} Every integral curve $C$ on $\\Fer$ with $C^2<0$ is either a $(-1)$-curve or the strict transform of $C_e$ on $\\Fer$.\n\t\t\\end{enumerate}", "eq-conjectures": "\\begin{proposition}\\label{eq-conjectures}\n\t\tWe have the following relationships among the above listed conjectures.\n\t\t\\begin{enumerate}\n\t\t\t\\item Conjecture \\ref{conj1} implies Conjecture \\ref{conj2}.\n\t\t\t\\item Conjecture \\ref{conj2} implies Conjecture \\ref{conj3}\\ref{conj3b}.\n\t\t\t\\item Conjecture \\ref{conj1} is equivalent to Conjecture \\ref{conj3}.\n\t\t\\end{enumerate}\n\t\\end{proposition}", "cond": "\\begin{equation}\\label{cond}\n \\text{$D= aH_e+bF_e-m_1E_1-\\dots -m_rE_r$ on $\\Fer$ where \n $a,b,m_1,\\dots ,m_r\\in \\mathbb Z_{\\geqslant 0}$.}\n\\end{equation}"}, "pre_theorem_intro_text_len": 6245, "pre_theorem_intro_text": "Throughout this article, we work over the field of complex numbers. \n The interpolation problem of plane curves of a given degree passing \n through given points with at least given multiplicities \n is a classical problem in algebraic geometry. \n A conjectural answer to this problem, assuming the points are \n very general, is given by the SHGH (Segre-Harbourne-Gimigliano-Hirschowitz) \n conjecture (see \\cite{Seg62,Har86,Gim87,Hir89}). \n\n One may pose a similar interpolation problem on arbitrary surfaces, \n with the notion of degree replaced by an appropriate numerical invariant. \n In this article, we focus on Hirzebruch surfaces.\n These surfaces are of particular interest because the minimal rational surfaces other than \n the projective plane are precisely the Hirzebruch surfaces. \n For each $e\\in \\mathbb Z_{\\ge 0}$, \n the surface $$\\mathbb F_e=\\mathbb P(\\mathcal O_{\\mathbb P^1}\\oplus \\mathcal O_{\\mathbb P^1}(-e))$$ \n is a Hirzebruch surface, and every Hirzebruch surface arises in this way.\n\tLet $C_e$ denote the divisor class corresponding to $\\mathcal O_{\\mb F_e}(1)$ and \n let $f$ denote the divisor class corresponding to the fiber of the map $\\mb F_e \\to \\mathbb P^1$. \n Then $\\text{Pic }\\mb F_e =\\mathbb Z C_e \\oplus \\mathbb Z f.$\n The interpolation problem on Hirzebruch surfaces is the following: \n \\begin{problem}\\label{prblm}\n Let $a,b, m_1,m_2,\\dots,m_r$ be non-negative integers. Given $r$ \n points $p_1,p_2,\\dots,p_r$ on $\\mb F_e$, does there exist a curve in \n the linear system $|aC_e+bf|$ passing through $p_i$ with multiplicity \n at least $m_i$ for all $1\\leqslant i \\leqslant r$?\n \\end{problem}\n More concretely, we are interested in determining the dimension of the \n linear system of curves in $|aC_e+bf|$. \n\n Consider $r$ very general points, say $p_1,p_2,\\dots,p_r$, on $\\mb F_e$.\n\t Let $\\pi: \\mb F_{e,r} \\to \\mb F_e$ denote the blow-up of $\\mb F_e $ at these points, \n and let $E_1,E_2,\\dots, E_r$ denote the exceptional divisors. Let \n $H_e=\\pi^{\\ast}(C_e)$ and $F_e=\\pi^{\\ast}(f)$; then the Picard group \n of $\\mb F_{e,r}$ is \n $$\\text{Pic }\\mb F_{e,r}=\\mathbb Z H_e\\oplus \\mathbb Z F_e \\oplus \\mathbb Z E_1 \\oplus \\mathbb Z E_2 \\oplus \\dots \\oplus \\mathbb Z E_r.$$\n Note that the linear system of curves in $|aC_e+bf|$ \n passing through $p_i$ with multiplicity at least $m_i$, for all \n $1\\leqslant i \\leqslant r$, \n is in bijective correspondence with the complete linear system \n $|aH_e+bF_e-m_1E_1-m_2E_2-\\dots-m_rE_r|$ on $\\mb F_{e,r}$. \n Hence, we are interested in finding the dimension of the \n linear system $|aH_e+bF_e-m_1E_1-m_2E_2-\\dots-m_rE_r|$. Hence, \n throughout the article, we are interested in divisors in the following form:\n\\begin{equation}\\label{cond}\n \\text{$D= aH_e+bF_e-m_1E_1-\\dots -m_rE_r$ on $\\mb F_{e,r}$ where \n $a,b,m_1,\\dots ,m_r\\in \\mathbb Z_{\\geqslant 0}$.}\n\\end{equation}\n \\begin{definition}\\label{defn}\n The \\textit{virtual dimension} of $D$ is defined as\n $$v(D)=\\frac{D^2-K_{\\mb F_{e,r}}\\cdot D}{2}\\,.$$\n The \\textit{expected dimension} of $D$ is defined as\n $e(D)=\\max\\left\\{ v(D),-1\\right\\}\\,.$\n \\end{definition}\n For divisors as in \\ref{cond}, we can see that $h^2(\\mb F_{e,r},\\mathcal{O}_{\\mb F_{e,r}}(D))=0$.\n From the Riemann-Roch theorem for surfaces, we see that \n $$\\dim |D|=\\frac{D^2-K_{\\mb F_{e,r}}\\cdot D}{2}+h^1(\\mb F_{e,r},\\mathcal{O}_{\\mb F_{e,r}}(D))=\n v(D)+h^1(\\mb F_{e,r},\\mathcal{O}_{\\mb F_{e,r}}(D))\\geqslant v(D)\\,.$$\n Since ${\\rm dim}|D|\\geqslant -1$, it follows that \n ${\\rm dim}|D|\\geqslant e(D)\\geqslant v(D)$.\n \\begin{definition}\\label{non-spcl defn}\n Let $D$ be a divisor as in \\eqref{cond}. \n We say $D$ is \\textit{non-special} if $\\dim |D|=e(D)$ and \\textit{special} otherwise $(\\dim |D|>e(D))$.\n \\end{definition}\n Thus, it follows that for $D$ as in \\eqref{cond}, if \n $h^1(\\mb F_{e,r},\\mathcal{O}_{\\mb F_{e,r}}(D))=0$,\n then $D$ is non-special. Further, if $D$ is effective and non-special then \n $h^1(\\mb F_{e,r},\\mathcal{O}_{\\mb F_{e,r}}(D))=0$. Thus, for an effective divisor as in \\eqref{cond},\n it follows that being non-special is equivalent to $h^1(\\mb F_{e,r},\\mathcal{O}_{\\mb F_{e,r}}(D))=0$.\n\n In \\cite[Conjecture 2.6]{Laf2002}, Laface proposed a conjecture (see Conjecture \\ref{conj1}) \n which provides a characterization for special divisors using the notion of \n $(-1)$-special divisors (see Algorithm \\ref{algo}). \n We propose the following conjecture (see the next section for the motivation for this \n conjecture and similarities with conjectures on $\\mathbb P^2$): \n \\begin{conjecture}\\label{conj3}\n\t\tThe following holds on $\\mb F_{e,r}$:\n\t\t\\begin{enumerate}[label=(\\alph*)]\n\t\t\t\\item\\label{conj3a} Effective nef divisors on $\\mb F_{e,r}$ are non-special.\n\t\t\t\\item\\label{conj3b} Every integral curve $C$ on $\\mb F_{e,r}$ with $C^2<0$ is either a $(-1)$-curve or the strict transform of $C_e$ on $\\mb F_{e,r}$.\n\t\t\\end{enumerate}\n\t\\end{conjecture}\n\n Under the hypothesis that \n Conjecture \\ref{conj3}\\ref{conj3b} is true for all $r>0$, \n \\cite[Theorem 3.6]{HJNS24} proves the existence of irrational \n Seshadri constants. Moreover, Conjecture \\ref{conj3}\\ref{conj3b} clearly\n implies the bounded negativity conjecture for $\\mb F_{e,r}$ (which states that for a smooth\n projective surface $X$, there is a constant $b(X)$, such that for all integral\n curves $C\\subset X$, we have $C^2>b(X)$). \n\n It is well known that the analogues of the above \n conjecture for $\\mathbb P^2$ (the SHGH conjecture) \n is true when $r \\leqslant 9$ \\cite{Nag61}. \n The main goal of this article is to obtain \n similar results for Conjecture \\ref{conj3}. \n It is known that the Conjecture \\ref{conj3}\\ref{conj3b} \n is true when $r\\leqslant e+4$ (see Conjecture \\cite[Theorem 5.5]{JKS25}). \n Here we show that \n Conjecture \\ref{conj3}\\ref{conj3a} is true when $r\\leqslant e+4$. \n In Proposition \\ref{eq-conjectures} we prove that Conjecture \\ref{conj3}\n is equivalent to \\cite[Conjecture 2.6]{Laf2002}.\n The following is the main result of this article.", "context": "Throughout this article, we work over the field of complex numbers. \n The interpolation problem of plane curves of a given degree passing \n through given points with at least given multiplicities \n is a classical problem in algebraic geometry. \n A conjectural answer to this problem, assuming the points are \n very general, is given by the SHGH (Segre-Harbourne-Gimigliano-Hirschowitz) \n conjecture (see \\cite{Seg62,Har86,Gim87,Hir89}).\n\nOne may pose a similar interpolation problem on arbitrary surfaces, \n with the notion of degree replaced by an appropriate numerical invariant. \n In this article, we focus on Hirzebruch surfaces.\n These surfaces are of particular interest because the minimal rational surfaces other than \n the projective plane are precisely the Hirzebruch surfaces. \n For each $e\\in \\mathbb Z_{\\ge 0}$, \n the surface $$\\mathbb F_e=\\mathbb P(\\mathcal O_{\\mathbb P^1}\\oplus \\mathcal O_{\\mathbb P^1}(-e))$$ \n is a Hirzebruch surface, and every Hirzebruch surface arises in this way.\n Let $C_e$ denote the divisor class corresponding to $\\mathcal O_{\\mb F_e}(1)$ and \n let $f$ denote the divisor class corresponding to the fiber of the map $\\mb F_e \\to \\mathbb P^1$. \n Then $\\text{Pic }\\mb F_e =\\mathbb Z C_e \\oplus \\mathbb Z f.$\n The interpolation problem on Hirzebruch surfaces is the following: \n \\begin{problem}\\label{prblm}\n Let $a,b, m_1,m_2,\\dots,m_r$ be non-negative integers. Given $r$ \n points $p_1,p_2,\\dots,p_r$ on $\\mb F_e$, does there exist a curve in \n the linear system $|aC_e+bf|$ passing through $p_i$ with multiplicity \n at least $m_i$ for all $1\\leqslant i \\leqslant r$?\n \\end{problem}\n More concretely, we are interested in determining the dimension of the \n linear system of curves in $|aC_e+bf|$.\n\nConsider $r$ very general points, say $p_1,p_2,\\dots,p_r$, on $\\mb F_e$.\n Let $\\pi: \\mb F_{e,r} \\to \\mb F_e$ denote the blow-up of $\\mb F_e $ at these points, \n and let $E_1,E_2,\\dots, E_r$ denote the exceptional divisors. Let \n $H_e=\\pi^{\\ast}(C_e)$ and $F_e=\\pi^{\\ast}(f)$; then the Picard group \n of $\\mb F_{e,r}$ is \n $$\\text{Pic }\\mb F_{e,r}=\\mathbb Z H_e\\oplus \\mathbb Z F_e \\oplus \\mathbb Z E_1 \\oplus \\mathbb Z E_2 \\oplus \\dots \\oplus \\mathbb Z E_r.$$\n Note that the linear system of curves in $|aC_e+bf|$ \n passing through $p_i$ with multiplicity at least $m_i$, for all \n $1\\leqslant i \\leqslant r$, \n is in bijective correspondence with the complete linear system \n $|aH_e+bF_e-m_1E_1-m_2E_2-\\dots-m_rE_r|$ on $\\mb F_{e,r}$. \n Hence, we are interested in finding the dimension of the \n linear system $|aH_e+bF_e-m_1E_1-m_2E_2-\\dots-m_rE_r|$. Hence, \n throughout the article, we are interested in divisors in the following form:\n\\begin{equation}\\label{cond}\n \\text{$D= aH_e+bF_e-m_1E_1-\\dots -m_rE_r$ on $\\mb F_{e,r}$ where \n $a,b,m_1,\\dots ,m_r\\in \\mathbb Z_{\\geqslant 0}$.}\n\\end{equation}\n \\begin{definition}\\label{defn}\n The \\textit{virtual dimension} of $D$ is defined as\n $$v(D)=\\frac{D^2-K_{\\mb F_{e,r}}\\cdot D}{2}\\,.$$\n The \\textit{expected dimension} of $D$ is defined as\n $e(D)=\\max\\left\\{ v(D),-1\\right\\}\\,.$\n \\end{definition}\n For divisors as in \\ref{cond}, we can see that $h^2(\\mb F_{e,r},\\mathcal{O}_{\\mb F_{e,r}}(D))=0$.\n From the Riemann-Roch theorem for surfaces, we see that \n $$\\dim |D|=\\frac{D^2-K_{\\mb F_{e,r}}\\cdot D}{2}+h^1(\\mb F_{e,r},\\mathcal{O}_{\\mb F_{e,r}}(D))=\n v(D)+h^1(\\mb F_{e,r},\\mathcal{O}_{\\mb F_{e,r}}(D))\\geqslant v(D)\\,.$$\n Since ${\\rm dim}|D|\\geqslant -1$, it follows that \n ${\\rm dim}|D|\\geqslant e(D)\\geqslant v(D)$.\n \\begin{definition}\\label{non-spcl defn}\n Let $D$ be a divisor as in \\eqref{cond}. \n We say $D$ is \\textit{non-special} if $\\dim |D|=e(D)$ and \\textit{special} otherwise $(\\dim |D|>e(D))$.\n \\end{definition}\n Thus, it follows that for $D$ as in \\eqref{cond}, if \n $h^1(\\mb F_{e,r},\\mathcal{O}_{\\mb F_{e,r}}(D))=0$,\n then $D$ is non-special. Further, if $D$ is effective and non-special then \n $h^1(\\mb F_{e,r},\\mathcal{O}_{\\mb F_{e,r}}(D))=0$. Thus, for an effective divisor as in \\eqref{cond},\n it follows that being non-special is equivalent to $h^1(\\mb F_{e,r},\\mathcal{O}_{\\mb F_{e,r}}(D))=0$.\n\nIn \\cite[Conjecture 2.6]{Laf2002}, Laface proposed a conjecture (see Conjecture \\ref{conj1}) \n which provides a characterization for special divisors using the notion of \n $(-1)$-special divisors (see Algorithm \\ref{algo}). \n We propose the following conjecture (see the next section for the motivation for this \n conjecture and similarities with conjectures on $\\mathbb P^2$): \n \\begin{conjecture}\\label{conj3}\n The following holds on $\\mb F_{e,r}$:\n \\begin{enumerate}[label=(\\alph*)]\n \\item\\label{conj3a} Effective nef divisors on $\\mb F_{e,r}$ are non-special.\n \\item\\label{conj3b} Every integral curve $C$ on $\\mb F_{e,r}$ with $C^2<0$ is either a $(-1)$-curve or the strict transform of $C_e$ on $\\mb F_{e,r}$.\n \\end{enumerate}\n \\end{conjecture}\n\nUnder the hypothesis that \n Conjecture \\ref{conj3}\\ref{conj3b} is true for all $r>0$, \n \\cite[Theorem 3.6]{HJNS24} proves the existence of irrational \n Seshadri constants. Moreover, Conjecture \\ref{conj3}\\ref{conj3b} clearly\n implies the bounded negativity conjecture for $\\mb F_{e,r}$ (which states that for a smooth\n projective surface $X$, there is a constant $b(X)$, such that for all integral\n curves $C\\subset X$, we have $C^2>b(X)$).\n\nIt is well known that the analogues of the above \n conjecture for $\\mathbb P^2$ (the SHGH conjecture) \n is true when $r \\leqslant 9$ \\cite{Nag61}. \n The main goal of this article is to obtain \n similar results for Conjecture \\ref{conj3}. \n It is known that the Conjecture \\ref{conj3}\\ref{conj3b} \n is true when $r\\leqslant e+4$ (see Conjecture \\cite[Theorem 5.5]{JKS25}). \n Here we show that \n Conjecture \\ref{conj3}\\ref{conj3a} is true when $r\\leqslant e+4$. \n In Proposition \\ref{eq-conjectures} we prove that Conjecture \\ref{conj3}\n is equivalent to \\cite[Conjecture 2.6]{Laf2002}.\n The following is the main result of this article.", "full_context": "Throughout this article, we work over the field of complex numbers. \n The interpolation problem of plane curves of a given degree passing \n through given points with at least given multiplicities \n is a classical problem in algebraic geometry. \n A conjectural answer to this problem, assuming the points are \n very general, is given by the SHGH (Segre-Harbourne-Gimigliano-Hirschowitz) \n conjecture (see \\cite{Seg62,Har86,Gim87,Hir89}).\n\nOne may pose a similar interpolation problem on arbitrary surfaces, \n with the notion of degree replaced by an appropriate numerical invariant. \n In this article, we focus on Hirzebruch surfaces.\n These surfaces are of particular interest because the minimal rational surfaces other than \n the projective plane are precisely the Hirzebruch surfaces. \n For each $e\\in \\mathbb Z_{\\ge 0}$, \n the surface $$\\mathbb F_e=\\mathbb P(\\mathcal O_{\\mathbb P^1}\\oplus \\mathcal O_{\\mathbb P^1}(-e))$$ \n is a Hirzebruch surface, and every Hirzebruch surface arises in this way.\n Let $C_e$ denote the divisor class corresponding to $\\mathcal O_{\\mb F_e}(1)$ and \n let $f$ denote the divisor class corresponding to the fiber of the map $\\mb F_e \\to \\mathbb P^1$. \n Then $\\text{Pic }\\mb F_e =\\mathbb Z C_e \\oplus \\mathbb Z f.$\n The interpolation problem on Hirzebruch surfaces is the following: \n \\begin{problem}\\label{prblm}\n Let $a,b, m_1,m_2,\\dots,m_r$ be non-negative integers. Given $r$ \n points $p_1,p_2,\\dots,p_r$ on $\\mb F_e$, does there exist a curve in \n the linear system $|aC_e+bf|$ passing through $p_i$ with multiplicity \n at least $m_i$ for all $1\\leqslant i \\leqslant r$?\n \\end{problem}\n More concretely, we are interested in determining the dimension of the \n linear system of curves in $|aC_e+bf|$.\n\nConsider $r$ very general points, say $p_1,p_2,\\dots,p_r$, on $\\mb F_e$.\n Let $\\pi: \\mb F_{e,r} \\to \\mb F_e$ denote the blow-up of $\\mb F_e $ at these points, \n and let $E_1,E_2,\\dots, E_r$ denote the exceptional divisors. Let \n $H_e=\\pi^{\\ast}(C_e)$ and $F_e=\\pi^{\\ast}(f)$; then the Picard group \n of $\\mb F_{e,r}$ is \n $$\\text{Pic }\\mb F_{e,r}=\\mathbb Z H_e\\oplus \\mathbb Z F_e \\oplus \\mathbb Z E_1 \\oplus \\mathbb Z E_2 \\oplus \\dots \\oplus \\mathbb Z E_r.$$\n Note that the linear system of curves in $|aC_e+bf|$ \n passing through $p_i$ with multiplicity at least $m_i$, for all \n $1\\leqslant i \\leqslant r$, \n is in bijective correspondence with the complete linear system \n $|aH_e+bF_e-m_1E_1-m_2E_2-\\dots-m_rE_r|$ on $\\mb F_{e,r}$. \n Hence, we are interested in finding the dimension of the \n linear system $|aH_e+bF_e-m_1E_1-m_2E_2-\\dots-m_rE_r|$. Hence, \n throughout the article, we are interested in divisors in the following form:\n\\begin{equation}\\label{cond}\n \\text{$D= aH_e+bF_e-m_1E_1-\\dots -m_rE_r$ on $\\mb F_{e,r}$ where \n $a,b,m_1,\\dots ,m_r\\in \\mathbb Z_{\\geqslant 0}$.}\n\\end{equation}\n \\begin{definition}\\label{defn}\n The \\textit{virtual dimension} of $D$ is defined as\n $$v(D)=\\frac{D^2-K_{\\mb F_{e,r}}\\cdot D}{2}\\,.$$\n The \\textit{expected dimension} of $D$ is defined as\n $e(D)=\\max\\left\\{ v(D),-1\\right\\}\\,.$\n \\end{definition}\n For divisors as in \\ref{cond}, we can see that $h^2(\\mb F_{e,r},\\mathcal{O}_{\\mb F_{e,r}}(D))=0$.\n From the Riemann-Roch theorem for surfaces, we see that \n $$\\dim |D|=\\frac{D^2-K_{\\mb F_{e,r}}\\cdot D}{2}+h^1(\\mb F_{e,r},\\mathcal{O}_{\\mb F_{e,r}}(D))=\n v(D)+h^1(\\mb F_{e,r},\\mathcal{O}_{\\mb F_{e,r}}(D))\\geqslant v(D)\\,.$$\n Since ${\\rm dim}|D|\\geqslant -1$, it follows that \n ${\\rm dim}|D|\\geqslant e(D)\\geqslant v(D)$.\n \\begin{definition}\\label{non-spcl defn}\n Let $D$ be a divisor as in \\eqref{cond}. \n We say $D$ is \\textit{non-special} if $\\dim |D|=e(D)$ and \\textit{special} otherwise $(\\dim |D|>e(D))$.\n \\end{definition}\n Thus, it follows that for $D$ as in \\eqref{cond}, if \n $h^1(\\mb F_{e,r},\\mathcal{O}_{\\mb F_{e,r}}(D))=0$,\n then $D$ is non-special. Further, if $D$ is effective and non-special then \n $h^1(\\mb F_{e,r},\\mathcal{O}_{\\mb F_{e,r}}(D))=0$. Thus, for an effective divisor as in \\eqref{cond},\n it follows that being non-special is equivalent to $h^1(\\mb F_{e,r},\\mathcal{O}_{\\mb F_{e,r}}(D))=0$.\n\nIn \\cite[Conjecture 2.6]{Laf2002}, Laface proposed a conjecture (see Conjecture \\ref{conj1}) \n which provides a characterization for special divisors using the notion of \n $(-1)$-special divisors (see Algorithm \\ref{algo}). \n We propose the following conjecture (see the next section for the motivation for this \n conjecture and similarities with conjectures on $\\mathbb P^2$): \n \\begin{conjecture}\\label{conj3}\n The following holds on $\\mb F_{e,r}$:\n \\begin{enumerate}[label=(\\alph*)]\n \\item\\label{conj3a} Effective nef divisors on $\\mb F_{e,r}$ are non-special.\n \\item\\label{conj3b} Every integral curve $C$ on $\\mb F_{e,r}$ with $C^2<0$ is either a $(-1)$-curve or the strict transform of $C_e$ on $\\mb F_{e,r}$.\n \\end{enumerate}\n \\end{conjecture}\n\nUnder the hypothesis that \n Conjecture \\ref{conj3}\\ref{conj3b} is true for all $r>0$, \n \\cite[Theorem 3.6]{HJNS24} proves the existence of irrational \n Seshadri constants. Moreover, Conjecture \\ref{conj3}\\ref{conj3b} clearly\n implies the bounded negativity conjecture for $\\mb F_{e,r}$ (which states that for a smooth\n projective surface $X$, there is a constant $b(X)$, such that for all integral\n curves $C\\subset X$, we have $C^2>b(X)$).\n\nIt is well known that the analogues of the above \n conjecture for $\\mathbb P^2$ (the SHGH conjecture) \n is true when $r \\leqslant 9$ \\cite{Nag61}. \n The main goal of this article is to obtain \n similar results for Conjecture \\ref{conj3}. \n It is known that the Conjecture \\ref{conj3}\\ref{conj3b} \n is true when $r\\leqslant e+4$ (see Conjecture \\cite[Theorem 5.5]{JKS25}). \n Here we show that \n Conjecture \\ref{conj3}\\ref{conj3a} is true when $r\\leqslant e+4$. \n In Proposition \\ref{eq-conjectures} we prove that Conjecture \\ref{conj3}\n is equivalent to \\cite[Conjecture 2.6]{Laf2002}.\n The following is the main result of this article.\n\nIt is well known that the analogues of the above \n conjecture for $\\mb P^2$ (the SHGH conjecture) \n is true when $r \\leqslant 9$ \\cite{Nag61}. \n The main goal of this article is to obtain \n similar results for Conjecture \\ref{conj3}. \n It is known that the Conjecture \\ref{conj3}\\ref{conj3b} \n is true when $r\\leqslant e+4$ (see Conjecture \\cite[Theorem 5.5]{JKS25}). \n Here we show that \n Conjecture \\ref{conj3}\\ref{conj3a} is true when $r\\leqslant e+4$. \n In Proposition \\ref{eq-conjectures} we prove that Conjecture \\ref{conj3}\n is equivalent to \\cite[Conjecture 2.6]{Laf2002}.\n The following is the main result of this article.\n\n\\section{Motivation and Preliminaries}\nConsider the blow-up of $\\mathbb P^2$ at $r$ very general points, denote it by $X_r$. \nWe define the \\emph{virtual dimension}, \\emph{expected dimension}, \nand \\emph{non-specialness} of a divisor on $X_r$ analogously to \nDefinitions~\\ref{defn} and~\\ref{non-spcl defn}.\nRecall that an integral curve $C$ on a smooth projective surface is a $(-1)$-curve if \nit is a smooth rational curve with $C^2=-1$. From the genus formula it follows \nthat $C$ is a $(-1)$-curve if and only if $C^2=-1$ and $K\\cdot C=-1$, \nwhere $K$ is the canonical divisor of the smooth surface.\nLet $D$ be an effective \ndivisor on $X_r$. We say that $D$ is \\emph{$(-1)$-special} if there \nexists a $(-1)$-curve $E$ on $X_r$ such that\n$D \\cdot E < -1$.\nThe following statements are equivalent\nand are referred to as the SHGH conjecture for $X_r$:\n\\begin{enumerate}\n \\item (Segre~\\cite{Seg62}) Let $D$ be an effective divisor on $X_r$. If the general \n curve of the linear system $|D|$ on $X_r$ is reduced, then $D$ is non-special.\n \\item (Hirschowitz~\\cite{Hir89}) Let $D$ be an effective divisor on $X_r$. Then $D$ is \n special if and only if $D$ is $(-1)$-special.\n \\item (Harbourne~\\cite{Har92}) \\label{shgh-harbourne}The following statements hold on $X_r$:\n \\begin{enumerate}\n \\item Every effective nef divisor on $X_r$ is non-special.\n \\item Every integral curve $C \\subset X_r$ with $C^2 < 0$ is a $(-1)$-curve.\n \\end{enumerate}\n\\end{enumerate}\nThere are more equivalent versions of the above conjectures. For the statements of \nthese and for the equivalences, we refer the reader to \\cite{Har86}, \\cite{Gim87} \nand \\cite{CM01}. In an interesting recent article, \n\\cite[Theorem 2]{Laf24}, the authors prove that on the blow-up of $\\mathbb P^2$\nat $r$ very general points, (3a) implies (3b).\n\nLet $M$ denote the divisor that we obtain after applying \n Algorithm \\ref{algo} to $D$. So we have $M\\cdot E\\geqslant 0$ \n for all $(-1)$-curves $E$ and $M\\cdot \\widetilde{C}_e\\geqslant 0$. \n Note that by Conjecture \\ref{conj3}\\ref{conj3b}, \n if $C$ is an integral curve with $C^2<0$, then $C$ is either a \n $(-1)$-curve or it is $\\widetilde{C}_e$. \n Hence, $M$ is nef by the elementary result stated above.\n It is clear from Remark \\ref{rem1} that $M$ is effective. \n By Conjecture \\ref{conj3}\\ref{conj3a}, $M$ is non-special. \n So $\\dim|M|=e(M)=\\max\\{v(M),-1\\}=v(M)$. Since $D$ is special, we have\n $$v(M)=\\dim|M|=\\dim|D|>e(D)\\geqslant v(D).$$\n So $D$ is a $(-1)$-special divisor.\n \\end{proof}\n\n\\section{Conjecture \\ref{conj3} when $r\\leqslant e+4$}\n In this section, we prove the Conjecture \\ref{conj3} when $r\\leqslant e+4$. \n Throughout this section, unless stated otherwise, we assume $e>0$ and $r\\leqslant e+4$. \n \\begin{lemma} \n $h^0(\\Fe, \\mc O_{\\Fe}(-K_{\\Fe}))\\geqslant e+6$.\n \\end{lemma}\n \\begin{proof}\n From \\cite[Lemma 2.7]{JKS25}, we have \n $$\\pi_{\\ast}(2C_e)=\\mc {O}_{\\mb {P}^1}\\oplus\\mc {O}_{\\mb {P}^1}(-e)\n \\oplus\\mc {O}_{\\mb {P}^1}(-2e)\\,.$$\n Hence, using projection formula we have\n \\begin{equation*}\n \\begin{split}\n h^0(\\Fe,O_{\\Fe}(-K_{\\Fe}))&=h^0(\\Fe,O_{\\Fe}(2C_e+(e+2)f)\\\\\n &=h^0(\\mb P^1,\\pi_{\\ast}(\\mc O_{\\Fer}(2C_e))\\otimes \\mc O_{\\mb P^1}(e+2))\\\\\n &=h^0(\\mb P^1,\\mc O_{\\mb P^1}(e+2)\\oplus \\mc O_{\\mb P^1}(2) \\oplus \\mc O_{\\mb P^1}(-e+2))\\\\\n &\\geqslant e+3+3+0=e+6\\,.\n \\end{split}\n \\end{equation*}\n \\end{proof}\n\nNow, suppose that $C^2\\geqslant 0$. Since $C$ is an integral curve, \n we have $C$ is a nef divisor. \n Hence if $-K_{\\Fer}\\cdot C\\geqslant 1$, Theorem \\ref{harbourne} enables us to conclude $h^1(\\Fer, \\mc O_{\\Fer}(C))=0$.\n Now we claim that there is no integral curve $C$ with $-K_{\\Fer}\\cdot C=0$.\n Suppose that such a $C$ exists. Then from \\eqref{eqn-dagger}, we can write \n $-K_{\\Fer}=\\widetilde{C}_e+C'$. \n $$-K_{\\Fer}\\cdot C=(\\widetilde{C}_e+C')\\cdot C=0$$\n Since $C$ is nef, we have $C'\\cdot C=\\widetilde{C}_e\\cdot C=0$.\n Write $$C=aH_e+bF_e-\\sum_{i=1}^rm_iE_i.$$ \n $\\widetilde{C}_e\\cdot C=0$ gives $b=ae$ and $-K_{\\Fer}\\cdot C=0$ gives\n \\begin{equation}\\label{eqn 2.1}\n ae+2a=\\sum_{i=1}^r m_i\n \\end{equation}\n and\n $C^2\\geqslant 0$ gives\n \\begin{equation}\\label{eqn 2.2}\n a^2e\\geqslant\\sum_{i=1}^r m_i^2\n \\end{equation}\n Using $\\eqref{eqn 2.1}$, $\\eqref{eqn 2.2}$ and Cauchy-Schwarz inequality we have\n $$(ae+2a)^2=\\left(\\sum_{i=1}^r m_i\\right)^2\\leqslant r\\left(\\sum_{i=1}^r m_i^2\\right) \\leqslant (e+4)\\left(a^2e\\right).$$\n This is a contradiction.\n \\end{proof}\n\n\\begin{theorem}\\label{e+4}\n Let $e\\geqslant 0$ and let $r\\leqslant e+4$. Then \n Conjecture \\ref{conj3} holds on $\\Fer$. \n \\end{theorem}\n \\begin{proof}\n When $e=0$, see Remark \\ref{rem e=0}. So assume $e>0$.\n By Proposition \\ref{2.1 is true}, when $r\\leqslant e+4$, for every \n integral curve $C\\neq \\widetilde{C}_e$ we have \n $h^1(\\Fer, \\mc O_{\\Fer}(C))=0$. \n Hence by Proposition \\ref{main thm1}, if $D$ is nef and effective and \n if $\\widetilde{C}_e$ is not a fixed component of $|D|$, then it is \n non-special. If $\\widetilde{C}_e$ is a fixed component of $|D|$, \n then Corollary \\ref{C_e is fixed component}, says $D$ is non-special. \n This proves that Conjecture \\ref{conj3}\\ref{conj3a} holds. \n Conjecture \\ref{conj3}\\ref{conj3b} is proved in \\cite[Theorem 5.5]{JKS25} when \n $r\\leqslant e+4$. \n \\end{proof}\n \\bibliographystyle{alpha}\n \\bibliography{Hir}\n\\end{document}\n\n\\begin{lemma}\\label{general curve}\n Assume Conjecture \\ref{conj2.1} is true. \n Let $D$ be a divisor as in \\eqref{cond}, which is effective. \n Suppose that $\\widetilde{C}_e$ is not a fixed component of $|D|$. \n Write a general curve in $|D|$ as follows:\n $$h_1C_1+h_2C_2+\\dots+h_kC_k+D_1+D_2+\\dots+D_h+M, \\hspace{.5cm}k\\geqslant 0, h\\geqslant 0$$\n where $C_1,C_2,\\dots,C_k,D_1,D_2,\\dots,D_h$ are the distinct, integral, fixed components of $|D|$ \n and $M$ is general curve in the moving part of $|D|$. \n Also assume $h_i\\geqslant 2$ for all $1\\leqslant i \\leqslant k$. Then we have the following:\n \\begin{enumerate}\n \\item \\label{A1} $C_i^2=K_{\\Fer}\\cdot C_i$, $C_i^2\\leqslant 0$, for all $1\\leqslant i \\leqslant k$,\n \\item \\label{A2} $D_j^2=K_{\\Fer}\\cdot D_j$, for all $1\\leqslant j \\leqslant h$,\n \\end{enumerate}", "post_theorem_intro_text_len": 393, "post_theorem_intro_text": "\\begin{remark}\\label{rem e=0}\n When $e=0$, the blow-up of $\\mathbb F_0$\n at $r$ points is isomorphic to the blow-up of $\\mathbb P^2$ at $r+1$ points. \n Thus, when $e=0$, the above Theorem is a consequence of the fact mentioned before, that \n SHGH conjecture for $\\mathbb P^2$ is true when $r\\leqslant 9$. In view of \n this, throughout we assume that $e>0$. \n \\end{remark}", "sketch": "When $e=0$, the result is reduced to the known case of $\u001b F_0$: the blow-up of $\\mathbb F_0$ at $r$ points is isomorphic to the blow-up of $\\mathbb P^2$ at $r+1$ points. Hence the theorem for $e=0$ follows from the previously mentioned fact that the SHGH conjecture for $\\mathbb P^2$ is true when $r\\leqslant 9$. Therefore the discussion proceeds assuming $e>0$.", "expanded_sketch": "When $e=0$, the result is reduced to the known case of $\u001b F_0$: the blow-up of $\\mathbb F_0$ at $r$ points is isomorphic to the blow-up of $\\mathbb P^2$ at $r+1$ points. Hence the theorem for $e=0$ follows from the previously mentioned fact that the SHGH conjecture for $\\mathbb P^2$ is true when $r\\leqslant 9$. Therefore the discussion proceeds assuming $e>0$.", "expanded_theorem": "\\begin{theorem}\\label{e+4}\n Let $e\\geqslant 0$ and let $r\\leqslant e+4$. Then \n Conjecture \\ref{conj3} holds on $\\Fer$. \n \\end{theorem}\nLet $e\\geqslant 0$ and let $r\\leqslant e+4$. We use the following conjecture:\n\\begin{conjecture}\\label{conj3}\n\t\tThe following holds on $\\Fer$:\n\t\t\\begin{enumerate}[label=(\\alph*)]\n\t\t\t\\item\\label{conj3a} Effective nef divisors on $\\Fer$ are non-special.\n\t\t\t\\item\\label{conj3b} Every integral curve $C$ on $\\Fer$ with $C^2<0$ is either a $(-1)$-curve or the strict transform of $C_e$ on $\\Fer$.\n\t\t\\end{enumerate}\n\t\\end{conjecture}\nThen this conjecture holds for $\\mb F_{e,r}$.", "theorem_type": "unknown", "mcq": {"question": "Let $e\\ge 0$, let $\\mathbb F_e=\\mathbb P\\big(\\mathcal O_{\\mathbb P^1}\\oplus \\mathcal O_{\\mathbb P^1}(-e)\\big)$ be the Hirzebruch surface, and let $\\pi:\\mathbb F_{e,r}\\to \\mathbb F_e$ be the blow-up of $r$ very general points, where $r\\le e+4$. Let $C_e$ be the divisor class of $\\mathcal O_{\\mathbb F_e}(1)$, and let $\\widetilde C_e$ denote its strict transform on $\\mathbb F_{e,r}$. For a divisor $D$ on $\\mathbb F_{e,r}$, write its virtual dimension as $v(D)=\\frac{D^2-K_{\\mathbb F_{e,r}}\\cdot D}{2}$ and its expected dimension as $e(D)=\\max\\{v(D),-1\\}$. A divisor is called non-special if $\\dim|D|=e(D)$. Which statement holds on $\\mathbb F_{e,r}$ under these assumptions?", "correct_choice": {"label": "A", "text": "Both of the following hold on $\\mathbb F_{e,r}$: (i) every effective nef divisor is non-special, i.e. for every effective nef divisor $D$ one has $\\dim|D|=\\max\\left\\{\\frac{D^2-K_{\\mathbb F_{e,r}}\\cdot D}{2},-1\\right\\}$; and (ii) every integral curve $C$ on $\\mathbb F_{e,r}$ with $C^2<0$ is either a $(-1)$-curve or the strict transform $\\widetilde C_e$ of $C_e$."}, "choices": [{"label": "B", "text": "Both of the following hold on $\\mathbb F_{e,r}$: (i) every effective divisor $D$ is non-special, i.e. for every effective divisor $D$ one has $\\dim|D|=\\max\\left\\{\\frac{D^2-K_{\\mathbb F_{e,r}}\\cdot D}{2},-1\\right\\}$; and (ii) every integral curve $C$ on $\\mathbb F_{e,r}$ with $C^2<0$ is either a $(-1)$-curve or the strict transform $\\widetilde C_e$ of $C_e$."}, {"label": "C", "text": "Every effective nef divisor on $\\mathbb F_{e,r}$ is non-special, i.e. for every effective nef divisor $D$ one has $\\dim|D|=\\max\\left\\{\\frac{D^2-K_{\\mathbb F_{e,r}}\\cdot D}{2},-1\\right\\}$."}, {"label": "D", "text": "Both of the following hold on $\\mathbb F_{e,r}$: (i) every effective nef divisor is non-special, i.e. for every effective nef divisor $D$ one has $\\dim|D|=\\max\\left\\{\\frac{D^2-K_{\\mathbb F_{e,r}}\\cdot D}{2},-1\\right\\}$; and (ii) every integral curve $C$ on $\\mathbb F_{e,r}$ with $C^2\\le 0$ is either a $(-1)$-curve or the strict transform $\\widetilde C_e$ of $C_e$."}, {"label": "E", "text": "Both of the following hold on $\\mathbb F_{e,r}$ whenever $r\\le e+5$: (i) every effective nef divisor is non-special, i.e. for every effective nef divisor $D$ one has $\\dim|D|=\\max\\left\\{\\frac{D^2-K_{\\mathbb F_{e,r}}\\cdot D}{2},-1\\right\\}$; and (ii) every integral curve $C$ on $\\mathbb F_{e,r}$ with $C^2<0$ is either a $(-1)$-curve or the strict transform $\\widetilde C_e$ of $C_e$."}], "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": "nef hypothesis in part (i)", "template_used": "wildcard"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "dropped part (ii) of the conjunction", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "other", "tampered_component": "strict negativity condition $C^2<0$ replaced by $C^2\\le 0$", "template_used": "boundary_range"}, {"label": "E", "sketch_hook_type": "case_split", "tampered_component": "numerical range bound on the number of blown-up points", "template_used": "stronger_trap"}]}} +{"id": "2601.22072v1", "paper_link": "http://arxiv.org/abs/2601.22072v1", "theorems_cnt": 3, "theorem": {"env_name": "thm", "content": "\\label{thm1_main}\nWith the above notation, the following hold:\n\\begin{enumerate}\n\\item[i)] If ${\\rm lct}(X,Z_A)\\geq c$, then ${\\rm lct}(Y,W_A)\\geq \\min\\{rc,r-1+c\\}$.\n\\item[ii)] If ${\\rm lct}(Y,W_A)\\geq r-1+c'$, then ${\\rm lct}(X,Z_A)\\geq \\min\\big\\{c',\\tfrac{c'-1}{r}+1\\big\\}$.\n\\end{enumerate}", "start_pos": 11633, "end_pos": 11946, "label": "thm1_main"}, "ref_dict": {"rmk:KempfDirectImage": "\\begin{rmk} \\label{rmk:KempfDirectImage}\nOne can give another proof of the ``if'' implication in Theorem \\ref{thm2_main}, at least when we know \\emph{a priori} that $Z_A$ is reduced,\nusing Kempf's argument for \\cite[Proposition~2]{Kempf}, as follows. \nRecall that the projection $p\\colon Y = X \\times {\\mathbf P}^{r-1} \\to X$ onto the first factor induces $W_A \\to Z_A$. On $Y$ we have the line bundle $\\cO_Y(1)$,\nthe pull-back of $\\cO_{{\\mathbf P}^{r-1}}(1)$ via the second projection.\n\n Suppose first only that $W_A$ is a local complete intersection of pure codimension $r$ in $Y$. Consider the $r$ global sections $q_i = \\sum_{1 \\leq j \\leq r} a_{ij} y_j$\n of $\\cO_Y(1)$, for $1 \\leq t \\leq r$. These give $r$ morphisms $\\cO_Y(-1) \\to \\cO_Y$ and a corresponding Koszul complex \n $$0 \\to K_r=\\cO_Y(-r)\\overset{\\varphi_r}\\to \\ldots \\to K_1=\\cO_Y(-1)^{\\oplus r}\\overset{\\varphi_1}\\to K_0=\\cO_Y\\to 0.$$ \n The assumption on $W_A$ implies that this is an acyclic complex, giving a resolution of\n $\\cO_{W_A}$. By breaking the Koszul complex into short exact sequences, taking the corresponding long exact sequences for higher direct images, and using the standard\n computation of cohomology of the projective space, it follows by descending induction on $j$ that\n $R^qp_*{\\rm Im}(\\varphi_j)=0$ for $1\\leq j\\leq r$ and $q>j-1$. Using this for $j=1$, we deduce that \n $R^qp_*\\cO_{W_A}=0$ for all $q>0$ and the induced morphism\n $\\cO_X\\to p_*\\cO_{W_A}$ is surjective. \n\n Suppose now that $W_A$ is a local complete intersection of pure codimension $r$ in $Y$, with rational singularities. \n This implies that $W_A$ is reduced and, as in the proof of Theorem \\ref{thm2_main}, we see that we may assume that $W_A$ is also irreducible.\n This implies that $Z_A$ is irreducible, too. If we assume that $Z_A$ is reduced (for example, this follows using Theorem~\\ref{thm1_main}), then we see \n that $W_A\\to Z_A$ is a proper birational morphism of algebraic varieties. What we have seen so far about $R^qp_*\\cO_{W_A}$ says that the canonical morphism \n $\\cO_{Z_A}\\to {\\mathbf R}p_*\\cO_{W_A}$ is an isomorphism.\n\n If $\\tau\\colon V\\to W_A$ is a resolution of singularities, then the canonical morphism\n $\\cO_{W_A}\\to {\\mathbf R}\\tau_*\\cO_V$ is an isomorphism by the assumption on the singularities of $W_A$, hence \n the morphism\n $$\\cO_{Z_A}\\to {\\mathbf R}(p\\circ\\tau)_*\\cO_V\\simeq {\\mathbf R}p_*({\\mathbf R}\\tau_*\\cO_V)\\simeq {\\mathbf R}p_*\\cO_{W_A}$$\n is an isomorphism. Since $p\\circ \\tau$ is a resolution of singularities of $Z_A$, we conclude that $Z_A$ has rational singularities. \n \\end{rmk}", "rmk:charp": "\\begin{rmk} \\label{rmk:charp}\nIt is shown in \\cite{BDSW} that if $M_U$ is connected, then $W_U$ has $F$-rational type (which implies it has rational singularities) and, in fact, it is strongly $F$-regular when \nthe configuration is defined over an $F$-finite field of positive characteristic. The argument in loc. cit. utilizes the matroid-theoretic structure in an essential way, hence it does not apply to arbitrary $Z_A$ and $W_A$.\n\\end{rmk}", "application": "\\begin{rmk}\n Any hypersurface can be defined by a determinant in a trivial way: simply consider a $1 \\times 1$ determinant. When $X = {\\mathbf A}^n$, every hypersurface can be defined by a determinant in a much less trivial way: it is classical (see, for instance, \\cite{HMV}), that given $f \\in {\\mathbf C}[x_1, \\dots, x_n]$, there exists a square matrix $A = (a_{ij})$ such that\n $f={\\rm det}(A)$ and every $a_{ij}$ is a polynomial in ${\\mathbf C}[x_1, \\dots, x_n]$ of degree at most $1$. Note that $A$ is usually quite large and often contains constants and inhomogeneous entries, even if $f$ itself is homogeneous. Therefore, Theorems~\\ref{thm1_main} and \\ref{thm2_main}\napply to all hypersurfaces in ${\\mathbf A}^n$ in a nontrivial way, modulo constructing the matrix $A$.\n\\end{rmk}\n\n\\section{Configuration hypersurfaces}\\label{application}\n\nWe begin by recalling, following the original \\cite{BEK} and the more recent \\cite{DSW}, the definition of configuration hypersurfaces. A \\emph{configuration} is a nonzero linear subspace $U$ in a finite-dimensional $\\C$-vector space \n$V=\\C^E$, with a fixed basis $E$. We denote by $V^*$ the dual vector space and by $E^*=\\{e^*\\mid e\\in E\\}$ the dual basis. For $S\\subseteq E$, we put\n$S^*=\\{e^*\\mid e\\in S\\}$. \n\nGiven a configuration $U$ as above, the corresponding \\emph{configuration matroid} $M_U$ is the matroid on $E$, whose independent sets are those \n$S\\subseteq E$ such that the restrictions of the elements of $S^*$ to $U$ are linearly independent. For basic facts about matroids, we refer to \\cite{matroids};\nhowever, we will not need any of this in what follows. \n\nThe \\emph{Hadamard product} on $V$ is given by $Q_E\\colon V\\times V\\to V$,\n$$Q_E\\left(\\sum_{e\\in E}a_ee,\\sum_{e\\in E}b_ee\\right)=\\sum_{e\\in E}a_eb_ee.$$\nGiven a configuration $U\\subseteq V$, we denote by $Q_U$ the restriction of $Q_E$ to $U\\times U$.\nSet-theoretically, the \\emph{configuration hypersurface} $Z_U\\subseteq V^*$ consists of those\n$\\beta\\in V^*$ such that $\\beta\\circ Q_U$ is degenerate. \n\nIn order to give an explicit equation of $Z_U$, let us order the elements of $E$ as $e_1,\\ldots,e_n$, so ${\\rm Sym}(V)\\simeq \\C[x_1,\\ldots,x_n]$.\nWe choose a basis $B$ of $U$, so that with respect to $B$ and $E$, the subspace $U$\nis described as the span of the row vectors of an $r\\times n$ matrix $D$, where $r=\\dim(U)$. Therefore, an equation\nof $Z_U$ is given by \n$${\\rm det}(A),\\quad\\text{where}\\quad A=D\\cdot {\\rm diag}(x_1,\\ldots,x_n)\\cdot D^T.$$\nThe matrix $A$ is the \\emph{Patterson} matrix of the configuration (with respect to the basis $B$). \nAs picking a different basis of $U$ changes the determinant of the Patterson matrix by a nonzero scalar, this defining equation depends (up to scalar) only on the configuration \n$U \\subseteq \\mathbf{C}^E$.\nAn application of the Cauchy-Binet formula\ngives\n\\begin{equation}\\label{eq_Patterson}\n{\\rm det}(A)=\\sum_{I\\subseteq E; \\#I=r}{\\rm det}(D\\vert_I)x^I,\n\\end{equation}\nwhere we denote by $D\\vert_I$ the submatrix of $D$ on the columns indexed by $I$ and $x^I=\\prod_{i\\in I}x_i$.\nNote that ${\\rm det}(D\\vert_I)\\neq 0$ if and only if $I$ is a basis of $M_U$, hence ${\\rm det}(A)$ is a matroid support polynomial of $M_U$ is the sense of \n\\cite[Definition~2.11]{BW}. \n\nLet ${\\mathbf P}(U)$ denote the projective space of lines in $U$.\nSet-theoretically, the \\emph{configuration incidence variety} $W_U$ is the subset of $V^*\\times {\\mathbf P}(U)$ consisting of those \n$\\big(\\beta,[u]\\big)$ such that $\\beta\\circ Q(u,-)$ vanishes on $U$. This incidence variety was first studied by Bloch \\cite{Bloch}. In order to give explicit equations for $W_U$, we choose a basis $B$\nof $U$ as above. In this case, the ideal defining the subscheme $W_U$ in $V^*\\times {\\mathbf P}(U)$ is generated by the entries of \n$A\\cdot (y_1,\\ldots,y_r)^T$, where $y_1,\\ldots,y_r$ are the homogeneous coordinates on ${\\mathbf P}(U)$\ncorresponding to the basis $B$. \n\n\\begin{cor}\nThe configuration incidence variety $W_U$ satisfies \n\\begin{equation}\\label{eq_cor}\n{\\rm lct}\\big(V^*\\times {\\mathbf P}(U), W_U)=r.\n\\end{equation}\nIn particular, $W_U$ is a local complete intersection of pure codimension $r$. \nIf, moreover, the matroid $M_U$ is connected, then the configuration incidence variety $W_U$ has rational singularities.\n\\end{cor}\n\n\\begin{proof}\nSince the polynomial in (\\ref{eq_Patterson}) is clearly square-free (in the sense that it is of degree $\\leq 1$ with respect to each variable),\nit follows from \\cite[Lemma~3.3]{BMW} that ${\\rm lct}(V^*,Z_U)=1$, and we deduce the formula in (\\ref{eq_cor}) from Theorem~\\ref{thm1_main}.\nThe fact that $W_U$ is a local complete intersection of pure codimension $r$ then follows from Proposition~\\ref{max_lct}.\n\nIf we assume that the matroid $M_U$ is connected, then it follows easily that the polynomial in (\\ref{eq_Patterson}) is irreducible (see, for example \\cite[Corollary~2.19]{BW}\nfor a more general statement). Using again the fact that the polynomial is square-free, we deduce from \\cite[Theorem~1.1]{BMW} that the hypersurface $Z_U$ has rational singularities. The fact that $W_U$ has rational singularities now follows from\nTheorem~\\ref{thm2_main}. \n\\end{proof}\n\n\\begin{rmk}\nWe may also consider the subscheme $\\widetilde{W}_U$ of $V^*\\times U$ given by the same equations as $W_U$. It follows from Remark~\\ref{rmk_affine_version}\nthat this is a local complete intersection of pure codimension $r$ in $V^*\\times U$ and ${\\rm lct}(V^*\\times U, \\widetilde{W}_U)=r$. However, $\\widetilde{W}_U$ never has\nrational singularities.\n\\end{rmk}", "thm1_main": "\\begin{thm}\\label{thm1_main}\nWith the above notation, the following hold:\n\\begin{enumerate}\n\\item[i)] If ${\\rm lct}(X,Z_A)\\geq c$, then ${\\rm lct}(Y,W_A)\\geq \\min\\{rc,r-1+c\\}$.\n\\item[ii)] If ${\\rm lct}(Y,W_A)\\geq r-1+c'$, then ${\\rm lct}(X,Z_A)\\geq \\min\\big\\{c',\\tfrac{c'-1}{r}+1\\big\\}$.\n\\end{enumerate}\n\\end{thm}", "thm2_main": "\\begin{thm}\\label{thm2_main}\nIf $r=s$, then $Z_A$ has rational singularities if and only if $W_A$ is a local complete intersection of pure codimension $r$ with rational singularities.\n\\end{thm}"}, "pre_theorem_intro_text_len": 1711, "pre_theorem_intro_text": "Let $X$ be a smooth, irreducible, complex algebraic variety, and let $A=(a_{ij})_{1\\leq i\\leq s,1\\leq j\\leq r}$ be a matrix with $s\\geq r$ and $a_{ij}\\in\\cO_X(X)$ for all $i$ and $j$.\nWe consider the closed subscheme $Z_A$ of $X$ defined by the ideal generated by the $r$-minors of $A$, which we assume is a proper subscheme. We also consider \nthe incidence correspondence $W_A\\hookrightarrow Y=X\\times {\\mathbf P}^{r-1}$, defined by $\\big\\{\\sum_{1\\leq j\\leq r}a_{ij}y_j\\mid 1\\leq i\\leq s\\big\\}$, where\n$y_1,\\ldots,y_r$ are the homogeneous coordinates on ${\\mathbf P}^{r-1}$.\nOur main interest is in the case of a square matrix $A$, when $Z_A$ is a hypersurface in $X$, but our first main result holds in the general setting. \nOur goal is to relate the singularities of the pairs $(X,Z_A)$ and $(Y,W_A)$. \n\nRecall that if $\\Gamma$ is a closed subscheme of a smooth variety $T$, then we may consider the \\emph{log canonical threshold} ${\\rm lct}(T,\\Gamma)$, which\ncan be defined in terms of a log resolution of the pair $(T,\\Gamma)$ (see \\cite[Chapter~9.3.B]{Lazarsfeld} for an introduction to log canonical thresholds). This is a fundamental invariant\nof singularities of pairs, which can be defined in a more general setting (only assuming that the pair $(T,\\Gamma)$ has mild singularities), and which plays an important role\nin birational geometry.\nIt is well-known and easy to see\nthat if $\\Gamma$ is locally defined by $d$ equations, then ${\\rm lct}(T,\\Gamma)\\leq d$. Moreover,\nif equality holds, then $\\Gamma$ is a local complete intersection of pure codimension $d$ in $T$.\nIn the setting of interest for us, we prove the following result relating the log canonical thresholds of $(X,Z_A)$ and $(Y,W_A)$:", "context": "Let $X$ be a smooth, irreducible, complex algebraic variety, and let $A=(a_{ij})_{1\\leq i\\leq s,1\\leq j\\leq r}$ be a matrix with $s\\geq r$ and $a_{ij}\\in\\cO_X(X)$ for all $i$ and $j$.\nWe consider the closed subscheme $Z_A$ of $X$ defined by the ideal generated by the $r$-minors of $A$, which we assume is a proper subscheme. We also consider \nthe incidence correspondence $W_A\\hookrightarrow Y=X\\times {\\mathbf P}^{r-1}$, defined by $\\big\\{\\sum_{1\\leq j\\leq r}a_{ij}y_j\\mid 1\\leq i\\leq s\\big\\}$, where\n$y_1,\\ldots,y_r$ are the homogeneous coordinates on ${\\mathbf P}^{r-1}$.\nOur main interest is in the case of a square matrix $A$, when $Z_A$ is a hypersurface in $X$, but our first main result holds in the general setting. \nOur goal is to relate the singularities of the pairs $(X,Z_A)$ and $(Y,W_A)$.\n\nRecall that if $\\Gamma$ is a closed subscheme of a smooth variety $T$, then we may consider the \\emph{log canonical threshold} ${\\rm lct}(T,\\Gamma)$, which\ncan be defined in terms of a log resolution of the pair $(T,\\Gamma)$ (see \\cite[Chapter~9.3.B]{Lazarsfeld} for an introduction to log canonical thresholds). This is a fundamental invariant\nof singularities of pairs, which can be defined in a more general setting (only assuming that the pair $(T,\\Gamma)$ has mild singularities), and which plays an important role\nin birational geometry.\nIt is well-known and easy to see\nthat if $\\Gamma$ is locally defined by $d$ equations, then ${\\rm lct}(T,\\Gamma)\\leq d$. Moreover,\nif equality holds, then $\\Gamma$ is a local complete intersection of pure codimension $d$ in $T$.\nIn the setting of interest for us, we prove the following result relating the log canonical thresholds of $(X,Z_A)$ and $(Y,W_A)$:", "full_context": "Let $X$ be a smooth, irreducible, complex algebraic variety, and let $A=(a_{ij})_{1\\leq i\\leq s,1\\leq j\\leq r}$ be a matrix with $s\\geq r$ and $a_{ij}\\in\\cO_X(X)$ for all $i$ and $j$.\nWe consider the closed subscheme $Z_A$ of $X$ defined by the ideal generated by the $r$-minors of $A$, which we assume is a proper subscheme. We also consider \nthe incidence correspondence $W_A\\hookrightarrow Y=X\\times {\\mathbf P}^{r-1}$, defined by $\\big\\{\\sum_{1\\leq j\\leq r}a_{ij}y_j\\mid 1\\leq i\\leq s\\big\\}$, where\n$y_1,\\ldots,y_r$ are the homogeneous coordinates on ${\\mathbf P}^{r-1}$.\nOur main interest is in the case of a square matrix $A$, when $Z_A$ is a hypersurface in $X$, but our first main result holds in the general setting. \nOur goal is to relate the singularities of the pairs $(X,Z_A)$ and $(Y,W_A)$.\n\nRecall that if $\\Gamma$ is a closed subscheme of a smooth variety $T$, then we may consider the \\emph{log canonical threshold} ${\\rm lct}(T,\\Gamma)$, which\ncan be defined in terms of a log resolution of the pair $(T,\\Gamma)$ (see \\cite[Chapter~9.3.B]{Lazarsfeld} for an introduction to log canonical thresholds). This is a fundamental invariant\nof singularities of pairs, which can be defined in a more general setting (only assuming that the pair $(T,\\Gamma)$ has mild singularities), and which plays an important role\nin birational geometry.\nIt is well-known and easy to see\nthat if $\\Gamma$ is locally defined by $d$ equations, then ${\\rm lct}(T,\\Gamma)\\leq d$. Moreover,\nif equality holds, then $\\Gamma$ is a local complete intersection of pure codimension $d$ in $T$.\nIn the setting of interest for us, we prove the following result relating the log canonical thresholds of $(X,Z_A)$ and $(Y,W_A)$:\n\nRecall that if $\\Gamma$ is a closed subscheme of a smooth variety $T$, then we may consider the \\emph{log canonical threshold} ${\\rm lct}(T,\\Gamma)$, which\ncan be defined in terms of a log resolution of the pair $(T,\\Gamma)$ (see \\cite[Chapter~9.3.B]{Lazarsfeld} for an introduction to log canonical thresholds). This is a fundamental invariant\nof singularities of pairs, which can be defined in a more general setting (only assuming that the pair $(T,\\Gamma)$ has mild singularities), and which plays an important role\nin birational geometry.\nIt is well-known and easy to see\nthat if $\\Gamma$ is locally defined by $d$ equations, then ${\\rm lct}(T,\\Gamma)\\leq d$. Moreover,\nif equality holds, then $\\Gamma$ is a local complete intersection of pure codimension $d$ in $T$.\nIn the setting of interest for us, we prove the following result relating the log canonical thresholds of $(X,Z_A)$ and $(Y,W_A)$:\n\nNote that we always have ${\\rm lct}(X,Z_A)\\leq {\\rm codim}_X(Z_A)\\leq s-r+1$, and both inequalities are equalities in the generic case, that is, \nwhen $X$ is the affine space of $s\\times r$ matrices and $A=(x_{ij})_{1\\leq i\\leq s,1\\leq j\\leq r}$\n(see \\cite[Theorem~D]{Docampo}). If ${\\rm lct}(X,Z_A)=s-r+1$, then assertion i) in Theorem~\\ref{thm1_main} implies that ${\\rm lct}(Y,W_A)=s$.\nIn particular, in this case $W_A$ is a complete intersection in $Y$, of pure codimension $s$. Of course, in the generic case, we know that\n$W_A$ is smooth of codimension $s$ in $Y$. This case shows that the inequality ii) in Theorem~\\ref{thm1_main} is not optimal when $s>r\\geq 2$.\nHowever, for square matrices, we get\n\n\\begin{thm}\\label{thm_old1}\nIf $Z$ is a proper closed subscheme in the smooth, irreducible complex algebraic variety $X$, and if $c>0$, then the following are equivalent:\n\\begin{enumerate}\n\\item[i)] ${\\rm lct}(X,Z)\\geq c$.\n\\item[ii)] ${\\rm codim}\\big({\\rm Cont}^m(Z)\\big)\\geq cm$ for all $m\\geq 0$.\n\\item[iii)] ${\\rm codim}\\big({\\rm Cont}^{\\geq m}(Z)\\big)\\geq cm$ for all $m\\geq 0$.\n\\item[iv)] ${\\rm codim}(C)\\geq c\\cdot {\\rm ord}_C(Z)$ for every locally closed irreducible cylinder $C\\subseteq X_{\\infty}$. \n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm1_main}]\nIt follows from the previous discussion and Theorem~\\ref{thm_old1} that\n\\begin{equation}\\label{eq_char_lct_Z}\n{\\rm lct}(X,Z_A)\\geq c\\quad\\text{if and only if}\\quad {\\rm codim}\\big(C_A(\\lambda)\\big)\\geq c|\\lambda|\\,\\,\\text{for all}\\,\\,\\lambda.\n\\end{equation}\nWe next need to translate the condition ${\\rm lct}(Y,W_A)\\geq c'$.\n\nNote that we have a canonical isomorphism $Y_{\\infty}\\simeq X_{\\infty}\\times {\\mathbf P}^{r-1}_{\\infty}$. \nSince we have the decomposition \n$${\\rm Cont}^m(W_A)\\smallsetminus \\big((Z_A)_{\\infty}\\times {\\mathbf P}^{r-1}_{\\infty}\\big)=\\bigsqcup_{\\lambda}{\\rm Cont}^m(W_A)_{\\lambda},$$\nwhere\n$${\\rm Cont}^m(W_A)_{\\lambda}=\n {\\rm Cont}^m(W_A)\\cap \\big(C_A(\\lambda)\\times {\\mathbf P}^{r-1}_{\\infty}\\big),$$\n it follows from \\cite[Proposition~1.10]{ELM} that\n $${\\rm codim}\\big({\\rm Cont}^m(W_A)\\big)=\\min_{\\lambda}{\\rm codim}\\big({\\rm Cont}^m(W_A)_{\\lambda}\\big).$$\nWe conclude using Theorem~\\ref{thm_old1} that\n \\begin{equation}\\label{eq_char_lct_W1}\n {\\rm lct}(Y,W_A)\\geq c'\\quad\\text{if and only if}\\quad {\\rm codim}\\big({\\rm Cont}^m(W_A)_{\\lambda}\\big)\\geq c'm\\,\\,\\text{for all}\\,\\,m\\,\\,\\text{and all}\\,\\,\\lambda.\n \\end{equation}\n\nTherefore, in order to compute ${\\rm codim}\\big({\\rm Cont}^{\\geq m}(W_A)_{\\gamma}\\big)$, we may and will assume that\n$\\gamma^*(A)={\\rm diag}(t^{\\lambda_1},\\ldots,t^{\\lambda_r})$. \nLet us consider ${\\rm Cont}^{\\geq m}(W_A)_{\\gamma}\\cap (U_i)_{\\infty}$. This consists of those\n$u=(u_1,\\ldots,1,\\ldots,u_r)\\in \\big(\\C\\llbracket t\\rrbracket\\big)^{r-1}$ \n(with $u_j=1$ for $j=i$) such that \n${\\rm ord}(t^{\\lambda_j}u_j)\\geq m$ for all $j$. This is equivalent to $\\lambda_i\\geq m$ and ${\\rm ord}(u_k)\\geq m-\\lambda_k$ for $1\\leq k\\leq i-1$.\nWe see that this is nonempty if and only if $\\lambda_i\\geq m$, and in this case ${\\rm Cont}^{\\geq m}(W_A)_{\\gamma}\\cap (U_i)_{\\infty}$ is\nan irreducible closed cylinder in $(U_i)_{\\infty}$, of codimension\n$\\sum_{j;\\lambda_jr\\geq 2$.\nHowever, for square matrices, we get\n\n\\begin{cor}\\label{cor}\nIf $r=s$, then ${\\rm lct}(X,Z_A)=1$ if and only ${\\rm lct}(Y,W_A)=r$.\n\\end{cor}\n\nFor square matrices, we can go further and relate the property of having rational singularities for $Z_A$ and $W_A$:\n\n\\begin{thm}\\label{thm2_main}\nIf $r=s$, then $Z_A$ has rational singularities if and only if $W_A$ is a local complete intersection of pure codimension $r$ with rational singularities.\n\\end{thm}\n\nFor example, in the case of the generic determinantal hypersurface, \n$W_A$ is smooth, of codimension $r$ in $X\\times {\\mathbf P}^{r-1}$, and we recover Kempf's result \\cite{Kempf} that $Z_A$ has rational singularities. \nIn fact, it is not hard to see that the ``if\" implication in Theorem~\\ref{thm2_main} can be deduced via Kempf's approach in \\emph{loc. cit}., at least if we know \\emph{a priori} that \n$Z_A$ is reduced (see\nRemark~\\ref{rmk:KempfDirectImage}).\n\nA more interesting example occurs for configuration hypersurfaces, in the sense of \\cite{BEK} (we recall the definition in Section~\\ref{application} below). In this case, \nthe equation $f={\\rm det}(A)$ is a square-free polynomial. Moreover, if the matroid corresponding to the hypersurface is connected, then $f$ is irreducible, and therefore\nthe hypersurface $Z_A$ has rational singularities by \\cite[Theorem~1.1]{BMW}, and therefore $W_A$ has rational singularities by Theorem~\\ref{thm2_main}. This will be revisited in \\cite{BDSW} in arbitrary characteristic, see Remark \\ref{rmk:charp}.\n\nThe proofs of Theorems~\\ref{thm1_main} and \\ref{thm2_main} make use of the description of local complete intersection rational singularities and of the log canonical threshold via the codimension \nof certain contact loci in the space of arcs of the ambient variety (see \\cite{Mustata1} and \\cite{Mustata2}, and also \\cite{ELM}). The space of arcs $T_{\\infty}$ of a smooth\nvariety $T$ parametrizes morphisms ${\\rm Spec}\\,{\\mathbf C}[\\negthinspace[ t]\\negthinspace]\\to T$. Given a proper closed subscheme $\\Gamma$ of $T$ defined by the ideal $I_{\\Gamma}$, the log canonical threshold of $(T,\\Gamma)$ can be described \nin terms of the codimensions of the contact loci\n$${\\rm Cont}^{m}(\\Gamma):=\\big\\{\\gamma\\in T_{\\infty}\\mid {\\rm ord}_t\\gamma^{-1}(I_T)=m\\big\\}.$$\nSimilarly, if $\\Gamma$ is a local complete intersection in $T$, then one can describe when $\\Gamma$ has rational singularities in terms of the contact loci. \nIn our setting, given an arc $\\gamma\\in X_{\\infty}\\smallsetminus (Z_A)_{\\infty}$, we can consider the pull-back $\\gamma^*(A)$ of $A$ via $\\gamma$, which is an $s\\times r$ matrix with \nentries in ${\\mathbf C}[\\negthinspace[ t]\\negthinspace]$, which has a nonzero $r$-minor. \nBy the structure theorem for matrices over PIDs, we see that up to the left action of ${\\rm GL}_s\\big({\\mathbf C}[\\negthinspace[ t]\\negthinspace]\\big)$\nand the right action of ${\\rm GL}_r\\big({\\mathbf C}[\\negthinspace[ t]\\negthinspace]\\big)$, the matrix $\\gamma^*(A)$ can be written as a matrix $(c_{ij})$, with $c_{ij}=0$ for $i\\neq j$ and\n$c_{ii}=t^{\\lambda_i}$ for $1\\leq i\\leq r$,\nfor\nsome $\\lambda=(\\lambda_1,\\ldots,\\lambda_r)$, with $\\lambda_1\\leq\\ldots\\leq\\lambda_r$. The key point is to consider the stratification of $X_{\\infty}\\smallsetminus (Z_A)_{\\infty}$\nby the loci $C_A(\\lambda)$ corresponding to a given $\\lambda$. We have $C_A(\\lambda)\\subseteq {\\rm Cont}^{|\\lambda|}(Z_A)$, where $|\\lambda|=\\sum_i\\lambda_i$, and\ninside each $C_A(\\lambda)\\times {\\mathbf P}^{r-1}_{\\infty}\\subseteq Y_{\\infty}$, one can easily describe \nthe codimension of the set of arcs with order $m$ along $W_A$. We note that such a stratification of the space of arcs by the type of the pull-back of a matrix was first considered by Docampo in \n\\cite{Docampo} in order to describe the contact loci of generic determinantal varieties, and it was later used by Zhu in \\cite{Zhu} in order to give a jet-theoretic proof of Kempf's result \\cite{Kempf}\nsaying that the theta divisor on the Jacobian of a smooth projective curve has rational singularities. \n\nThe paper is organized as follows. In the next section, we recall the connection between singularities of pairs and contact loci in arc spaces, following \\cite{ELM}. In\nSection~\\ref{section_main} we give the proofs of Theorems~\\ref{thm1_main} and \\ref{thm2_main}, and in Section~\\ref{application} we discuss the application to \nconfiguration hypersurfaces.\n\n\\subsection*{\\bf Acknowledgment} We are indebted to Uli Walther for many discussions related to the topic of this note.", "sketch": "The proofs of Theorems~\\ref{thm1_main} and \\ref{thm2_main} use “the description of local complete intersection rational singularities and of the log canonical threshold via the codimension of certain contact loci in the space of arcs of the ambient variety.” Concretely, for a smooth variety $T$ and a closed subscheme $\\Gamma\\subset T$, one describes ${\\rm lct}(T,\\Gamma)$ in terms of codimensions of the contact loci\n$$ {\\rm Cont}^{m}(\\Gamma):=\\{\\gamma\\in T_{\\infty}\\mid {\\rm ord}_t\\gamma^{-1}(I_T)=m\\}. $$\nSimilarly, when $\\Gamma$ is a local complete intersection, one can characterize “when $\\Gamma$ has rational singularities in terms of the contact loci.”\n\nIn the determinantal setting, given an arc $\\gamma\\in X_{\\infty}\\smallsetminus (Z_A)_{\\infty}$, one studies the pull-back matrix $\\gamma^*(A)$ over ${\\mathbf C}[\\negthinspace[ t]\\negthinspace]$. Using “the structure theorem for matrices over PIDs,” up to left/right actions of ${\\rm GL}_s({\\mathbf C}[\\negthinspace[ t]\\negthinspace])$ and ${\\rm GL}_r({\\mathbf C}[\\negthinspace[ t]\\negthinspace])$, one can diagonalize to $c_{ii}=t^{\\lambda_i}$ with $\\lambda_1\\le\\cdots\\le\\lambda_r$. The “key point” is a stratification of $X_{\\infty}\\smallsetminus (Z_A)_{\\infty}$ by loci $C_A(\\lambda)$ of fixed type $\\lambda$, with $C_A(\\lambda)\\subseteq {\\rm Cont}^{|\\lambda|}(Z_A)$ for $|\\lambda|=\\sum_i\\lambda_i$. Then, inside each $C_A(\\lambda)\\times {\\mathbf P}^{r-1}_{\\infty}\\subseteq Y_{\\infty}$, one “can easily describe the codimension of the set of arcs with order $m$ along $W_A$,” which is what is used to compare thresholds (Theorem~\\ref{thm1_main}) and rational singularities (Theorem~\\ref{thm2_main}).", "expanded_sketch": "The proofs of the main theorem and \\begin{thm}\\label{thm2_main}\nIf $r=s$, then $Z_A$ has rational singularities if and only if $W_A$ is a local complete intersection of pure codimension $r$ with rational singularities.\n\\end{thm}\nuse “the description of local complete intersection rational singularities and of the log canonical threshold via the codimension of certain contact loci in the space of arcs of the ambient variety.” Concretely, for a smooth variety $T$ and a closed subscheme $\\Gamma\\subset T$, one describes ${\\rm lct}(T,\\Gamma)$ in terms of codimensions of the contact loci\n$$ {\\rm Cont}^{m}(\\Gamma):=\\{\\gamma\\in T_{\\infty}\\mid {\\rm ord}_t\\gamma^{-1}(I_T)=m\\}. $$\nSimilarly, when $\\Gamma$ is a local complete intersection, one can characterize “when $\\Gamma$ has rational singularities in terms of the contact loci.”\n\nIn the determinantal setting, given an arc $\\gamma\\in X_{\\infty}\\smallsetminus (Z_A)_{\\infty}$, one studies the pull-back matrix $\\gamma^*(A)$ over ${\\mathbf C}[\\negthinspace[ t]\\negthinspace]$. Using “the structure theorem for matrices over PIDs,” up to left/right actions of ${\\rm GL}_s({\\mathbf C}[\\negthinspace[ t]\\negthinspace])$ and ${\\rm GL}_r({\\mathbf C}[\\negthinspace[ t]\\negthinspace])$, one can diagonalize to $c_{ii}=t^{\\lambda_i}$ with $\\lambda_1\\le\\cdots\\le\\lambda_r$. The “key point” is a stratification of $X_{\\infty}\\smallsetminus (Z_A)_{\\infty}$ by loci $C_A(\\lambda)$ of fixed type $\\lambda$, with $C_A(\\lambda)\\subseteq {\\rm Cont}^{|\\lambda|}(Z_A)$ for $|\\lambda|=\\sum_i\\lambda_i$. Then, inside each $C_A(\\lambda)\\times {\\mathbf P}^{r-1}_{\\infty}\\subseteq Y_{\\infty}$, one “can easily describe the codimension of the set of arcs with order $m$ along $W_A$,” which is what is used to compare thresholds (in establishing the main theorem) and rational singularities (the theorem above).", "expanded_theorem": "\\label{thm1_main}\nWith the above notation, the following hold:\n\\begin{enumerate}\n\\item[i)] If ${\\rm lct}(X,Z_A)\\geq c$, then ${\\rm lct}(Y,W_A)\\geq \\min\\{rc,r-1+c\\}$.\n\\item[ii)] If ${\\rm lct}(Y,W_A)\\geq r-1+c'$, then ${\\rm lct}(X,Z_A)\\geq \\min\\big\\{c',\\tfrac{c'-1}{r}+1\\big\\}$.\n\\end{enumerate}", "theorem_type": ["Implication", "Inequality or Bound"], "mcq": {"question": "Let $X$ be a smooth, irreducible complex algebraic variety, let $A=(a_{ij})_{1\\le i\\le s,\\,1\\le j\\le r}$ with $s\\ge r$ and $a_{ij}\\in \\mathcal O_X(X)$, and let $Z_A\\subsetneq X$ be the closed subscheme defined by the ideal generated by the $r\\times r$ minors of $A$. Set $Y=X\\times \\mathbf P^{r-1}$, and let $W_A\\hookrightarrow Y$ be the closed subscheme cut out by the $s$ equations $\\sum_{j=1}^r a_{ij}y_j=0$ for $1\\le i\\le s$, where $[y_1:\\cdots:y_r]$ are homogeneous coordinates on $\\mathbf P^{r-1}$. Denote by ${\\rm lct}(T,\\Gamma)$ the log canonical threshold of a closed subscheme $\\Gamma$ of a smooth variety $T$. Under this setup, which quantitative implication between ${\\rm lct}(X,Z_A)$ and ${\\rm lct}(Y,W_A)$ holds?", "correct_choice": {"label": "A", "text": "For any real numbers $c$ and $c'$, if ${\\rm lct}(X,Z_A)\\ge c$, then ${\\rm lct}(Y,W_A)\\ge \\min\\{rc,\\,r-1+c\\}$; and if ${\\rm lct}(Y,W_A)\\ge r-1+c'$, then ${\\rm lct}(X,Z_A)\\ge \\min\\!\\left\\{c',\\,\\tfrac{c'-1}{r}+1\\right\\}$."}, "choices": [{"label": "B", "text": "For any real numbers $c$ and $c'$, if ${\\rm lct}(X,Z_A)\\ge c$, then ${\\rm lct}(Y,W_A)\\ge \\min\\{rc,\\,r+c\\}$; and if ${\\rm lct}(Y,W_A)\\ge r+c'$, then ${\\rm lct}(X,Z_A)\\ge \\min\\!\\left\\{c',\\,\\tfrac{c'-1}{r}+1\\right\\}$."}, {"label": "C", "text": "If ${\\rm lct}(X,Z_A)\\ge c$, then ${\\rm lct}(Y,W_A)\\ge rc$."}, {"label": "D", "text": "For any real numbers $c$ and $c'$, if ${\\rm lct}(X,Z_A)\\ge c$, then ${\\rm lct}(Y,W_A)\\ge \\max\\{rc,\\,r-1+c\\}$; and if ${\\rm lct}(Y,W_A)\\ge r-1+c'$, then ${\\rm lct}(X,Z_A)\\ge \\max\\!\\left\\{c',\\,\\tfrac{c'-1}{r}+1\\right\\}$."}, {"label": "E", "text": "For any real numbers $c$ and $c'$, if ${\\rm lct}(X,Z_A)\\ge c$, then ${\\rm lct}(Y,W_A)\\ge \\min\\{rc,\\,r-1+c\\}$; and if ${\\rm lct}(Y,W_A)\\ge c'$, then ${\\rm lct}(X,Z_A)\\ge \\min\\!\\left\\{c',\\,\\tfrac{c'-1}{r}+1\\right\\}$."}], "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": "critical shift by r-1 in the threshold comparison", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "counting_estimate", "tampered_component": "dropped the second branch $r-1+c$ from the lower bound", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "counting_estimate", "tampered_component": "piecewise minimum coming from the two cases $c\\le 1$ and $c\\ge 1$ replaced by a stronger maximum", "template_used": "wildcard"}, {"label": "E", "sketch_hook_type": "counting_estimate", "tampered_component": "quantifier threshold on ${\\rm lct}(Y,W_A)$ changed from $r-1+c'$ to $c'$", "template_used": "quantifier_dependence"}]}} +{"id": "2601.22303v1", "paper_link": "http://arxiv.org/abs/2601.22303v1", "theorems_cnt": 6, "theorem": {"env_name": "theorem", "content": "\\label{theorem:intro-A}\n\tThere is an $E_*^A$-algebra isomorphism\n\t\\[\n\t\tE_*^A(\\MUP_A)\\cong \\operatorname{Sym}(R_E[-2])[\\vartheta_\\alpha^{-1}\\mid \\alpha\\in A^*].\n\t\\]", "start_pos": 8787, "end_pos": 8965, "label": "theorem:intro-A"}, "ref_dict": {"theorem:intro-B": "\\begin{theorem}\\label{theorem:intro-B}\n\tThere are $E_*^A$-algebra isomorphisms\n\t\\begin{align*}\n\t\tE_*^A(\\MUP_A)&\\cong \\Sym(R_E[2])[\\vartheta_{\\alpha}^{-1}\\mid \\alpha\\in A^*],\\\\\n\t\tE_*^A(\\mUP_A)&\\cong \\Sym(R_E[2])[\\vartheta_\\eps^{-1}].\n\t\\end{align*}\n\\end{theorem}", "theorem:B": "\\begin{theorem}\\label{theorem:B}\n\tThe ring map\n\t\\begin{align*}\n\t\t& \\Sym(R_E[2])\\xrightarrow[\\cong]{C} E_*^A(\\MGr_A)\\xrightarrow{i} E_*^A(\\mUP_A)\\xrightarrow{j} E_*^A(\\MUP_A)\n\t\\end{align*}\n\tinduces equivalences of $E_*^A$-algebras\n\t\\begin{align*}\n\t\tE_*^A(\\MUP_A)&\\cong \\Sym(R_E[2])[\\vartheta_{\\alpha}^{-1}\\mid \\alpha\\in A^*],\\\\\n\t\tE_*^A(\\mUP_A)&\\cong \\Sym(R_E[2])[\\vartheta_\\eps^{-1}].\n\t\\end{align*}\n\\end{theorem}", "corollary:B-coordinatized": "\\begin{corollary}\\label{corollary:B-coordinatized}\n\tGiven a flag $\\cF = \\{V_i\\}_i$ of a complete universe, the associated basis $\\{\\beta_i^\\cF\\}_i$ of the $E_*^A$-algebra $R_E = E_*^A(\\bC P(\\cU_A)_+)$ gives rise to equivalences\n\t\\begin{align*}\n\t\tE_*^A(\\MUP_A)&\\cong E_*^A[C(\\beta_0^\\cF),C(\\beta_1^\\cF),C(\\beta_2^\\cF),\\cdots][C(\\vartheta_\\alpha)^{-1}\\mid \\alpha\\in A^*],\\\\\n\t\tE_*^A(\\mUP_A)&\\cong E_*^A[C(\\beta_0^\\cF),C(\\beta_1^\\cF),C(\\beta_2^\\cF),\\cdots][C(\\vartheta_\\epsilon)^{-1}].\n\t\\end{align*}\n\tFor $\\alpha = V_1$, write $c(\\beta_i^\\cF) \\coloneqq C(\\vartheta_{\\alpha})^{-1}C(\\beta_i^\\cF)$ and $c(\\vartheta_{\\alpha'}) = C(\\vartheta_{\\alpha})^{-1}C(\\vartheta_{\\alpha'})$.\n\tThese new classes live in dimension-grading 0 and there is an isomorphism\n\t\\[\n\t\tE_*^A(\\MU_A)\\cong E_*^A[c(\\beta_1^\\cF),c(\\beta_2^\\cF),c(\\beta_3^\\cF),\\cdots ][c(\\vartheta_{\\alpha'})^{-1}\\mid \\alpha'\\in A^*].\n\t\\]\n\tIf $V_1 = \\eps$, there is an equivalence\n\t\\[\n\t\tE_*^A(\\mU_A)\\cong E_*^A[c(\\beta_1^\\cF),c(\\beta_2^\\cF),c(\\beta_3^\\cF),\\cdots ].\n\t\\]\n\\end{corollary}", "appendix": "\\label{appendix}\nThe \\emph{canonical (unitary) homotopy presentation} gives a more homotopy theoretic description of the underlying genuine $G$-spectrum of a unitary $G$-spectrum.\nWe adapt arguments f", "corollary:A-coordinatized": "\\begin{corollary}\\label{corollary:A-coordinatized}\n\tGiven a flag $\\cF = \\{V_i\\}_i$ of a complete universe, the associated $E_*^A$-basis $\\{\\beta_i^\\cF\\}_i$ of the algebra $R_E = E_*^A(\\bC P(\\cU_A)_+)$ gives rise to an equivalence \n\t\\[\n\t\tE_*^A(\\MUP_A)\\cong E_*^A[B(\\beta_0^\\cF), B(\\beta_1^\\cF),B(\\beta_2^\\cF),\\cdots ][B(\\vartheta_\\alpha)^{-1}\\mid \\alpha\\in A^*].\n\t\\]\n\tLet $V_1 = \\alpha$.\n\tFor $b(\\beta_i^\\cF) = B(\\vartheta_\\alpha)^{-1}B(\\beta_i^\\cF)$, there is an isomorphism of rings\n\t\\[\n\tE_*^A(\\MU_A)\\cong E_*^A[b(\\beta_1^\\cF),b(\\beta_2^\\cF),b(\\beta_3^\\cF)\\cdots ][b(\\vartheta_{\\alpha'})^{-1}\\mid \\alpha'\\in A^*].\n\t\\]\n\\end{corollary}", "theorem:A": "\\begin{theorem}\\label{theorem:A}\n\tThe ring homomorphism \n\t\\[\n\t\t\\Sym(R_E[-2])\\xrightarrow[\\cong]{B} E_*^A(\\MRep_A)\\xrightarrow{\\iota} E_*^A(\\MUP_A) \n\t\\]\n\tinduces an equivalence of $E_*^A$-algebras\n\t\\[\n\t\tE_*^A(\\MUP_A)\\cong \\Sym(R_E[-2])[\\vartheta_\\alpha^{-1}\\mid \\alpha\\in A^*].\n\t\\]\n\\end{theorem}", "remark:thom-spectra-map": "\\begin{remark}\\label{remark:thom-spectra-map}\n\tLet us quickly summarize how the spectra above correspond.\n\t\\[\n\t\t\\begin{tikzcd}\n\t\t{\\M(-n)^\\bfU_G} & \\MGr_G^\\bfU & \\mUP_G^\\bfU & \\mU_G^\\bfU \\\\\n\t\t{\\M(n)_G^\\bfU} & {\\MRep_G^\\bfU} & \\MUP_G^\\bfU & \\MU_G^\\bfU\n\t\t\\arrow[\"\\incl\", from=1-1, to=1-2]\n\t\t\\arrow[\"i\", from=1-2, to=1-3]\n\t\t\\arrow[\"j(k)\"', from=1-3, to=2-3]\n\t\t\\arrow[\"\\incl\"', from=1-4, to=1-3]\n\t\t\\arrow[\"\\incl\", from=2-1, to=2-2]\n\t\t\\arrow[\"{\\overline i}\", from=2-2, to=2-3]\n\t\t\\arrow[\"\\incl\"', from=2-4, to=2-3]\n\t\t\\end{tikzcd}\n\t\\]\n\tThe tautological bundle over the $G$-classifying space $\\Gr_n(\\cU_A)$ gives rise to the Thom spectrum $\\M(n)^\\bfU_G$.\n\tTaking the negative (virtual) bundle yields the Thom spectrum $\\M(-n)^\\bfU_G$.\n\tSumming over $n$ yield the (negative) tautological bundle on $\\Rep_G\\coloneqq \\coprod_{k\\geq 0} \\Gr_n(\\cU_A)\\simeq \\coprod_{k\\geq 0} \\mathrm{B}\\mathrm{U}(n)_G$, leading to the Thom spectra $\\MRep_G^\\bfU$ and $\\MGr_G^\\bfU$, respectively.\n\tThe bundle sum equips $\\Rep_G$ with the structure of a $G$-monoid object.\n\tAn element in $\\pi^G_0(\\Rep_G)$ corresponds to an isotopy class of $G$-representation.\n\t(Sequentially) inverting either only the trivial representation, or all representations, yields $G$-monoid spaces $\\bUP_G$ and $\\BUP_G$. \n\tThe two spectra $\\mUP^{\\bfU}_G$ and $\\MUP_G^\\bfU$ are Thom spectra over these spaces, respectively.\n\tFinally, we note that these two base spaces come with a $\\bZ$-grading, and the Thom spectra over the $0$-th component are $\\mU_G^\\bfU$ and $\\MU_G^\\bfU$.\n\tThis can be explained by the fact that the monoid $\\Rep_G$ is itself $\\bN$-graded.\n\tFinally, we will see that upon passing to underlying genuine $G$-spectra, $j(k)$ will not depend on the choice of $k$. \n\\end{remark}", "theorem:intro-A": "\\begin{theorem}\\label{theorem:intro-A}\n\tThere is an $E_*^A$-algebra isomorphism\n\t\\[\n\t\tE_*^A(\\MUP_A)\\cong \\Sym(R_E[-2])[\\vartheta_\\alpha^{-1}\\mid \\alpha\\in A^*].\n\t\\]\n\\end{theorem}"}, "pre_theorem_intro_text_len": 4377, "pre_theorem_intro_text": "The bordism spectrum $\\mathrm{MU}$ is at the center of many active areas of research.\nIt is the initial orientable spectrum with the universal formal group law \\cite{Q69}.\nThe study of orientable spectra and their formal group laws has led to proofs of celebrated results, such as the classification of thick subcategories of the category of $p$-local finite spectra \\cite{HS98}.\nThis article is part of an effort to extend these methods to the equivariant world.\nFor a compact Lie group $G$ the \\emph{homotopical bordism $G$-spectrum} $\\MU_G$ is a genuine $G$-spectrum introduced as an equivariant cohomology theory by tom Dieck \\cite{D70}.\nEarly advances in the equivariant setting include work by Okonek \\cite{O82}, who showed that the cohomology theory $\\MU_G^*$ is the initial cohomology theory with Thom classes for $G$-vector bundles.\nMore recently, Cole, Greenlees and Kriz \\cite{CGK00} discuss equivariant orientations and define equivariant formal group laws for \\emph{abelian} compact Lie groups.\nFor an abelian compact Lie group $A$, Hausmann \\cite{H22} proved that the orientation of $\\MU_A$ induces an isomorphism from $\\pi_*^A(\\MU_A)$ to the \\emph{equivariant Lazard ring}, the representing ring for equivariant formal group laws.\n\nThe article \\cite{CGK02} contains an inaccuracy which leads to a mistake in \\cite[Thm. 8.2.]{CGK02}, the calculation of the $E$-homology of $\\MU_A$ for an orientable $A$-spectrum $E$.\nThe present article corrects this.\nBy obtaining two colimit formulae for the $A$-spectrum $\\MU_A$, we derive its homology in two ways, in \\cref{theorem:intro-A} and \\cref{theorem:intro-B}.\nWe further calculate the homology of the \\emph{geometric bordism $A$-spectrum} $\\mU_A$, representing bordism of $A$-manifolds with a normally almost complex structure, see \\cite{Com96}.\n\n\\subsection{Orientations of $A$-Spectra}\nLet $A$ be an abelian compact Lie group. \nThere is a notion of \\emph{orientation} for $A$-spectra due to \\cite{C96}.\nWe let $\\cU_A$ be a (complex) complete $A$-universe, a unitary $A$-representation in which any other countably infnite-dimensional $A$-representation may be embedded.\nFor $E$ a commutative $A$-ring spectrum, an \\emph{orientation} of $E$ is a class $x(\\epsilon)\\in E^2_A(\\mathbb{C} P (\\cU_A),\\mathbb{C} P(\\epsilon))$, where $\\epsilon$ is the trivial representation on $\\mathbb{C}$, such that:\n\\begin{enumerate}[label = (\\roman*)]\n\t\\item\n\t\tFor any character $\\alpha$ of $A$, we view $\\alpha$ and $\\alpha^{-1}$ as one-dimensional $A$-representations, and pulling back $x(\\epsilon)$ along an embedding $i \\colon \\mathbb{C} P(\\alpha\\oplus \\epsilon)\\to \\mathbb{C} P(\\cU_A)$ yields an $\\operatorname{RO}(A)$-graded unit in $E^2_A(S^{\\alpha^{-1}})$: \n\t\t\\[\n\t\t\ti^*\\colon E^2_A(\\mathbb{C} P(\\cU_A),\\mathbb{C} P(\\epsilon))\\to E^2_A(\\mathbb{C} P(\\alpha\\oplus \\epsilon), \\mathbb{C} P(\\epsilon))\\cong E^2_A(S^{\\alpha^{-1}}).\n\t\t\\]\n\t\\item\n\t\tFor $\\alpha = \\epsilon$, the class $x(\\epsilon)$ restricts to $\\Sigma^2 1$ in $E^2_A(S^{\\epsilon^{-1}}) = E^2_A(S^2)$.\n\\end{enumerate}\n\n\\subsection{Calculation of Homology}\nLet $E$ be an oriented $A$-spectrum.\nWe will write $R_E\\coloneqq E_*^A(\\mathbb{C} P(\\cU_A)_+)$.\nThe tautological bundle over $\\coprod_{n\\geq 0} \\Gr_n(\\cU_A)$ and its (virtual) negative yield the Thom spectra $\\MRep_A$ and $\\MGr_A$, respectively.\nIn \\cref{section:4} and \\cref{section:5} we define isomorphisms\n\\[\n\tB\\colon \\operatorname{Sym}(R_E[-2])\\xrightarrow{\\cong} E_*^A(\\MRep_A), \\quad C\\colon \\operatorname{Sym}(R_E[2])\\xrightarrow{\\cong} E_*^A(\\MGr_A),\n\\]\nwhere the brackets $[-]$ refer to shifts of graded $E_*^A$-modules.\nWe study $A$-spectra $\\MUP_A$ and $\\mUP_A$, periodic versions of $\\MU_A$ and $\\mU_A$.\nThese all admit ring structures, and there are ring maps of the form $\\MU_A\\to \\MUP_A$ and $\\mU_A\\to \\mUP_A$ which are summand inclusions on underlying $A$-spectra.\nFor an overview of the various Thom spectra and their relations, see \\cref{remark:thom-spectra-map}.\nViewing a character $\\alpha\\in A^*$ of $A$ as a one-dimensional representation, we obtain an inclusion $\\alpha\\to \\cU_A$, which induces a map $E_*^A(\\mathbb{C} P(\\alpha)_+)\\to R_E$.\nWe refer to the image of 1 under this map as the \\emph{$\\alpha$-coaugmentation class} $\\vartheta_\\alpha\\in R_E$.\nWith this in mind, we show that the map $B$ and a natural morphism of $A$-spectra $ \\MRep_A\\to \\MUP_A$ lead to the following result.", "context": "The bordism spectrum $\\mathrm{MU}$ is at the center of many active areas of research.\nIt is the initial orientable spectrum with the universal formal group law \\cite{Q69}.\nThe study of orientable spectra and their formal group laws has led to proofs of celebrated results, such as the classification of thick subcategories of the category of $p$-local finite spectra \\cite{HS98}.\nThis article is part of an effort to extend these methods to the equivariant world.\nFor a compact Lie group $G$ the \\emph{homotopical bordism $G$-spectrum} $\\MU_G$ is a genuine $G$-spectrum introduced as an equivariant cohomology theory by tom Dieck \\cite{D70}.\nEarly advances in the equivariant setting include work by Okonek \\cite{O82}, who showed that the cohomology theory $\\MU_G^*$ is the initial cohomology theory with Thom classes for $G$-vector bundles.\nMore recently, Cole, Greenlees and Kriz \\cite{CGK00} discuss equivariant orientations and define equivariant formal group laws for \\emph{abelian} compact Lie groups.\nFor an abelian compact Lie group $A$, Hausmann \\cite{H22} proved that the orientation of $\\MU_A$ induces an isomorphism from $\\pi_*^A(\\MU_A)$ to the \\emph{equivariant Lazard ring}, the representing ring for equivariant formal group laws.\n\nThe article \\cite{CGK02} contains an inaccuracy which leads to a mistake in \\cite[Thm. 8.2.]{CGK02}, the calculation of the $E$-homology of $\\MU_A$ for an orientable $A$-spectrum $E$.\nThe present article corrects this.\nBy obtaining two colimit formulae for the $A$-spectrum $\\MU_A$, we derive its homology in two ways, in \\cref{theorem:intro-A} and \\cref{theorem:intro-B}.\nWe further calculate the homology of the \\emph{geometric bordism $A$-spectrum} $\\mU_A$, representing bordism of $A$-manifolds with a normally almost complex structure, see \\cite{Com96}.\n\n\\subsection{Orientations of $A$-Spectra}\nLet $A$ be an abelian compact Lie group. \nThere is a notion of \\emph{orientation} for $A$-spectra due to \\cite{C96}.\nWe let $\\cU_A$ be a (complex) complete $A$-universe, a unitary $A$-representation in which any other countably infnite-dimensional $A$-representation may be embedded.\nFor $E$ a commutative $A$-ring spectrum, an \\emph{orientation} of $E$ is a class $x(\\epsilon)\\in E^2_A(\\mathbb{C} P (\\cU_A),\\mathbb{C} P(\\epsilon))$, where $\\epsilon$ is the trivial representation on $\\mathbb{C}$, such that:\n\\begin{enumerate}[label = (\\roman*)]\n \\item\n For any character $\\alpha$ of $A$, we view $\\alpha$ and $\\alpha^{-1}$ as one-dimensional $A$-representations, and pulling back $x(\\epsilon)$ along an embedding $i \\colon \\mathbb{C} P(\\alpha\\oplus \\epsilon)\\to \\mathbb{C} P(\\cU_A)$ yields an $\\operatorname{RO}(A)$-graded unit in $E^2_A(S^{\\alpha^{-1}})$: \n \\[\n i^*\\colon E^2_A(\\mathbb{C} P(\\cU_A),\\mathbb{C} P(\\epsilon))\\to E^2_A(\\mathbb{C} P(\\alpha\\oplus \\epsilon), \\mathbb{C} P(\\epsilon))\\cong E^2_A(S^{\\alpha^{-1}}).\n \\]\n \\item\n For $\\alpha = \\epsilon$, the class $x(\\epsilon)$ restricts to $\\Sigma^2 1$ in $E^2_A(S^{\\epsilon^{-1}}) = E^2_A(S^2)$.\n\\end{enumerate}\n\n\\subsection{Calculation of Homology}\nLet $E$ be an oriented $A$-spectrum.\nWe will write $R_E\\coloneqq E_*^A(\\mathbb{C} P(\\cU_A)_+)$.\nThe tautological bundle over $\\coprod_{n\\geq 0} \\Gr_n(\\cU_A)$ and its (virtual) negative yield the Thom spectra $\\MRep_A$ and $\\MGr_A$, respectively.\nIn \\cref{section:4} and \\cref{section:5} we define isomorphisms\n\\[\n B\\colon \\operatorname{Sym}(R_E[-2])\\xrightarrow{\\cong} E_*^A(\\MRep_A), \\quad C\\colon \\operatorname{Sym}(R_E[2])\\xrightarrow{\\cong} E_*^A(\\MGr_A),\n\\]\nwhere the brackets $[-]$ refer to shifts of graded $E_*^A$-modules.\nWe study $A$-spectra $\\MUP_A$ and $\\mUP_A$, periodic versions of $\\MU_A$ and $\\mU_A$.\nThese all admit ring structures, and there are ring maps of the form $\\MU_A\\to \\MUP_A$ and $\\mU_A\\to \\mUP_A$ which are summand inclusions on underlying $A$-spectra.\nFor an overview of the various Thom spectra and their relations, see \\cref{remark:thom-spectra-map}.\nViewing a character $\\alpha\\in A^*$ of $A$ as a one-dimensional representation, we obtain an inclusion $\\alpha\\to \\cU_A$, which induces a map $E_*^A(\\mathbb{C} P(\\alpha)_+)\\to R_E$.\nWe refer to the image of 1 under this map as the \\emph{$\\alpha$-coaugmentation class} $\\vartheta_\\alpha\\in R_E$.\nWith this in mind, we show that the map $B$ and a natural morphism of $A$-spectra $ \\MRep_A\\to \\MUP_A$ lead to the following result.", "full_context": "The bordism spectrum $\\mathrm{MU}$ is at the center of many active areas of research.\nIt is the initial orientable spectrum with the universal formal group law \\cite{Q69}.\nThe study of orientable spectra and their formal group laws has led to proofs of celebrated results, such as the classification of thick subcategories of the category of $p$-local finite spectra \\cite{HS98}.\nThis article is part of an effort to extend these methods to the equivariant world.\nFor a compact Lie group $G$ the \\emph{homotopical bordism $G$-spectrum} $\\MU_G$ is a genuine $G$-spectrum introduced as an equivariant cohomology theory by tom Dieck \\cite{D70}.\nEarly advances in the equivariant setting include work by Okonek \\cite{O82}, who showed that the cohomology theory $\\MU_G^*$ is the initial cohomology theory with Thom classes for $G$-vector bundles.\nMore recently, Cole, Greenlees and Kriz \\cite{CGK00} discuss equivariant orientations and define equivariant formal group laws for \\emph{abelian} compact Lie groups.\nFor an abelian compact Lie group $A$, Hausmann \\cite{H22} proved that the orientation of $\\MU_A$ induces an isomorphism from $\\pi_*^A(\\MU_A)$ to the \\emph{equivariant Lazard ring}, the representing ring for equivariant formal group laws.\n\nThe article \\cite{CGK02} contains an inaccuracy which leads to a mistake in \\cite[Thm. 8.2.]{CGK02}, the calculation of the $E$-homology of $\\MU_A$ for an orientable $A$-spectrum $E$.\nThe present article corrects this.\nBy obtaining two colimit formulae for the $A$-spectrum $\\MU_A$, we derive its homology in two ways, in \\cref{theorem:intro-A} and \\cref{theorem:intro-B}.\nWe further calculate the homology of the \\emph{geometric bordism $A$-spectrum} $\\mU_A$, representing bordism of $A$-manifolds with a normally almost complex structure, see \\cite{Com96}.\n\n\\subsection{Orientations of $A$-Spectra}\nLet $A$ be an abelian compact Lie group. \nThere is a notion of \\emph{orientation} for $A$-spectra due to \\cite{C96}.\nWe let $\\cU_A$ be a (complex) complete $A$-universe, a unitary $A$-representation in which any other countably infnite-dimensional $A$-representation may be embedded.\nFor $E$ a commutative $A$-ring spectrum, an \\emph{orientation} of $E$ is a class $x(\\epsilon)\\in E^2_A(\\mathbb{C} P (\\cU_A),\\mathbb{C} P(\\epsilon))$, where $\\epsilon$ is the trivial representation on $\\mathbb{C}$, such that:\n\\begin{enumerate}[label = (\\roman*)]\n \\item\n For any character $\\alpha$ of $A$, we view $\\alpha$ and $\\alpha^{-1}$ as one-dimensional $A$-representations, and pulling back $x(\\epsilon)$ along an embedding $i \\colon \\mathbb{C} P(\\alpha\\oplus \\epsilon)\\to \\mathbb{C} P(\\cU_A)$ yields an $\\operatorname{RO}(A)$-graded unit in $E^2_A(S^{\\alpha^{-1}})$: \n \\[\n i^*\\colon E^2_A(\\mathbb{C} P(\\cU_A),\\mathbb{C} P(\\epsilon))\\to E^2_A(\\mathbb{C} P(\\alpha\\oplus \\epsilon), \\mathbb{C} P(\\epsilon))\\cong E^2_A(S^{\\alpha^{-1}}).\n \\]\n \\item\n For $\\alpha = \\epsilon$, the class $x(\\epsilon)$ restricts to $\\Sigma^2 1$ in $E^2_A(S^{\\epsilon^{-1}}) = E^2_A(S^2)$.\n\\end{enumerate}\n\n\\subsection{Calculation of Homology}\nLet $E$ be an oriented $A$-spectrum.\nWe will write $R_E\\coloneqq E_*^A(\\mathbb{C} P(\\cU_A)_+)$.\nThe tautological bundle over $\\coprod_{n\\geq 0} \\Gr_n(\\cU_A)$ and its (virtual) negative yield the Thom spectra $\\MRep_A$ and $\\MGr_A$, respectively.\nIn \\cref{section:4} and \\cref{section:5} we define isomorphisms\n\\[\n B\\colon \\operatorname{Sym}(R_E[-2])\\xrightarrow{\\cong} E_*^A(\\MRep_A), \\quad C\\colon \\operatorname{Sym}(R_E[2])\\xrightarrow{\\cong} E_*^A(\\MGr_A),\n\\]\nwhere the brackets $[-]$ refer to shifts of graded $E_*^A$-modules.\nWe study $A$-spectra $\\MUP_A$ and $\\mUP_A$, periodic versions of $\\MU_A$ and $\\mU_A$.\nThese all admit ring structures, and there are ring maps of the form $\\MU_A\\to \\MUP_A$ and $\\mU_A\\to \\mUP_A$ which are summand inclusions on underlying $A$-spectra.\nFor an overview of the various Thom spectra and their relations, see \\cref{remark:thom-spectra-map}.\nViewing a character $\\alpha\\in A^*$ of $A$ as a one-dimensional representation, we obtain an inclusion $\\alpha\\to \\cU_A$, which induces a map $E_*^A(\\mathbb{C} P(\\alpha)_+)\\to R_E$.\nWe refer to the image of 1 under this map as the \\emph{$\\alpha$-coaugmentation class} $\\vartheta_\\alpha\\in R_E$.\nWith this in mind, we show that the map $B$ and a natural morphism of $A$-spectra $ \\MRep_A\\to \\MUP_A$ lead to the following result.\n\nThe multiplication on $E_*^A(\\MUP_A)$ is graded for the dimension decomposition.\nAs for any irreducible representation $\\alpha$, the element $B(\\vartheta_\\alpha)$ is invertible, we find that:\n\\begin{lemma}\\label{lemma:periodicity}\n For $\\alpha$ an irreducible representation, multiplication with $B(\\vartheta_\\alpha)$ on $E_*^A(\\MUP_A)$ defines an equivalence \n \\[\n E_*^A(\\MUP_A^{[k]})\\to E_*^A(\\MUP_A^{[k+1]})[-2].\n \\] \n\\end{lemma}\n\nNote that the above can be seen as a periodicity phenomenon for $E\\otimes \\MUP_A$, which in fact can be derived from a periodicity result of $\\MUP_A$ itself, see \\cite[Proposition 6.1.20]{S18} for a related discussion.\nThe following is a restatement and proof of \\cref{corollary:intro-A-coordinatized}.\n\\begin{corollary}\\label{corollary:A-coordinatized}\n Given a flag $\\cF = \\{V_i\\}_i$ of a complete universe, the associated $E_*^A$-basis $\\{\\beta_i^\\cF\\}_i$ of the algebra $R_E = E_*^A(\\bC P(\\cU_A)_+)$ gives rise to an equivalence \n \\[\n E_*^A(\\MUP_A)\\cong E_*^A[B(\\beta_0^\\cF), B(\\beta_1^\\cF),B(\\beta_2^\\cF),\\cdots ][B(\\vartheta_\\alpha)^{-1}\\mid \\alpha\\in A^*].\n \\]\n Let $V_1 = \\alpha$.\n For $b(\\beta_i^\\cF) = B(\\vartheta_\\alpha)^{-1}B(\\beta_i^\\cF)$, there is an isomorphism of rings\n \\[\n E_*^A(\\MU_A)\\cong E_*^A[b(\\beta_1^\\cF),b(\\beta_2^\\cF),b(\\beta_3^\\cF)\\cdots ][b(\\vartheta_{\\alpha'})^{-1}\\mid \\alpha'\\in A^*].\n \\]\n\\end{corollary}\n\\begin{proof}\n The presentation of $E_*^A(\\MUP_A)$ is immediate.\n Continuing, note that $E_*^A(\\MU_A)\\subseteq E_*^A(\\MUP_A)$ is precisely the subring making up the zero summand in the dimension-decomposition.\n A monomial in $E_*^A(\\MUP_A)$ can be uniquely rewritten so as to contain only elements $b_i^\\cF$ and $b(\\vartheta_{\\alpha'})$, by multiplying with $B(\\vartheta_\\alpha)^{-p}$, where $p$ is the dimension-degree of the denominator (or that of the numerator, they are equal.)\n \\[\n \\frac{B(\\beta_{i_1}^\\cF)^{e_1}\\cdots B(\\beta_{i_k}^\\cF)^{e_k}}{B(\\vartheta_{\\alpha_1})^{f_1}\\cdots B(\\vartheta_{\\alpha_l})^{f_l}} = \\frac{B(\\vartheta_\\alpha)^{-p}\\cdot B(\\beta_{i_1}^\\cF)^{e_1}\\cdots B(\\beta_{i_k}^\\cF)^{e_k}}{B(\\vartheta_\\alpha)^{-p}\\cdot B(\\vartheta_{\\alpha_1})^{f_1}\\cdots B(\\vartheta_{\\alpha_l})^{f_l}}= \\frac{(b_{i_1}^\\cF)^{e_1}\\cdots (b_{i_k}^\\cF)^{e_k}}{b(\\vartheta_{\\alpha_1})^{f_1}\\cdots b(\\vartheta_{\\alpha_l})^{f_l}} \n \\]\n Hence, the proposed ring map is indeed an isomorphism.\n\\end{proof}\n\nThe map $\\tilde b$ takes elements in $R_E \\cong E_*^A(\\bC P(\\alpha\\oplus \\cU_A)_+)$ first to $E_*^A(\\Omega^\\alpha \\M(1)^\\alpha_A)\\subseteq E_*^A(\\Omega^\\alpha\\MRep_A^\\alpha)$, and then further to $E_*^A(\\MU_A)\\subseteq E_*^A(\\MUP_A)$ via $(p_\\alpha)_*$.\n Hence, $\\beta_i^\\cF$ is mapped to the class $\\Omega^\\alpha \\cT_{\\gamma_1}(\\beta_i^\\cF)\\in E_*^A(\\Omega^\\alpha \\M(1)^\\alpha_A)$.\n As the colimit maps are $E_*^A(\\MRep_A)$-module maps, we can already multiply by $B(\\vartheta_\\alpha)$ in $E_*^A(\\Omega^\\alpha\\MRep_A^\\alpha)$, which takes $\\Omega^\\alpha \\cT_{\\gamma_1}(\\beta_i^\\cF)$ to $\\Omega^\\alpha \\cT_{\\gamma_2}(\\vartheta_\\alpha\\otimes \\beta_i^\\cF)$.\n We see that already at this stage, it holds that $ B(\\beta_i^\\cF) = B(\\vartheta_\\alpha)\\tilde b(\\beta_i^\\cF) $.\n Finally, we conclude that $\\tilde b(\\beta_i^\\cF) = b(\\beta_i^\\cF)$.\n\\end{remark}\n\nWe restate and prove \\cref{theorem:intro-B}.\n\\begin{theorem}\\label{theorem:B}\n The ring map\n \\begin{align*}\n & \\Sym(R_E[2])\\xrightarrow[\\cong]{C} E_*^A(\\MGr_A)\\xrightarrow{i} E_*^A(\\mUP_A)\\xrightarrow{j} E_*^A(\\MUP_A)\n \\end{align*}\n induces equivalences of $E_*^A$-algebras\n \\begin{align*}\n E_*^A(\\MUP_A)&\\cong \\Sym(R_E[2])[\\vartheta_{\\alpha}^{-1}\\mid \\alpha\\in A^*],\\\\\n E_*^A(\\mUP_A)&\\cong \\Sym(R_E[2])[\\vartheta_\\eps^{-1}].\n \\end{align*}\n\\end{theorem} \n\\begin{proof}\n We will deal with $\\MUP_A$, the case of $\\mUP_A$ is analogous.\n Similarly to the proof of \\cref{theorem:A}, we make use of a colimit formula.\n This time, it will be \\cref{corollary:MGr-colimit}.\n We show that the maps $C(V)$ define a natural transformation of $s(\\cU_A)$-diagrams, meaning that for all codimension one maps $i\\colon V\\to W$ with $\\alpha\\coloneqq i(V)^\\bot$, we show that the squares\n \\[\n \\begin{tikzcd}[column sep = huge]\n {\\Sym(R_E[2])[-2|V|]} & {\\Sym(R_E[2])[-2|W|]} \\\\\n {E_*^A(\\MGr_A^V)} & {E_*^A(\\MGr_A^W)}\n \\arrow[\"{\\vartheta_{\\alpha}\\otimes -}\", from=1-1, to=1-2]\n \\arrow[\"{C(V)}\", from=1-1, to=2-1]\n \\arrow[\"{C(W)}\", from=1-2, to=2-2]\n \\arrow[\"(j^i)_*\", from=2-1, to=2-2]\n \\end{tikzcd}\n \\]\n commute.\n This is once again an exercise in expanding the definition of the vertical maps and showing that the individual squares commute.\n The following depicts an expanded version of a summand of the above square, using the dimension-decomposition.\n \\[\n \\begin{tikzcd}[column sep = huge, row sep = 1em]\n {\\Sym_n(R_E[2])[-2|V|]} & {\\Sym_m(R_E[2])[-2|W|]} \\\\\n {\\displaystyle \\colim_{Z\\in s(\\cU_A)} E_*^A(\\Gr_n(V\\oplus Z))[-2|V|]} & {\\displaystyle \\colim_{Z\\in s(\\cU_A)} E_*^A(\\Gr_m(W\\oplus Z))[-2|W|]} \\\\\n {\\displaystyle \\colim_{Z\\in s(\\cU_A)} E_*^A(\\M(-n)_A^{V,\\bfU}(Z))[2|Z|]} & {\\displaystyle \\colim_{Z\\in s(\\cU_A)} E_*^A(\\M(-m)_A^{W,\\bfU}(Z))[2|Z|]} \\\\\n {\\displaystyle \\colim_{Z\\in s(\\cU_A)} E_*^A(\\Omega^Z\\Sigma^\\infty l(\\M(-n)_A^{V,\\bfU}(Z)))} & {\\displaystyle \\colim_{Z\\in s(\\cU_A)} E_*^A(\\Omega^Z\\Sigma^\\infty l(\\M(-m)_A^{W,\\bfU}(Z)))} \\\\\n {E_*^A(\\MGr^V_A)} & {E_*^A(\\MGr^W_A)}\n \\arrow[\"{{\\vartheta_{\\alpha}\\otimes -}}\", from=1-1, to=1-2]\n \\arrow[from=1-1, to=2-1]\n \\arrow[from=1-2, to=2-2]\n \\arrow[\"{\\colim_Z(\\alpha\\oplus-)_*}\", from=2-1, to=2-2]\n \\arrow[\"{\\cT_{\\gamma_n^\\bot}}\", from=2-1, to=3-1]\n \\arrow[\"{\\cT_{\\gamma_m^\\bot}}\", from=2-2, to=3-2]\n \\arrow[\"{(j^i)_*}\", from=3-1, to=3-2]\n \\arrow[\"{\\widetilde \\chi(Z)}\", from=3-1, to=4-1]\n \\arrow[\"{\\widetilde \\chi(Z)}\", from=3-2, to=4-2]\n \\arrow[\"{\\Omega^Z\\Sigma^\\infty l(j^i)_*}\", from=4-1, to=4-2]\n \\arrow[\"p\", from=4-1, to=5-1]\n \\arrow[\"p\", from=4-2, to=5-2]\n \\arrow[\"{(j^i)_*}\", from=5-1, to=5-2]\n \\end{tikzcd}\n \\]\n The first square commutes by \\cref{proposition:sym-ring-multiplication}.\n The second square commutes, as $j^i = \\Th(\\alpha\\oplus -)$.\n The last squares commute by the naturality of $\\widetilde \\chi(Z)$ and the canonical presentation.\n\\end{proof}\n\nWe now prove \\cref{corollary:intro-B-coordinatized} from the introduction.\n\\begin{corollary}\\label{corollary:B-coordinatized}\n Given a flag $\\cF = \\{V_i\\}_i$ of a complete universe, the associated basis $\\{\\beta_i^\\cF\\}_i$ of the $E_*^A$-algebra $R_E = E_*^A(\\bC P(\\cU_A)_+)$ gives rise to equivalences\n \\begin{align*}\n E_*^A(\\MUP_A)&\\cong E_*^A[C(\\beta_0^\\cF),C(\\beta_1^\\cF),C(\\beta_2^\\cF),\\cdots][C(\\vartheta_\\alpha)^{-1}\\mid \\alpha\\in A^*],\\\\\n E_*^A(\\mUP_A)&\\cong E_*^A[C(\\beta_0^\\cF),C(\\beta_1^\\cF),C(\\beta_2^\\cF),\\cdots][C(\\vartheta_\\epsilon)^{-1}].\n \\end{align*}\n For $\\alpha = V_1$, write $c(\\beta_i^\\cF) \\coloneqq C(\\vartheta_{\\alpha})^{-1}C(\\beta_i^\\cF)$ and $c(\\vartheta_{\\alpha'}) = C(\\vartheta_{\\alpha})^{-1}C(\\vartheta_{\\alpha'})$.\n These new classes live in dimension-grading 0 and there is an isomorphism\n \\[\n E_*^A(\\MU_A)\\cong E_*^A[c(\\beta_1^\\cF),c(\\beta_2^\\cF),c(\\beta_3^\\cF),\\cdots ][c(\\vartheta_{\\alpha'})^{-1}\\mid \\alpha'\\in A^*].\n \\]\n If $V_1 = \\eps$, there is an equivalence\n \\[\n E_*^A(\\mU_A)\\cong E_*^A[c(\\beta_1^\\cF),c(\\beta_2^\\cF),c(\\beta_3^\\cF),\\cdots ].\n \\]\n\\end{corollary}\n\\begin{proof}\n The proof is analogous to \\cref{lemma:periodicity} and \\cref{corollary:A-coordinatized}.\n\\end{proof}", "post_theorem_intro_text_len": 7050, "post_theorem_intro_text": "The above theorem, proven as \\cref{theorem:A}, makes use of the \\emph{canonical homotopy presentation} of $\\MUP_A$, an equivalence\n\\[\n\t\\colim_{V\\subseteq \\cU_A} \\Omega^V \\MRep_A \\xrightarrow{\\simeq}\\MUP_A\n\\]\nfrom a (homotopy) colimit to $\\MUP_A$ itself, and we use the fact that $E$-homology commutes with this colimit to find $E_*^A(\\MUP_A)$.\nThe mistake in \\cite{CGK02} occurs in the analysis of the colimit diagram above, after applying $E_*^A(-)$.\nContrary to \\cite[7.3]{CGK02}, for representations $V\\subseteq W\\subseteq \\cU_A$, the transition map $E_*^A(\\Omega^V\\MRep_A)\\to E_*^A(\\Omega^W\\MRep_A)$ depends on $W-V$ as a representation, not only on its dimension.\nAs a result, for $\\alpha$ an irreducible representation, $\\vartheta_\\alpha$ is invertible in the homology ring.\nUsing $C$ and the natural maps $ \\MGr_A\\to \\mUP_A$ and $ \\MGr_A\\to \\MUP_A$ we show:\n\\begin{theorem}\\label{theorem:intro-B}\n\tThere are $E_*^A$-algebra isomorphisms\n\t\\begin{align*}\n\t\tE_*^A(\\MUP_A)&\\cong \\operatorname{Sym}(R_E[2])[\\vartheta_{\\alpha}^{-1}\\mid \\alpha\\in A^*],\\\\\n\t\tE_*^A(\\mUP_A)&\\cong \\operatorname{Sym}(R_E[2])[\\vartheta_\\epsilon^{-1}].\n\t\\end{align*}\n\\end{theorem}\nFor the proof of the above in \\cref{theorem:B} we again rely on colimit formulae:\n\\[\n\t\\colim_{n\\in \\mathbb{N}} \\Omega^{\\epsilon^n}\\MGr_A \\xrightarrow{\\simeq} \\mUP_A, \\quad \\colim_{V\\subseteq \\cU_A} \\Omega^{V}\\MGr_A \\xrightarrow{\\simeq} \\MUP_A.\n\\]\n\n\\subsection{Coordinatized Statements}\nWe are able to give a more explicit description of the above results.\nLet $\\mathcal{F} = \\{V_i\\}_i$ be a flag of $\\cU_A$, an exhaustive sequence of increasing invariant subspaces, starting with $V_1 = \\epsilon$.\nAs shown in \\cite{C96}, for an oriented $A$-spectrum $E$, there is an $E_*^A$-module isomorphism\n\\[\n \tR_E=E_*^A(\\mathbb{C} P(\\cU_A)_+)\\cong \\bigoplus_{i=0}^\\infty E_*^A\\{\\beta_i^\\mathcal{F}\\},\n\\]\nwith $\\beta_0^\\mathcal{F} = \\vartheta_\\epsilon$.\nBy using the map $E_*^A(\\MRep_A)\\to E_*^A(\\MUP_A)$, we specialize \\cref{theorem:intro-A} to\n\\[\n\tE_*^A(\\MUP_A) \\cong E_*^A[B(\\beta_0^\\mathcal{F}), B(\\beta_1^\\mathcal{F}),B(\\beta_2^\\mathcal{F}),\\cdots ][B(\\vartheta_\\alpha)^{-1}\\mid \\alpha\\in A^*].\n\\]\nThe summand inclusion $\\MU_A\\to \\MUP_A$ defines a sub-algebra in homology.\n\\begin{corollary}\\label{corollary:intro-A-coordinatized}\n\tLet $b(\\beta_i^\\mathcal{F}) \\coloneqq B(\\vartheta_{\\epsilon})^{-1}B(\\beta_i^\\mathcal{F})$ as well as $b(\\vartheta_{\\alpha}) \\coloneqq B(\\vartheta_{\\epsilon})^{-1}B(\\vartheta_{\\alpha})$. The above restricts to an $E_*^A$-algebra isomorphism\n\t\\[\n\t\tE_*^A(\\MU_A)\\cong E_*^A[b(\\beta_1^\\mathcal{F}),b(\\beta_2^\\mathcal{F}),b(\\beta_3^\\mathcal{F})\\cdots ][b(\\vartheta_{\\alpha})^{-1}\\mid \\alpha\\in A^*].\n\t\\]\n\\end{corollary}\nThe above is proven in \\cref{corollary:A-coordinatized}.\nThe notation is consistent with \\cite{CGK02}, and as mentioned, the difference to their result is the inversion of the elements $b(\\vartheta_\\alpha)$.\nWe can also give coordinatized versions of the results obtained from the negative Thom spectrum $\\MGr_A$.\nFirstly, we find that \n\\[\n\tE_*^A(\\mUP_A) \\cong E_*^A[C(\\beta_0^\\mathcal{F}), C(\\beta_1^\\mathcal{F}),C(\\beta_2^\\mathcal{F}),\\cdots ][C(\\vartheta_\\epsilon)^{-1}],\n\\] \nSecondly, recall that $\\vartheta_\\epsilon = \\beta_0^\\mathcal{F}$, and we have \n\\[\n\tE_*^A(\\MUP_A) \\cong E_*^A[C(\\beta_0^\\mathcal{F}), C(\\beta_1^\\mathcal{F}),C(\\beta_2^\\mathcal{F}),\\cdots ][C(\\vartheta_\\alpha)^{-1}\\mid \\alpha\\in A^*].\n\\]\nLastly, we prove the following in \\cref{corollary:B-coordinatized}.\n\\begin{corollary}\\label{corollary:intro-B-coordinatized}\n\tFor $c(\\beta_i^\\mathcal{F}) \\coloneqq C(\\vartheta_{\\epsilon})^{-1}C(\\beta_i^\\mathcal{F})$ as well as $c(\\vartheta_{\\alpha}) \\coloneqq C(\\vartheta_{\\epsilon})^{-1}C(\\vartheta_{\\alpha})$, the above restricts to $E_*^A$-algebra isomorphisms\n\t\\begin{align*}\n\t\tE_*^A(\\mU_A)&\\cong E_*^A[c(\\beta_1^\\mathcal{F}),c(\\beta_2^\\mathcal{F}),c(\\beta_3^\\mathcal{F}),\\cdots ],\\\\\n\t\tE_*^A(\\MU_A)&\\cong E_*^A[c(\\beta_1^\\mathcal{F}),c(\\beta_2^\\mathcal{F}),c(\\beta_3^\\mathcal{F}), \\cdots ][c(\\vartheta_{\\alpha})^{-1}\\mid \\alpha\\in A^*].\n\t\\end{align*}\n\n\\end{corollary}\n\n\\subsection{Universality of $\\MU_A$}\nWe define the universal orientation $x^{\\mathrm{uni}}(\\epsilon) \\in \\MU_A^2(\\mathbb{C} P(\\cU_A),\\mathbb{C} P(\\epsilon))$ of $\\MU_A$.\nOrientations can be pushed forward along a map of commutative (homotopy) $A$-ring spectra.\nIn \\cref{section:6} we recover \\cite[Thm. 1.2.]{CGK02}:\n\\begin{theorem}\\label{theorem:intro-universality}\nLet $E$ be a commutative (homotopy) $A$-ring spectrum.\nThere is a natural bijection\n\\[\n\t[\\MU_A,E]_{\\CRing_A}\\to \\mathrm{Or}(E), \\quad f\\mapsto f_*(x^\\mathrm{uni}(\\epsilon))\n\\]\nbetween orientations of $E$ and homotopy classes of ring maps $\\MU_A\\to E$.\n\\end{theorem}\nHowever, due to the new calculation of $E_*^A(\\MU_A)$, the proof requires an additional argument.\nWe must show that the canonical homotopy presentation does not have a $\\lim^1$-term:\n\\begin{proposition}\\label{proposition:intro-lim1}\n\tLet $E$ be an oriented $A$-spectrum.\n\tThe assembly maps \n\t\\[\n\t\tp_V\\colon \\Omega^V\\Sigma^\\infty \\Gr_{|V|}(V\\oplus\\cU_A)^{\\gamma_{|V|}}\\to \\MU_A\n\t\\]\n\tin the canonical homotopy presentation of $\\MU_A$ yield an isomorphism of $E_*^A$-modules\n\t\\[\n\t\t\\lim_{V\\in s(\\cU_A)^{\\mathrm{op}}} {E^*_A}(\\Omega^V\\Sigma^\\infty l(\\Gr_{|V|}(V\\oplus \\cU_A)^{\\gamma_{|V|}}))\\xrightarrow{\\simeq} {E^*_A}(\\MU_A).\n\t\\]\n\\end{proposition}\nOne can check that all subsequent results of \\cite{CGK02} remain true.\n\n\\subsection*{Outline of Document}\nIn \\cref{section:1} we define orientations of $A$-spectra and give names to many important (co)homology classes existing for oriented spectra.\nThen, in \\cref{section:2} we recall the calculation of the homology of Grassmannians of complete universes.\nIt is in \\cref{section:3} that we introduce the relevant bordism spectra as the genuine $A$-spectra underlying certain unitary $A$-spectra.\nMoving on, in \\cref{section:4} and \\cref{section:5} we calculate the homology of the bordism spectra $\\MU_A$ and $\\mU_A$, proving theorems A and B, as well as deriving corollaries for orientable spectra and the cohomology of the aforementioned bordism spectra.\nLastly, in \\cref{section:6}, we comment on the cohomology of $\\MU_A$ and prove that it is the universal $A$-spectrum with an orientation.\nFinally, \\cref{appendix} contains a recollection of the canonical homotopy presentation.\n\n\\subsection*{Acknowledgements}\nThis work emerged from the Master's thesis of the author.\nHe would like to thank Markus Hausmann for support while and after writing the thesis, and John Greenlees for an encouraging and helpful conversation.\nFurther, he thanks Steffen Sagave for comments on an earlier draft, and his friends for continuous encouragement.\nThe author would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme Equivariant homotopy theory in context where work on this paper was undertaken.\nThis work was supported by EPSRC grant no EP/Z000580/1 and the Dutch Research Council (NWO) grant number OCENW.M.22.358.", "sketch": "The post-theorem introduction explains that the theorem (proven as \\cref{theorem:A}) uses the \\emph{canonical homotopy presentation} of $\\MUP_A$, namely an equivalence\n\\[\n\\t\\colim_{V\\subseteq \\cU_A} \\Omega^V \\MRep_A \\xrightarrow{\\simeq}\\MUP_A,\n\\]\nfrom a (homotopy) colimit to $\\MUP_A$, and then uses that “$E$-homology commutes with this colimit” to compute $E_*^A(\\MUP_A)$.\n\nIt further notes that, after applying $E_*^A(-)$, the transition maps in the resulting colimit diagram are subtler than in \\cite{CGK02}: for $V\\subseteq W\\subseteq \\cU_A$, the map $E_*^A(\\Omega^V\\MRep_A)\\to E_*^A(\\Omega^W\\MRep_A)$ “depends on $W-V$ as a representation, not only on its dimension.” From this, it concludes that “for $\\alpha$ an irreducible representation, $\\vartheta_\\alpha$ is invertible in the homology ring,” explaining why the final answer involves inverting the elements $\\vartheta_\\alpha$.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{theorem:intro-A}\n\tThere is an $E_*^A$-algebra isomorphism\n\t\\[\n\t\tE_*^A(\\MUP_A)\\cong \\operatorname{Sym}(R_E[-2])[\\vartheta_\\alpha^{-1}\\mid \\alpha\\in A^*].\n\t\\]", "theorem_type": ["Existence", "Classification or Bijection"], "mcq": {"question": "Let $A$ be an abelian compact Lie group and let $E$ be an oriented $A$-spectrum. Write $R_E:=E_*^A(\\mathbb{C}P(\\mathcal U_A)_+)$, where $\\mathcal U_A$ is a complete complex $A$-universe. For each character $\\alpha\\in A^*$, the inclusion of the one-dimensional $A$-representation $\\alpha\\hookrightarrow \\mathcal U_A$ induces a map $E_*^A(\\mathbb{C}P(\\alpha)_+)\\to R_E$, and the image of $1$ is the $\\alpha$-coaugmentation class $\\vartheta_\\alpha\\in R_E$. If $\\MUP_A$ denotes the periodic version of the homotopical bordism $A$-spectrum $\\MU_A$, which $E_*^A$-algebra is isomorphic to $E_*^A(\\MUP_A)$? (Here $\\operatorname{Sym}(R_E[-2])$ is the symmetric algebra on the graded $E_*^A$-module $R_E$ shifted by $-2$.)", "correct_choice": {"label": "A", "text": "$E_*^A(\\MUP_A)\\cong \\operatorname{Sym}(R_E[-2])[\\vartheta_\\alpha^{-1}\\mid \\alpha\\in A^*]$ as an $E_*^A$-algebra."}, "choices": [{"label": "B", "text": "$E_*^A(\\MUP_A)\\cong \\operatorname{Sym}(R_E[2])[\\vartheta_\\alpha^{-1}\\mid \\alpha\\in A^*]$ as an $E_*^A$-algebra."}, {"label": "C", "text": "$E_*^A(\\MUP_A)$ is isomorphic, as an $E_*^A$-algebra, to a localization of $\\operatorname{Sym}(R_E[-2])$ obtained by inverting the classes $\\vartheta_\\alpha$ for all $\\alpha\\in A^*$."}, {"label": "D", "text": "$E_*^A(\\MUP_A)\\cong \\operatorname{Sym}(R_E[-2])[\\vartheta_\\epsilon^{-1}]$ as an $E_*^A$-algebra."}, {"label": "E", "text": "$E_*^A(\\MUP_A)\\cong \\operatorname{Sym}(R_E[-2])[\\vartheta_\\alpha^{-1}\\mid \\alpha\\in A^*\\setminus\\{\\epsilon\\}]$ as an $E_*^A$-algebra."}], "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": "grading shift on the symmetric algebra", "template_used": "boundary_range"}, {"label": "C", "sketch_hook_type": "regularity", "tampered_component": "explicit identification of the localization with adjoined inverses written out as generators", "template_used": "weaker_true"}, {"label": "D", "sketch_hook_type": "regularity", "tampered_component": "which coaugmentation classes must be inverted", "template_used": "quantifier_dependence"}, {"label": "E", "sketch_hook_type": "regularity", "tampered_component": "inversion set omitting the trivial character", "template_used": "wildcard"}]}} +{"id": "2601.22822v1", "paper_link": "http://arxiv.org/abs/2601.22822v1", "theorems_cnt": 1, "theorem": {"env_name": "theorem", "content": "\\label{thm-unc}\nLet $j \\ge k \\ge 2$ be fixed integers.\nThen, for every $\\varepsilon > 0$, there exists a constant $C = C(\\varepsilon) > 0$,\nindependent of $k$ and $j$, such that\n\\[\n \\sum_{n=N+1}^{N+H} R_{\\phi,j}(n)\n =\n \\frac{\\gamma_{k} ^j}{\\gamma_{k,j}}\\cdot HN^{(j-k)/k}\n +\n \\Odi{H N^{(j - k) / k} A(N; -C)},\n\\]\nas $N\\to\\infty$, uniformly for $N^{1-13/15k+\\varepsilon} < H < N^{1-\\varepsilon}$.", "start_pos": 4709, "end_pos": 5111, "label": "thm-unc"}, "ref_dict": {"lemma-tolev": "\\begin{lemma}\n\\label{lemma-tolev}\nFor $k>1$ and $0 < \\tau \\le \\dfrac{1}{2}$, we have that\n\\[\n \\int_{-\\tau}^{\\tau} |\\Stilde_{\\phi}(\\alpha)|^2 \\, \\dx \\alpha\n \\ll_{\\phi}\n (\\tau N^{1/k} + N^{2/k\\,-1}) L^4.\n\\]\n\\end{lemma}", "lemma-L2": "\\begin{lemma}\n\\label{lemma-L2}\nLet $\\eps$ be an arbitrarily small positive constant, let $k\\ge 1$ be\nan integer, consider $\\phi(n)\\in\\mathbb{Z}[n]$ such that\n$\\partial\\phi=k$ and $\\lead(\\phi)=1$; let $N$ be a sufficiently large\ninteger. Then there exists a positive constant\n$c_{1}=c_{1}(\\eps)$, which does not depend on $\\phi$, such that\n\\[\n \\int_{-\\xi}^{\\xi} | \\Etilde_{\\phi}(\\alpha)|^2 \\, \\dx \\alpha\n :=\n \\int_{-\\xi}^{\\xi}\n \\vert \\Stilde_{\\phi}(\\alpha)-\\gamma_{k}z^{-1/k} \\vert^2 \\, \\dx\\alpha\n \\ll_{\\phi} N^{2/k\\,-1} A(N; - 2c_{1}),\n\\]\nuniformly for $0\\le \\xi < N^{-1+13/15k\\,-\\eps}$.\n\\end{lemma}", "thm-unc": "\\begin{theorem}\n\\label{thm-unc}\nLet $j \\ge k \\ge 2$ be fixed integers.\nThen, for every $\\eps > 0$, there exists a constant $C = C(\\eps) > 0$,\nindependent of $k$ and $j$, such that\n\\[\n \\sum_{n=N+1}^{N+H} R_{\\phi,j}(n)\n =\n \\frac{\\gamma_{k} ^j}{\\gamma_{k,j}}\\cdot HN^{(j-k)/k}\n +\n \\Odi{H N^{(j - k) / k} A(N; -C)},\n\\]\nas $N\\to\\infty$, uniformly for $N^{1-13/15k+\\eps} < H < N^{1-\\eps}$.\n\\end{theorem}"}, "pre_theorem_intro_text_len": 2467, "pre_theorem_intro_text": "Our goal is the study of the average number of representations of an\ninteger as the sum of values of a polynomial with integer coefficients\ncomputed at prime powers.\nMore precisely, we will consider the following setting: we take\n\\[\n \\phi\\in\\mathbb{Z}[n]\n \\quad\\text{with}\\quad\n \\phi(n)=\\sum_{h=1}^k a_{h}n^h\n \\quad\\text{and}\\quad\n a_{k} = 1.\n\\]\nThe choice $a_{k}=1$ simplifies the generalization of previous\nresults, but we will remove this constraint later; we also assume that\n$\\phi(0) = 0$ and that $\\phi(n) \\not \\equiv n^k$ identically.\nLet us define\n\\begin{equation}\n\\label{sec1:1}\n R_{\\phi, j}(n)\n =\n \\sum_{n=\\phi(n_{1})+\\dots+\\phi(n_{j})}\n \\Lambda(n_{1}) \\dots \\Lambda(n_{j}),\n\\end{equation}\nfor $j\\ge k$, where $\\Lambda(n)$ is the von Mangoldt function defined\nas\n\\[\n \\Lambda(n)\n =\n \\begin{cases}\n \\log p & \\text{if $\\exists\\,t\\in\\mathbb{N}^*\\,:\\, n=p^t$} \\\\\n 0 & \\text{otherwise}.\n \\end{cases}\n\\]\nWe study the average number of representations $R_{\\phi, j}(n)$ of a\npositive integer $n$, as sums of values of this kind of polynomials,\nin short intervals of the form $[N+1, N+H]$, for $H\\ge 1$ as small as\npossible.\n\nThe main unconditional results of Cantarini, Gambini and Zaccagnini in\n\\cite{bib1}, extending earlier work by Languasco and Zaccagnini in\n\\cite{bib12}, were collected in their Theorems 1.1 and 1.3.\nThese Authors studied the weighted average number of representations\n$R_{k}$ and $R'_{k}$ of an integer $n$ as a sum of prime powers,\nwhere, in our notation, $R_{k}(n)=R_{\\phi,k+1}(n)$ and\n$R'_{k}(n)=R_{\\phi,k}(n)$, with $\\phi(n)=n^k$.\nWe prove that estimates similar to those in \\cite{bib1} still hold for\naverages of $R_{\\phi,j}(n)$, up to constants depending on the\npolynomial $\\phi$.\nIn contrast to the papers \\cite{bib12} and \\cite{bib1}, we give a\nsingle statement and proof valid for all $j \\ge k$, the most\ninteresting case being when $j$ is small, that is, $j = k$.\nFor positive integers $j$ and $k$ it is convenient to write\n\\[\n \\gamma_{k}:=\\Gamma\\left(1+\\frac{1}{k}\\right)\n \\quad \\text{and}\\quad\n \\gamma_{k,j}:=\\Gamma\\left( \\frac{j}{k}\\right),\n\\]\nwhere $\\Gamma$ is the Euler Gamma function.\nThroughout the paper, implicit constants may depend on the polynomial\n$\\phi$, or its degree $k$, and on the integer $j$.\nThroughout the paper we write $L = \\log(N)$ for brevity.\nFinally, let\n\\begin{equation}\n\\label{sec1:2}\n A(N; C)\n =\n \\exp\\left(C\\left(\\frac{\\log N}{\\log\\log N}\\right)^{1/3}\\right).\n\\end{equation}", "context": "Our goal is the study of the average number of representations of an\ninteger as the sum of values of a polynomial with integer coefficients\ncomputed at prime powers.\nMore precisely, we will consider the following setting: we take\n\\[\n \\phi\\in\\mathbb{Z}[n]\n \\quad\\text{with}\\quad\n \\phi(n)=\\sum_{h=1}^k a_{h}n^h\n \\quad\\text{and}\\quad\n a_{k} = 1.\n\\]\nThe choice $a_{k}=1$ simplifies the generalization of previous\nresults, but we will remove this constraint later; we also assume that\n$\\phi(0) = 0$ and that $\\phi(n) \\not \\equiv n^k$ identically.\nLet us define\n\\begin{equation}\n\\label{sec1:1}\n R_{\\phi, j}(n)\n =\n \\sum_{n=\\phi(n_{1})+\\dots+\\phi(n_{j})}\n \\Lambda(n_{1}) \\dots \\Lambda(n_{j}),\n\\end{equation}\nfor $j\\ge k$, where $\\Lambda(n)$ is the von Mangoldt function defined\nas\n\\[\n \\Lambda(n)\n =\n \\begin{cases}\n \\log p & \\text{if $\\exists\\,t\\in\\mathbb{N}^*\\,:\\, n=p^t$} \\\\\n 0 & \\text{otherwise}.\n \\end{cases}\n\\]\nWe study the average number of representations $R_{\\phi, j}(n)$ of a\npositive integer $n$, as sums of values of this kind of polynomials,\nin short intervals of the form $[N+1, N+H]$, for $H\\ge 1$ as small as\npossible.\n\nThe main unconditional results of Cantarini, Gambini and Zaccagnini in\n\\cite{bib1}, extending earlier work by Languasco and Zaccagnini in\n\\cite{bib12}, were collected in their Theorems 1.1 and 1.3.\nThese Authors studied the weighted average number of representations\n$R_{k}$ and $R'_{k}$ of an integer $n$ as a sum of prime powers,\nwhere, in our notation, $R_{k}(n)=R_{\\phi,k+1}(n)$ and\n$R'_{k}(n)=R_{\\phi,k}(n)$, with $\\phi(n)=n^k$.\nWe prove that estimates similar to those in \\cite{bib1} still hold for\naverages of $R_{\\phi,j}(n)$, up to constants depending on the\npolynomial $\\phi$.\nIn contrast to the papers \\cite{bib12} and \\cite{bib1}, we give a\nsingle statement and proof valid for all $j \\ge k$, the most\ninteresting case being when $j$ is small, that is, $j = k$.\nFor positive integers $j$ and $k$ it is convenient to write\n\\[\n \\gamma_{k}:=\\Gamma\\left(1+\\frac{1}{k}\\right)\n \\quad \\text{and}\\quad\n \\gamma_{k,j}:=\\Gamma\\left( \\frac{j}{k}\\right),\n\\]\nwhere $\\Gamma$ is the Euler Gamma function.\nThroughout the paper, implicit constants may depend on the polynomial\n$\\phi$, or its degree $k$, and on the integer $j$.\nThroughout the paper we write $L = \\log(N)$ for brevity.\nFinally, let\n\\begin{equation}\n\\label{sec1:2}\n A(N; C)\n =\n \\exp\\left(C\\left(\\frac{\\log N}{\\log\\log N}\\right)^{1/3}\\right).\n\\end{equation}", "full_context": "Our goal is the study of the average number of representations of an\ninteger as the sum of values of a polynomial with integer coefficients\ncomputed at prime powers.\nMore precisely, we will consider the following setting: we take\n\\[\n \\phi\\in\\mathbb{Z}[n]\n \\quad\\text{with}\\quad\n \\phi(n)=\\sum_{h=1}^k a_{h}n^h\n \\quad\\text{and}\\quad\n a_{k} = 1.\n\\]\nThe choice $a_{k}=1$ simplifies the generalization of previous\nresults, but we will remove this constraint later; we also assume that\n$\\phi(0) = 0$ and that $\\phi(n) \\not \\equiv n^k$ identically.\nLet us define\n\\begin{equation}\n\\label{sec1:1}\n R_{\\phi, j}(n)\n =\n \\sum_{n=\\phi(n_{1})+\\dots+\\phi(n_{j})}\n \\Lambda(n_{1}) \\dots \\Lambda(n_{j}),\n\\end{equation}\nfor $j\\ge k$, where $\\Lambda(n)$ is the von Mangoldt function defined\nas\n\\[\n \\Lambda(n)\n =\n \\begin{cases}\n \\log p & \\text{if $\\exists\\,t\\in\\mathbb{N}^*\\,:\\, n=p^t$} \\\\\n 0 & \\text{otherwise}.\n \\end{cases}\n\\]\nWe study the average number of representations $R_{\\phi, j}(n)$ of a\npositive integer $n$, as sums of values of this kind of polynomials,\nin short intervals of the form $[N+1, N+H]$, for $H\\ge 1$ as small as\npossible.\n\nThe main unconditional results of Cantarini, Gambini and Zaccagnini in\n\\cite{bib1}, extending earlier work by Languasco and Zaccagnini in\n\\cite{bib12}, were collected in their Theorems 1.1 and 1.3.\nThese Authors studied the weighted average number of representations\n$R_{k}$ and $R'_{k}$ of an integer $n$ as a sum of prime powers,\nwhere, in our notation, $R_{k}(n)=R_{\\phi,k+1}(n)$ and\n$R'_{k}(n)=R_{\\phi,k}(n)$, with $\\phi(n)=n^k$.\nWe prove that estimates similar to those in \\cite{bib1} still hold for\naverages of $R_{\\phi,j}(n)$, up to constants depending on the\npolynomial $\\phi$.\nIn contrast to the papers \\cite{bib12} and \\cite{bib1}, we give a\nsingle statement and proof valid for all $j \\ge k$, the most\ninteresting case being when $j$ is small, that is, $j = k$.\nFor positive integers $j$ and $k$ it is convenient to write\n\\[\n \\gamma_{k}:=\\Gamma\\left(1+\\frac{1}{k}\\right)\n \\quad \\text{and}\\quad\n \\gamma_{k,j}:=\\Gamma\\left( \\frac{j}{k}\\right),\n\\]\nwhere $\\Gamma$ is the Euler Gamma function.\nThroughout the paper, implicit constants may depend on the polynomial\n$\\phi$, or its degree $k$, and on the integer $j$.\nThroughout the paper we write $L = \\log(N)$ for brevity.\nFinally, let\n\\begin{equation}\n\\label{sec1:2}\n A(N; C)\n =\n \\exp\\left(C\\left(\\frac{\\log N}{\\log\\log N}\\right)^{1/3}\\right).\n\\end{equation}\n\nThe improvement on the range of uniformity for $H$ in our\nTheorem~\\ref{thm-unc} is due to the recent result of Guth and Maynard\nin \\cite{bib4} on the density of the zeros of the Riemann zeta-function,\nwhich would also affect all previous unconditional results in the\npapers \\cite{bib12} and \\cite{bib1}.\n\nIt will spare us some repetition to notice at the outset that for\nfixed $\\alpha \\ge 0$ and $k \\ge 1$ we have\n\\begin{align}\n\\notag\n \\int_0^{+\\infty} t^\\alpha \\e^{- t^k / M} \\, \\dx t\n &=\n \\frac1k M^{(\\alpha + 1) / k}\n \\int_0^{+\\infty} u^{(\\alpha + 1) / k - 1} \\e^{- u} \\, \\dx u \\\\\n\\label{bound-int}\n &\\ll_{\\alpha, k}\n M^{(\\alpha + 1) / k},\n\\end{align}\nas $M \\to +\\infty$, which can be seen by the obvious change of variable.\nWe also recall the Prime Number Theorem in the weak form\n\\begin{equation}\n\\label{PNT}\n \\psi(x)\n =\n \\sum_{n \\le x} \\Lambda(n)\n \\sim x\n \\qquad\\text{as $x \\to +\\infty$.}\n\\end{equation}\n\n\\begin{lemma}\n\\label{lemma-L2}\nLet $\\eps$ be an arbitrarily small positive constant, let $k\\ge 1$ be\nan integer, consider $\\phi(n)\\in\\mathbb{Z}[n]$ such that\n$\\partial\\phi=k$ and $\\lead(\\phi)=1$; let $N$ be a sufficiently large\ninteger. Then there exists a positive constant\n$c_{1}=c_{1}(\\eps)$, which does not depend on $\\phi$, such that\n\\[\n \\int_{-\\xi}^{\\xi} | \\Etilde_{\\phi}(\\alpha)|^2 \\, \\dx \\alpha\n :=\n \\int_{-\\xi}^{\\xi}\n \\vert \\Stilde_{\\phi}(\\alpha)-\\gamma_{k}z^{-1/k} \\vert^2 \\, \\dx\\alpha\n \\ll_{\\phi} N^{2/k\\,-1} A(N; - 2c_{1}),\n\\]\nuniformly for $0\\le \\xi < N^{-1+13/15k\\,-\\eps}$.\n\\end{lemma}\n\nWe recall the decomposition in \\eqref{eq1:lemma-L2}.\nOur goal is the estimate of\n\\[\n \\Sigma_{2}\n \\ll_{\\phi}\n \\int_{-\\xi}^{\\xi}\n |S_{2,M}(\\alpha)|^2 \\, \\dx \\alpha\n +\n \\int_{-\\xi}^{\\xi}|S_{2,\\infty}(\\alpha)|^2 \\, \\dx \\alpha.\n\\]\nWe will eventually choose $M = N^{1 / k} L$; for the time being we\njust assume assume that $M^d = o(N)$, so that\n$n^d \\le M^d = o(N)$ for all the summands in $S_{2, M}$.\nHence, by the Taylor expansion of the exponential function, we get\n\\begin{align*}\n \\e^{-\\phi(n) / N} - \\e^{-n^k / N}\n &=\n \\e^{-n^k /N} (\\e^{-\\eta(n) / N} - 1)\n =\n \\e^{-n^k /N}\n \\Bigl( - \\frac{\\eta(n)}{N} + \\Odig{\\frac{\\eta^2(n)}{N^2}} \\Bigr) \\\\\n &=\n \\e^{-n^k /N}\n \\Bigl( - \\frac{\\eta(n)}{N} + \\Odig{\\frac{n^{2 d}}{N^2}} \\Bigr)\n\\end{align*}\nand therefore we can rewrite $S_{2,M}$ as\n\\begin{align}\n\\notag\n S_{2,M}(\\alpha)\n &=\n - N^{-1}\n \\sum_{n \\le M}\n \\Lambda(n) \\e^{-n^k / N} \\eta(n) \\e(\\phi(n) \\alpha)\n +\n \\Odig{N^{-2} \\sum_{n \\le M} \\Lambda(n) \\, \\e^{-n^k/N} n^{2 d}} \\\\\n\\label{lemma-5.0}\n &=:\n \\Sigma_{2, M, 1} + \\Odi{N^{-2} M^{2 d + 1}},\n\\end{align}\nsay, by a weak form of the PNT.\nThen, by Corollary 2 of \\cite{mon74}, we obtain that\n\\begin{align*}\n \\int_{-\\xi}^{\\xi} |\\Sigma_{2, M, 1}(\\alpha)|^2 \\, \\dx \\alpha\n &=\n \\sum_{n \\le M}\n \\Lambda(n)^2 \\e^{-2n^k /N}\\frac{\\eta^2(n)}{N^2}\n (2 \\xi + \\Odim{\\delta_{n}^{-1}}) \\\\\n &\\ll\n \\xi N^{-2}\n \\sum_{n\\le M} \\Lambda(n)^2 \\e^{-2n^k /N} n^{2 d}\n +\n N^{-2}\n \\sum_{n \\le M}\\Lambda(n)^2 \\e^{-2n^k /N} n^{2d+1-k}.\n\\end{align*}\nSince we assumed that $M^d = o(N)$, we have that\n\\begin{align}\n\\notag\n \\xi N^{-2}\n \\sum_{n \\le M} \\Lambda(n)^2 \\e^{-2n^k /N} n^{2d}\n &\\ll\n \\xi N^{-2} L \\sum_{n \\le M} \\Lambda(n) n^{2 d} \\\\\n\\label{lemma-5.1}\n &\\ll\n \\xi N^{-2} L M^{2 d + 1}.\n\\end{align}\n\nAs before, we bound $\\Sigma_{3}$ with two integrals that come from the\ndecomposition in \\eqref{eq2:lemma-L2}:\n\\[\n \\Sigma_{3}\n \\ll\n \\int_{-\\xi}^{\\xi} |S_{3,M}(\\alpha)|^2 \\, \\dx \\alpha\n +\n \\int_{-\\xi}^{\\xi} |S_{3,\\infty}(\\alpha)|^2 \\, \\dx \\alpha.\n\\]\nWe split the Taylor series of $\\e(\\eta(n)\\alpha)$ into two parts, for\na finite $j_{0}>1$, depending on $\\eps$, (thus, from now on, $\\ll$\nmeans $\\ll_{\\eps}$):\n\\begin{align*}\n S_{3,M}(\\alpha)\n &=\n \\sum_{n\\le M}\n \\Lambda(n) \\e^{-n^k /N} \\e(n^k \\alpha) (\\e(\\eta(n)\\alpha) - 1) \\\\ \n &=\n \\sum_{n\\le M}\n \\Lambda(n) \\e^{-n^k /N} \\e(n^k \\alpha)\n \\left(\n \\sum_{j=1}^{j_{0}} \\frac{(\\eta(n)\\alpha)^j}{j!}\n +\n \\mathcal{O}\\left(\\frac{\\eta(n)^t |\\alpha|^t}{t!}\\right)\n \\right) \\\\\n &=:\n (C) + (D), \n\\end{align*}\nfor $t= j_{0}+1$.\nWe apply the triangle inequality and then Corollary $2$ of\n\\cite{mon74} to the $L^2$-norm of $(C)$, for every $j\\in [1,j_{0}]$:\nfirst we get that the $L^2$-norm of $(C)$ is bounded by\n\\begin{align*}\n \\int_{-\\xi}^{\\xi}\n &\\left| \\sum_{j=1}^{j_{0}}\n \\sum_{n\\le M} \\Lambda(n) \\e^{-n^k /N} \\e(n^k \\alpha)\n \\frac{(\\eta(n)\\alpha)^j}{j!}\\right|^2 \\, \\dx \\alpha \\\\\n &\\ll\n \\sum_{j = 1}^{j_0}\n \\int_{-\\xi}^{\\xi}\n \\left| \\sum_{n\\le M} \\Lambda(n) \\e^{-n^k / N} \\e(n^k \\alpha)\n \\frac{\\eta(n)^j \\alpha^j}{j!} \n \\right|^2 \\, \\dx \\alpha \\\\\n &=:\n C_{1}+\\dots+C_{j_{0}}.\n\\end{align*}\nBy the Corollary just quoted we get that\n\\begin{align*}\n C_{j}\n &=\n \\int_{-\\xi}^{\\xi}\n \\vert \\alpha \\vert^{2 j}\n \\cdot\n \\left|\n \\sum_{n\\le M}\\Lambda(n)\n \\e^{-n^k /N} \\e(n^k \\alpha) \\frac{\\eta(n)^j}{j!}\n \\right|^2 \\, \\dx \\alpha \\\\\n &\\ll\n \\xi^{2j+1}\\sum_{n\\le M} \\Lambda(n)^2 \\e^{-2n^k /N} n^{2dj}\n +\n \\xi^{2j}\\sum_{n\\le M}\\Lambda(n)^2 \\e^{-2n^k /N} n^{2dj+1-k} \\\\\n &=:\n (C_{j,1}) + (C_{j,2}).\n\\end{align*}\nOur choice $M = N^{1/k} L$ and the fact that $d \\le k - 1$ imply that\n\\begin{align*}\n (C_{j,1})\n &\\ll\n L^2\\xi^{2j+1}M^{2dj+1}\n \\le\n L^2 N^{(-1+1/k\\,-\\eps)(2j+1)} (N^{1/k}L)^{2dj+1} \\\\ \n &\\ll\n N^{2/k\\,-1-\\eps-2j\\eps} L^{2 j (k - 1) + 3} \n \\ll\n N^{2/k\\,-1}A(N;-2c_{1})\n\\end{align*}\nsince $\\eps > 0$. \nFor $(C_{j,2})$ we need only consider the case when $d$ is large, so\nthat $2dj+1-k\\ge 0$; we find that\n\\begin{align*}\n (C_{j,2})\n &\\ll\n \\xi^{2j} L^2 M^{2dj+2-k}\n \\le\n N^{-2j+2j/k\\,-2j\\eps}L^2 (N^{1/k}L)^{2kj-2j+2-k} \\\\ \n &\\ll\n N^{-2j\\eps+2/k\\,-1} L^{2kj-2j+4-k}\n \\ll\n N^{2/k\\,-1}A(N;-2c_{1}),\n\\end{align*}\nas above.\nTherefore, for every $1\\le j\\le j_{0}$, we can estimate $C_{j}$.\n\nSumming the three terms $\\gamma_k^j I_1$, $I_2$ and $I_3$ we arrive at\n\\begin{align}\n\\notag\n \\sum_{n=N+1}^{N+H} \\e^{-n/N}R_{\\phi,j}(n)\n &=\n \\frac{\\gamma_{k}^j}{\\gamma_{k,j}} \\cdot \\, \\e^{-1}HN^{(j-k)/k} \\\\\n\\label{final-est}\n &\\qquad+\n \\mathcal{O}\\bigl(N^{(j-k)/k} H \\cdot A(N; -c_1)\\bigr).\n\\end{align}\nTo complete the proof, we remove the exponential factor at the\nleft-hand side: in fact, as $\\e^{-n/N} \\in [\\e^{-2}, \\e^{-1}]$,\n\\begin{align*}\n \\e^{-2} \\sum_{n=N+1}^{N+H} R_{\\phi,j}(n)\n &\\le\n \\sum_{n=N+1}^{N+H} \\e^{-n/N}R_{\\phi,j}(n) \\ll_{\\phi} HN^{(j-k)/k}\n\\end{align*}\nso that\n\\[\n \\sum_{n=N+1}^{N+H} R_{\\phi,j}(n)\n \\ll_{\\phi} H N^{(j-k)/k}.\n\\]\nWe use this inequality for an estimate of the error term that appears\nfrom the development\n$\\e^{-n/N} = \\e^{-1-(n-N)/N} = \\e^{-1}(1+\\mathcal{O}((n-N)N^{-1}))$:\n\\begin{align*}\n \\sum_{n=N+1}^{N+H} \\e^{-n/N}R_{\\phi,j}(n)\n &=\n \\sum_{n=N+1}^{N+H} \\e^{-1}(1+\\mathcal{O}((n-N)N^{-1}))R_{\\phi,j}(n) \\\\\n &=\n \\e^{-1} \\sum_{n=N+1}^{N+H} R_{\\phi,j}(n)\n +\n \\mathcal{O}_{\\phi}(H^2N^{(j-k)/k\\,-1}),\n\\end{align*}\nand Theorem~\\ref{thm-unc} follows from~\\eqref{final-est} since we\nassumed that $H < N^{1 - \\eps}$ so that the last error term above is\nsmaller than the one in~\\eqref{final-est}.", "post_theorem_intro_text_len": 3858, "post_theorem_intro_text": "The improvement on the range of uniformity for $H$ in our\nTheorem~\\ref{thm-unc} is due to the recent result of Guth and Maynard\nin \\cite{bib4} on the density of the zeros of the Riemann zeta-function,\nwhich would also affect all previous unconditional results in the\npapers \\cite{bib12} and \\cite{bib1}.\n\n\\subsection{Outline of the proof}\n\nWe briefly sum up the strategy used in \\cite{bib1} and \\cite{bib12}.\nWe consider $j \\ge k \\ge 2$ fixed.\nFor $\\phi$ as above, we introduce the exponential sum\n\\begin{equation}\n\\label{sec1:3}\n \\Stilde_{\\phi}(\\alpha)\n =\n \\sum_{n\\ge 1} \\Lambda(n) \\mathrm{e}^{-\\phi(n)/N} \\mathrm{e}(\\phi(n)\\alpha)\n =\n \\sum_{n\\ge 1} \\Lambda(n) \\mathrm{e}^{-\\phi(n)z},\n\\end{equation}\nwhere $\\mathrm{e}(\\alpha) = \\mathrm{e}^{2 \\pi i \\alpha}$,\n$z = \\displaystyle{\\frac{1}{N}-2\\pi i\\alpha}$, and the finite sum\n\\[\n U(\\alpha, H)\n =\n \\sum_{m=1}^H \\mathrm{e}(m\\alpha), \\qquad\\text{for}\\quad H\\le N.\n\\]\nThus, we can rewrite $R_{\\phi,j}(n)$ as\n\\[\n R_{\\phi,j}(n)\n =\n \\mathrm{e}^{n/N}\n \\int_{-1/2}^{1/2} \\Stilde_{\\phi}(\\alpha)^j \\mathrm{e}(-n\\alpha) \\, \\mathrm{d} \\alpha,\n\\]\nby definition $\\eqref{sec1:1}$ of $R_{\\phi,j}(n)$ and by\northogonality.\nSumming $R_{\\phi,j}(n)$ over $n$, we obtain that\n\\[\n \\sum_{n=N+1}^{N+H} R_{\\phi,j}(n) \\mathrm{e}^{-n/N}\n =\n \\int_{-1/2}^{1/2}\n \\Stilde_{\\phi}(\\alpha)^j U(H, -\\alpha) \\, \\mathrm{e}(-N\\alpha) \\, \\mathrm{d}\\alpha.\n\\]\nThe expected main term comes from a comparatively short arc around 0.\nFurthermore, $\\Stilde_{\\phi}(\\alpha)$ is of size\n$\\gamma_{k}z^{-1/k}$, for ``small\" $\\alpha$.\nHence, we set $\\tau = B H^{-1}$ and decompose the quantity we are\nstudying as follows:\n\\begin{align*}\n \\sum_{n=N+1}^{N+H} R_{\\phi,j}(n) \\mathrm{e}^{-n/N}\n &=\n \\gamma_{k}^j\n \\int_{-\\tau}^{\\tau} z^{-j/k}U(-\\alpha, H) \\mathrm{e}(-N\\alpha) \\, \\mathrm{d} \\alpha \\\\\n &\\quad+\n \\int_{-\\tau}^{\\tau}\n (\\Stilde_{\\phi}(\\alpha)^j-\\gamma_{k}^jz^{-j/k})U(-\\alpha, H)\n \\mathrm{e}(-N\\alpha) \\, \\mathrm{d} \\alpha \\\\\n &\\quad+\n \\int_{\\mathcal{C}} \\Stilde_{\\phi}(\\alpha)^j U(-\\alpha, H) \\mathrm{e}(-N\\alpha)\n \\, \\mathrm{d} \\alpha \\,\n =:\n \\gamma_{k}^jI_{1} + I_{2} + I_{3},\n\\end{align*}\nsay, where $\\mathcal{C} = [-1/2, -\\tau] \\cup [\\tau, 1/2]$ and\n$B = N^{2 \\varepsilon}$.\nThe parameter $\\tau$ controls the width of the ``major arc\" around 0,\nwhere it is easy to obtain a good approximation to $\\Stilde_{\\phi}$.\n\nIn fact, for small $\\alpha$ the exponential sum $\\Stilde_\\phi$\nbehaves pretty much like\n\\begin{equation}\n\\label{sec1:2.5}\n \\Stilde_{k}(\\alpha)\n =\n \\sum_{n \\ge 1} \\Lambda(n) \\mathrm{e}^{-n^k /N} \\, \\mathrm{e}(n^k \\alpha).\n\\end{equation}\nThe evaluation of $I_{1}$ is in Lemma~4 of \\cite{lan16jnt}.\nThe bound for $I_{2}$ relies on our Lemma~\\ref{lemma-L2} below, which\ngeneralizes Lemma~4 of \\cite{bib12} since we have $\\Stilde_\\phi$ in\nplace of $\\Stilde_k$.\nThe term $I_{3}$ requires a slightly more general version of Lemma~7\nof Tolev \\cite{bib11}, which is our Lemma~\\ref{lemma-tolev}.\nFinally, we can conclude, using the same method of \\S3 of\n\\cite{bib12}.\n\nTwo final remarks: the lower bound for $H$ in our Theorem~\\ref{thm-unc}\nis only due to the limitations in Lemma~\\ref{lemma-L2}; at all other\nplaces we just need the more natural bound $H > N^{1 - 1 / k + \\varepsilon}$.\nOur proof does not rely on the generalization of Lemmas~6 and~7 of\n\\cite{bib12} to our case: indeed, we do not need bounds for the $L^4$\nnorm of $\\Stilde_\\phi$, but show that $L^2$ estimates suffice.\nApparently this simplification, which depends on the use of identity\n\\eqref{sec3:8} below introduced in \\cite{bib1}, was overlooked by its\nAuthors.\n\nIn contrast to previous papers, we do not have any conditional result\nbecause the proof of Lemma~\\ref{lemma-L2} on the whole unit interval\n$[-\\frac12, \\frac12]$ seems to be quite hard: the Riemann Hypothesis\ndoes not seem to help on the complement of $[-\\tau, \\tau]$.", "sketch": "To prove Theorem~\\ref{thm-unc} the authors “briefly sum up the strategy” of \\cite{bib1,bib12}. They introduce the exponential sum\n\\[\n\\Stilde_{\\phi}(\\alpha)=\\sum_{n\\ge 1}\\Lambda(n)e^{-\\phi(n)/N}e(\\phi(n)\\alpha)=\\sum_{n\\ge 1}\\Lambda(n)e^{-\\phi(n)z},\\qquad z=\\frac1N-2\\pi i\\alpha,\n\\]\nand the finite sum \\(U(\\alpha,H)=\\sum_{m=1}^H e(m\\alpha)\\). By orthogonality they rewrite\n\\[\nR_{\\phi,j}(n)=e^{n/N}\\int_{-1/2}^{1/2}\\Stilde_{\\phi}(\\alpha)^j e(-n\\alpha)\\,d\\alpha,\n\\]\nso that, summing in \\(n\\),\n\\[\n\\sum_{n=N+1}^{N+H}R_{\\phi,j}(n)e^{-n/N}=\\int_{-1/2}^{1/2}\\Stilde_{\\phi}(\\alpha)^j U(H,-\\alpha)e(-N\\alpha)\\,d\\alpha.\n\\]\nThe “expected main term comes from a comparatively short arc around 0”, where \\(\\Stilde_{\\phi}(\\alpha)\\) has size \\(\\gamma_k z^{-1/k}\\) for “small” \\(\\alpha\\). They set \\(\\tau=BH^{-1}\\) with \\(B=N^{2\\varepsilon}\\) and decompose into a major arc \\([ -\\tau,\\tau]\\) and its complement \\(\\mathcal C=[-1/2,-\\tau]\\cup[\\tau,1/2]\\):\n\\[\n\\sum_{n=N+1}^{N+H}R_{\\phi,j}(n)e^{-n/N}=\\gamma_k^j I_1+I_2+I_3,\n\\]\nwhere\n\\[\nI_1=\\int_{-\\tau}^{\\tau} z^{-j/k}U(-\\alpha,H)e(-N\\alpha)\\,d\\alpha,\n\\]\n\\(I_2\\) is the same integral over \\([-\\tau,\\tau]\\) with \\(\\Stilde_{\\phi}(\\alpha)^j-\\gamma_k^j z^{-j/k}\\), and\n\\(I_3=\\int_{\\mathcal C}\\Stilde_{\\phi}(\\alpha)^jU(-\\alpha,H)e(-N\\alpha)\\,d\\alpha\\).\nThey note that for small \\(\\alpha\\), \\(\\Stilde_\\phi\\) “behaves pretty much like”\n\\(\\Stilde_k(\\alpha)=\\sum_{n\\ge 1}\\Lambda(n)e^{-n^k/N}e(n^k\\alpha)\\).\nThe evaluation of \\(I_1\\) is given by Lemma~4 of \\cite{lan16jnt}; the bound for \\(I_2\\) “relies on” Lemma~\\ref{lemma-L2} (an \\(L^2\\) estimate generalizing Lemma~4 of \\cite{bib12} to \\(\\Stilde_\\phi\\)); and \\(I_3\\) uses “a slightly more general version” of Tolev’s Lemma~7, stated as Lemma~\\ref{lemma-tolev}. With these three pieces, they “conclude, using the same method of \\S3 of \\cite{bib12}.” They add that the lower bound on \\(H\\) in Theorem~\\ref{thm-unc} is “only due to the limitations in Lemma~\\ref{lemma-L2}”, and that they “do not need bounds for the \\(L^4\\) norm of \\(\\Stilde_\\phi\\), but show that \\(L^2\\) estimates suffice,” via identity \\eqref{sec3:8} from \\cite{bib1}.", "expanded_sketch": "No expanded sketch found.", "expanded_theorem": "\\label{thm-unc}\nLet $j \\ge k \\ge 2$ be fixed integers.\nThen, for every $\\varepsilon > 0$, there exists a constant $C = C(\\varepsilon) > 0$,\nindependent of $k$ and $j$, such that\n\\[\n \\sum_{n=N+1}^{N+H} R_{\\phi,j}(n)\n =\n \\frac{\\gamma_{k} ^j}{\\gamma_{k,j}}\\cdot HN^{(j-k)/k}\n +\n \\Odi{H N^{(j - k) / k} A(N; -C)},\n\\]\nas $N\\to\\infty$, uniformly for $N^{1-13/15k+\\varepsilon} < H < N^{1-\\varepsilon}.", "theorem_type": ["Asymptotic or Limit", "Existential–Universal"], "mcq": {"question": "Let \\(\\phi(n)=\\sum_{h=1}^k a_h n^h\\in \\mathbb Z[n]\\) be a monic polynomial of degree \\(k\\) with \\(\\phi(0)=0\\) and \\(\\phi(n)\\not\\equiv n^k\\). For integers \\(j\\ge k\\), define\n\\[\nR_{\\phi,j}(n)=\\sum_{n=\\phi(n_1)+\\cdots+\\phi(n_j)} \\Lambda(n_1)\\cdots \\Lambda(n_j),\n\\]\nwhere \\(\\Lambda(m)=\\log p\\) if \\(m=p^t\\) for some prime \\(p\\) and integer \\(t\\ge 1\\), and \\(\\Lambda(m)=0\\) otherwise. Also set\n\\[\n\\gamma_k=\\Gamma\\!\\left(1+\\frac1k\\right),\\qquad \\gamma_{k,j}=\\Gamma\\!\\left(\\frac{j}{k}\\right),\\qquad A(N;C)=\\exp\\!\\left(C\\left(\\frac{\\log N}{\\log\\log N}\\right)^{1/3}\\right).\n\\]\nFix integers \\(j\\ge k\\ge 2\\). As \\(N\\to\\infty\\), uniformly for \\(H\\) in the range \\(N^{1-13/(15k)+\\varepsilon}0\\), there exists a constant \\(C=C(\\varepsilon)>0\\), independent of \\(k\\) and \\(j\\), such that\n\\[\n\\sum_{n=N+1}^{N+H} R_{\\phi,j}(n)\n=\n\\frac{\\gamma_k^j}{\\gamma_{k,j}}\\, H N^{(j-k)/k}\n+ O\\!\\left(H N^{(j-k)/k} A(N;-C)\\right)\n\\]\nas \\(N\\to\\infty\\), uniformly for \\(N^{1-13/(15k)+\\varepsilon}0\\), there exists a constant \\(C=C(\\varepsilon)>0\\), independent of \\(k\\) and \\(j\\), such that\n\\[\n\\sum_{n=N+1}^{N+H} R_{\\phi,j}(n)\n=\n\\frac{\\gamma_k^j}{\\gamma_{k,j}}\\, H N^{(j-k)/k}\n+ O\\!\\left(H N^{(j-k)/k} A(N;-C)\\right)\n\\]\nas \\(N\\to\\infty\\), uniformly for \\(N^{1-1/k+\\varepsilon}0\\) such that\n\\[\n\\sum_{n=N+1}^{N+H} R_{\\phi,j}(n)\n=\n\\frac{\\gamma_k^j}{\\gamma_{k,j}}\\, H N^{(j-k)/k}\n+ O\\!\\left(H N^{(j-k)/k} A(N;-C)\\right)\n\\]\nas \\(N\\to\\infty\\), uniformly for \\(N^{1-13/(15k)+\\varepsilon}0\\), there exists a constant \\(C=C(\\varepsilon,k,j)>0\\) such that\n\\[\n\\sum_{n=N+1}^{N+H} R_{\\phi,j}(n)\n=\n\\frac{\\gamma_k^j}{\\gamma_{k,j}}\\, H N^{(j-k)/k}\n+ O\\!\\left(H N^{(j-k)/k} A(N;-C)\\right)\n\\]\nas \\(N\\to\\infty\\), uniformly for \\(N^{1-13/(15k)+\\varepsilon}0\\), there exists a constant \\(C=C(\\varepsilon)>0\\), independent of \\(k\\) and \\(j\\), such that\n\\[\n\\sum_{n=N+1}^{N+H} R_{\\phi,j}(n)\n=\n\\frac{\\gamma_k^j}{\\gamma_{k,j}}\\, H N^{(j-k)/k}\n+ O\\!\\left(H N^{(j-k)/k-1} A(N;-C)\\right)\n\\]\nas \\(N\\to\\infty\\), uniformly for \\(N^{1-13/(15k)+\\varepsilon}